Properties

Label 5225.2.a.h.1.1
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $5$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.71457\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.65432 q^{2} -0.121872 q^{3} +5.04540 q^{4} +0.323487 q^{6} +3.36889 q^{7} -8.08346 q^{8} -2.98515 q^{9} +O(q^{10})\) \(q-2.65432 q^{2} -0.121872 q^{3} +5.04540 q^{4} +0.323487 q^{6} +3.36889 q^{7} -8.08346 q^{8} -2.98515 q^{9} +1.00000 q^{11} -0.614893 q^{12} +2.15993 q^{13} -8.94209 q^{14} +11.3653 q^{16} -3.67288 q^{17} +7.92353 q^{18} -1.00000 q^{19} -0.410573 q^{21} -2.65432 q^{22} -3.15468 q^{23} +0.985147 q^{24} -5.73313 q^{26} +0.729422 q^{27} +16.9974 q^{28} +7.17849 q^{29} +4.65295 q^{31} -14.0001 q^{32} -0.121872 q^{33} +9.74900 q^{34} -15.0613 q^{36} -2.27446 q^{37} +2.65432 q^{38} -0.263235 q^{39} -11.3852 q^{41} +1.08979 q^{42} -9.38838 q^{43} +5.04540 q^{44} +8.37352 q^{46} +5.77094 q^{47} -1.38511 q^{48} +4.34940 q^{49} +0.447622 q^{51} +10.8977 q^{52} +5.65820 q^{53} -1.93612 q^{54} -27.2322 q^{56} +0.121872 q^{57} -19.0540 q^{58} -13.7944 q^{59} +6.98152 q^{61} -12.3504 q^{62} -10.0566 q^{63} +14.4301 q^{64} +0.323487 q^{66} +4.81332 q^{67} -18.5312 q^{68} +0.384467 q^{69} +15.2629 q^{71} +24.1303 q^{72} +8.08806 q^{73} +6.03713 q^{74} -5.04540 q^{76} +3.36889 q^{77} +0.698709 q^{78} +13.4291 q^{79} +8.86655 q^{81} +30.2199 q^{82} -9.96666 q^{83} -2.07150 q^{84} +24.9197 q^{86} -0.874858 q^{87} -8.08346 q^{88} -4.61626 q^{89} +7.27655 q^{91} -15.9166 q^{92} -0.567064 q^{93} -15.3179 q^{94} +1.70622 q^{96} +4.09907 q^{97} -11.5447 q^{98} -2.98515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 14 q^{14} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 5 q^{19} + 10 q^{21} - 2 q^{22} - 3 q^{23} - 14 q^{24} - 6 q^{26} + 11 q^{27} + 10 q^{28} + 10 q^{29} + 11 q^{31} - 14 q^{32} - q^{33} - 4 q^{34} - 26 q^{36} - q^{37} + 2 q^{38} + 2 q^{39} + 2 q^{41} + 16 q^{42} - 20 q^{43} + 6 q^{44} - 4 q^{46} + 20 q^{47} - 4 q^{48} + 3 q^{49} + 24 q^{51} - 6 q^{52} + 14 q^{53} + 16 q^{54} - 38 q^{56} + q^{57} + 6 q^{58} + 3 q^{59} - 10 q^{61} + 6 q^{62} - 24 q^{63} - 2 q^{66} - 9 q^{67} - 24 q^{68} - 5 q^{69} + 23 q^{71} + 12 q^{72} + 8 q^{74} - 6 q^{76} - 6 q^{77} + 22 q^{78} + 44 q^{79} + q^{81} + 30 q^{82} + 14 q^{83} + 14 q^{84} + 52 q^{86} - 28 q^{87} - 6 q^{88} - 27 q^{89} + 24 q^{91} - 58 q^{92} + 27 q^{93} - 8 q^{94} + 50 q^{96} - 15 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.65432 −1.87689 −0.938443 0.345435i \(-0.887732\pi\)
−0.938443 + 0.345435i \(0.887732\pi\)
\(3\) −0.121872 −0.0703629 −0.0351814 0.999381i \(-0.511201\pi\)
−0.0351814 + 0.999381i \(0.511201\pi\)
\(4\) 5.04540 2.52270
\(5\) 0 0
\(6\) 0.323487 0.132063
\(7\) 3.36889 1.27332 0.636660 0.771145i \(-0.280316\pi\)
0.636660 + 0.771145i \(0.280316\pi\)
\(8\) −8.08346 −2.85793
\(9\) −2.98515 −0.995049
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −0.614893 −0.177504
\(13\) 2.15993 0.599056 0.299528 0.954087i \(-0.403171\pi\)
0.299528 + 0.954087i \(0.403171\pi\)
\(14\) −8.94209 −2.38987
\(15\) 0 0
\(16\) 11.3653 2.84131
\(17\) −3.67288 −0.890805 −0.445402 0.895330i \(-0.646939\pi\)
−0.445402 + 0.895330i \(0.646939\pi\)
\(18\) 7.92353 1.86759
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.410573 −0.0895944
\(22\) −2.65432 −0.565902
\(23\) −3.15468 −0.657797 −0.328898 0.944365i \(-0.606677\pi\)
−0.328898 + 0.944365i \(0.606677\pi\)
\(24\) 0.985147 0.201092
\(25\) 0 0
\(26\) −5.73313 −1.12436
\(27\) 0.729422 0.140377
\(28\) 16.9974 3.21220
\(29\) 7.17849 1.33301 0.666506 0.745499i \(-0.267789\pi\)
0.666506 + 0.745499i \(0.267789\pi\)
\(30\) 0 0
\(31\) 4.65295 0.835694 0.417847 0.908517i \(-0.362785\pi\)
0.417847 + 0.908517i \(0.362785\pi\)
\(32\) −14.0001 −2.47489
\(33\) −0.121872 −0.0212152
\(34\) 9.74900 1.67194
\(35\) 0 0
\(36\) −15.0613 −2.51021
\(37\) −2.27446 −0.373918 −0.186959 0.982368i \(-0.559863\pi\)
−0.186959 + 0.982368i \(0.559863\pi\)
\(38\) 2.65432 0.430587
\(39\) −0.263235 −0.0421513
\(40\) 0 0
\(41\) −11.3852 −1.77807 −0.889034 0.457841i \(-0.848623\pi\)
−0.889034 + 0.457841i \(0.848623\pi\)
\(42\) 1.08979 0.168158
\(43\) −9.38838 −1.43171 −0.715857 0.698247i \(-0.753964\pi\)
−0.715857 + 0.698247i \(0.753964\pi\)
\(44\) 5.04540 0.760623
\(45\) 0 0
\(46\) 8.37352 1.23461
\(47\) 5.77094 0.841778 0.420889 0.907112i \(-0.361718\pi\)
0.420889 + 0.907112i \(0.361718\pi\)
\(48\) −1.38511 −0.199923
\(49\) 4.34940 0.621342
\(50\) 0 0
\(51\) 0.447622 0.0626796
\(52\) 10.8977 1.51124
\(53\) 5.65820 0.777213 0.388607 0.921404i \(-0.372957\pi\)
0.388607 + 0.921404i \(0.372957\pi\)
\(54\) −1.93612 −0.263472
\(55\) 0 0
\(56\) −27.2322 −3.63906
\(57\) 0.121872 0.0161423
\(58\) −19.0540 −2.50191
\(59\) −13.7944 −1.79588 −0.897939 0.440121i \(-0.854936\pi\)
−0.897939 + 0.440121i \(0.854936\pi\)
\(60\) 0 0
\(61\) 6.98152 0.893892 0.446946 0.894561i \(-0.352512\pi\)
0.446946 + 0.894561i \(0.352512\pi\)
\(62\) −12.3504 −1.56850
\(63\) −10.0566 −1.26702
\(64\) 14.4301 1.80377
\(65\) 0 0
\(66\) 0.323487 0.0398185
\(67\) 4.81332 0.588041 0.294020 0.955799i \(-0.405007\pi\)
0.294020 + 0.955799i \(0.405007\pi\)
\(68\) −18.5312 −2.24723
\(69\) 0.384467 0.0462844
\(70\) 0 0
\(71\) 15.2629 1.81137 0.905685 0.423951i \(-0.139357\pi\)
0.905685 + 0.423951i \(0.139357\pi\)
\(72\) 24.1303 2.84378
\(73\) 8.08806 0.946636 0.473318 0.880892i \(-0.343056\pi\)
0.473318 + 0.880892i \(0.343056\pi\)
\(74\) 6.03713 0.701802
\(75\) 0 0
\(76\) −5.04540 −0.578747
\(77\) 3.36889 0.383920
\(78\) 0.698709 0.0791132
\(79\) 13.4291 1.51090 0.755448 0.655209i \(-0.227419\pi\)
0.755448 + 0.655209i \(0.227419\pi\)
\(80\) 0 0
\(81\) 8.86655 0.985172
\(82\) 30.2199 3.33723
\(83\) −9.96666 −1.09398 −0.546992 0.837138i \(-0.684227\pi\)
−0.546992 + 0.837138i \(0.684227\pi\)
\(84\) −2.07150 −0.226020
\(85\) 0 0
\(86\) 24.9197 2.68716
\(87\) −0.874858 −0.0937946
\(88\) −8.08346 −0.861699
\(89\) −4.61626 −0.489323 −0.244661 0.969609i \(-0.578677\pi\)
−0.244661 + 0.969609i \(0.578677\pi\)
\(90\) 0 0
\(91\) 7.27655 0.762790
\(92\) −15.9166 −1.65942
\(93\) −0.567064 −0.0588018
\(94\) −15.3179 −1.57992
\(95\) 0 0
\(96\) 1.70622 0.174140
\(97\) 4.09907 0.416197 0.208099 0.978108i \(-0.433273\pi\)
0.208099 + 0.978108i \(0.433273\pi\)
\(98\) −11.5447 −1.16619
\(99\) −2.98515 −0.300019
\(100\) 0 0
\(101\) −10.8730 −1.08190 −0.540950 0.841055i \(-0.681935\pi\)
−0.540950 + 0.841055i \(0.681935\pi\)
\(102\) −1.18813 −0.117642
\(103\) −8.85864 −0.872867 −0.436434 0.899736i \(-0.643759\pi\)
−0.436434 + 0.899736i \(0.643759\pi\)
\(104\) −17.4597 −1.71206
\(105\) 0 0
\(106\) −15.0186 −1.45874
\(107\) 18.1669 1.75626 0.878131 0.478421i \(-0.158791\pi\)
0.878131 + 0.478421i \(0.158791\pi\)
\(108\) 3.68023 0.354130
\(109\) −0.211303 −0.0202392 −0.0101196 0.999949i \(-0.503221\pi\)
−0.0101196 + 0.999949i \(0.503221\pi\)
\(110\) 0 0
\(111\) 0.277193 0.0263100
\(112\) 38.2883 3.61790
\(113\) 0.198345 0.0186587 0.00932935 0.999956i \(-0.497030\pi\)
0.00932935 + 0.999956i \(0.497030\pi\)
\(114\) −0.323487 −0.0302973
\(115\) 0 0
\(116\) 36.2184 3.36279
\(117\) −6.44770 −0.596090
\(118\) 36.6147 3.37066
\(119\) −12.3735 −1.13428
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −18.5312 −1.67773
\(123\) 1.38754 0.125110
\(124\) 23.4760 2.10821
\(125\) 0 0
\(126\) 26.6935 2.37804
\(127\) 3.60331 0.319742 0.159871 0.987138i \(-0.448892\pi\)
0.159871 + 0.987138i \(0.448892\pi\)
\(128\) −10.3020 −0.910578
\(129\) 1.14418 0.100739
\(130\) 0 0
\(131\) 16.8741 1.47430 0.737150 0.675729i \(-0.236171\pi\)
0.737150 + 0.675729i \(0.236171\pi\)
\(132\) −0.614893 −0.0535196
\(133\) −3.36889 −0.292119
\(134\) −12.7761 −1.10369
\(135\) 0 0
\(136\) 29.6896 2.54586
\(137\) 6.10839 0.521875 0.260937 0.965356i \(-0.415968\pi\)
0.260937 + 0.965356i \(0.415968\pi\)
\(138\) −1.02050 −0.0868706
\(139\) 11.3347 0.961398 0.480699 0.876886i \(-0.340383\pi\)
0.480699 + 0.876886i \(0.340383\pi\)
\(140\) 0 0
\(141\) −0.703316 −0.0592299
\(142\) −40.5125 −3.39973
\(143\) 2.15993 0.180622
\(144\) −33.9270 −2.82725
\(145\) 0 0
\(146\) −21.4683 −1.77673
\(147\) −0.530070 −0.0437194
\(148\) −11.4755 −0.943284
\(149\) 6.75698 0.553554 0.276777 0.960934i \(-0.410734\pi\)
0.276777 + 0.960934i \(0.410734\pi\)
\(150\) 0 0
\(151\) 6.46231 0.525895 0.262948 0.964810i \(-0.415305\pi\)
0.262948 + 0.964810i \(0.415305\pi\)
\(152\) 8.08346 0.655655
\(153\) 10.9641 0.886395
\(154\) −8.94209 −0.720574
\(155\) 0 0
\(156\) −1.32813 −0.106335
\(157\) −0.248382 −0.0198230 −0.00991152 0.999951i \(-0.503155\pi\)
−0.00991152 + 0.999951i \(0.503155\pi\)
\(158\) −35.6452 −2.83578
\(159\) −0.689576 −0.0546869
\(160\) 0 0
\(161\) −10.6278 −0.837585
\(162\) −23.5346 −1.84905
\(163\) 16.9710 1.32927 0.664637 0.747167i \(-0.268586\pi\)
0.664637 + 0.747167i \(0.268586\pi\)
\(164\) −57.4428 −4.48553
\(165\) 0 0
\(166\) 26.4547 2.05328
\(167\) 0.865538 0.0669773 0.0334887 0.999439i \(-0.489338\pi\)
0.0334887 + 0.999439i \(0.489338\pi\)
\(168\) 3.31885 0.256055
\(169\) −8.33471 −0.641132
\(170\) 0 0
\(171\) 2.98515 0.228280
\(172\) −47.3681 −3.61178
\(173\) 4.14483 0.315125 0.157563 0.987509i \(-0.449636\pi\)
0.157563 + 0.987509i \(0.449636\pi\)
\(174\) 2.32215 0.176042
\(175\) 0 0
\(176\) 11.3653 0.856688
\(177\) 1.68115 0.126363
\(178\) 12.2530 0.918403
\(179\) 2.04584 0.152913 0.0764567 0.997073i \(-0.475639\pi\)
0.0764567 + 0.997073i \(0.475639\pi\)
\(180\) 0 0
\(181\) 5.27519 0.392101 0.196051 0.980594i \(-0.437188\pi\)
0.196051 + 0.980594i \(0.437188\pi\)
\(182\) −19.3143 −1.43167
\(183\) −0.850852 −0.0628968
\(184\) 25.5007 1.87994
\(185\) 0 0
\(186\) 1.50517 0.110364
\(187\) −3.67288 −0.268588
\(188\) 29.1167 2.12355
\(189\) 2.45734 0.178745
\(190\) 0 0
\(191\) −16.1899 −1.17146 −0.585729 0.810507i \(-0.699192\pi\)
−0.585729 + 0.810507i \(0.699192\pi\)
\(192\) −1.75863 −0.126918
\(193\) 20.0620 1.44409 0.722045 0.691846i \(-0.243202\pi\)
0.722045 + 0.691846i \(0.243202\pi\)
\(194\) −10.8802 −0.781155
\(195\) 0 0
\(196\) 21.9444 1.56746
\(197\) 23.8398 1.69851 0.849257 0.527979i \(-0.177050\pi\)
0.849257 + 0.527979i \(0.177050\pi\)
\(198\) 7.92353 0.563101
\(199\) −1.90194 −0.134825 −0.0674125 0.997725i \(-0.521474\pi\)
−0.0674125 + 0.997725i \(0.521474\pi\)
\(200\) 0 0
\(201\) −0.586609 −0.0413762
\(202\) 28.8603 2.03060
\(203\) 24.1835 1.69735
\(204\) 2.25843 0.158122
\(205\) 0 0
\(206\) 23.5136 1.63827
\(207\) 9.41719 0.654540
\(208\) 24.5481 1.70211
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 12.3048 0.847100 0.423550 0.905873i \(-0.360784\pi\)
0.423550 + 0.905873i \(0.360784\pi\)
\(212\) 28.5479 1.96068
\(213\) −1.86012 −0.127453
\(214\) −48.2207 −3.29630
\(215\) 0 0
\(216\) −5.89625 −0.401189
\(217\) 15.6753 1.06411
\(218\) 0.560865 0.0379866
\(219\) −0.985709 −0.0666080
\(220\) 0 0
\(221\) −7.93316 −0.533642
\(222\) −0.735757 −0.0493808
\(223\) −22.8473 −1.52997 −0.764985 0.644048i \(-0.777254\pi\)
−0.764985 + 0.644048i \(0.777254\pi\)
\(224\) −47.1647 −3.15132
\(225\) 0 0
\(226\) −0.526470 −0.0350202
\(227\) −4.00654 −0.265923 −0.132962 0.991121i \(-0.542449\pi\)
−0.132962 + 0.991121i \(0.542449\pi\)
\(228\) 0.614893 0.0407223
\(229\) −24.6024 −1.62577 −0.812887 0.582422i \(-0.802105\pi\)
−0.812887 + 0.582422i \(0.802105\pi\)
\(230\) 0 0
\(231\) −0.410573 −0.0270137
\(232\) −58.0270 −3.80966
\(233\) −16.2994 −1.06781 −0.533903 0.845545i \(-0.679275\pi\)
−0.533903 + 0.845545i \(0.679275\pi\)
\(234\) 17.1143 1.11879
\(235\) 0 0
\(236\) −69.5982 −4.53046
\(237\) −1.63664 −0.106311
\(238\) 32.8433 2.12891
\(239\) −15.4182 −0.997324 −0.498662 0.866797i \(-0.666175\pi\)
−0.498662 + 0.866797i \(0.666175\pi\)
\(240\) 0 0
\(241\) 24.1441 1.55526 0.777629 0.628723i \(-0.216422\pi\)
0.777629 + 0.628723i \(0.216422\pi\)
\(242\) −2.65432 −0.170626
\(243\) −3.26885 −0.209697
\(244\) 35.2245 2.25502
\(245\) 0 0
\(246\) −3.68296 −0.234817
\(247\) −2.15993 −0.137433
\(248\) −37.6119 −2.38836
\(249\) 1.21466 0.0769758
\(250\) 0 0
\(251\) −19.9099 −1.25670 −0.628350 0.777931i \(-0.716269\pi\)
−0.628350 + 0.777931i \(0.716269\pi\)
\(252\) −50.7397 −3.19630
\(253\) −3.15468 −0.198333
\(254\) −9.56433 −0.600119
\(255\) 0 0
\(256\) −1.51547 −0.0947166
\(257\) 22.3126 1.39182 0.695911 0.718128i \(-0.255001\pi\)
0.695911 + 0.718128i \(0.255001\pi\)
\(258\) −3.03702 −0.189076
\(259\) −7.66239 −0.476118
\(260\) 0 0
\(261\) −21.4289 −1.32641
\(262\) −44.7893 −2.76709
\(263\) −13.7963 −0.850716 −0.425358 0.905025i \(-0.639852\pi\)
−0.425358 + 0.905025i \(0.639852\pi\)
\(264\) 0.985147 0.0606316
\(265\) 0 0
\(266\) 8.94209 0.548275
\(267\) 0.562593 0.0344301
\(268\) 24.2851 1.48345
\(269\) −11.2028 −0.683048 −0.341524 0.939873i \(-0.610943\pi\)
−0.341524 + 0.939873i \(0.610943\pi\)
\(270\) 0 0
\(271\) 17.1234 1.04017 0.520085 0.854115i \(-0.325900\pi\)
0.520085 + 0.854115i \(0.325900\pi\)
\(272\) −41.7433 −2.53106
\(273\) −0.886808 −0.0536721
\(274\) −16.2136 −0.979499
\(275\) 0 0
\(276\) 1.93979 0.116762
\(277\) 13.7127 0.823919 0.411960 0.911202i \(-0.364844\pi\)
0.411960 + 0.911202i \(0.364844\pi\)
\(278\) −30.0859 −1.80443
\(279\) −13.8897 −0.831557
\(280\) 0 0
\(281\) 25.8599 1.54267 0.771337 0.636427i \(-0.219588\pi\)
0.771337 + 0.636427i \(0.219588\pi\)
\(282\) 1.86682 0.111168
\(283\) −28.8711 −1.71621 −0.858103 0.513477i \(-0.828357\pi\)
−0.858103 + 0.513477i \(0.828357\pi\)
\(284\) 77.0073 4.56954
\(285\) 0 0
\(286\) −5.73313 −0.339007
\(287\) −38.3554 −2.26405
\(288\) 41.7923 2.46264
\(289\) −3.50993 −0.206467
\(290\) 0 0
\(291\) −0.499562 −0.0292848
\(292\) 40.8075 2.38808
\(293\) −6.37971 −0.372707 −0.186353 0.982483i \(-0.559667\pi\)
−0.186353 + 0.982483i \(0.559667\pi\)
\(294\) 1.40697 0.0820563
\(295\) 0 0
\(296\) 18.3855 1.06863
\(297\) 0.729422 0.0423254
\(298\) −17.9352 −1.03896
\(299\) −6.81389 −0.394057
\(300\) 0 0
\(301\) −31.6284 −1.82303
\(302\) −17.1530 −0.987045
\(303\) 1.32511 0.0761256
\(304\) −11.3653 −0.651842
\(305\) 0 0
\(306\) −29.1022 −1.66366
\(307\) −10.4299 −0.595264 −0.297632 0.954681i \(-0.596197\pi\)
−0.297632 + 0.954681i \(0.596197\pi\)
\(308\) 16.9974 0.968515
\(309\) 1.07962 0.0614174
\(310\) 0 0
\(311\) −8.26224 −0.468508 −0.234254 0.972175i \(-0.575265\pi\)
−0.234254 + 0.972175i \(0.575265\pi\)
\(312\) 2.12785 0.120466
\(313\) 24.1167 1.36315 0.681577 0.731746i \(-0.261294\pi\)
0.681577 + 0.731746i \(0.261294\pi\)
\(314\) 0.659285 0.0372056
\(315\) 0 0
\(316\) 67.7554 3.81154
\(317\) 11.7330 0.658989 0.329495 0.944157i \(-0.393122\pi\)
0.329495 + 0.944157i \(0.393122\pi\)
\(318\) 1.83035 0.102641
\(319\) 7.17849 0.401918
\(320\) 0 0
\(321\) −2.21404 −0.123576
\(322\) 28.2095 1.57205
\(323\) 3.67288 0.204365
\(324\) 44.7353 2.48529
\(325\) 0 0
\(326\) −45.0465 −2.49489
\(327\) 0.0257519 0.00142409
\(328\) 92.0317 5.08160
\(329\) 19.4416 1.07185
\(330\) 0 0
\(331\) −6.95223 −0.382129 −0.191065 0.981577i \(-0.561194\pi\)
−0.191065 + 0.981577i \(0.561194\pi\)
\(332\) −50.2858 −2.75979
\(333\) 6.78959 0.372067
\(334\) −2.29741 −0.125709
\(335\) 0 0
\(336\) −4.66627 −0.254566
\(337\) −29.4416 −1.60379 −0.801893 0.597468i \(-0.796174\pi\)
−0.801893 + 0.597468i \(0.796174\pi\)
\(338\) 22.1230 1.20333
\(339\) −0.0241727 −0.00131288
\(340\) 0 0
\(341\) 4.65295 0.251971
\(342\) −7.92353 −0.428455
\(343\) −8.92959 −0.482152
\(344\) 75.8905 4.09174
\(345\) 0 0
\(346\) −11.0017 −0.591454
\(347\) 17.2180 0.924311 0.462155 0.886799i \(-0.347076\pi\)
0.462155 + 0.886799i \(0.347076\pi\)
\(348\) −4.41401 −0.236616
\(349\) 4.32405 0.231461 0.115731 0.993281i \(-0.463079\pi\)
0.115731 + 0.993281i \(0.463079\pi\)
\(350\) 0 0
\(351\) 1.57550 0.0840939
\(352\) −14.0001 −0.746207
\(353\) −23.4421 −1.24770 −0.623848 0.781545i \(-0.714432\pi\)
−0.623848 + 0.781545i \(0.714432\pi\)
\(354\) −4.46231 −0.237169
\(355\) 0 0
\(356\) −23.2909 −1.23441
\(357\) 1.50799 0.0798111
\(358\) −5.43031 −0.287001
\(359\) 8.20544 0.433066 0.216533 0.976275i \(-0.430525\pi\)
0.216533 + 0.976275i \(0.430525\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −14.0020 −0.735930
\(363\) −0.121872 −0.00639662
\(364\) 36.7131 1.92429
\(365\) 0 0
\(366\) 2.25843 0.118050
\(367\) −33.1494 −1.73039 −0.865193 0.501439i \(-0.832804\pi\)
−0.865193 + 0.501439i \(0.832804\pi\)
\(368\) −35.8538 −1.86901
\(369\) 33.9865 1.76926
\(370\) 0 0
\(371\) 19.0618 0.989640
\(372\) −2.86107 −0.148339
\(373\) 16.5840 0.858685 0.429343 0.903142i \(-0.358745\pi\)
0.429343 + 0.903142i \(0.358745\pi\)
\(374\) 9.74900 0.504109
\(375\) 0 0
\(376\) −46.6492 −2.40575
\(377\) 15.5050 0.798550
\(378\) −6.52256 −0.335484
\(379\) 5.79467 0.297652 0.148826 0.988863i \(-0.452451\pi\)
0.148826 + 0.988863i \(0.452451\pi\)
\(380\) 0 0
\(381\) −0.439143 −0.0224980
\(382\) 42.9730 2.19869
\(383\) −5.89016 −0.300973 −0.150487 0.988612i \(-0.548084\pi\)
−0.150487 + 0.988612i \(0.548084\pi\)
\(384\) 1.25553 0.0640709
\(385\) 0 0
\(386\) −53.2508 −2.71039
\(387\) 28.0257 1.42463
\(388\) 20.6814 1.04994
\(389\) 13.8055 0.699964 0.349982 0.936756i \(-0.386188\pi\)
0.349982 + 0.936756i \(0.386188\pi\)
\(390\) 0 0
\(391\) 11.5868 0.585968
\(392\) −35.1581 −1.77575
\(393\) −2.05649 −0.103736
\(394\) −63.2784 −3.18792
\(395\) 0 0
\(396\) −15.0613 −0.756857
\(397\) −6.48471 −0.325458 −0.162729 0.986671i \(-0.552030\pi\)
−0.162729 + 0.986671i \(0.552030\pi\)
\(398\) 5.04835 0.253051
\(399\) 0.410573 0.0205544
\(400\) 0 0
\(401\) 3.70806 0.185172 0.0925858 0.995705i \(-0.470487\pi\)
0.0925858 + 0.995705i \(0.470487\pi\)
\(402\) 1.55705 0.0776584
\(403\) 10.0500 0.500628
\(404\) −54.8584 −2.72931
\(405\) 0 0
\(406\) −64.1908 −3.18573
\(407\) −2.27446 −0.112741
\(408\) −3.61833 −0.179134
\(409\) 31.0569 1.53567 0.767833 0.640650i \(-0.221335\pi\)
0.767833 + 0.640650i \(0.221335\pi\)
\(410\) 0 0
\(411\) −0.744442 −0.0367206
\(412\) −44.6954 −2.20198
\(413\) −46.4717 −2.28673
\(414\) −24.9962 −1.22850
\(415\) 0 0
\(416\) −30.2392 −1.48260
\(417\) −1.38138 −0.0676467
\(418\) 2.65432 0.129827
\(419\) −11.2871 −0.551410 −0.275705 0.961242i \(-0.588911\pi\)
−0.275705 + 0.961242i \(0.588911\pi\)
\(420\) 0 0
\(421\) 34.5194 1.68237 0.841186 0.540746i \(-0.181858\pi\)
0.841186 + 0.540746i \(0.181858\pi\)
\(422\) −32.6609 −1.58991
\(423\) −17.2271 −0.837611
\(424\) −45.7378 −2.22122
\(425\) 0 0
\(426\) 4.93734 0.239215
\(427\) 23.5199 1.13821
\(428\) 91.6593 4.43052
\(429\) −0.263235 −0.0127091
\(430\) 0 0
\(431\) 18.1654 0.874996 0.437498 0.899219i \(-0.355865\pi\)
0.437498 + 0.899219i \(0.355865\pi\)
\(432\) 8.29007 0.398856
\(433\) −10.7804 −0.518075 −0.259037 0.965867i \(-0.583405\pi\)
−0.259037 + 0.965867i \(0.583405\pi\)
\(434\) −41.6071 −1.99720
\(435\) 0 0
\(436\) −1.06611 −0.0510573
\(437\) 3.15468 0.150909
\(438\) 2.61638 0.125016
\(439\) −1.38742 −0.0662180 −0.0331090 0.999452i \(-0.510541\pi\)
−0.0331090 + 0.999452i \(0.510541\pi\)
\(440\) 0 0
\(441\) −12.9836 −0.618266
\(442\) 21.0571 1.00159
\(443\) 7.26792 0.345310 0.172655 0.984982i \(-0.444766\pi\)
0.172655 + 0.984982i \(0.444766\pi\)
\(444\) 1.39855 0.0663722
\(445\) 0 0
\(446\) 60.6441 2.87158
\(447\) −0.823487 −0.0389496
\(448\) 48.6135 2.29677
\(449\) 21.4298 1.01134 0.505668 0.862728i \(-0.331246\pi\)
0.505668 + 0.862728i \(0.331246\pi\)
\(450\) 0 0
\(451\) −11.3852 −0.536108
\(452\) 1.00073 0.0470703
\(453\) −0.787575 −0.0370035
\(454\) 10.6346 0.499108
\(455\) 0 0
\(456\) −0.985147 −0.0461337
\(457\) 17.6312 0.824754 0.412377 0.911013i \(-0.364699\pi\)
0.412377 + 0.911013i \(0.364699\pi\)
\(458\) 65.3026 3.05139
\(459\) −2.67908 −0.125049
\(460\) 0 0
\(461\) 34.2073 1.59320 0.796598 0.604510i \(-0.206631\pi\)
0.796598 + 0.604510i \(0.206631\pi\)
\(462\) 1.08979 0.0507017
\(463\) 11.0009 0.511253 0.255626 0.966776i \(-0.417718\pi\)
0.255626 + 0.966776i \(0.417718\pi\)
\(464\) 81.5854 3.78751
\(465\) 0 0
\(466\) 43.2637 2.00415
\(467\) −4.38453 −0.202892 −0.101446 0.994841i \(-0.532347\pi\)
−0.101446 + 0.994841i \(0.532347\pi\)
\(468\) −32.5312 −1.50376
\(469\) 16.2155 0.748764
\(470\) 0 0
\(471\) 0.0302708 0.00139481
\(472\) 111.506 5.13250
\(473\) −9.38838 −0.431678
\(474\) 4.34415 0.199534
\(475\) 0 0
\(476\) −62.4294 −2.86145
\(477\) −16.8905 −0.773365
\(478\) 40.9249 1.87186
\(479\) −14.3683 −0.656505 −0.328253 0.944590i \(-0.606460\pi\)
−0.328253 + 0.944590i \(0.606460\pi\)
\(480\) 0 0
\(481\) −4.91266 −0.223998
\(482\) −64.0861 −2.91904
\(483\) 1.29523 0.0589349
\(484\) 5.04540 0.229336
\(485\) 0 0
\(486\) 8.67657 0.393577
\(487\) −1.01712 −0.0460899 −0.0230449 0.999734i \(-0.507336\pi\)
−0.0230449 + 0.999734i \(0.507336\pi\)
\(488\) −56.4348 −2.55468
\(489\) −2.06829 −0.0935314
\(490\) 0 0
\(491\) −2.41045 −0.108782 −0.0543910 0.998520i \(-0.517322\pi\)
−0.0543910 + 0.998520i \(0.517322\pi\)
\(492\) 7.00067 0.315615
\(493\) −26.3658 −1.18745
\(494\) 5.73313 0.257946
\(495\) 0 0
\(496\) 52.8820 2.37447
\(497\) 51.4189 2.30645
\(498\) −3.22409 −0.144475
\(499\) 4.07426 0.182389 0.0911945 0.995833i \(-0.470931\pi\)
0.0911945 + 0.995833i \(0.470931\pi\)
\(500\) 0 0
\(501\) −0.105485 −0.00471272
\(502\) 52.8471 2.35868
\(503\) 19.9553 0.889762 0.444881 0.895590i \(-0.353246\pi\)
0.444881 + 0.895590i \(0.353246\pi\)
\(504\) 81.2923 3.62104
\(505\) 0 0
\(506\) 8.37352 0.372249
\(507\) 1.01577 0.0451118
\(508\) 18.1801 0.806613
\(509\) −8.79239 −0.389716 −0.194858 0.980831i \(-0.562425\pi\)
−0.194858 + 0.980831i \(0.562425\pi\)
\(510\) 0 0
\(511\) 27.2478 1.20537
\(512\) 24.6266 1.08835
\(513\) −0.729422 −0.0322048
\(514\) −59.2247 −2.61229
\(515\) 0 0
\(516\) 5.77285 0.254135
\(517\) 5.77094 0.253806
\(518\) 20.3384 0.893618
\(519\) −0.505138 −0.0221731
\(520\) 0 0
\(521\) 27.3483 1.19815 0.599075 0.800693i \(-0.295535\pi\)
0.599075 + 0.800693i \(0.295535\pi\)
\(522\) 56.8790 2.48953
\(523\) 41.7043 1.82360 0.911800 0.410634i \(-0.134693\pi\)
0.911800 + 0.410634i \(0.134693\pi\)
\(524\) 85.1368 3.71922
\(525\) 0 0
\(526\) 36.6197 1.59670
\(527\) −17.0897 −0.744441
\(528\) −1.38511 −0.0602790
\(529\) −13.0480 −0.567304
\(530\) 0 0
\(531\) 41.1783 1.78699
\(532\) −16.9974 −0.736930
\(533\) −24.5912 −1.06516
\(534\) −1.49330 −0.0646214
\(535\) 0 0
\(536\) −38.9083 −1.68058
\(537\) −0.249331 −0.0107594
\(538\) 29.7358 1.28200
\(539\) 4.34940 0.187342
\(540\) 0 0
\(541\) −20.7584 −0.892472 −0.446236 0.894915i \(-0.647236\pi\)
−0.446236 + 0.894915i \(0.647236\pi\)
\(542\) −45.4508 −1.95228
\(543\) −0.642898 −0.0275894
\(544\) 51.4207 2.20464
\(545\) 0 0
\(546\) 2.35387 0.100736
\(547\) −5.39396 −0.230629 −0.115315 0.993329i \(-0.536788\pi\)
−0.115315 + 0.993329i \(0.536788\pi\)
\(548\) 30.8193 1.31653
\(549\) −20.8409 −0.889466
\(550\) 0 0
\(551\) −7.17849 −0.305814
\(552\) −3.10783 −0.132278
\(553\) 45.2412 1.92385
\(554\) −36.3980 −1.54640
\(555\) 0 0
\(556\) 57.1881 2.42532
\(557\) 38.0076 1.61043 0.805217 0.592981i \(-0.202049\pi\)
0.805217 + 0.592981i \(0.202049\pi\)
\(558\) 36.8678 1.56074
\(559\) −20.2782 −0.857677
\(560\) 0 0
\(561\) 0.447622 0.0188986
\(562\) −68.6404 −2.89542
\(563\) 26.4313 1.11395 0.556973 0.830530i \(-0.311963\pi\)
0.556973 + 0.830530i \(0.311963\pi\)
\(564\) −3.54851 −0.149419
\(565\) 0 0
\(566\) 76.6330 3.22112
\(567\) 29.8704 1.25444
\(568\) −123.377 −5.17677
\(569\) 40.6105 1.70248 0.851239 0.524778i \(-0.175852\pi\)
0.851239 + 0.524778i \(0.175852\pi\)
\(570\) 0 0
\(571\) 8.44299 0.353328 0.176664 0.984271i \(-0.443469\pi\)
0.176664 + 0.984271i \(0.443469\pi\)
\(572\) 10.8977 0.455656
\(573\) 1.97309 0.0824271
\(574\) 101.807 4.24936
\(575\) 0 0
\(576\) −43.0761 −1.79484
\(577\) −20.9771 −0.873288 −0.436644 0.899634i \(-0.643833\pi\)
−0.436644 + 0.899634i \(0.643833\pi\)
\(578\) 9.31648 0.387514
\(579\) −2.44499 −0.101610
\(580\) 0 0
\(581\) −33.5766 −1.39299
\(582\) 1.32600 0.0549643
\(583\) 5.65820 0.234339
\(584\) −65.3795 −2.70542
\(585\) 0 0
\(586\) 16.9338 0.699528
\(587\) 21.1452 0.872756 0.436378 0.899764i \(-0.356261\pi\)
0.436378 + 0.899764i \(0.356261\pi\)
\(588\) −2.67441 −0.110291
\(589\) −4.65295 −0.191721
\(590\) 0 0
\(591\) −2.90540 −0.119512
\(592\) −25.8498 −1.06242
\(593\) −27.7731 −1.14051 −0.570253 0.821469i \(-0.693155\pi\)
−0.570253 + 0.821469i \(0.693155\pi\)
\(594\) −1.93612 −0.0794399
\(595\) 0 0
\(596\) 34.0917 1.39645
\(597\) 0.231793 0.00948667
\(598\) 18.0862 0.739600
\(599\) 23.0129 0.940281 0.470140 0.882592i \(-0.344203\pi\)
0.470140 + 0.882592i \(0.344203\pi\)
\(600\) 0 0
\(601\) 23.3322 0.951738 0.475869 0.879516i \(-0.342134\pi\)
0.475869 + 0.879516i \(0.342134\pi\)
\(602\) 83.9517 3.42162
\(603\) −14.3685 −0.585129
\(604\) 32.6049 1.32668
\(605\) 0 0
\(606\) −3.51726 −0.142879
\(607\) 3.22857 0.131044 0.0655218 0.997851i \(-0.479129\pi\)
0.0655218 + 0.997851i \(0.479129\pi\)
\(608\) 14.0001 0.567778
\(609\) −2.94730 −0.119430
\(610\) 0 0
\(611\) 12.4648 0.504273
\(612\) 55.3182 2.23611
\(613\) 17.2202 0.695519 0.347759 0.937584i \(-0.386943\pi\)
0.347759 + 0.937584i \(0.386943\pi\)
\(614\) 27.6842 1.11724
\(615\) 0 0
\(616\) −27.2322 −1.09722
\(617\) 27.4776 1.10621 0.553103 0.833113i \(-0.313443\pi\)
0.553103 + 0.833113i \(0.313443\pi\)
\(618\) −2.86565 −0.115274
\(619\) −9.53870 −0.383393 −0.191696 0.981454i \(-0.561399\pi\)
−0.191696 + 0.981454i \(0.561399\pi\)
\(620\) 0 0
\(621\) −2.30109 −0.0923397
\(622\) 21.9306 0.879337
\(623\) −15.5517 −0.623064
\(624\) −2.99173 −0.119765
\(625\) 0 0
\(626\) −64.0133 −2.55848
\(627\) 0.121872 0.00486710
\(628\) −1.25319 −0.0500076
\(629\) 8.35381 0.333088
\(630\) 0 0
\(631\) 1.16647 0.0464363 0.0232182 0.999730i \(-0.492609\pi\)
0.0232182 + 0.999730i \(0.492609\pi\)
\(632\) −108.554 −4.31804
\(633\) −1.49962 −0.0596044
\(634\) −31.1430 −1.23685
\(635\) 0 0
\(636\) −3.47919 −0.137959
\(637\) 9.39438 0.372219
\(638\) −19.0540 −0.754355
\(639\) −45.5619 −1.80240
\(640\) 0 0
\(641\) 10.0249 0.395960 0.197980 0.980206i \(-0.436562\pi\)
0.197980 + 0.980206i \(0.436562\pi\)
\(642\) 5.87676 0.231937
\(643\) 40.7197 1.60583 0.802915 0.596094i \(-0.203281\pi\)
0.802915 + 0.596094i \(0.203281\pi\)
\(644\) −53.6213 −2.11298
\(645\) 0 0
\(646\) −9.74900 −0.383569
\(647\) 16.6460 0.654423 0.327211 0.944951i \(-0.393891\pi\)
0.327211 + 0.944951i \(0.393891\pi\)
\(648\) −71.6723 −2.81555
\(649\) −13.7944 −0.541477
\(650\) 0 0
\(651\) −1.91038 −0.0748735
\(652\) 85.6256 3.35336
\(653\) 34.7416 1.35954 0.679771 0.733424i \(-0.262079\pi\)
0.679771 + 0.733424i \(0.262079\pi\)
\(654\) −0.0683538 −0.00267285
\(655\) 0 0
\(656\) −129.396 −5.05205
\(657\) −24.1441 −0.941949
\(658\) −51.6043 −2.01175
\(659\) −22.9537 −0.894148 −0.447074 0.894497i \(-0.647534\pi\)
−0.447074 + 0.894497i \(0.647534\pi\)
\(660\) 0 0
\(661\) −22.7141 −0.883476 −0.441738 0.897144i \(-0.645638\pi\)
−0.441738 + 0.897144i \(0.645638\pi\)
\(662\) 18.4534 0.717213
\(663\) 0.966831 0.0375486
\(664\) 80.5651 3.12653
\(665\) 0 0
\(666\) −18.0217 −0.698328
\(667\) −22.6459 −0.876851
\(668\) 4.36698 0.168964
\(669\) 2.78445 0.107653
\(670\) 0 0
\(671\) 6.98152 0.269518
\(672\) 5.74806 0.221736
\(673\) −20.9448 −0.807364 −0.403682 0.914899i \(-0.632270\pi\)
−0.403682 + 0.914899i \(0.632270\pi\)
\(674\) 78.1473 3.01012
\(675\) 0 0
\(676\) −42.0519 −1.61738
\(677\) −11.4153 −0.438726 −0.219363 0.975643i \(-0.570398\pi\)
−0.219363 + 0.975643i \(0.570398\pi\)
\(678\) 0.0641619 0.00246412
\(679\) 13.8093 0.529952
\(680\) 0 0
\(681\) 0.488285 0.0187111
\(682\) −12.3504 −0.472921
\(683\) 34.9550 1.33752 0.668759 0.743479i \(-0.266826\pi\)
0.668759 + 0.743479i \(0.266826\pi\)
\(684\) 15.0613 0.575882
\(685\) 0 0
\(686\) 23.7020 0.904945
\(687\) 2.99835 0.114394
\(688\) −106.701 −4.06795
\(689\) 12.2213 0.465594
\(690\) 0 0
\(691\) −15.8717 −0.603789 −0.301894 0.953341i \(-0.597619\pi\)
−0.301894 + 0.953341i \(0.597619\pi\)
\(692\) 20.9123 0.794967
\(693\) −10.0566 −0.382019
\(694\) −45.7020 −1.73483
\(695\) 0 0
\(696\) 7.07187 0.268059
\(697\) 41.8165 1.58391
\(698\) −11.4774 −0.434426
\(699\) 1.98644 0.0751339
\(700\) 0 0
\(701\) −18.5830 −0.701869 −0.350935 0.936400i \(-0.614136\pi\)
−0.350935 + 0.936400i \(0.614136\pi\)
\(702\) −4.18188 −0.157835
\(703\) 2.27446 0.0857828
\(704\) 14.4301 0.543857
\(705\) 0 0
\(706\) 62.2228 2.34178
\(707\) −36.6298 −1.37760
\(708\) 8.48208 0.318776
\(709\) 23.8063 0.894064 0.447032 0.894518i \(-0.352481\pi\)
0.447032 + 0.894518i \(0.352481\pi\)
\(710\) 0 0
\(711\) −40.0880 −1.50342
\(712\) 37.3153 1.39845
\(713\) −14.6786 −0.549717
\(714\) −4.00267 −0.149796
\(715\) 0 0
\(716\) 10.3221 0.385755
\(717\) 1.87905 0.0701745
\(718\) −21.7798 −0.812816
\(719\) 37.0144 1.38040 0.690202 0.723616i \(-0.257522\pi\)
0.690202 + 0.723616i \(0.257522\pi\)
\(720\) 0 0
\(721\) −29.8437 −1.11144
\(722\) −2.65432 −0.0987835
\(723\) −2.94249 −0.109432
\(724\) 26.6154 0.989154
\(725\) 0 0
\(726\) 0.323487 0.0120057
\(727\) 22.0744 0.818694 0.409347 0.912379i \(-0.365757\pi\)
0.409347 + 0.912379i \(0.365757\pi\)
\(728\) −58.8197 −2.18000
\(729\) −26.2013 −0.970417
\(730\) 0 0
\(731\) 34.4824 1.27538
\(732\) −4.29289 −0.158670
\(733\) 18.7090 0.691031 0.345515 0.938413i \(-0.387704\pi\)
0.345515 + 0.938413i \(0.387704\pi\)
\(734\) 87.9891 3.24774
\(735\) 0 0
\(736\) 44.1658 1.62797
\(737\) 4.81332 0.177301
\(738\) −90.2109 −3.32071
\(739\) −5.91630 −0.217635 −0.108817 0.994062i \(-0.534706\pi\)
−0.108817 + 0.994062i \(0.534706\pi\)
\(740\) 0 0
\(741\) 0.263235 0.00967017
\(742\) −50.5961 −1.85744
\(743\) −14.0422 −0.515159 −0.257579 0.966257i \(-0.582925\pi\)
−0.257579 + 0.966257i \(0.582925\pi\)
\(744\) 4.58384 0.168052
\(745\) 0 0
\(746\) −44.0191 −1.61165
\(747\) 29.7520 1.08857
\(748\) −18.5312 −0.677566
\(749\) 61.2023 2.23628
\(750\) 0 0
\(751\) 4.91166 0.179229 0.0896145 0.995977i \(-0.471437\pi\)
0.0896145 + 0.995977i \(0.471437\pi\)
\(752\) 65.5882 2.39176
\(753\) 2.42646 0.0884250
\(754\) −41.1553 −1.49879
\(755\) 0 0
\(756\) 12.3983 0.450920
\(757\) −49.2829 −1.79122 −0.895609 0.444843i \(-0.853259\pi\)
−0.895609 + 0.444843i \(0.853259\pi\)
\(758\) −15.3809 −0.558659
\(759\) 0.384467 0.0139553
\(760\) 0 0
\(761\) −7.25922 −0.263146 −0.131573 0.991306i \(-0.542003\pi\)
−0.131573 + 0.991306i \(0.542003\pi\)
\(762\) 1.16562 0.0422261
\(763\) −0.711856 −0.0257709
\(764\) −81.6843 −2.95524
\(765\) 0 0
\(766\) 15.6344 0.564892
\(767\) −29.7949 −1.07583
\(768\) 0.184693 0.00666453
\(769\) −21.5123 −0.775755 −0.387877 0.921711i \(-0.626792\pi\)
−0.387877 + 0.921711i \(0.626792\pi\)
\(770\) 0 0
\(771\) −2.71928 −0.0979325
\(772\) 101.221 3.64301
\(773\) −4.54038 −0.163306 −0.0816530 0.996661i \(-0.526020\pi\)
−0.0816530 + 0.996661i \(0.526020\pi\)
\(774\) −74.3891 −2.67386
\(775\) 0 0
\(776\) −33.1346 −1.18946
\(777\) 0.933831 0.0335010
\(778\) −36.6440 −1.31375
\(779\) 11.3852 0.407917
\(780\) 0 0
\(781\) 15.2629 0.546149
\(782\) −30.7550 −1.09980
\(783\) 5.23615 0.187125
\(784\) 49.4320 1.76543
\(785\) 0 0
\(786\) 5.45856 0.194701
\(787\) 7.25028 0.258444 0.129222 0.991616i \(-0.458752\pi\)
0.129222 + 0.991616i \(0.458752\pi\)
\(788\) 120.281 4.28484
\(789\) 1.68138 0.0598588
\(790\) 0 0
\(791\) 0.668201 0.0237585
\(792\) 24.1303 0.857433
\(793\) 15.0796 0.535491
\(794\) 17.2125 0.610848
\(795\) 0 0
\(796\) −9.59605 −0.340123
\(797\) −18.8674 −0.668319 −0.334159 0.942517i \(-0.608452\pi\)
−0.334159 + 0.942517i \(0.608452\pi\)
\(798\) −1.08979 −0.0385782
\(799\) −21.1960 −0.749860
\(800\) 0 0
\(801\) 13.7802 0.486900
\(802\) −9.84236 −0.347546
\(803\) 8.08806 0.285422
\(804\) −2.95968 −0.104380
\(805\) 0 0
\(806\) −26.6760 −0.939622
\(807\) 1.36531 0.0480612
\(808\) 87.8911 3.09200
\(809\) −19.4948 −0.685399 −0.342700 0.939445i \(-0.611341\pi\)
−0.342700 + 0.939445i \(0.611341\pi\)
\(810\) 0 0
\(811\) 4.28635 0.150514 0.0752570 0.997164i \(-0.476022\pi\)
0.0752570 + 0.997164i \(0.476022\pi\)
\(812\) 122.016 4.28191
\(813\) −2.08686 −0.0731893
\(814\) 6.03713 0.211601
\(815\) 0 0
\(816\) 5.08734 0.178092
\(817\) 9.38838 0.328458
\(818\) −82.4350 −2.88227
\(819\) −21.7216 −0.759014
\(820\) 0 0
\(821\) 5.37419 0.187561 0.0937803 0.995593i \(-0.470105\pi\)
0.0937803 + 0.995593i \(0.470105\pi\)
\(822\) 1.97598 0.0689204
\(823\) −26.3196 −0.917445 −0.458723 0.888580i \(-0.651693\pi\)
−0.458723 + 0.888580i \(0.651693\pi\)
\(824\) 71.6084 2.49460
\(825\) 0 0
\(826\) 123.351 4.29192
\(827\) 8.11186 0.282077 0.141039 0.990004i \(-0.454956\pi\)
0.141039 + 0.990004i \(0.454956\pi\)
\(828\) 47.5135 1.65121
\(829\) −21.5948 −0.750017 −0.375009 0.927021i \(-0.622360\pi\)
−0.375009 + 0.927021i \(0.622360\pi\)
\(830\) 0 0
\(831\) −1.67120 −0.0579733
\(832\) 31.1681 1.08056
\(833\) −15.9748 −0.553495
\(834\) 3.66663 0.126965
\(835\) 0 0
\(836\) −5.04540 −0.174499
\(837\) 3.39396 0.117313
\(838\) 29.9595 1.03493
\(839\) −4.35109 −0.150216 −0.0751082 0.997175i \(-0.523930\pi\)
−0.0751082 + 0.997175i \(0.523930\pi\)
\(840\) 0 0
\(841\) 22.5308 0.776923
\(842\) −91.6254 −3.15762
\(843\) −3.15160 −0.108547
\(844\) 62.0828 2.13698
\(845\) 0 0
\(846\) 45.7262 1.57210
\(847\) 3.36889 0.115756
\(848\) 64.3069 2.20831
\(849\) 3.51858 0.120757
\(850\) 0 0
\(851\) 7.17519 0.245962
\(852\) −9.38504 −0.321526
\(853\) −19.5022 −0.667744 −0.333872 0.942618i \(-0.608355\pi\)
−0.333872 + 0.942618i \(0.608355\pi\)
\(854\) −62.4294 −2.13629
\(855\) 0 0
\(856\) −146.851 −5.01928
\(857\) −13.3462 −0.455898 −0.227949 0.973673i \(-0.573202\pi\)
−0.227949 + 0.973673i \(0.573202\pi\)
\(858\) 0.698709 0.0238535
\(859\) 21.6016 0.737036 0.368518 0.929621i \(-0.379865\pi\)
0.368518 + 0.929621i \(0.379865\pi\)
\(860\) 0 0
\(861\) 4.67445 0.159305
\(862\) −48.2167 −1.64227
\(863\) 2.72425 0.0927345 0.0463673 0.998924i \(-0.485236\pi\)
0.0463673 + 0.998924i \(0.485236\pi\)
\(864\) −10.2120 −0.347418
\(865\) 0 0
\(866\) 28.6147 0.972367
\(867\) 0.427763 0.0145276
\(868\) 79.0879 2.68442
\(869\) 13.4291 0.455552
\(870\) 0 0
\(871\) 10.3964 0.352269
\(872\) 1.70806 0.0578422
\(873\) −12.2363 −0.414137
\(874\) −8.37352 −0.283239
\(875\) 0 0
\(876\) −4.97329 −0.168032
\(877\) −1.29130 −0.0436041 −0.0218021 0.999762i \(-0.506940\pi\)
−0.0218021 + 0.999762i \(0.506940\pi\)
\(878\) 3.68266 0.124284
\(879\) 0.777508 0.0262247
\(880\) 0 0
\(881\) 24.9873 0.841844 0.420922 0.907097i \(-0.361707\pi\)
0.420922 + 0.907097i \(0.361707\pi\)
\(882\) 34.4626 1.16041
\(883\) −15.5046 −0.521772 −0.260886 0.965370i \(-0.584015\pi\)
−0.260886 + 0.965370i \(0.584015\pi\)
\(884\) −40.0260 −1.34622
\(885\) 0 0
\(886\) −19.2914 −0.648107
\(887\) 9.98029 0.335105 0.167553 0.985863i \(-0.446414\pi\)
0.167553 + 0.985863i \(0.446414\pi\)
\(888\) −2.24068 −0.0751921
\(889\) 12.1391 0.407134
\(890\) 0 0
\(891\) 8.86655 0.297040
\(892\) −115.274 −3.85966
\(893\) −5.77094 −0.193117
\(894\) 2.18580 0.0731040
\(895\) 0 0
\(896\) −34.7063 −1.15946
\(897\) 0.830422 0.0277270
\(898\) −56.8815 −1.89816
\(899\) 33.4012 1.11399
\(900\) 0 0
\(901\) −20.7819 −0.692345
\(902\) 30.2199 1.00621
\(903\) 3.85461 0.128274
\(904\) −1.60331 −0.0533253
\(905\) 0 0
\(906\) 2.09047 0.0694513
\(907\) −11.8887 −0.394758 −0.197379 0.980327i \(-0.563243\pi\)
−0.197379 + 0.980327i \(0.563243\pi\)
\(908\) −20.2146 −0.670845
\(909\) 32.4574 1.07654
\(910\) 0 0
\(911\) −23.7166 −0.785767 −0.392883 0.919588i \(-0.628522\pi\)
−0.392883 + 0.919588i \(0.628522\pi\)
\(912\) 1.38511 0.0458655
\(913\) −9.96666 −0.329848
\(914\) −46.7989 −1.54797
\(915\) 0 0
\(916\) −124.129 −4.10134
\(917\) 56.8471 1.87726
\(918\) 7.11113 0.234702
\(919\) 18.1426 0.598470 0.299235 0.954179i \(-0.403269\pi\)
0.299235 + 0.954179i \(0.403269\pi\)
\(920\) 0 0
\(921\) 1.27111 0.0418845
\(922\) −90.7972 −2.99024
\(923\) 32.9667 1.08511
\(924\) −2.07150 −0.0681475
\(925\) 0 0
\(926\) −29.1997 −0.959563
\(927\) 26.4443 0.868546
\(928\) −100.500 −3.29906
\(929\) −50.5019 −1.65691 −0.828457 0.560053i \(-0.810781\pi\)
−0.828457 + 0.560053i \(0.810781\pi\)
\(930\) 0 0
\(931\) −4.34940 −0.142546
\(932\) −82.2368 −2.69376
\(933\) 1.00694 0.0329656
\(934\) 11.6379 0.380805
\(935\) 0 0
\(936\) 52.1197 1.70359
\(937\) 39.6587 1.29559 0.647797 0.761813i \(-0.275691\pi\)
0.647797 + 0.761813i \(0.275691\pi\)
\(938\) −43.0412 −1.40534
\(939\) −2.93915 −0.0959154
\(940\) 0 0
\(941\) −34.8809 −1.13709 −0.568543 0.822653i \(-0.692493\pi\)
−0.568543 + 0.822653i \(0.692493\pi\)
\(942\) −0.0803484 −0.00261789
\(943\) 35.9166 1.16961
\(944\) −156.777 −5.10265
\(945\) 0 0
\(946\) 24.9197 0.810210
\(947\) 30.3067 0.984835 0.492417 0.870359i \(-0.336113\pi\)
0.492417 + 0.870359i \(0.336113\pi\)
\(948\) −8.25748 −0.268191
\(949\) 17.4696 0.567088
\(950\) 0 0
\(951\) −1.42992 −0.0463684
\(952\) 100.021 3.24169
\(953\) 6.65804 0.215675 0.107838 0.994169i \(-0.465607\pi\)
0.107838 + 0.994169i \(0.465607\pi\)
\(954\) 44.8329 1.45152
\(955\) 0 0
\(956\) −77.7912 −2.51595
\(957\) −0.874858 −0.0282801
\(958\) 38.1381 1.23219
\(959\) 20.5785 0.664513
\(960\) 0 0
\(961\) −9.35006 −0.301615
\(962\) 13.0398 0.420419
\(963\) −54.2309 −1.74757
\(964\) 121.817 3.92345
\(965\) 0 0
\(966\) −3.43794 −0.110614
\(967\) 20.0020 0.643221 0.321611 0.946872i \(-0.395776\pi\)
0.321611 + 0.946872i \(0.395776\pi\)
\(968\) −8.08346 −0.259812
\(969\) −0.447622 −0.0143797
\(970\) 0 0
\(971\) −0.491941 −0.0157871 −0.00789357 0.999969i \(-0.502513\pi\)
−0.00789357 + 0.999969i \(0.502513\pi\)
\(972\) −16.4927 −0.529002
\(973\) 38.1853 1.22417
\(974\) 2.69975 0.0865055
\(975\) 0 0
\(976\) 79.3467 2.53983
\(977\) −23.6515 −0.756679 −0.378339 0.925667i \(-0.623505\pi\)
−0.378339 + 0.925667i \(0.623505\pi\)
\(978\) 5.48991 0.175548
\(979\) −4.61626 −0.147536
\(980\) 0 0
\(981\) 0.630771 0.0201390
\(982\) 6.39809 0.204171
\(983\) −39.5609 −1.26180 −0.630899 0.775865i \(-0.717314\pi\)
−0.630899 + 0.775865i \(0.717314\pi\)
\(984\) −11.2161 −0.357556
\(985\) 0 0
\(986\) 69.9831 2.22872
\(987\) −2.36939 −0.0754186
\(988\) −10.8977 −0.346702
\(989\) 29.6173 0.941777
\(990\) 0 0
\(991\) −13.6096 −0.432322 −0.216161 0.976358i \(-0.569354\pi\)
−0.216161 + 0.976358i \(0.569354\pi\)
\(992\) −65.1417 −2.06825
\(993\) 0.847283 0.0268877
\(994\) −136.482 −4.32895
\(995\) 0 0
\(996\) 6.12843 0.194187
\(997\) −41.1821 −1.30425 −0.652125 0.758111i \(-0.726122\pi\)
−0.652125 + 0.758111i \(0.726122\pi\)
\(998\) −10.8144 −0.342323
\(999\) −1.65904 −0.0524897
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 5225.2.a.h.1.1 5
5.4 even 2 209.2.a.c.1.5 5
15.14 odd 2 1881.2.a.k.1.1 5
20.19 odd 2 3344.2.a.t.1.3 5
55.54 odd 2 2299.2.a.n.1.1 5
95.94 odd 2 3971.2.a.h.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.5 5 5.4 even 2
1881.2.a.k.1.1 5 15.14 odd 2
2299.2.a.n.1.1 5 55.54 odd 2
3344.2.a.t.1.3 5 20.19 odd 2
3971.2.a.h.1.1 5 95.94 odd 2
5225.2.a.h.1.1 5 1.1 even 1 trivial