Properties

Label 201.2.a.d.1.2
Level $201$
Weight $2$
Character 201.1
Self dual yes
Analytic conductor $1.605$
Analytic rank $0$
Dimension $3$
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] = [201,2,Mod(1,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, 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("201.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 201.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: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 201.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.53919 q^{2} -1.00000 q^{3} +0.369102 q^{4} +2.17009 q^{5} -1.53919 q^{6} +2.70928 q^{7} -2.51026 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.53919 q^{2} -1.00000 q^{3} +0.369102 q^{4} +2.17009 q^{5} -1.53919 q^{6} +2.70928 q^{7} -2.51026 q^{8} +1.00000 q^{9} +3.34017 q^{10} +4.63090 q^{11} -0.369102 q^{12} -1.36910 q^{13} +4.17009 q^{14} -2.17009 q^{15} -4.60197 q^{16} -6.04945 q^{17} +1.53919 q^{18} -0.447480 q^{19} +0.800984 q^{20} -2.70928 q^{21} +7.12783 q^{22} -5.58864 q^{23} +2.51026 q^{24} -0.290725 q^{25} -2.10731 q^{26} -1.00000 q^{27} +1.00000 q^{28} +8.68035 q^{29} -3.34017 q^{30} -4.75872 q^{31} -2.06278 q^{32} -4.63090 q^{33} -9.31124 q^{34} +5.87936 q^{35} +0.369102 q^{36} -7.97107 q^{37} -0.688756 q^{38} +1.36910 q^{39} -5.44748 q^{40} -1.72261 q^{41} -4.17009 q^{42} -1.00000 q^{43} +1.70928 q^{44} +2.17009 q^{45} -8.60197 q^{46} +13.1278 q^{47} +4.60197 q^{48} +0.340173 q^{49} -0.447480 q^{50} +6.04945 q^{51} -0.505339 q^{52} -4.03612 q^{53} -1.53919 q^{54} +10.0494 q^{55} -6.80098 q^{56} +0.447480 q^{57} +13.3607 q^{58} +7.69594 q^{59} -0.800984 q^{60} -10.3896 q^{61} -7.32457 q^{62} +2.70928 q^{63} +6.02893 q^{64} -2.97107 q^{65} -7.12783 q^{66} +1.00000 q^{67} -2.23287 q^{68} +5.58864 q^{69} +9.04945 q^{70} +10.9711 q^{71} -2.51026 q^{72} -3.73820 q^{73} -12.2690 q^{74} +0.290725 q^{75} -0.165166 q^{76} +12.5464 q^{77} +2.10731 q^{78} +5.65983 q^{79} -9.98667 q^{80} +1.00000 q^{81} -2.65142 q^{82} +2.95774 q^{83} -1.00000 q^{84} -13.1278 q^{85} -1.53919 q^{86} -8.68035 q^{87} -11.6248 q^{88} +0.814315 q^{89} +3.34017 q^{90} -3.70928 q^{91} -2.06278 q^{92} +4.75872 q^{93} +20.2062 q^{94} -0.971071 q^{95} +2.06278 q^{96} +13.9649 q^{97} +0.523590 q^{98} +4.63090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} + q^{5} - 3 q^{6} + q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} + q^{5} - 3 q^{6} + q^{7} + 9 q^{8} + 3 q^{9} - q^{10} + 10 q^{11} - 5 q^{12} - 8 q^{13} + 7 q^{14} - q^{15} + 5 q^{16} + 3 q^{18} - 2 q^{19} - 7 q^{20} - q^{21} + 3 q^{23} - 9 q^{24} - 8 q^{25} - 18 q^{26} - 3 q^{27} + 3 q^{28} + 4 q^{29} + q^{30} + 11 q^{31} + 11 q^{32} - 10 q^{33} - 2 q^{34} + 5 q^{35} + 5 q^{36} - 9 q^{37} - 28 q^{38} + 8 q^{39} - 17 q^{40} + q^{41} - 7 q^{42} - 3 q^{43} - 2 q^{44} + q^{45} - 7 q^{46} + 18 q^{47} - 5 q^{48} - 10 q^{49} - 2 q^{50} - 32 q^{52} + 7 q^{53} - 3 q^{54} + 12 q^{55} - 11 q^{56} + 2 q^{57} - 4 q^{58} + 15 q^{59} + 7 q^{60} - 2 q^{61} + 3 q^{62} + q^{63} + 33 q^{64} + 6 q^{65} + 3 q^{67} + 16 q^{68} - 3 q^{69} + 9 q^{70} + 18 q^{71} + 9 q^{72} - 19 q^{73} + 5 q^{74} + 8 q^{75} - 42 q^{76} + 2 q^{77} + 18 q^{78} + 28 q^{79} - 29 q^{80} + 3 q^{81} + 29 q^{82} - 7 q^{83} - 3 q^{84} - 18 q^{85} - 3 q^{86} - 4 q^{87} + 4 q^{88} - 6 q^{89} - q^{90} - 4 q^{91} + 11 q^{92} - 11 q^{93} + 36 q^{94} + 12 q^{95} - 11 q^{96} - 8 q^{97} - 14 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.53919 1.08837 0.544185 0.838965i \(-0.316839\pi\)
0.544185 + 0.838965i \(0.316839\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.369102 0.184551
\(5\) 2.17009 0.970492 0.485246 0.874378i \(-0.338730\pi\)
0.485246 + 0.874378i \(0.338730\pi\)
\(6\) −1.53919 −0.628371
\(7\) 2.70928 1.02401 0.512005 0.858983i \(-0.328903\pi\)
0.512005 + 0.858983i \(0.328903\pi\)
\(8\) −2.51026 −0.887511
\(9\) 1.00000 0.333333
\(10\) 3.34017 1.05626
\(11\) 4.63090 1.39627 0.698134 0.715967i \(-0.254014\pi\)
0.698134 + 0.715967i \(0.254014\pi\)
\(12\) −0.369102 −0.106551
\(13\) −1.36910 −0.379721 −0.189860 0.981811i \(-0.560804\pi\)
−0.189860 + 0.981811i \(0.560804\pi\)
\(14\) 4.17009 1.11450
\(15\) −2.17009 −0.560314
\(16\) −4.60197 −1.15049
\(17\) −6.04945 −1.46721 −0.733603 0.679578i \(-0.762163\pi\)
−0.733603 + 0.679578i \(0.762163\pi\)
\(18\) 1.53919 0.362790
\(19\) −0.447480 −0.102659 −0.0513295 0.998682i \(-0.516346\pi\)
−0.0513295 + 0.998682i \(0.516346\pi\)
\(20\) 0.800984 0.179105
\(21\) −2.70928 −0.591212
\(22\) 7.12783 1.51966
\(23\) −5.58864 −1.16531 −0.582656 0.812719i \(-0.697986\pi\)
−0.582656 + 0.812719i \(0.697986\pi\)
\(24\) 2.51026 0.512405
\(25\) −0.290725 −0.0581449
\(26\) −2.10731 −0.413277
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 8.68035 1.61190 0.805950 0.591984i \(-0.201655\pi\)
0.805950 + 0.591984i \(0.201655\pi\)
\(30\) −3.34017 −0.609829
\(31\) −4.75872 −0.854692 −0.427346 0.904088i \(-0.640551\pi\)
−0.427346 + 0.904088i \(0.640551\pi\)
\(32\) −2.06278 −0.364651
\(33\) −4.63090 −0.806136
\(34\) −9.31124 −1.59687
\(35\) 5.87936 0.993794
\(36\) 0.369102 0.0615171
\(37\) −7.97107 −1.31044 −0.655218 0.755440i \(-0.727423\pi\)
−0.655218 + 0.755440i \(0.727423\pi\)
\(38\) −0.688756 −0.111731
\(39\) 1.36910 0.219232
\(40\) −5.44748 −0.861322
\(41\) −1.72261 −0.269026 −0.134513 0.990912i \(-0.542947\pi\)
−0.134513 + 0.990912i \(0.542947\pi\)
\(42\) −4.17009 −0.643458
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 1.70928 0.257683
\(45\) 2.17009 0.323497
\(46\) −8.60197 −1.26829
\(47\) 13.1278 1.91489 0.957445 0.288615i \(-0.0931949\pi\)
0.957445 + 0.288615i \(0.0931949\pi\)
\(48\) 4.60197 0.664237
\(49\) 0.340173 0.0485961
\(50\) −0.447480 −0.0632832
\(51\) 6.04945 0.847092
\(52\) −0.505339 −0.0700779
\(53\) −4.03612 −0.554403 −0.277202 0.960812i \(-0.589407\pi\)
−0.277202 + 0.960812i \(0.589407\pi\)
\(54\) −1.53919 −0.209457
\(55\) 10.0494 1.35507
\(56\) −6.80098 −0.908820
\(57\) 0.447480 0.0592702
\(58\) 13.3607 1.75434
\(59\) 7.69594 1.00193 0.500963 0.865469i \(-0.332979\pi\)
0.500963 + 0.865469i \(0.332979\pi\)
\(60\) −0.800984 −0.103407
\(61\) −10.3896 −1.33025 −0.665127 0.746730i \(-0.731623\pi\)
−0.665127 + 0.746730i \(0.731623\pi\)
\(62\) −7.32457 −0.930222
\(63\) 2.70928 0.341337
\(64\) 6.02893 0.753616
\(65\) −2.97107 −0.368516
\(66\) −7.12783 −0.877375
\(67\) 1.00000 0.122169
\(68\) −2.23287 −0.270775
\(69\) 5.58864 0.672793
\(70\) 9.04945 1.08162
\(71\) 10.9711 1.30203 0.651013 0.759066i \(-0.274344\pi\)
0.651013 + 0.759066i \(0.274344\pi\)
\(72\) −2.51026 −0.295837
\(73\) −3.73820 −0.437524 −0.218762 0.975778i \(-0.570202\pi\)
−0.218762 + 0.975778i \(0.570202\pi\)
\(74\) −12.2690 −1.42624
\(75\) 0.290725 0.0335700
\(76\) −0.165166 −0.0189458
\(77\) 12.5464 1.42979
\(78\) 2.10731 0.238606
\(79\) 5.65983 0.636780 0.318390 0.947960i \(-0.396858\pi\)
0.318390 + 0.947960i \(0.396858\pi\)
\(80\) −9.98667 −1.11654
\(81\) 1.00000 0.111111
\(82\) −2.65142 −0.292800
\(83\) 2.95774 0.324654 0.162327 0.986737i \(-0.448100\pi\)
0.162327 + 0.986737i \(0.448100\pi\)
\(84\) −1.00000 −0.109109
\(85\) −13.1278 −1.42391
\(86\) −1.53919 −0.165975
\(87\) −8.68035 −0.930631
\(88\) −11.6248 −1.23920
\(89\) 0.814315 0.0863172 0.0431586 0.999068i \(-0.486258\pi\)
0.0431586 + 0.999068i \(0.486258\pi\)
\(90\) 3.34017 0.352085
\(91\) −3.70928 −0.388838
\(92\) −2.06278 −0.215060
\(93\) 4.75872 0.493457
\(94\) 20.2062 2.08411
\(95\) −0.971071 −0.0996297
\(96\) 2.06278 0.210532
\(97\) 13.9649 1.41792 0.708962 0.705247i \(-0.249164\pi\)
0.708962 + 0.705247i \(0.249164\pi\)
\(98\) 0.523590 0.0528906
\(99\) 4.63090 0.465423
\(100\) −0.107307 −0.0107307
\(101\) −17.5669 −1.74797 −0.873986 0.485952i \(-0.838473\pi\)
−0.873986 + 0.485952i \(0.838473\pi\)
\(102\) 9.31124 0.921950
\(103\) −10.5464 −1.03917 −0.519583 0.854420i \(-0.673913\pi\)
−0.519583 + 0.854420i \(0.673913\pi\)
\(104\) 3.43680 0.337006
\(105\) −5.87936 −0.573767
\(106\) −6.21235 −0.603396
\(107\) 0.630898 0.0609912 0.0304956 0.999535i \(-0.490291\pi\)
0.0304956 + 0.999535i \(0.490291\pi\)
\(108\) −0.369102 −0.0355169
\(109\) −3.86603 −0.370299 −0.185149 0.982710i \(-0.559277\pi\)
−0.185149 + 0.982710i \(0.559277\pi\)
\(110\) 15.4680 1.47482
\(111\) 7.97107 0.756581
\(112\) −12.4680 −1.17812
\(113\) 10.5464 0.992120 0.496060 0.868288i \(-0.334780\pi\)
0.496060 + 0.868288i \(0.334780\pi\)
\(114\) 0.688756 0.0645080
\(115\) −12.1278 −1.13093
\(116\) 3.20394 0.297478
\(117\) −1.36910 −0.126574
\(118\) 11.8455 1.09047
\(119\) −16.3896 −1.50243
\(120\) 5.44748 0.497285
\(121\) 10.4452 0.949565
\(122\) −15.9916 −1.44781
\(123\) 1.72261 0.155322
\(124\) −1.75646 −0.157734
\(125\) −11.4813 −1.02692
\(126\) 4.17009 0.371501
\(127\) 19.8082 1.75769 0.878846 0.477106i \(-0.158314\pi\)
0.878846 + 0.477106i \(0.158314\pi\)
\(128\) 13.4052 1.18487
\(129\) 1.00000 0.0880451
\(130\) −4.57304 −0.401082
\(131\) 0.326842 0.0285563 0.0142782 0.999898i \(-0.495455\pi\)
0.0142782 + 0.999898i \(0.495455\pi\)
\(132\) −1.70928 −0.148773
\(133\) −1.21235 −0.105124
\(134\) 1.53919 0.132966
\(135\) −2.17009 −0.186771
\(136\) 15.1857 1.30216
\(137\) 8.40295 0.717913 0.358956 0.933354i \(-0.383133\pi\)
0.358956 + 0.933354i \(0.383133\pi\)
\(138\) 8.60197 0.732248
\(139\) −0.735937 −0.0624214 −0.0312107 0.999513i \(-0.509936\pi\)
−0.0312107 + 0.999513i \(0.509936\pi\)
\(140\) 2.17009 0.183406
\(141\) −13.1278 −1.10556
\(142\) 16.8865 1.41709
\(143\) −6.34017 −0.530192
\(144\) −4.60197 −0.383497
\(145\) 18.8371 1.56434
\(146\) −5.75380 −0.476188
\(147\) −0.340173 −0.0280570
\(148\) −2.94214 −0.241843
\(149\) 13.5753 1.11213 0.556066 0.831138i \(-0.312310\pi\)
0.556066 + 0.831138i \(0.312310\pi\)
\(150\) 0.447480 0.0365366
\(151\) 21.0433 1.71248 0.856240 0.516578i \(-0.172794\pi\)
0.856240 + 0.516578i \(0.172794\pi\)
\(152\) 1.12329 0.0911110
\(153\) −6.04945 −0.489069
\(154\) 19.3112 1.55614
\(155\) −10.3268 −0.829472
\(156\) 0.505339 0.0404595
\(157\) −7.24128 −0.577917 −0.288958 0.957342i \(-0.593309\pi\)
−0.288958 + 0.957342i \(0.593309\pi\)
\(158\) 8.71154 0.693053
\(159\) 4.03612 0.320085
\(160\) −4.47641 −0.353891
\(161\) −15.1412 −1.19329
\(162\) 1.53919 0.120930
\(163\) −18.3402 −1.43651 −0.718257 0.695778i \(-0.755060\pi\)
−0.718257 + 0.695778i \(0.755060\pi\)
\(164\) −0.635818 −0.0496491
\(165\) −10.0494 −0.782348
\(166\) 4.55252 0.353344
\(167\) −17.7454 −1.37318 −0.686590 0.727045i \(-0.740893\pi\)
−0.686590 + 0.727045i \(0.740893\pi\)
\(168\) 6.80098 0.524707
\(169\) −11.1256 −0.855812
\(170\) −20.2062 −1.54975
\(171\) −0.447480 −0.0342197
\(172\) −0.369102 −0.0281438
\(173\) −7.89269 −0.600070 −0.300035 0.953928i \(-0.596998\pi\)
−0.300035 + 0.953928i \(0.596998\pi\)
\(174\) −13.3607 −1.01287
\(175\) −0.787653 −0.0595410
\(176\) −21.3112 −1.60640
\(177\) −7.69594 −0.578463
\(178\) 1.25338 0.0939452
\(179\) 1.55252 0.116041 0.0580204 0.998315i \(-0.481521\pi\)
0.0580204 + 0.998315i \(0.481521\pi\)
\(180\) 0.800984 0.0597018
\(181\) 9.72979 0.723210 0.361605 0.932331i \(-0.382229\pi\)
0.361605 + 0.932331i \(0.382229\pi\)
\(182\) −5.70928 −0.423200
\(183\) 10.3896 0.768023
\(184\) 14.0289 1.03423
\(185\) −17.2979 −1.27177
\(186\) 7.32457 0.537064
\(187\) −28.0144 −2.04861
\(188\) 4.84551 0.353395
\(189\) −2.70928 −0.197071
\(190\) −1.49466 −0.108434
\(191\) 12.0989 0.875445 0.437723 0.899110i \(-0.355785\pi\)
0.437723 + 0.899110i \(0.355785\pi\)
\(192\) −6.02893 −0.435101
\(193\) 12.6020 0.907110 0.453555 0.891228i \(-0.350156\pi\)
0.453555 + 0.891228i \(0.350156\pi\)
\(194\) 21.4947 1.54323
\(195\) 2.97107 0.212763
\(196\) 0.125559 0.00896848
\(197\) −4.46081 −0.317820 −0.158910 0.987293i \(-0.550798\pi\)
−0.158910 + 0.987293i \(0.550798\pi\)
\(198\) 7.12783 0.506553
\(199\) −9.07838 −0.643549 −0.321775 0.946816i \(-0.604279\pi\)
−0.321775 + 0.946816i \(0.604279\pi\)
\(200\) 0.729794 0.0516042
\(201\) −1.00000 −0.0705346
\(202\) −27.0388 −1.90244
\(203\) 23.5174 1.65060
\(204\) 2.23287 0.156332
\(205\) −3.73820 −0.261088
\(206\) −16.2329 −1.13100
\(207\) −5.58864 −0.388437
\(208\) 6.30057 0.436866
\(209\) −2.07223 −0.143339
\(210\) −9.04945 −0.624471
\(211\) 15.3919 1.05962 0.529811 0.848116i \(-0.322263\pi\)
0.529811 + 0.848116i \(0.322263\pi\)
\(212\) −1.48974 −0.102316
\(213\) −10.9711 −0.751725
\(214\) 0.971071 0.0663810
\(215\) −2.17009 −0.147999
\(216\) 2.51026 0.170802
\(217\) −12.8927 −0.875213
\(218\) −5.95055 −0.403022
\(219\) 3.73820 0.252604
\(220\) 3.70928 0.250079
\(221\) 8.28231 0.557129
\(222\) 12.2690 0.823440
\(223\) −25.5174 −1.70877 −0.854387 0.519637i \(-0.826067\pi\)
−0.854387 + 0.519637i \(0.826067\pi\)
\(224\) −5.58864 −0.373407
\(225\) −0.290725 −0.0193816
\(226\) 16.2329 1.07979
\(227\) −23.6381 −1.56891 −0.784457 0.620183i \(-0.787058\pi\)
−0.784457 + 0.620183i \(0.787058\pi\)
\(228\) 0.165166 0.0109384
\(229\) 3.75872 0.248383 0.124192 0.992258i \(-0.460366\pi\)
0.124192 + 0.992258i \(0.460366\pi\)
\(230\) −18.6670 −1.23087
\(231\) −12.5464 −0.825491
\(232\) −21.7899 −1.43058
\(233\) 24.6670 1.61599 0.807995 0.589189i \(-0.200553\pi\)
0.807995 + 0.589189i \(0.200553\pi\)
\(234\) −2.10731 −0.137759
\(235\) 28.4885 1.85839
\(236\) 2.84059 0.184907
\(237\) −5.65983 −0.367645
\(238\) −25.2267 −1.63521
\(239\) 2.08452 0.134836 0.0674182 0.997725i \(-0.478524\pi\)
0.0674182 + 0.997725i \(0.478524\pi\)
\(240\) 9.98667 0.644637
\(241\) −10.1012 −0.650673 −0.325337 0.945598i \(-0.605478\pi\)
−0.325337 + 0.945598i \(0.605478\pi\)
\(242\) 16.0772 1.03348
\(243\) −1.00000 −0.0641500
\(244\) −3.83483 −0.245500
\(245\) 0.738205 0.0471622
\(246\) 2.65142 0.169048
\(247\) 0.612646 0.0389817
\(248\) 11.9456 0.758548
\(249\) −2.95774 −0.187439
\(250\) −17.6719 −1.11767
\(251\) −10.8371 −0.684032 −0.342016 0.939694i \(-0.611110\pi\)
−0.342016 + 0.939694i \(0.611110\pi\)
\(252\) 1.00000 0.0629941
\(253\) −25.8804 −1.62709
\(254\) 30.4885 1.91302
\(255\) 13.1278 0.822096
\(256\) 8.57531 0.535957
\(257\) 25.3607 1.58196 0.790978 0.611844i \(-0.209572\pi\)
0.790978 + 0.611844i \(0.209572\pi\)
\(258\) 1.53919 0.0958257
\(259\) −21.5958 −1.34190
\(260\) −1.09663 −0.0680101
\(261\) 8.68035 0.537300
\(262\) 0.503072 0.0310799
\(263\) −12.9311 −0.797364 −0.398682 0.917089i \(-0.630532\pi\)
−0.398682 + 0.917089i \(0.630532\pi\)
\(264\) 11.6248 0.715454
\(265\) −8.75872 −0.538044
\(266\) −1.86603 −0.114414
\(267\) −0.814315 −0.0498353
\(268\) 0.369102 0.0225465
\(269\) 22.0638 1.34526 0.672628 0.739981i \(-0.265166\pi\)
0.672628 + 0.739981i \(0.265166\pi\)
\(270\) −3.34017 −0.203276
\(271\) 16.1834 0.983073 0.491536 0.870857i \(-0.336436\pi\)
0.491536 + 0.870857i \(0.336436\pi\)
\(272\) 27.8394 1.68801
\(273\) 3.70928 0.224496
\(274\) 12.9337 0.781355
\(275\) −1.34632 −0.0811859
\(276\) 2.06278 0.124165
\(277\) −2.86991 −0.172436 −0.0862180 0.996276i \(-0.527478\pi\)
−0.0862180 + 0.996276i \(0.527478\pi\)
\(278\) −1.13275 −0.0679376
\(279\) −4.75872 −0.284897
\(280\) −14.7587 −0.882002
\(281\) −6.63090 −0.395566 −0.197783 0.980246i \(-0.563374\pi\)
−0.197783 + 0.980246i \(0.563374\pi\)
\(282\) −20.2062 −1.20326
\(283\) 16.2329 0.964944 0.482472 0.875911i \(-0.339739\pi\)
0.482472 + 0.875911i \(0.339739\pi\)
\(284\) 4.04945 0.240291
\(285\) 0.971071 0.0575213
\(286\) −9.75872 −0.577045
\(287\) −4.66701 −0.275485
\(288\) −2.06278 −0.121550
\(289\) 19.5958 1.15270
\(290\) 28.9939 1.70258
\(291\) −13.9649 −0.818639
\(292\) −1.37978 −0.0807455
\(293\) 33.3340 1.94739 0.973697 0.227845i \(-0.0731680\pi\)
0.973697 + 0.227845i \(0.0731680\pi\)
\(294\) −0.523590 −0.0305364
\(295\) 16.7009 0.972362
\(296\) 20.0095 1.16303
\(297\) −4.63090 −0.268712
\(298\) 20.8950 1.21041
\(299\) 7.65142 0.442493
\(300\) 0.107307 0.00619538
\(301\) −2.70928 −0.156160
\(302\) 32.3896 1.86381
\(303\) 17.5669 1.00919
\(304\) 2.05929 0.118108
\(305\) −22.5464 −1.29100
\(306\) −9.31124 −0.532288
\(307\) 24.5113 1.39893 0.699467 0.714665i \(-0.253421\pi\)
0.699467 + 0.714665i \(0.253421\pi\)
\(308\) 4.63090 0.263870
\(309\) 10.5464 0.599962
\(310\) −15.8950 −0.902773
\(311\) −3.21235 −0.182155 −0.0910777 0.995844i \(-0.529031\pi\)
−0.0910777 + 0.995844i \(0.529031\pi\)
\(312\) −3.43680 −0.194571
\(313\) 7.55252 0.426894 0.213447 0.976955i \(-0.431531\pi\)
0.213447 + 0.976955i \(0.431531\pi\)
\(314\) −11.1457 −0.628988
\(315\) 5.87936 0.331265
\(316\) 2.08906 0.117519
\(317\) −6.72979 −0.377983 −0.188991 0.981979i \(-0.560522\pi\)
−0.188991 + 0.981979i \(0.560522\pi\)
\(318\) 6.21235 0.348371
\(319\) 40.1978 2.25064
\(320\) 13.0833 0.731379
\(321\) −0.630898 −0.0352133
\(322\) −23.3051 −1.29874
\(323\) 2.70701 0.150622
\(324\) 0.369102 0.0205057
\(325\) 0.398032 0.0220788
\(326\) −28.2290 −1.56346
\(327\) 3.86603 0.213792
\(328\) 4.32419 0.238763
\(329\) 35.5669 1.96087
\(330\) −15.4680 −0.851485
\(331\) 1.49693 0.0822786 0.0411393 0.999153i \(-0.486901\pi\)
0.0411393 + 0.999153i \(0.486901\pi\)
\(332\) 1.09171 0.0599153
\(333\) −7.97107 −0.436812
\(334\) −27.3135 −1.49453
\(335\) 2.17009 0.118564
\(336\) 12.4680 0.680185
\(337\) −19.7503 −1.07587 −0.537934 0.842987i \(-0.680795\pi\)
−0.537934 + 0.842987i \(0.680795\pi\)
\(338\) −17.1243 −0.931441
\(339\) −10.5464 −0.572801
\(340\) −4.84551 −0.262785
\(341\) −22.0372 −1.19338
\(342\) −0.688756 −0.0372437
\(343\) −18.0433 −0.974247
\(344\) 2.51026 0.135344
\(345\) 12.1278 0.652940
\(346\) −12.1483 −0.653099
\(347\) −23.2495 −1.24810 −0.624050 0.781385i \(-0.714514\pi\)
−0.624050 + 0.781385i \(0.714514\pi\)
\(348\) −3.20394 −0.171749
\(349\) −9.25953 −0.495651 −0.247826 0.968805i \(-0.579716\pi\)
−0.247826 + 0.968805i \(0.579716\pi\)
\(350\) −1.21235 −0.0648027
\(351\) 1.36910 0.0730773
\(352\) −9.55252 −0.509151
\(353\) −16.1701 −0.860647 −0.430323 0.902675i \(-0.641600\pi\)
−0.430323 + 0.902675i \(0.641600\pi\)
\(354\) −11.8455 −0.629582
\(355\) 23.8082 1.26361
\(356\) 0.300566 0.0159299
\(357\) 16.3896 0.867431
\(358\) 2.38962 0.126295
\(359\) 4.30018 0.226955 0.113477 0.993541i \(-0.463801\pi\)
0.113477 + 0.993541i \(0.463801\pi\)
\(360\) −5.44748 −0.287107
\(361\) −18.7998 −0.989461
\(362\) 14.9760 0.787121
\(363\) −10.4452 −0.548231
\(364\) −1.36910 −0.0717605
\(365\) −8.11223 −0.424613
\(366\) 15.9916 0.835894
\(367\) 35.4329 1.84958 0.924792 0.380473i \(-0.124239\pi\)
0.924792 + 0.380473i \(0.124239\pi\)
\(368\) 25.7187 1.34068
\(369\) −1.72261 −0.0896753
\(370\) −26.6248 −1.38416
\(371\) −10.9350 −0.567714
\(372\) 1.75646 0.0910680
\(373\) −15.2846 −0.791406 −0.395703 0.918379i \(-0.629499\pi\)
−0.395703 + 0.918379i \(0.629499\pi\)
\(374\) −43.1194 −2.22965
\(375\) 11.4813 0.592893
\(376\) −32.9542 −1.69949
\(377\) −11.8843 −0.612072
\(378\) −4.17009 −0.214486
\(379\) −14.5402 −0.746882 −0.373441 0.927654i \(-0.621822\pi\)
−0.373441 + 0.927654i \(0.621822\pi\)
\(380\) −0.358424 −0.0183868
\(381\) −19.8082 −1.01480
\(382\) 18.6225 0.952809
\(383\) 3.13624 0.160254 0.0801271 0.996785i \(-0.474467\pi\)
0.0801271 + 0.996785i \(0.474467\pi\)
\(384\) −13.4052 −0.684082
\(385\) 27.2267 1.38760
\(386\) 19.3968 0.987272
\(387\) −1.00000 −0.0508329
\(388\) 5.15449 0.261679
\(389\) 26.9399 1.36591 0.682953 0.730462i \(-0.260695\pi\)
0.682953 + 0.730462i \(0.260695\pi\)
\(390\) 4.57304 0.231565
\(391\) 33.8082 1.70975
\(392\) −0.853922 −0.0431296
\(393\) −0.326842 −0.0164870
\(394\) −6.86603 −0.345906
\(395\) 12.2823 0.617990
\(396\) 1.70928 0.0858943
\(397\) −17.7854 −0.892623 −0.446311 0.894878i \(-0.647263\pi\)
−0.446311 + 0.894878i \(0.647263\pi\)
\(398\) −13.9733 −0.700420
\(399\) 1.21235 0.0606933
\(400\) 1.33791 0.0668953
\(401\) 9.93722 0.496241 0.248121 0.968729i \(-0.420187\pi\)
0.248121 + 0.968729i \(0.420187\pi\)
\(402\) −1.53919 −0.0767678
\(403\) 6.51518 0.324544
\(404\) −6.48398 −0.322590
\(405\) 2.17009 0.107832
\(406\) 36.1978 1.79647
\(407\) −36.9132 −1.82972
\(408\) −15.1857 −0.751803
\(409\) −18.3090 −0.905321 −0.452660 0.891683i \(-0.649525\pi\)
−0.452660 + 0.891683i \(0.649525\pi\)
\(410\) −5.75380 −0.284160
\(411\) −8.40295 −0.414487
\(412\) −3.89269 −0.191779
\(413\) 20.8504 1.02598
\(414\) −8.60197 −0.422764
\(415\) 6.41855 0.315074
\(416\) 2.82416 0.138466
\(417\) 0.735937 0.0360390
\(418\) −3.18956 −0.156007
\(419\) −15.9821 −0.780778 −0.390389 0.920650i \(-0.627660\pi\)
−0.390389 + 0.920650i \(0.627660\pi\)
\(420\) −2.17009 −0.105889
\(421\) 36.7731 1.79221 0.896106 0.443841i \(-0.146384\pi\)
0.896106 + 0.443841i \(0.146384\pi\)
\(422\) 23.6910 1.15326
\(423\) 13.1278 0.638297
\(424\) 10.1317 0.492039
\(425\) 1.75872 0.0853106
\(426\) −16.8865 −0.818156
\(427\) −28.1483 −1.36219
\(428\) 0.232866 0.0112560
\(429\) 6.34017 0.306106
\(430\) −3.34017 −0.161077
\(431\) 21.1412 1.01833 0.509167 0.860668i \(-0.329954\pi\)
0.509167 + 0.860668i \(0.329954\pi\)
\(432\) 4.60197 0.221412
\(433\) 13.3607 0.642074 0.321037 0.947067i \(-0.395969\pi\)
0.321037 + 0.947067i \(0.395969\pi\)
\(434\) −19.8443 −0.952556
\(435\) −18.8371 −0.903170
\(436\) −1.42696 −0.0683390
\(437\) 2.50080 0.119630
\(438\) 5.75380 0.274927
\(439\) −25.5441 −1.21915 −0.609577 0.792727i \(-0.708661\pi\)
−0.609577 + 0.792727i \(0.708661\pi\)
\(440\) −25.2267 −1.20264
\(441\) 0.340173 0.0161987
\(442\) 12.7480 0.606363
\(443\) −19.3958 −0.921521 −0.460760 0.887525i \(-0.652423\pi\)
−0.460760 + 0.887525i \(0.652423\pi\)
\(444\) 2.94214 0.139628
\(445\) 1.76713 0.0837702
\(446\) −39.2762 −1.85978
\(447\) −13.5753 −0.642090
\(448\) 16.3340 0.771710
\(449\) −21.3112 −1.00574 −0.502870 0.864362i \(-0.667723\pi\)
−0.502870 + 0.864362i \(0.667723\pi\)
\(450\) −0.447480 −0.0210944
\(451\) −7.97721 −0.375632
\(452\) 3.89269 0.183097
\(453\) −21.0433 −0.988701
\(454\) −36.3835 −1.70756
\(455\) −8.04945 −0.377364
\(456\) −1.12329 −0.0526029
\(457\) −24.6681 −1.15392 −0.576962 0.816771i \(-0.695762\pi\)
−0.576962 + 0.816771i \(0.695762\pi\)
\(458\) 5.78539 0.270333
\(459\) 6.04945 0.282364
\(460\) −4.47641 −0.208714
\(461\) 24.5958 1.14554 0.572771 0.819716i \(-0.305868\pi\)
0.572771 + 0.819716i \(0.305868\pi\)
\(462\) −19.3112 −0.898440
\(463\) 0.793796 0.0368908 0.0184454 0.999830i \(-0.494128\pi\)
0.0184454 + 0.999830i \(0.494128\pi\)
\(464\) −39.9467 −1.85448
\(465\) 10.3268 0.478896
\(466\) 37.9672 1.75880
\(467\) 5.80817 0.268770 0.134385 0.990929i \(-0.457094\pi\)
0.134385 + 0.990929i \(0.457094\pi\)
\(468\) −0.505339 −0.0233593
\(469\) 2.70928 0.125103
\(470\) 43.8492 2.02261
\(471\) 7.24128 0.333660
\(472\) −19.3188 −0.889221
\(473\) −4.63090 −0.212929
\(474\) −8.71154 −0.400134
\(475\) 0.130094 0.00596910
\(476\) −6.04945 −0.277276
\(477\) −4.03612 −0.184801
\(478\) 3.20847 0.146752
\(479\) 6.27739 0.286821 0.143411 0.989663i \(-0.454193\pi\)
0.143411 + 0.989663i \(0.454193\pi\)
\(480\) 4.47641 0.204319
\(481\) 10.9132 0.497600
\(482\) −15.5476 −0.708174
\(483\) 15.1412 0.688947
\(484\) 3.85535 0.175243
\(485\) 30.3051 1.37608
\(486\) −1.53919 −0.0698190
\(487\) 9.26794 0.419970 0.209985 0.977705i \(-0.432658\pi\)
0.209985 + 0.977705i \(0.432658\pi\)
\(488\) 26.0806 1.18062
\(489\) 18.3402 0.829371
\(490\) 1.13624 0.0513299
\(491\) −24.4608 −1.10390 −0.551950 0.833877i \(-0.686116\pi\)
−0.551950 + 0.833877i \(0.686116\pi\)
\(492\) 0.635818 0.0286649
\(493\) −52.5113 −2.36499
\(494\) 0.942978 0.0424266
\(495\) 10.0494 0.451689
\(496\) 21.8995 0.983316
\(497\) 29.7237 1.33329
\(498\) −4.55252 −0.204003
\(499\) 23.4329 1.04900 0.524501 0.851410i \(-0.324252\pi\)
0.524501 + 0.851410i \(0.324252\pi\)
\(500\) −4.23779 −0.189520
\(501\) 17.7454 0.792806
\(502\) −16.6803 −0.744480
\(503\) 15.9337 0.710450 0.355225 0.934781i \(-0.384404\pi\)
0.355225 + 0.934781i \(0.384404\pi\)
\(504\) −6.80098 −0.302940
\(505\) −38.1217 −1.69639
\(506\) −39.8348 −1.77087
\(507\) 11.1256 0.494103
\(508\) 7.31124 0.324384
\(509\) −27.9071 −1.23696 −0.618480 0.785801i \(-0.712251\pi\)
−0.618480 + 0.785801i \(0.712251\pi\)
\(510\) 20.2062 0.894746
\(511\) −10.1278 −0.448029
\(512\) −13.6114 −0.601546
\(513\) 0.447480 0.0197567
\(514\) 39.0349 1.72176
\(515\) −22.8865 −1.00850
\(516\) 0.369102 0.0162488
\(517\) 60.7936 2.67370
\(518\) −33.2401 −1.46048
\(519\) 7.89269 0.346451
\(520\) 7.45816 0.327062
\(521\) 19.7237 0.864109 0.432054 0.901848i \(-0.357789\pi\)
0.432054 + 0.901848i \(0.357789\pi\)
\(522\) 13.3607 0.584782
\(523\) −24.2739 −1.06142 −0.530712 0.847552i \(-0.678075\pi\)
−0.530712 + 0.847552i \(0.678075\pi\)
\(524\) 0.120638 0.00527010
\(525\) 0.787653 0.0343760
\(526\) −19.9034 −0.867828
\(527\) 28.7877 1.25401
\(528\) 21.3112 0.927453
\(529\) 8.23287 0.357951
\(530\) −13.4813 −0.585592
\(531\) 7.69594 0.333976
\(532\) −0.447480 −0.0194007
\(533\) 2.35842 0.102155
\(534\) −1.25338 −0.0542393
\(535\) 1.36910 0.0591915
\(536\) −2.51026 −0.108427
\(537\) −1.55252 −0.0669962
\(538\) 33.9604 1.46414
\(539\) 1.57531 0.0678532
\(540\) −0.800984 −0.0344689
\(541\) 37.1194 1.59589 0.797944 0.602731i \(-0.205921\pi\)
0.797944 + 0.602731i \(0.205921\pi\)
\(542\) 24.9093 1.06995
\(543\) −9.72979 −0.417545
\(544\) 12.4787 0.535019
\(545\) −8.38962 −0.359372
\(546\) 5.70928 0.244334
\(547\) 25.0494 1.07104 0.535519 0.844523i \(-0.320116\pi\)
0.535519 + 0.844523i \(0.320116\pi\)
\(548\) 3.10155 0.132492
\(549\) −10.3896 −0.443418
\(550\) −2.07223 −0.0883604
\(551\) −3.88428 −0.165476
\(552\) −14.0289 −0.597111
\(553\) 15.3340 0.652069
\(554\) −4.41733 −0.187674
\(555\) 17.2979 0.734255
\(556\) −0.271636 −0.0115199
\(557\) −23.0121 −0.975054 −0.487527 0.873108i \(-0.662101\pi\)
−0.487527 + 0.873108i \(0.662101\pi\)
\(558\) −7.32457 −0.310074
\(559\) 1.36910 0.0579069
\(560\) −27.0566 −1.14335
\(561\) 28.0144 1.18277
\(562\) −10.2062 −0.430523
\(563\) −35.3835 −1.49124 −0.745618 0.666374i \(-0.767846\pi\)
−0.745618 + 0.666374i \(0.767846\pi\)
\(564\) −4.84551 −0.204033
\(565\) 22.8865 0.962844
\(566\) 24.9854 1.05022
\(567\) 2.70928 0.113779
\(568\) −27.5402 −1.15556
\(569\) −27.5136 −1.15343 −0.576714 0.816946i \(-0.695665\pi\)
−0.576714 + 0.816946i \(0.695665\pi\)
\(570\) 1.49466 0.0626045
\(571\) 25.2800 1.05794 0.528969 0.848641i \(-0.322579\pi\)
0.528969 + 0.848641i \(0.322579\pi\)
\(572\) −2.34017 −0.0978475
\(573\) −12.0989 −0.505439
\(574\) −7.18342 −0.299830
\(575\) 1.62475 0.0677569
\(576\) 6.02893 0.251205
\(577\) −33.6475 −1.40077 −0.700383 0.713767i \(-0.746987\pi\)
−0.700383 + 0.713767i \(0.746987\pi\)
\(578\) 30.1617 1.25456
\(579\) −12.6020 −0.523720
\(580\) 6.95282 0.288700
\(581\) 8.01333 0.332449
\(582\) −21.4947 −0.890982
\(583\) −18.6908 −0.774096
\(584\) 9.38386 0.388307
\(585\) −2.97107 −0.122839
\(586\) 51.3074 2.11949
\(587\) −22.7838 −0.940387 −0.470194 0.882563i \(-0.655816\pi\)
−0.470194 + 0.882563i \(0.655816\pi\)
\(588\) −0.125559 −0.00517795
\(589\) 2.12943 0.0877418
\(590\) 25.7058 1.05829
\(591\) 4.46081 0.183493
\(592\) 36.6826 1.50765
\(593\) −25.6647 −1.05392 −0.526962 0.849889i \(-0.676669\pi\)
−0.526962 + 0.849889i \(0.676669\pi\)
\(594\) −7.12783 −0.292458
\(595\) −35.5669 −1.45810
\(596\) 5.01068 0.205245
\(597\) 9.07838 0.371553
\(598\) 11.7770 0.481596
\(599\) 1.44134 0.0588914 0.0294457 0.999566i \(-0.490626\pi\)
0.0294457 + 0.999566i \(0.490626\pi\)
\(600\) −0.729794 −0.0297937
\(601\) 28.4391 1.16005 0.580027 0.814597i \(-0.303042\pi\)
0.580027 + 0.814597i \(0.303042\pi\)
\(602\) −4.17009 −0.169960
\(603\) 1.00000 0.0407231
\(604\) 7.76713 0.316040
\(605\) 22.6670 0.921545
\(606\) 27.0388 1.09837
\(607\) 1.97721 0.0802526 0.0401263 0.999195i \(-0.487224\pi\)
0.0401263 + 0.999195i \(0.487224\pi\)
\(608\) 0.923053 0.0374347
\(609\) −23.5174 −0.952975
\(610\) −34.7031 −1.40509
\(611\) −17.9733 −0.727123
\(612\) −2.23287 −0.0902583
\(613\) −30.6020 −1.23600 −0.618001 0.786177i \(-0.712057\pi\)
−0.618001 + 0.786177i \(0.712057\pi\)
\(614\) 37.7275 1.52256
\(615\) 3.73820 0.150739
\(616\) −31.4947 −1.26896
\(617\) −37.5981 −1.51364 −0.756821 0.653622i \(-0.773249\pi\)
−0.756821 + 0.653622i \(0.773249\pi\)
\(618\) 16.2329 0.652982
\(619\) −31.3568 −1.26034 −0.630168 0.776458i \(-0.717014\pi\)
−0.630168 + 0.776458i \(0.717014\pi\)
\(620\) −3.81166 −0.153080
\(621\) 5.58864 0.224264
\(622\) −4.94441 −0.198253
\(623\) 2.20620 0.0883897
\(624\) −6.30057 −0.252224
\(625\) −23.4619 −0.938474
\(626\) 11.6248 0.464619
\(627\) 2.07223 0.0827571
\(628\) −2.67277 −0.106655
\(629\) 48.2206 1.92268
\(630\) 9.04945 0.360539
\(631\) −16.5814 −0.660097 −0.330049 0.943964i \(-0.607065\pi\)
−0.330049 + 0.943964i \(0.607065\pi\)
\(632\) −14.2076 −0.565149
\(633\) −15.3919 −0.611773
\(634\) −10.3584 −0.411386
\(635\) 42.9854 1.70583
\(636\) 1.48974 0.0590721
\(637\) −0.465732 −0.0184530
\(638\) 61.8720 2.44954
\(639\) 10.9711 0.434009
\(640\) 29.0905 1.14990
\(641\) −34.4524 −1.36079 −0.680394 0.732847i \(-0.738191\pi\)
−0.680394 + 0.732847i \(0.738191\pi\)
\(642\) −0.971071 −0.0383251
\(643\) −22.6765 −0.894273 −0.447136 0.894466i \(-0.647556\pi\)
−0.447136 + 0.894466i \(0.647556\pi\)
\(644\) −5.58864 −0.220223
\(645\) 2.17009 0.0854471
\(646\) 4.16660 0.163933
\(647\) 16.1399 0.634526 0.317263 0.948338i \(-0.397236\pi\)
0.317263 + 0.948338i \(0.397236\pi\)
\(648\) −2.51026 −0.0986123
\(649\) 35.6391 1.39896
\(650\) 0.612646 0.0240300
\(651\) 12.8927 0.505304
\(652\) −6.76940 −0.265110
\(653\) −14.6358 −0.572744 −0.286372 0.958119i \(-0.592449\pi\)
−0.286372 + 0.958119i \(0.592449\pi\)
\(654\) 5.95055 0.232685
\(655\) 0.709275 0.0277137
\(656\) 7.92738 0.309512
\(657\) −3.73820 −0.145841
\(658\) 54.7442 2.13415
\(659\) −46.4934 −1.81113 −0.905564 0.424210i \(-0.860552\pi\)
−0.905564 + 0.424210i \(0.860552\pi\)
\(660\) −3.70928 −0.144383
\(661\) −38.3812 −1.49286 −0.746428 0.665466i \(-0.768233\pi\)
−0.746428 + 0.665466i \(0.768233\pi\)
\(662\) 2.30406 0.0895497
\(663\) −8.28231 −0.321658
\(664\) −7.42469 −0.288134
\(665\) −2.63090 −0.102022
\(666\) −12.2690 −0.475413
\(667\) −48.5113 −1.87837
\(668\) −6.54987 −0.253422
\(669\) 25.5174 0.986562
\(670\) 3.34017 0.129042
\(671\) −48.1133 −1.85739
\(672\) 5.58864 0.215586
\(673\) 41.6658 1.60610 0.803049 0.595913i \(-0.203210\pi\)
0.803049 + 0.595913i \(0.203210\pi\)
\(674\) −30.3995 −1.17094
\(675\) 0.290725 0.0111900
\(676\) −4.10647 −0.157941
\(677\) 43.6875 1.67905 0.839524 0.543322i \(-0.182834\pi\)
0.839524 + 0.543322i \(0.182834\pi\)
\(678\) −16.2329 −0.623419
\(679\) 37.8348 1.45197
\(680\) 32.9542 1.26374
\(681\) 23.6381 0.905813
\(682\) −33.9194 −1.29884
\(683\) −40.3584 −1.54427 −0.772136 0.635457i \(-0.780812\pi\)
−0.772136 + 0.635457i \(0.780812\pi\)
\(684\) −0.165166 −0.00631528
\(685\) 18.2351 0.696729
\(686\) −27.7721 −1.06034
\(687\) −3.75872 −0.143404
\(688\) 4.60197 0.175448
\(689\) 5.52586 0.210518
\(690\) 18.6670 0.710641
\(691\) −0.255652 −0.00972547 −0.00486273 0.999988i \(-0.501548\pi\)
−0.00486273 + 0.999988i \(0.501548\pi\)
\(692\) −2.91321 −0.110744
\(693\) 12.5464 0.476597
\(694\) −35.7854 −1.35839
\(695\) −1.59705 −0.0605795
\(696\) 21.7899 0.825945
\(697\) 10.4208 0.394717
\(698\) −14.2522 −0.539452
\(699\) −24.6670 −0.932992
\(700\) −0.290725 −0.0109884
\(701\) 15.2218 0.574920 0.287460 0.957793i \(-0.407189\pi\)
0.287460 + 0.957793i \(0.407189\pi\)
\(702\) 2.10731 0.0795352
\(703\) 3.56690 0.134528
\(704\) 27.9194 1.05225
\(705\) −28.4885 −1.07294
\(706\) −24.8888 −0.936703
\(707\) −47.5936 −1.78994
\(708\) −2.84059 −0.106756
\(709\) −9.56916 −0.359377 −0.179689 0.983724i \(-0.557509\pi\)
−0.179689 + 0.983724i \(0.557509\pi\)
\(710\) 36.6453 1.37527
\(711\) 5.65983 0.212260
\(712\) −2.04414 −0.0766075
\(713\) 26.5948 0.995982
\(714\) 25.2267 0.944086
\(715\) −13.7587 −0.514547
\(716\) 0.573039 0.0214155
\(717\) −2.08452 −0.0778479
\(718\) 6.61879 0.247011
\(719\) 21.9516 0.818656 0.409328 0.912387i \(-0.365763\pi\)
0.409328 + 0.912387i \(0.365763\pi\)
\(720\) −9.98667 −0.372181
\(721\) −28.5730 −1.06412
\(722\) −28.9364 −1.07690
\(723\) 10.1012 0.375666
\(724\) 3.59129 0.133469
\(725\) −2.52359 −0.0937238
\(726\) −16.0772 −0.596679
\(727\) 6.70540 0.248690 0.124345 0.992239i \(-0.460317\pi\)
0.124345 + 0.992239i \(0.460317\pi\)
\(728\) 9.31124 0.345098
\(729\) 1.00000 0.0370370
\(730\) −12.4863 −0.462137
\(731\) 6.04945 0.223747
\(732\) 3.83483 0.141740
\(733\) −38.9939 −1.44027 −0.720135 0.693833i \(-0.755920\pi\)
−0.720135 + 0.693833i \(0.755920\pi\)
\(734\) 54.5380 2.01303
\(735\) −0.738205 −0.0272291
\(736\) 11.5281 0.424932
\(737\) 4.63090 0.170581
\(738\) −2.65142 −0.0976000
\(739\) 26.9565 0.991612 0.495806 0.868433i \(-0.334873\pi\)
0.495806 + 0.868433i \(0.334873\pi\)
\(740\) −6.38470 −0.234706
\(741\) −0.612646 −0.0225061
\(742\) −16.8310 −0.617884
\(743\) 31.3428 1.14986 0.574928 0.818204i \(-0.305030\pi\)
0.574928 + 0.818204i \(0.305030\pi\)
\(744\) −11.9456 −0.437948
\(745\) 29.4596 1.07932
\(746\) −23.5259 −0.861343
\(747\) 2.95774 0.108218
\(748\) −10.3402 −0.378074
\(749\) 1.70928 0.0624556
\(750\) 17.6719 0.645288
\(751\) 6.01438 0.219468 0.109734 0.993961i \(-0.465000\pi\)
0.109734 + 0.993961i \(0.465000\pi\)
\(752\) −60.4138 −2.20307
\(753\) 10.8371 0.394926
\(754\) −18.2922 −0.666161
\(755\) 45.6658 1.66195
\(756\) −1.00000 −0.0363696
\(757\) −3.57531 −0.129947 −0.0649734 0.997887i \(-0.520696\pi\)
−0.0649734 + 0.997887i \(0.520696\pi\)
\(758\) −22.3802 −0.812884
\(759\) 25.8804 0.939399
\(760\) 2.43764 0.0884225
\(761\) −7.81205 −0.283187 −0.141593 0.989925i \(-0.545223\pi\)
−0.141593 + 0.989925i \(0.545223\pi\)
\(762\) −30.4885 −1.10448
\(763\) −10.4741 −0.379189
\(764\) 4.46573 0.161565
\(765\) −13.1278 −0.474638
\(766\) 4.82726 0.174416
\(767\) −10.5365 −0.380452
\(768\) −8.57531 −0.309435
\(769\) −21.2990 −0.768060 −0.384030 0.923321i \(-0.625464\pi\)
−0.384030 + 0.923321i \(0.625464\pi\)
\(770\) 41.9071 1.51023
\(771\) −25.3607 −0.913343
\(772\) 4.65142 0.167408
\(773\) −20.1522 −0.724825 −0.362412 0.932018i \(-0.618047\pi\)
−0.362412 + 0.932018i \(0.618047\pi\)
\(774\) −1.53919 −0.0553250
\(775\) 1.38348 0.0496960
\(776\) −35.0556 −1.25842
\(777\) 21.5958 0.774746
\(778\) 41.4656 1.48661
\(779\) 0.770832 0.0276179
\(780\) 1.09663 0.0392656
\(781\) 50.8059 1.81798
\(782\) 52.0372 1.86085
\(783\) −8.68035 −0.310210
\(784\) −1.56547 −0.0559095
\(785\) −15.7142 −0.560864
\(786\) −0.503072 −0.0179440
\(787\) 17.9048 0.638237 0.319119 0.947715i \(-0.396613\pi\)
0.319119 + 0.947715i \(0.396613\pi\)
\(788\) −1.64650 −0.0586540
\(789\) 12.9311 0.460359
\(790\) 18.9048 0.672603
\(791\) 28.5730 1.01594
\(792\) −11.6248 −0.413068
\(793\) 14.2245 0.505125
\(794\) −27.3751 −0.971505
\(795\) 8.75872 0.310640
\(796\) −3.35085 −0.118768
\(797\) 25.3958 0.899564 0.449782 0.893138i \(-0.351502\pi\)
0.449782 + 0.893138i \(0.351502\pi\)
\(798\) 1.86603 0.0660568
\(799\) −79.4161 −2.80954
\(800\) 0.599701 0.0212026
\(801\) 0.814315 0.0287724
\(802\) 15.2953 0.540094
\(803\) −17.3112 −0.610901
\(804\) −0.369102 −0.0130172
\(805\) −32.8576 −1.15808
\(806\) 10.0281 0.353224
\(807\) −22.0638 −0.776683
\(808\) 44.0975 1.55134
\(809\) −13.8482 −0.486876 −0.243438 0.969917i \(-0.578275\pi\)
−0.243438 + 0.969917i \(0.578275\pi\)
\(810\) 3.34017 0.117362
\(811\) −15.4924 −0.544012 −0.272006 0.962296i \(-0.587687\pi\)
−0.272006 + 0.962296i \(0.587687\pi\)
\(812\) 8.68035 0.304620
\(813\) −16.1834 −0.567577
\(814\) −56.8164 −1.99141
\(815\) −39.7998 −1.39412
\(816\) −27.8394 −0.974573
\(817\) 0.447480 0.0156553
\(818\) −28.1810 −0.985325
\(819\) −3.70928 −0.129613
\(820\) −1.37978 −0.0481840
\(821\) −34.7670 −1.21338 −0.606688 0.794940i \(-0.707502\pi\)
−0.606688 + 0.794940i \(0.707502\pi\)
\(822\) −12.9337 −0.451116
\(823\) −42.4573 −1.47997 −0.739985 0.672624i \(-0.765167\pi\)
−0.739985 + 0.672624i \(0.765167\pi\)
\(824\) 26.4741 0.922270
\(825\) 1.34632 0.0468727
\(826\) 32.0928 1.11665
\(827\) −35.6802 −1.24072 −0.620361 0.784317i \(-0.713014\pi\)
−0.620361 + 0.784317i \(0.713014\pi\)
\(828\) −2.06278 −0.0716865
\(829\) −26.2823 −0.912823 −0.456411 0.889769i \(-0.650865\pi\)
−0.456411 + 0.889769i \(0.650865\pi\)
\(830\) 9.87936 0.342918
\(831\) 2.86991 0.0995560
\(832\) −8.25422 −0.286164
\(833\) −2.05786 −0.0713006
\(834\) 1.13275 0.0392238
\(835\) −38.5090 −1.33266
\(836\) −0.764867 −0.0264535
\(837\) 4.75872 0.164486
\(838\) −24.5995 −0.849776
\(839\) 12.0806 0.417070 0.208535 0.978015i \(-0.433130\pi\)
0.208535 + 0.978015i \(0.433130\pi\)
\(840\) 14.7587 0.509224
\(841\) 46.3484 1.59822
\(842\) 56.6007 1.95059
\(843\) 6.63090 0.228380
\(844\) 5.68118 0.195554
\(845\) −24.1434 −0.830559
\(846\) 20.2062 0.694704
\(847\) 28.2990 0.972364
\(848\) 18.5741 0.637837
\(849\) −16.2329 −0.557111
\(850\) 2.70701 0.0928496
\(851\) 44.5474 1.52707
\(852\) −4.04945 −0.138732
\(853\) −27.0472 −0.926078 −0.463039 0.886338i \(-0.653241\pi\)
−0.463039 + 0.886338i \(0.653241\pi\)
\(854\) −43.3256 −1.48257
\(855\) −0.971071 −0.0332099
\(856\) −1.58372 −0.0541303
\(857\) 44.2956 1.51311 0.756555 0.653930i \(-0.226881\pi\)
0.756555 + 0.653930i \(0.226881\pi\)
\(858\) 9.75872 0.333157
\(859\) 27.2990 0.931428 0.465714 0.884935i \(-0.345797\pi\)
0.465714 + 0.884935i \(0.345797\pi\)
\(860\) −0.800984 −0.0273133
\(861\) 4.66701 0.159051
\(862\) 32.5402 1.10832
\(863\) 10.7477 0.365855 0.182927 0.983126i \(-0.441443\pi\)
0.182927 + 0.983126i \(0.441443\pi\)
\(864\) 2.06278 0.0701772
\(865\) −17.1278 −0.582364
\(866\) 20.5646 0.698815
\(867\) −19.5958 −0.665509
\(868\) −4.75872 −0.161522
\(869\) 26.2101 0.889116
\(870\) −28.9939 −0.982984
\(871\) −1.36910 −0.0463903
\(872\) 9.70474 0.328644
\(873\) 13.9649 0.472641
\(874\) 3.84921 0.130201
\(875\) −31.1061 −1.05158
\(876\) 1.37978 0.0466185
\(877\) 45.0349 1.52072 0.760360 0.649502i \(-0.225022\pi\)
0.760360 + 0.649502i \(0.225022\pi\)
\(878\) −39.3172 −1.32689
\(879\) −33.3340 −1.12433
\(880\) −46.2472 −1.55899
\(881\) −39.9155 −1.34479 −0.672393 0.740194i \(-0.734734\pi\)
−0.672393 + 0.740194i \(0.734734\pi\)
\(882\) 0.523590 0.0176302
\(883\) −7.94828 −0.267481 −0.133741 0.991016i \(-0.542699\pi\)
−0.133741 + 0.991016i \(0.542699\pi\)
\(884\) 3.05702 0.102819
\(885\) −16.7009 −0.561393
\(886\) −29.8537 −1.00296
\(887\) 31.2934 1.05073 0.525364 0.850877i \(-0.323929\pi\)
0.525364 + 0.850877i \(0.323929\pi\)
\(888\) −20.0095 −0.671473
\(889\) 53.6658 1.79989
\(890\) 2.71995 0.0911730
\(891\) 4.63090 0.155141
\(892\) −9.41855 −0.315356
\(893\) −5.87444 −0.196581
\(894\) −20.8950 −0.698832
\(895\) 3.36910 0.112617
\(896\) 36.3184 1.21331
\(897\) −7.65142 −0.255473
\(898\) −32.8020 −1.09462
\(899\) −41.3074 −1.37768
\(900\) −0.107307 −0.00357691
\(901\) 24.4163 0.813424
\(902\) −12.2784 −0.408827
\(903\) 2.70928 0.0901590
\(904\) −26.4741 −0.880517
\(905\) 21.1145 0.701870
\(906\) −32.3896 −1.07607
\(907\) −17.9733 −0.596795 −0.298397 0.954442i \(-0.596452\pi\)
−0.298397 + 0.954442i \(0.596452\pi\)
\(908\) −8.72487 −0.289545
\(909\) −17.5669 −0.582657
\(910\) −12.3896 −0.410712
\(911\) 49.0121 1.62384 0.811922 0.583766i \(-0.198422\pi\)
0.811922 + 0.583766i \(0.198422\pi\)
\(912\) −2.05929 −0.0681899
\(913\) 13.6970 0.453304
\(914\) −37.9688 −1.25590
\(915\) 22.5464 0.745360
\(916\) 1.38735 0.0458395
\(917\) 0.885505 0.0292419
\(918\) 9.31124 0.307317
\(919\) 46.6307 1.53821 0.769103 0.639125i \(-0.220703\pi\)
0.769103 + 0.639125i \(0.220703\pi\)
\(920\) 30.4440 1.00371
\(921\) −24.5113 −0.807675
\(922\) 37.8576 1.24677
\(923\) −15.0205 −0.494406
\(924\) −4.63090 −0.152345
\(925\) 2.31739 0.0761952
\(926\) 1.22180 0.0401509
\(927\) −10.5464 −0.346388
\(928\) −17.9056 −0.587781
\(929\) 39.9214 1.30978 0.654890 0.755724i \(-0.272715\pi\)
0.654890 + 0.755724i \(0.272715\pi\)
\(930\) 15.8950 0.521216
\(931\) −0.152221 −0.00498883
\(932\) 9.10465 0.298233
\(933\) 3.21235 0.105168
\(934\) 8.93987 0.292522
\(935\) −60.7936 −1.98816
\(936\) 3.43680 0.112335
\(937\) 34.6888 1.13323 0.566616 0.823982i \(-0.308252\pi\)
0.566616 + 0.823982i \(0.308252\pi\)
\(938\) 4.17009 0.136158
\(939\) −7.55252 −0.246467
\(940\) 10.5152 0.342967
\(941\) −15.6248 −0.509352 −0.254676 0.967026i \(-0.581969\pi\)
−0.254676 + 0.967026i \(0.581969\pi\)
\(942\) 11.1457 0.363146
\(943\) 9.62702 0.313499
\(944\) −35.4165 −1.15271
\(945\) −5.87936 −0.191256
\(946\) −7.12783 −0.231746
\(947\) 29.9432 0.973023 0.486511 0.873674i \(-0.338269\pi\)
0.486511 + 0.873674i \(0.338269\pi\)
\(948\) −2.08906 −0.0678494
\(949\) 5.11799 0.166137
\(950\) 0.200238 0.00649659
\(951\) 6.72979 0.218229
\(952\) 41.1422 1.33343
\(953\) 42.3234 1.37099 0.685494 0.728078i \(-0.259586\pi\)
0.685494 + 0.728078i \(0.259586\pi\)
\(954\) −6.21235 −0.201132
\(955\) 26.2557 0.849613
\(956\) 0.769402 0.0248842
\(957\) −40.1978 −1.29941
\(958\) 9.66209 0.312168
\(959\) 22.7659 0.735150
\(960\) −13.0833 −0.422262
\(961\) −8.35455 −0.269502
\(962\) 16.7975 0.541573
\(963\) 0.630898 0.0203304
\(964\) −3.72836 −0.120083
\(965\) 27.3474 0.880343
\(966\) 23.3051 0.749829
\(967\) −15.4368 −0.496414 −0.248207 0.968707i \(-0.579841\pi\)
−0.248207 + 0.968707i \(0.579841\pi\)
\(968\) −26.2202 −0.842749
\(969\) −2.70701 −0.0869616
\(970\) 46.6453 1.49769
\(971\) 28.8143 0.924695 0.462348 0.886699i \(-0.347007\pi\)
0.462348 + 0.886699i \(0.347007\pi\)
\(972\) −0.369102 −0.0118390
\(973\) −1.99386 −0.0639201
\(974\) 14.2651 0.457084
\(975\) −0.398032 −0.0127472
\(976\) 47.8127 1.53045
\(977\) 9.78539 0.313062 0.156531 0.987673i \(-0.449969\pi\)
0.156531 + 0.987673i \(0.449969\pi\)
\(978\) 28.2290 0.902664
\(979\) 3.77101 0.120522
\(980\) 0.272473 0.00870384
\(981\) −3.86603 −0.123433
\(982\) −37.6498 −1.20145
\(983\) −27.2495 −0.869124 −0.434562 0.900642i \(-0.643097\pi\)
−0.434562 + 0.900642i \(0.643097\pi\)
\(984\) −4.32419 −0.137850
\(985\) −9.68035 −0.308441
\(986\) −80.8248 −2.57399
\(987\) −35.5669 −1.13211
\(988\) 0.226129 0.00719413
\(989\) 5.58864 0.177708
\(990\) 15.4680 0.491605
\(991\) −41.6186 −1.32206 −0.661029 0.750360i \(-0.729880\pi\)
−0.661029 + 0.750360i \(0.729880\pi\)
\(992\) 9.81620 0.311665
\(993\) −1.49693 −0.0475036
\(994\) 45.7503 1.45111
\(995\) −19.7009 −0.624559
\(996\) −1.09171 −0.0345921
\(997\) 1.03507 0.0327811 0.0163905 0.999866i \(-0.494782\pi\)
0.0163905 + 0.999866i \(0.494782\pi\)
\(998\) 36.0677 1.14170
\(999\) 7.97107 0.252194
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 201.2.a.d.1.2 3
3.2 odd 2 603.2.a.i.1.2 3
4.3 odd 2 3216.2.a.u.1.3 3
5.4 even 2 5025.2.a.m.1.2 3
7.6 odd 2 9849.2.a.ba.1.2 3
12.11 even 2 9648.2.a.bn.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
201.2.a.d.1.2 3 1.1 even 1 trivial
603.2.a.i.1.2 3 3.2 odd 2
3216.2.a.u.1.3 3 4.3 odd 2
5025.2.a.m.1.2 3 5.4 even 2
9648.2.a.bn.1.1 3 12.11 even 2
9849.2.a.ba.1.2 3 7.6 odd 2