Properties

Label 605.2.a.k.1.3
Level $605$
Weight $2$
Character 605.1
Self dual yes
Analytic conductor $4.831$
Analytic rank $0$
Dimension $4$
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] = [605,2,Mod(1,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, 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("605.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.737640\) of defining polynomial
Character \(\chi\) \(=\) 605.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.737640 q^{2} -2.81156 q^{3} -1.45589 q^{4} -1.00000 q^{5} -2.07392 q^{6} -1.03138 q^{7} -2.54920 q^{8} +4.90488 q^{9} +O(q^{10})\) \(q+0.737640 q^{2} -2.81156 q^{3} -1.45589 q^{4} -1.00000 q^{5} -2.07392 q^{6} -1.03138 q^{7} -2.54920 q^{8} +4.90488 q^{9} -0.737640 q^{10} +4.09331 q^{12} -3.44899 q^{13} -0.760787 q^{14} +2.81156 q^{15} +1.03138 q^{16} -2.39822 q^{17} +3.61803 q^{18} +7.66881 q^{19} +1.45589 q^{20} +2.89979 q^{21} +2.45589 q^{23} +7.16724 q^{24} +1.00000 q^{25} -2.54411 q^{26} -5.35567 q^{27} +1.50157 q^{28} +5.95431 q^{29} +2.07392 q^{30} -3.68820 q^{31} +5.85919 q^{32} -1.76902 q^{34} +1.03138 q^{35} -7.14094 q^{36} +5.95858 q^{37} +5.65682 q^{38} +9.69704 q^{39} +2.54920 q^{40} -3.93626 q^{41} +2.13900 q^{42} +7.64941 q^{43} -4.90488 q^{45} +1.81156 q^{46} +5.84294 q^{47} -2.89979 q^{48} -5.93626 q^{49} +0.737640 q^{50} +6.74273 q^{51} +5.02134 q^{52} -11.8480 q^{53} -3.95056 q^{54} +2.62920 q^{56} -21.5613 q^{57} +4.39214 q^{58} +2.94630 q^{59} -4.09331 q^{60} -2.48037 q^{61} -2.72057 q^{62} -5.05879 q^{63} +2.25922 q^{64} +3.44899 q^{65} -6.14702 q^{67} +3.49153 q^{68} -6.90488 q^{69} +0.760787 q^{70} +2.02315 q^{71} -12.5035 q^{72} +0.825867 q^{73} +4.39529 q^{74} -2.81156 q^{75} -11.1649 q^{76} +7.15293 q^{78} +12.0782 q^{79} -1.03138 q^{80} +0.343178 q^{81} -2.90354 q^{82} -1.61002 q^{83} -4.22176 q^{84} +2.39822 q^{85} +5.64252 q^{86} -16.7409 q^{87} +8.16116 q^{89} -3.61803 q^{90} +3.55722 q^{91} -3.57549 q^{92} +10.3696 q^{93} +4.30999 q^{94} -7.66881 q^{95} -16.4735 q^{96} +2.44278 q^{97} -4.37882 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{4} - 4 q^{5} + q^{6} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - q^{4} - 4 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} - q^{10} + 8 q^{12} + q^{13} + 2 q^{14} - 3 q^{16} - q^{17} + 10 q^{18} + 20 q^{19} + q^{20} + 10 q^{21} + 5 q^{23} + 11 q^{24} + 4 q^{25} - 15 q^{26} - 15 q^{27} + 13 q^{28} + 12 q^{29} - q^{30} - 5 q^{31} - 8 q^{32} + 2 q^{34} - 3 q^{35} + 7 q^{37} + 20 q^{38} + 7 q^{39} - 3 q^{40} + 11 q^{41} + 12 q^{42} + 19 q^{43} - 4 q^{46} + 5 q^{47} - 10 q^{48} + 3 q^{49} + q^{50} + 7 q^{51} - 11 q^{52} - 11 q^{53} - 8 q^{54} + 11 q^{56} - 5 q^{57} - 14 q^{58} + 9 q^{59} - 8 q^{60} + 12 q^{61} - 35 q^{62} - 5 q^{63} - 3 q^{64} - q^{65} - 19 q^{67} - 3 q^{68} - 8 q^{69} - 2 q^{70} + 5 q^{71} - 25 q^{72} + 11 q^{73} - 8 q^{78} + 34 q^{79} + 3 q^{80} + 4 q^{81} - 6 q^{82} - 11 q^{83} + 11 q^{84} + q^{85} + q^{86} - 19 q^{87} - 8 q^{89} - 10 q^{90} - 8 q^{91} - 12 q^{92} - 5 q^{93} - q^{94} - 20 q^{95} - 34 q^{96} + 32 q^{97} - 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.737640 0.521590 0.260795 0.965394i \(-0.416015\pi\)
0.260795 + 0.965394i \(0.416015\pi\)
\(3\) −2.81156 −1.62326 −0.811628 0.584175i \(-0.801418\pi\)
−0.811628 + 0.584175i \(0.801418\pi\)
\(4\) −1.45589 −0.727943
\(5\) −1.00000 −0.447214
\(6\) −2.07392 −0.846675
\(7\) −1.03138 −0.389825 −0.194912 0.980821i \(-0.562442\pi\)
−0.194912 + 0.980821i \(0.562442\pi\)
\(8\) −2.54920 −0.901279
\(9\) 4.90488 1.63496
\(10\) −0.737640 −0.233262
\(11\) 0 0
\(12\) 4.09331 1.18164
\(13\) −3.44899 −0.956577 −0.478289 0.878203i \(-0.658743\pi\)
−0.478289 + 0.878203i \(0.658743\pi\)
\(14\) −0.760787 −0.203329
\(15\) 2.81156 0.725942
\(16\) 1.03138 0.257845
\(17\) −2.39822 −0.581653 −0.290826 0.956776i \(-0.593930\pi\)
−0.290826 + 0.956776i \(0.593930\pi\)
\(18\) 3.61803 0.852779
\(19\) 7.66881 1.75935 0.879673 0.475580i \(-0.157762\pi\)
0.879673 + 0.475580i \(0.157762\pi\)
\(20\) 1.45589 0.325546
\(21\) 2.89979 0.632786
\(22\) 0 0
\(23\) 2.45589 0.512088 0.256044 0.966665i \(-0.417581\pi\)
0.256044 + 0.966665i \(0.417581\pi\)
\(24\) 7.16724 1.46301
\(25\) 1.00000 0.200000
\(26\) −2.54411 −0.498942
\(27\) −5.35567 −1.03070
\(28\) 1.50157 0.283770
\(29\) 5.95431 1.10569 0.552844 0.833285i \(-0.313542\pi\)
0.552844 + 0.833285i \(0.313542\pi\)
\(30\) 2.07392 0.378644
\(31\) −3.68820 −0.662421 −0.331210 0.943557i \(-0.607457\pi\)
−0.331210 + 0.943557i \(0.607457\pi\)
\(32\) 5.85919 1.03577
\(33\) 0 0
\(34\) −1.76902 −0.303384
\(35\) 1.03138 0.174335
\(36\) −7.14094 −1.19016
\(37\) 5.95858 0.979584 0.489792 0.871839i \(-0.337073\pi\)
0.489792 + 0.871839i \(0.337073\pi\)
\(38\) 5.65682 0.917658
\(39\) 9.69704 1.55277
\(40\) 2.54920 0.403064
\(41\) −3.93626 −0.614740 −0.307370 0.951590i \(-0.599449\pi\)
−0.307370 + 0.951590i \(0.599449\pi\)
\(42\) 2.13900 0.330055
\(43\) 7.64941 1.16652 0.583262 0.812284i \(-0.301776\pi\)
0.583262 + 0.812284i \(0.301776\pi\)
\(44\) 0 0
\(45\) −4.90488 −0.731176
\(46\) 1.81156 0.267100
\(47\) 5.84294 0.852281 0.426140 0.904657i \(-0.359873\pi\)
0.426140 + 0.904657i \(0.359873\pi\)
\(48\) −2.89979 −0.418548
\(49\) −5.93626 −0.848037
\(50\) 0.737640 0.104318
\(51\) 6.74273 0.944171
\(52\) 5.02134 0.696334
\(53\) −11.8480 −1.62745 −0.813726 0.581249i \(-0.802564\pi\)
−0.813726 + 0.581249i \(0.802564\pi\)
\(54\) −3.95056 −0.537603
\(55\) 0 0
\(56\) 2.62920 0.351341
\(57\) −21.5613 −2.85587
\(58\) 4.39214 0.576717
\(59\) 2.94630 0.383575 0.191788 0.981436i \(-0.438572\pi\)
0.191788 + 0.981436i \(0.438572\pi\)
\(60\) −4.09331 −0.528445
\(61\) −2.48037 −0.317579 −0.158789 0.987312i \(-0.550759\pi\)
−0.158789 + 0.987312i \(0.550759\pi\)
\(62\) −2.72057 −0.345512
\(63\) −5.05879 −0.637348
\(64\) 2.25922 0.282402
\(65\) 3.44899 0.427794
\(66\) 0 0
\(67\) −6.14702 −0.750978 −0.375489 0.926827i \(-0.622525\pi\)
−0.375489 + 0.926827i \(0.622525\pi\)
\(68\) 3.49153 0.423410
\(69\) −6.90488 −0.831249
\(70\) 0.760787 0.0909315
\(71\) 2.02315 0.240103 0.120052 0.992768i \(-0.461694\pi\)
0.120052 + 0.992768i \(0.461694\pi\)
\(72\) −12.5035 −1.47355
\(73\) 0.825867 0.0966604 0.0483302 0.998831i \(-0.484610\pi\)
0.0483302 + 0.998831i \(0.484610\pi\)
\(74\) 4.39529 0.510942
\(75\) −2.81156 −0.324651
\(76\) −11.1649 −1.28070
\(77\) 0 0
\(78\) 7.15293 0.809910
\(79\) 12.0782 1.35890 0.679451 0.733721i \(-0.262218\pi\)
0.679451 + 0.733721i \(0.262218\pi\)
\(80\) −1.03138 −0.115312
\(81\) 0.343178 0.0381309
\(82\) −2.90354 −0.320642
\(83\) −1.61002 −0.176722 −0.0883612 0.996088i \(-0.528163\pi\)
−0.0883612 + 0.996088i \(0.528163\pi\)
\(84\) −4.22176 −0.460632
\(85\) 2.39822 0.260123
\(86\) 5.64252 0.608448
\(87\) −16.7409 −1.79481
\(88\) 0 0
\(89\) 8.16116 0.865081 0.432541 0.901614i \(-0.357617\pi\)
0.432541 + 0.901614i \(0.357617\pi\)
\(90\) −3.61803 −0.381374
\(91\) 3.55722 0.372898
\(92\) −3.57549 −0.372771
\(93\) 10.3696 1.07528
\(94\) 4.30999 0.444541
\(95\) −7.66881 −0.786803
\(96\) −16.4735 −1.68132
\(97\) 2.44278 0.248027 0.124013 0.992281i \(-0.460423\pi\)
0.124013 + 0.992281i \(0.460423\pi\)
\(98\) −4.37882 −0.442328
\(99\) 0 0
\(100\) −1.45589 −0.145589
\(101\) 7.52373 0.748640 0.374320 0.927300i \(-0.377876\pi\)
0.374320 + 0.927300i \(0.377876\pi\)
\(102\) 4.97371 0.492471
\(103\) −9.48231 −0.934320 −0.467160 0.884173i \(-0.654723\pi\)
−0.467160 + 0.884173i \(0.654723\pi\)
\(104\) 8.79217 0.862143
\(105\) −2.89979 −0.282990
\(106\) −8.73958 −0.848863
\(107\) 4.64678 0.449221 0.224611 0.974449i \(-0.427889\pi\)
0.224611 + 0.974449i \(0.427889\pi\)
\(108\) 7.79726 0.750291
\(109\) 5.32826 0.510355 0.255178 0.966894i \(-0.417866\pi\)
0.255178 + 0.966894i \(0.417866\pi\)
\(110\) 0 0
\(111\) −16.7529 −1.59012
\(112\) −1.06374 −0.100514
\(113\) 0.304901 0.0286826 0.0143413 0.999897i \(-0.495435\pi\)
0.0143413 + 0.999897i \(0.495435\pi\)
\(114\) −15.9045 −1.48959
\(115\) −2.45589 −0.229013
\(116\) −8.66881 −0.804879
\(117\) −16.9169 −1.56396
\(118\) 2.17331 0.200069
\(119\) 2.47347 0.226743
\(120\) −7.16724 −0.654276
\(121\) 0 0
\(122\) −1.82962 −0.165646
\(123\) 11.0670 0.997880
\(124\) 5.36960 0.482205
\(125\) −1.00000 −0.0894427
\(126\) −3.73157 −0.332434
\(127\) 11.9100 1.05684 0.528419 0.848984i \(-0.322785\pi\)
0.528419 + 0.848984i \(0.322785\pi\)
\(128\) −10.0519 −0.888470
\(129\) −21.5068 −1.89357
\(130\) 2.54411 0.223133
\(131\) 11.1875 0.977452 0.488726 0.872437i \(-0.337462\pi\)
0.488726 + 0.872437i \(0.337462\pi\)
\(132\) 0 0
\(133\) −7.90945 −0.685837
\(134\) −4.53429 −0.391703
\(135\) 5.35567 0.460943
\(136\) 6.11353 0.524231
\(137\) −4.28124 −0.365771 −0.182886 0.983134i \(-0.558544\pi\)
−0.182886 + 0.983134i \(0.558544\pi\)
\(138\) −5.09331 −0.433572
\(139\) 5.95320 0.504943 0.252472 0.967604i \(-0.418757\pi\)
0.252472 + 0.967604i \(0.418757\pi\)
\(140\) −1.50157 −0.126906
\(141\) −16.4278 −1.38347
\(142\) 1.49235 0.125236
\(143\) 0 0
\(144\) 5.05879 0.421566
\(145\) −5.95431 −0.494479
\(146\) 0.609193 0.0504171
\(147\) 16.6901 1.37658
\(148\) −8.67501 −0.713082
\(149\) 3.78444 0.310034 0.155017 0.987912i \(-0.450457\pi\)
0.155017 + 0.987912i \(0.450457\pi\)
\(150\) −2.07392 −0.169335
\(151\) 24.3566 1.98211 0.991057 0.133437i \(-0.0426013\pi\)
0.991057 + 0.133437i \(0.0426013\pi\)
\(152\) −19.5493 −1.58566
\(153\) −11.7629 −0.950978
\(154\) 0 0
\(155\) 3.68820 0.296243
\(156\) −14.1178 −1.13033
\(157\) 7.23210 0.577184 0.288592 0.957452i \(-0.406813\pi\)
0.288592 + 0.957452i \(0.406813\pi\)
\(158\) 8.90936 0.708790
\(159\) 33.3115 2.64177
\(160\) −5.85919 −0.463210
\(161\) −2.53295 −0.199625
\(162\) 0.253142 0.0198887
\(163\) 18.6892 1.46385 0.731924 0.681386i \(-0.238623\pi\)
0.731924 + 0.681386i \(0.238623\pi\)
\(164\) 5.73074 0.447496
\(165\) 0 0
\(166\) −1.18761 −0.0921767
\(167\) −7.85328 −0.607705 −0.303852 0.952719i \(-0.598273\pi\)
−0.303852 + 0.952719i \(0.598273\pi\)
\(168\) −7.39214 −0.570316
\(169\) −1.10448 −0.0849597
\(170\) 1.76902 0.135678
\(171\) 37.6145 2.87646
\(172\) −11.1367 −0.849164
\(173\) −11.2047 −0.851877 −0.425938 0.904752i \(-0.640056\pi\)
−0.425938 + 0.904752i \(0.640056\pi\)
\(174\) −12.3488 −0.936158
\(175\) −1.03138 −0.0779650
\(176\) 0 0
\(177\) −8.28370 −0.622641
\(178\) 6.02000 0.451218
\(179\) 1.46463 0.109472 0.0547358 0.998501i \(-0.482568\pi\)
0.0547358 + 0.998501i \(0.482568\pi\)
\(180\) 7.14094 0.532255
\(181\) 9.28900 0.690446 0.345223 0.938521i \(-0.387803\pi\)
0.345223 + 0.938521i \(0.387803\pi\)
\(182\) 2.62395 0.194500
\(183\) 6.97371 0.515511
\(184\) −6.26055 −0.461534
\(185\) −5.95858 −0.438083
\(186\) 7.64904 0.560855
\(187\) 0 0
\(188\) −8.50666 −0.620412
\(189\) 5.52373 0.401793
\(190\) −5.65682 −0.410389
\(191\) −4.47296 −0.323652 −0.161826 0.986819i \(-0.551738\pi\)
−0.161826 + 0.986819i \(0.551738\pi\)
\(192\) −6.35192 −0.458410
\(193\) 22.6660 1.63154 0.815768 0.578380i \(-0.196315\pi\)
0.815768 + 0.578380i \(0.196315\pi\)
\(194\) 1.80189 0.129368
\(195\) −9.69704 −0.694420
\(196\) 8.64252 0.617323
\(197\) 11.2080 0.798535 0.399267 0.916835i \(-0.369265\pi\)
0.399267 + 0.916835i \(0.369265\pi\)
\(198\) 0 0
\(199\) −7.81979 −0.554330 −0.277165 0.960822i \(-0.589395\pi\)
−0.277165 + 0.960822i \(0.589395\pi\)
\(200\) −2.54920 −0.180256
\(201\) 17.2827 1.21903
\(202\) 5.54981 0.390483
\(203\) −6.14116 −0.431025
\(204\) −9.81665 −0.687303
\(205\) 3.93626 0.274920
\(206\) −6.99454 −0.487332
\(207\) 12.0458 0.837242
\(208\) −3.55722 −0.246649
\(209\) 0 0
\(210\) −2.13900 −0.147605
\(211\) 22.7670 1.56734 0.783672 0.621175i \(-0.213344\pi\)
0.783672 + 0.621175i \(0.213344\pi\)
\(212\) 17.2494 1.18469
\(213\) −5.68820 −0.389749
\(214\) 3.42765 0.234309
\(215\) −7.64941 −0.521686
\(216\) 13.6527 0.928948
\(217\) 3.80394 0.258228
\(218\) 3.93034 0.266196
\(219\) −2.32197 −0.156905
\(220\) 0 0
\(221\) 8.27142 0.556396
\(222\) −12.3576 −0.829389
\(223\) −16.0427 −1.07430 −0.537148 0.843488i \(-0.680499\pi\)
−0.537148 + 0.843488i \(0.680499\pi\)
\(224\) −6.04305 −0.403768
\(225\) 4.90488 0.326992
\(226\) 0.224907 0.0149606
\(227\) 24.1562 1.60330 0.801652 0.597791i \(-0.203955\pi\)
0.801652 + 0.597791i \(0.203955\pi\)
\(228\) 31.3908 2.07891
\(229\) −15.0143 −0.992172 −0.496086 0.868274i \(-0.665230\pi\)
−0.496086 + 0.868274i \(0.665230\pi\)
\(230\) −1.81156 −0.119451
\(231\) 0 0
\(232\) −15.1787 −0.996534
\(233\) −11.4259 −0.748538 −0.374269 0.927320i \(-0.622106\pi\)
−0.374269 + 0.927320i \(0.622106\pi\)
\(234\) −12.4786 −0.815749
\(235\) −5.84294 −0.381151
\(236\) −4.28948 −0.279221
\(237\) −33.9586 −2.20584
\(238\) 1.82453 0.118267
\(239\) −27.4067 −1.77279 −0.886397 0.462927i \(-0.846799\pi\)
−0.886397 + 0.462927i \(0.846799\pi\)
\(240\) 2.89979 0.187180
\(241\) −10.9387 −0.704624 −0.352312 0.935883i \(-0.614604\pi\)
−0.352312 + 0.935883i \(0.614604\pi\)
\(242\) 0 0
\(243\) 15.1022 0.968804
\(244\) 3.61114 0.231179
\(245\) 5.93626 0.379253
\(246\) 8.16348 0.520485
\(247\) −26.4496 −1.68295
\(248\) 9.40197 0.597026
\(249\) 4.52666 0.286866
\(250\) −0.737640 −0.0466525
\(251\) 17.2311 1.08762 0.543809 0.839209i \(-0.316982\pi\)
0.543809 + 0.839209i \(0.316982\pi\)
\(252\) 7.36503 0.463953
\(253\) 0 0
\(254\) 8.78527 0.551237
\(255\) −6.74273 −0.422246
\(256\) −11.9331 −0.745819
\(257\) 18.4954 1.15371 0.576856 0.816846i \(-0.304279\pi\)
0.576856 + 0.816846i \(0.304279\pi\)
\(258\) −15.8643 −0.987667
\(259\) −6.14556 −0.381866
\(260\) −5.02134 −0.311410
\(261\) 29.2052 1.80775
\(262\) 8.25232 0.509830
\(263\) −3.69135 −0.227618 −0.113809 0.993503i \(-0.536305\pi\)
−0.113809 + 0.993503i \(0.536305\pi\)
\(264\) 0 0
\(265\) 11.8480 0.727819
\(266\) −5.83433 −0.357726
\(267\) −22.9456 −1.40425
\(268\) 8.94936 0.546669
\(269\) −9.94510 −0.606363 −0.303182 0.952933i \(-0.598049\pi\)
−0.303182 + 0.952933i \(0.598049\pi\)
\(270\) 3.95056 0.240423
\(271\) −1.25365 −0.0761539 −0.0380770 0.999275i \(-0.512123\pi\)
−0.0380770 + 0.999275i \(0.512123\pi\)
\(272\) −2.47347 −0.149976
\(273\) −10.0013 −0.605308
\(274\) −3.15802 −0.190783
\(275\) 0 0
\(276\) 10.0527 0.605102
\(277\) 8.09331 0.486280 0.243140 0.969991i \(-0.421823\pi\)
0.243140 + 0.969991i \(0.421823\pi\)
\(278\) 4.39132 0.263374
\(279\) −18.0902 −1.08303
\(280\) −2.62920 −0.157124
\(281\) −25.3702 −1.51346 −0.756731 0.653726i \(-0.773205\pi\)
−0.756731 + 0.653726i \(0.773205\pi\)
\(282\) −12.1178 −0.721604
\(283\) 20.4424 1.21517 0.607587 0.794253i \(-0.292137\pi\)
0.607587 + 0.794253i \(0.292137\pi\)
\(284\) −2.94547 −0.174782
\(285\) 21.5613 1.27718
\(286\) 0 0
\(287\) 4.05977 0.239641
\(288\) 28.7386 1.69344
\(289\) −11.2486 −0.661680
\(290\) −4.39214 −0.257915
\(291\) −6.86803 −0.402611
\(292\) −1.20237 −0.0703633
\(293\) −2.54907 −0.148918 −0.0744591 0.997224i \(-0.523723\pi\)
−0.0744591 + 0.997224i \(0.523723\pi\)
\(294\) 12.3113 0.718011
\(295\) −2.94630 −0.171540
\(296\) −15.1896 −0.882878
\(297\) 0 0
\(298\) 2.79156 0.161711
\(299\) −8.47033 −0.489852
\(300\) 4.09331 0.236328
\(301\) −7.88945 −0.454740
\(302\) 17.9664 1.03385
\(303\) −21.1534 −1.21523
\(304\) 7.90945 0.453638
\(305\) 2.48037 0.142026
\(306\) −8.67682 −0.496021
\(307\) −8.99273 −0.513242 −0.256621 0.966512i \(-0.582609\pi\)
−0.256621 + 0.966512i \(0.582609\pi\)
\(308\) 0 0
\(309\) 26.6601 1.51664
\(310\) 2.72057 0.154518
\(311\) −20.1232 −1.14108 −0.570540 0.821270i \(-0.693266\pi\)
−0.570540 + 0.821270i \(0.693266\pi\)
\(312\) −24.7197 −1.39948
\(313\) 7.10483 0.401589 0.200794 0.979633i \(-0.435648\pi\)
0.200794 + 0.979633i \(0.435648\pi\)
\(314\) 5.33469 0.301054
\(315\) 5.05879 0.285031
\(316\) −17.5845 −0.989204
\(317\) 2.29323 0.128801 0.0644003 0.997924i \(-0.479487\pi\)
0.0644003 + 0.997924i \(0.479487\pi\)
\(318\) 24.5719 1.37792
\(319\) 0 0
\(320\) −2.25922 −0.126294
\(321\) −13.0647 −0.729201
\(322\) −1.86841 −0.104122
\(323\) −18.3915 −1.02333
\(324\) −0.499629 −0.0277572
\(325\) −3.44899 −0.191315
\(326\) 13.7859 0.763529
\(327\) −14.9807 −0.828437
\(328\) 10.0343 0.554052
\(329\) −6.02629 −0.332240
\(330\) 0 0
\(331\) 15.3951 0.846192 0.423096 0.906085i \(-0.360943\pi\)
0.423096 + 0.906085i \(0.360943\pi\)
\(332\) 2.34400 0.128644
\(333\) 29.2261 1.60158
\(334\) −5.79289 −0.316973
\(335\) 6.14702 0.335847
\(336\) 2.99078 0.163161
\(337\) 19.4968 1.06206 0.531030 0.847353i \(-0.321805\pi\)
0.531030 + 0.847353i \(0.321805\pi\)
\(338\) −0.814706 −0.0443141
\(339\) −0.857247 −0.0465592
\(340\) −3.49153 −0.189355
\(341\) 0 0
\(342\) 27.7460 1.50033
\(343\) 13.3422 0.720411
\(344\) −19.4999 −1.05136
\(345\) 6.90488 0.371746
\(346\) −8.26503 −0.444331
\(347\) 2.16905 0.116440 0.0582202 0.998304i \(-0.481457\pi\)
0.0582202 + 0.998304i \(0.481457\pi\)
\(348\) 24.3729 1.30652
\(349\) 25.0520 1.34100 0.670502 0.741908i \(-0.266079\pi\)
0.670502 + 0.741908i \(0.266079\pi\)
\(350\) −0.760787 −0.0406658
\(351\) 18.4717 0.985944
\(352\) 0 0
\(353\) −23.2532 −1.23764 −0.618821 0.785532i \(-0.712389\pi\)
−0.618821 + 0.785532i \(0.712389\pi\)
\(354\) −6.11039 −0.324764
\(355\) −2.02315 −0.107377
\(356\) −11.8817 −0.629730
\(357\) −6.95431 −0.368061
\(358\) 1.08037 0.0570993
\(359\) −10.1224 −0.534239 −0.267119 0.963663i \(-0.586072\pi\)
−0.267119 + 0.963663i \(0.586072\pi\)
\(360\) 12.5035 0.658993
\(361\) 39.8106 2.09530
\(362\) 6.85194 0.360130
\(363\) 0 0
\(364\) −5.17891 −0.271448
\(365\) −0.825867 −0.0432278
\(366\) 5.14409 0.268886
\(367\) 3.70925 0.193621 0.0968105 0.995303i \(-0.469136\pi\)
0.0968105 + 0.995303i \(0.469136\pi\)
\(368\) 2.53295 0.132039
\(369\) −19.3068 −1.00507
\(370\) −4.39529 −0.228500
\(371\) 12.2198 0.634421
\(372\) −15.0970 −0.782741
\(373\) 9.34017 0.483616 0.241808 0.970324i \(-0.422260\pi\)
0.241808 + 0.970324i \(0.422260\pi\)
\(374\) 0 0
\(375\) 2.81156 0.145188
\(376\) −14.8948 −0.768142
\(377\) −20.5364 −1.05768
\(378\) 4.07453 0.209571
\(379\) −9.81169 −0.503993 −0.251996 0.967728i \(-0.581087\pi\)
−0.251996 + 0.967728i \(0.581087\pi\)
\(380\) 11.1649 0.572748
\(381\) −33.4856 −1.71552
\(382\) −3.29944 −0.168814
\(383\) −18.0468 −0.922149 −0.461074 0.887362i \(-0.652536\pi\)
−0.461074 + 0.887362i \(0.652536\pi\)
\(384\) 28.2615 1.44221
\(385\) 0 0
\(386\) 16.7194 0.850993
\(387\) 37.5194 1.90722
\(388\) −3.55641 −0.180550
\(389\) 31.1915 1.58147 0.790737 0.612156i \(-0.209698\pi\)
0.790737 + 0.612156i \(0.209698\pi\)
\(390\) −7.15293 −0.362203
\(391\) −5.88974 −0.297857
\(392\) 15.1327 0.764317
\(393\) −31.4542 −1.58666
\(394\) 8.26745 0.416508
\(395\) −12.0782 −0.607719
\(396\) 0 0
\(397\) −10.6212 −0.533062 −0.266531 0.963826i \(-0.585877\pi\)
−0.266531 + 0.963826i \(0.585877\pi\)
\(398\) −5.76820 −0.289133
\(399\) 22.2379 1.11329
\(400\) 1.03138 0.0515690
\(401\) −27.5679 −1.37668 −0.688338 0.725390i \(-0.741659\pi\)
−0.688338 + 0.725390i \(0.741659\pi\)
\(402\) 12.7484 0.635834
\(403\) 12.7206 0.633657
\(404\) −10.9537 −0.544967
\(405\) −0.343178 −0.0170527
\(406\) −4.52997 −0.224818
\(407\) 0 0
\(408\) −17.1886 −0.850961
\(409\) −14.3682 −0.710460 −0.355230 0.934779i \(-0.615598\pi\)
−0.355230 + 0.934779i \(0.615598\pi\)
\(410\) 2.90354 0.143396
\(411\) 12.0370 0.593740
\(412\) 13.8052 0.680132
\(413\) −3.03875 −0.149527
\(414\) 8.88548 0.436698
\(415\) 1.61002 0.0790327
\(416\) −20.2083 −0.990793
\(417\) −16.7378 −0.819652
\(418\) 0 0
\(419\) −31.4707 −1.53744 −0.768722 0.639584i \(-0.779107\pi\)
−0.768722 + 0.639584i \(0.779107\pi\)
\(420\) 4.22176 0.206001
\(421\) 26.5712 1.29500 0.647500 0.762065i \(-0.275815\pi\)
0.647500 + 0.762065i \(0.275815\pi\)
\(422\) 16.7939 0.817512
\(423\) 28.6589 1.39344
\(424\) 30.2030 1.46679
\(425\) −2.39822 −0.116331
\(426\) −4.19585 −0.203289
\(427\) 2.55820 0.123800
\(428\) −6.76518 −0.327008
\(429\) 0 0
\(430\) −5.64252 −0.272106
\(431\) 3.86870 0.186349 0.0931744 0.995650i \(-0.470299\pi\)
0.0931744 + 0.995650i \(0.470299\pi\)
\(432\) −5.52373 −0.265761
\(433\) 40.1388 1.92895 0.964474 0.264179i \(-0.0851010\pi\)
0.964474 + 0.264179i \(0.0851010\pi\)
\(434\) 2.80594 0.134689
\(435\) 16.7409 0.802666
\(436\) −7.75735 −0.371510
\(437\) 18.8337 0.900939
\(438\) −1.71278 −0.0818399
\(439\) −1.02336 −0.0488425 −0.0244212 0.999702i \(-0.507774\pi\)
−0.0244212 + 0.999702i \(0.507774\pi\)
\(440\) 0 0
\(441\) −29.1166 −1.38650
\(442\) 6.10133 0.290211
\(443\) −16.0728 −0.763644 −0.381822 0.924236i \(-0.624703\pi\)
−0.381822 + 0.924236i \(0.624703\pi\)
\(444\) 24.3903 1.15751
\(445\) −8.16116 −0.386876
\(446\) −11.8337 −0.560343
\(447\) −10.6402 −0.503264
\(448\) −2.33011 −0.110087
\(449\) 35.8421 1.69149 0.845746 0.533585i \(-0.179156\pi\)
0.845746 + 0.533585i \(0.179156\pi\)
\(450\) 3.61803 0.170556
\(451\) 0 0
\(452\) −0.443901 −0.0208793
\(453\) −68.4802 −3.21748
\(454\) 17.8186 0.836268
\(455\) −3.55722 −0.166765
\(456\) 54.9641 2.57393
\(457\) 25.1525 1.17658 0.588291 0.808649i \(-0.299801\pi\)
0.588291 + 0.808649i \(0.299801\pi\)
\(458\) −11.0751 −0.517507
\(459\) 12.8441 0.599509
\(460\) 3.57549 0.166708
\(461\) −6.65631 −0.310015 −0.155008 0.987913i \(-0.549540\pi\)
−0.155008 + 0.987913i \(0.549540\pi\)
\(462\) 0 0
\(463\) 38.7730 1.80194 0.900968 0.433886i \(-0.142858\pi\)
0.900968 + 0.433886i \(0.142858\pi\)
\(464\) 6.14116 0.285096
\(465\) −10.3696 −0.480879
\(466\) −8.42823 −0.390430
\(467\) 22.5494 1.04346 0.521731 0.853110i \(-0.325286\pi\)
0.521731 + 0.853110i \(0.325286\pi\)
\(468\) 24.6290 1.13848
\(469\) 6.33991 0.292750
\(470\) −4.30999 −0.198805
\(471\) −20.3335 −0.936918
\(472\) −7.51071 −0.345708
\(473\) 0 0
\(474\) −25.0492 −1.15055
\(475\) 7.66881 0.351869
\(476\) −3.60109 −0.165056
\(477\) −58.1131 −2.66082
\(478\) −20.2163 −0.924672
\(479\) 1.63186 0.0745618 0.0372809 0.999305i \(-0.488130\pi\)
0.0372809 + 0.999305i \(0.488130\pi\)
\(480\) 16.4735 0.751908
\(481\) −20.5511 −0.937048
\(482\) −8.06883 −0.367525
\(483\) 7.12155 0.324042
\(484\) 0 0
\(485\) −2.44278 −0.110921
\(486\) 11.1400 0.505319
\(487\) 1.05030 0.0475936 0.0237968 0.999717i \(-0.492425\pi\)
0.0237968 + 0.999717i \(0.492425\pi\)
\(488\) 6.32296 0.286227
\(489\) −52.5457 −2.37620
\(490\) 4.37882 0.197815
\(491\) 13.9141 0.627934 0.313967 0.949434i \(-0.398342\pi\)
0.313967 + 0.949434i \(0.398342\pi\)
\(492\) −16.1123 −0.726400
\(493\) −14.2797 −0.643127
\(494\) −19.5103 −0.877811
\(495\) 0 0
\(496\) −3.80394 −0.170802
\(497\) −2.08663 −0.0935983
\(498\) 3.33905 0.149626
\(499\) 17.3673 0.777466 0.388733 0.921350i \(-0.372913\pi\)
0.388733 + 0.921350i \(0.372913\pi\)
\(500\) 1.45589 0.0651092
\(501\) 22.0800 0.986460
\(502\) 12.7104 0.567291
\(503\) −35.8536 −1.59863 −0.799316 0.600910i \(-0.794805\pi\)
−0.799316 + 0.600910i \(0.794805\pi\)
\(504\) 12.8959 0.574428
\(505\) −7.52373 −0.334802
\(506\) 0 0
\(507\) 3.10530 0.137911
\(508\) −17.3396 −0.769319
\(509\) 2.13878 0.0947999 0.0474000 0.998876i \(-0.484906\pi\)
0.0474000 + 0.998876i \(0.484906\pi\)
\(510\) −4.97371 −0.220240
\(511\) −0.851782 −0.0376806
\(512\) 11.3014 0.499458
\(513\) −41.0716 −1.81336
\(514\) 13.6430 0.601765
\(515\) 9.48231 0.417841
\(516\) 31.3115 1.37841
\(517\) 0 0
\(518\) −4.53321 −0.199178
\(519\) 31.5027 1.38281
\(520\) −8.79217 −0.385562
\(521\) −12.7352 −0.557940 −0.278970 0.960300i \(-0.589993\pi\)
−0.278970 + 0.960300i \(0.589993\pi\)
\(522\) 21.5429 0.942908
\(523\) −23.9088 −1.04546 −0.522729 0.852499i \(-0.675086\pi\)
−0.522729 + 0.852499i \(0.675086\pi\)
\(524\) −16.2877 −0.711530
\(525\) 2.89979 0.126557
\(526\) −2.72289 −0.118723
\(527\) 8.84510 0.385299
\(528\) 0 0
\(529\) −16.9686 −0.737766
\(530\) 8.73958 0.379623
\(531\) 14.4512 0.627130
\(532\) 11.5153 0.499250
\(533\) 13.5761 0.588046
\(534\) −16.9256 −0.732443
\(535\) −4.64678 −0.200898
\(536\) 15.6700 0.676840
\(537\) −4.11790 −0.177700
\(538\) −7.33590 −0.316273
\(539\) 0 0
\(540\) −7.79726 −0.335540
\(541\) −1.31921 −0.0567171 −0.0283586 0.999598i \(-0.509028\pi\)
−0.0283586 + 0.999598i \(0.509028\pi\)
\(542\) −0.924744 −0.0397212
\(543\) −26.1166 −1.12077
\(544\) −14.0516 −0.602457
\(545\) −5.32826 −0.228238
\(546\) −7.37739 −0.315723
\(547\) −9.32128 −0.398549 −0.199275 0.979944i \(-0.563859\pi\)
−0.199275 + 0.979944i \(0.563859\pi\)
\(548\) 6.23301 0.266261
\(549\) −12.1659 −0.519228
\(550\) 0 0
\(551\) 45.6625 1.94529
\(552\) 17.6019 0.749187
\(553\) −12.4572 −0.529734
\(554\) 5.96996 0.253639
\(555\) 16.7529 0.711121
\(556\) −8.66718 −0.367570
\(557\) −39.3172 −1.66592 −0.832961 0.553331i \(-0.813356\pi\)
−0.832961 + 0.553331i \(0.813356\pi\)
\(558\) −13.3440 −0.564898
\(559\) −26.3827 −1.11587
\(560\) 1.06374 0.0449514
\(561\) 0 0
\(562\) −18.7141 −0.789407
\(563\) −19.9810 −0.842097 −0.421048 0.907038i \(-0.638338\pi\)
−0.421048 + 0.907038i \(0.638338\pi\)
\(564\) 23.9170 1.00709
\(565\) −0.304901 −0.0128273
\(566\) 15.0791 0.633824
\(567\) −0.353947 −0.0148644
\(568\) −5.15741 −0.216400
\(569\) 34.5647 1.44903 0.724515 0.689260i \(-0.242064\pi\)
0.724515 + 0.689260i \(0.242064\pi\)
\(570\) 15.9045 0.666166
\(571\) −3.15090 −0.131861 −0.0659306 0.997824i \(-0.521002\pi\)
−0.0659306 + 0.997824i \(0.521002\pi\)
\(572\) 0 0
\(573\) 12.5760 0.525370
\(574\) 2.99465 0.124994
\(575\) 2.45589 0.102418
\(576\) 11.0812 0.461715
\(577\) 27.3226 1.13745 0.568727 0.822526i \(-0.307436\pi\)
0.568727 + 0.822526i \(0.307436\pi\)
\(578\) −8.29739 −0.345126
\(579\) −63.7269 −2.64840
\(580\) 8.66881 0.359953
\(581\) 1.66054 0.0688908
\(582\) −5.06614 −0.209998
\(583\) 0 0
\(584\) −2.10530 −0.0871180
\(585\) 16.9169 0.699426
\(586\) −1.88030 −0.0776743
\(587\) −46.1679 −1.90555 −0.952776 0.303675i \(-0.901786\pi\)
−0.952776 + 0.303675i \(0.901786\pi\)
\(588\) −24.2990 −1.00207
\(589\) −28.2841 −1.16543
\(590\) −2.17331 −0.0894737
\(591\) −31.5119 −1.29623
\(592\) 6.14556 0.252581
\(593\) −39.4265 −1.61905 −0.809525 0.587085i \(-0.800275\pi\)
−0.809525 + 0.587085i \(0.800275\pi\)
\(594\) 0 0
\(595\) −2.47347 −0.101402
\(596\) −5.50972 −0.225687
\(597\) 21.9858 0.899820
\(598\) −6.24805 −0.255502
\(599\) 1.04875 0.0428507 0.0214253 0.999770i \(-0.493180\pi\)
0.0214253 + 0.999770i \(0.493180\pi\)
\(600\) 7.16724 0.292601
\(601\) −27.2498 −1.11154 −0.555771 0.831336i \(-0.687577\pi\)
−0.555771 + 0.831336i \(0.687577\pi\)
\(602\) −5.81958 −0.237188
\(603\) −30.1504 −1.22782
\(604\) −35.4605 −1.44287
\(605\) 0 0
\(606\) −15.6036 −0.633854
\(607\) −21.9217 −0.889774 −0.444887 0.895587i \(-0.646756\pi\)
−0.444887 + 0.895587i \(0.646756\pi\)
\(608\) 44.9330 1.82227
\(609\) 17.2662 0.699664
\(610\) 1.82962 0.0740791
\(611\) −20.1522 −0.815272
\(612\) 17.1255 0.692258
\(613\) −10.5921 −0.427809 −0.213905 0.976855i \(-0.568618\pi\)
−0.213905 + 0.976855i \(0.568618\pi\)
\(614\) −6.63340 −0.267702
\(615\) −11.0670 −0.446265
\(616\) 0 0
\(617\) 4.60402 0.185351 0.0926755 0.995696i \(-0.470458\pi\)
0.0926755 + 0.995696i \(0.470458\pi\)
\(618\) 19.6656 0.791065
\(619\) 37.0037 1.48731 0.743653 0.668566i \(-0.233092\pi\)
0.743653 + 0.668566i \(0.233092\pi\)
\(620\) −5.36960 −0.215648
\(621\) −13.1529 −0.527809
\(622\) −14.8437 −0.595177
\(623\) −8.41726 −0.337230
\(624\) 10.0013 0.400374
\(625\) 1.00000 0.0400000
\(626\) 5.24081 0.209465
\(627\) 0 0
\(628\) −10.5291 −0.420157
\(629\) −14.2900 −0.569778
\(630\) 3.73157 0.148669
\(631\) −24.8406 −0.988889 −0.494444 0.869209i \(-0.664628\pi\)
−0.494444 + 0.869209i \(0.664628\pi\)
\(632\) −30.7897 −1.22475
\(633\) −64.0108 −2.54420
\(634\) 1.69158 0.0671812
\(635\) −11.9100 −0.472632
\(636\) −48.4977 −1.92306
\(637\) 20.4741 0.811213
\(638\) 0 0
\(639\) 9.92328 0.392559
\(640\) 10.0519 0.397336
\(641\) −44.4293 −1.75485 −0.877425 0.479714i \(-0.840740\pi\)
−0.877425 + 0.479714i \(0.840740\pi\)
\(642\) −9.63705 −0.380344
\(643\) 25.8610 1.01986 0.509929 0.860217i \(-0.329672\pi\)
0.509929 + 0.860217i \(0.329672\pi\)
\(644\) 3.68769 0.145315
\(645\) 21.5068 0.846829
\(646\) −13.5663 −0.533758
\(647\) 19.4865 0.766093 0.383047 0.923729i \(-0.374875\pi\)
0.383047 + 0.923729i \(0.374875\pi\)
\(648\) −0.874831 −0.0343666
\(649\) 0 0
\(650\) −2.54411 −0.0997883
\(651\) −10.6950 −0.419170
\(652\) −27.2093 −1.06560
\(653\) −16.7298 −0.654687 −0.327344 0.944905i \(-0.606153\pi\)
−0.327344 + 0.944905i \(0.606153\pi\)
\(654\) −11.0504 −0.432105
\(655\) −11.1875 −0.437130
\(656\) −4.05977 −0.158508
\(657\) 4.05077 0.158036
\(658\) −4.44524 −0.173293
\(659\) 1.66127 0.0647137 0.0323569 0.999476i \(-0.489699\pi\)
0.0323569 + 0.999476i \(0.489699\pi\)
\(660\) 0 0
\(661\) −44.0130 −1.71191 −0.855953 0.517053i \(-0.827029\pi\)
−0.855953 + 0.517053i \(0.827029\pi\)
\(662\) 11.3561 0.441365
\(663\) −23.2556 −0.903172
\(664\) 4.10426 0.159276
\(665\) 7.90945 0.306715
\(666\) 21.5583 0.835369
\(667\) 14.6231 0.566210
\(668\) 11.4335 0.442375
\(669\) 45.1050 1.74386
\(670\) 4.53429 0.175175
\(671\) 0 0
\(672\) 16.9904 0.655419
\(673\) 38.5949 1.48772 0.743862 0.668333i \(-0.232992\pi\)
0.743862 + 0.668333i \(0.232992\pi\)
\(674\) 14.3817 0.553960
\(675\) −5.35567 −0.206140
\(676\) 1.60799 0.0618458
\(677\) 38.5988 1.48347 0.741737 0.670691i \(-0.234002\pi\)
0.741737 + 0.670691i \(0.234002\pi\)
\(678\) −0.632340 −0.0242849
\(679\) −2.51944 −0.0966871
\(680\) −6.11353 −0.234443
\(681\) −67.9167 −2.60257
\(682\) 0 0
\(683\) −0.748158 −0.0286275 −0.0143137 0.999898i \(-0.504556\pi\)
−0.0143137 + 0.999898i \(0.504556\pi\)
\(684\) −54.7625 −2.09390
\(685\) 4.28124 0.163578
\(686\) 9.84174 0.375759
\(687\) 42.2136 1.61055
\(688\) 7.88945 0.300783
\(689\) 40.8637 1.55678
\(690\) 5.09331 0.193899
\(691\) 5.22184 0.198648 0.0993242 0.995055i \(-0.468332\pi\)
0.0993242 + 0.995055i \(0.468332\pi\)
\(692\) 16.3128 0.620118
\(693\) 0 0
\(694\) 1.59998 0.0607342
\(695\) −5.95320 −0.225818
\(696\) 42.6760 1.61763
\(697\) 9.43999 0.357565
\(698\) 18.4794 0.699455
\(699\) 32.1247 1.21507
\(700\) 1.50157 0.0567541
\(701\) 14.1580 0.534740 0.267370 0.963594i \(-0.413845\pi\)
0.267370 + 0.963594i \(0.413845\pi\)
\(702\) 13.6254 0.514259
\(703\) 45.6952 1.72343
\(704\) 0 0
\(705\) 16.4278 0.618706
\(706\) −17.1525 −0.645542
\(707\) −7.75983 −0.291838
\(708\) 12.0601 0.453247
\(709\) −17.2144 −0.646499 −0.323249 0.946314i \(-0.604775\pi\)
−0.323249 + 0.946314i \(0.604775\pi\)
\(710\) −1.49235 −0.0560071
\(711\) 59.2420 2.22175
\(712\) −20.8044 −0.779680
\(713\) −9.05781 −0.339217
\(714\) −5.12978 −0.191977
\(715\) 0 0
\(716\) −2.13233 −0.0796891
\(717\) 77.0557 2.87770
\(718\) −7.46667 −0.278654
\(719\) −26.5559 −0.990369 −0.495185 0.868788i \(-0.664900\pi\)
−0.495185 + 0.868788i \(0.664900\pi\)
\(720\) −5.05879 −0.188530
\(721\) 9.77987 0.364221
\(722\) 29.3659 1.09289
\(723\) 30.7548 1.14379
\(724\) −13.5237 −0.502606
\(725\) 5.95431 0.221138
\(726\) 0 0
\(727\) 44.1917 1.63898 0.819490 0.573094i \(-0.194257\pi\)
0.819490 + 0.573094i \(0.194257\pi\)
\(728\) −9.06806 −0.336085
\(729\) −43.4902 −1.61075
\(730\) −0.609193 −0.0225472
\(731\) −18.3449 −0.678512
\(732\) −10.1529 −0.375263
\(733\) −24.9604 −0.921935 −0.460968 0.887417i \(-0.652498\pi\)
−0.460968 + 0.887417i \(0.652498\pi\)
\(734\) 2.73609 0.100991
\(735\) −16.6901 −0.615625
\(736\) 14.3895 0.530404
\(737\) 0 0
\(738\) −14.2415 −0.524237
\(739\) −21.2342 −0.781114 −0.390557 0.920579i \(-0.627718\pi\)
−0.390557 + 0.920579i \(0.627718\pi\)
\(740\) 8.67501 0.318900
\(741\) 74.3648 2.73186
\(742\) 9.01383 0.330908
\(743\) 30.6527 1.12454 0.562270 0.826954i \(-0.309928\pi\)
0.562270 + 0.826954i \(0.309928\pi\)
\(744\) −26.4342 −0.969125
\(745\) −3.78444 −0.138651
\(746\) 6.88968 0.252249
\(747\) −7.89694 −0.288934
\(748\) 0 0
\(749\) −4.79259 −0.175118
\(750\) 2.07392 0.0757289
\(751\) −30.3073 −1.10593 −0.552965 0.833204i \(-0.686504\pi\)
−0.552965 + 0.833204i \(0.686504\pi\)
\(752\) 6.02629 0.219756
\(753\) −48.4463 −1.76548
\(754\) −15.1485 −0.551674
\(755\) −24.3566 −0.886429
\(756\) −8.04193 −0.292482
\(757\) −34.8694 −1.26735 −0.633674 0.773600i \(-0.718454\pi\)
−0.633674 + 0.773600i \(0.718454\pi\)
\(758\) −7.23750 −0.262878
\(759\) 0 0
\(760\) 19.5493 0.709129
\(761\) 2.68972 0.0975022 0.0487511 0.998811i \(-0.484476\pi\)
0.0487511 + 0.998811i \(0.484476\pi\)
\(762\) −24.7003 −0.894798
\(763\) −5.49546 −0.198949
\(764\) 6.51212 0.235600
\(765\) 11.7629 0.425290
\(766\) −13.3121 −0.480984
\(767\) −10.1617 −0.366920
\(768\) 33.5507 1.21066
\(769\) 32.5735 1.17463 0.587315 0.809359i \(-0.300185\pi\)
0.587315 + 0.809359i \(0.300185\pi\)
\(770\) 0 0
\(771\) −52.0010 −1.87277
\(772\) −32.9991 −1.18767
\(773\) 41.6637 1.49854 0.749270 0.662265i \(-0.230405\pi\)
0.749270 + 0.662265i \(0.230405\pi\)
\(774\) 27.6758 0.994788
\(775\) −3.68820 −0.132484
\(776\) −6.22714 −0.223541
\(777\) 17.2786 0.619867
\(778\) 23.0081 0.824882
\(779\) −30.1864 −1.08154
\(780\) 14.1178 0.505498
\(781\) 0 0
\(782\) −4.34451 −0.155359
\(783\) −31.8894 −1.13963
\(784\) −6.12253 −0.218662
\(785\) −7.23210 −0.258125
\(786\) −23.2019 −0.827584
\(787\) 35.9207 1.28044 0.640218 0.768193i \(-0.278844\pi\)
0.640218 + 0.768193i \(0.278844\pi\)
\(788\) −16.3175 −0.581288
\(789\) 10.3784 0.369482
\(790\) −8.90936 −0.316981
\(791\) −0.314468 −0.0111812
\(792\) 0 0
\(793\) 8.55476 0.303789
\(794\) −7.83461 −0.278040
\(795\) −33.3115 −1.18144
\(796\) 11.3847 0.403521
\(797\) −31.7197 −1.12357 −0.561785 0.827283i \(-0.689885\pi\)
−0.561785 + 0.827283i \(0.689885\pi\)
\(798\) 16.4036 0.580680
\(799\) −14.0126 −0.495731
\(800\) 5.85919 0.207154
\(801\) 40.0295 1.41437
\(802\) −20.3352 −0.718061
\(803\) 0 0
\(804\) −25.1617 −0.887384
\(805\) 2.53295 0.0892748
\(806\) 9.38320 0.330509
\(807\) 27.9612 0.984283
\(808\) −19.1795 −0.674733
\(809\) 8.51572 0.299397 0.149698 0.988732i \(-0.452170\pi\)
0.149698 + 0.988732i \(0.452170\pi\)
\(810\) −0.253142 −0.00889451
\(811\) −9.03030 −0.317097 −0.158548 0.987351i \(-0.550681\pi\)
−0.158548 + 0.987351i \(0.550681\pi\)
\(812\) 8.94083 0.313762
\(813\) 3.52472 0.123617
\(814\) 0 0
\(815\) −18.6892 −0.654653
\(816\) 6.95431 0.243450
\(817\) 58.6619 2.05232
\(818\) −10.5985 −0.370569
\(819\) 17.4477 0.609672
\(820\) −5.73074 −0.200126
\(821\) 54.3225 1.89587 0.947935 0.318465i \(-0.103167\pi\)
0.947935 + 0.318465i \(0.103167\pi\)
\(822\) 8.87896 0.309689
\(823\) 17.8594 0.622541 0.311271 0.950321i \(-0.399245\pi\)
0.311271 + 0.950321i \(0.399245\pi\)
\(824\) 24.1723 0.842083
\(825\) 0 0
\(826\) −2.24151 −0.0779920
\(827\) 51.9494 1.80646 0.903229 0.429159i \(-0.141190\pi\)
0.903229 + 0.429159i \(0.141190\pi\)
\(828\) −17.5373 −0.609465
\(829\) −19.7460 −0.685807 −0.342904 0.939371i \(-0.611410\pi\)
−0.342904 + 0.939371i \(0.611410\pi\)
\(830\) 1.18761 0.0412227
\(831\) −22.7548 −0.789357
\(832\) −7.79201 −0.270139
\(833\) 14.2364 0.493263
\(834\) −12.3465 −0.427523
\(835\) 7.85328 0.271774
\(836\) 0 0
\(837\) 19.7528 0.682757
\(838\) −23.2140 −0.801916
\(839\) −4.11650 −0.142117 −0.0710586 0.997472i \(-0.522638\pi\)
−0.0710586 + 0.997472i \(0.522638\pi\)
\(840\) 7.39214 0.255053
\(841\) 6.45386 0.222547
\(842\) 19.6000 0.675460
\(843\) 71.3300 2.45674
\(844\) −33.1462 −1.14094
\(845\) 1.10448 0.0379951
\(846\) 21.1400 0.726807
\(847\) 0 0
\(848\) −12.2198 −0.419630
\(849\) −57.4751 −1.97254
\(850\) −1.76902 −0.0606769
\(851\) 14.6336 0.501633
\(852\) 8.28138 0.283715
\(853\) −5.50285 −0.188414 −0.0942070 0.995553i \(-0.530032\pi\)
−0.0942070 + 0.995553i \(0.530032\pi\)
\(854\) 1.88703 0.0645729
\(855\) −37.6145 −1.28639
\(856\) −11.8456 −0.404873
\(857\) 26.9281 0.919847 0.459924 0.887959i \(-0.347877\pi\)
0.459924 + 0.887959i \(0.347877\pi\)
\(858\) 0 0
\(859\) 19.1519 0.653456 0.326728 0.945118i \(-0.394054\pi\)
0.326728 + 0.945118i \(0.394054\pi\)
\(860\) 11.1367 0.379758
\(861\) −11.4143 −0.388998
\(862\) 2.85371 0.0971977
\(863\) 4.96151 0.168892 0.0844458 0.996428i \(-0.473088\pi\)
0.0844458 + 0.996428i \(0.473088\pi\)
\(864\) −31.3799 −1.06757
\(865\) 11.2047 0.380971
\(866\) 29.6080 1.00612
\(867\) 31.6260 1.07408
\(868\) −5.53810 −0.187975
\(869\) 0 0
\(870\) 12.3488 0.418663
\(871\) 21.2010 0.718368
\(872\) −13.5828 −0.459972
\(873\) 11.9815 0.405514
\(874\) 13.8925 0.469921
\(875\) 1.03138 0.0348670
\(876\) 3.38053 0.114218
\(877\) 27.2053 0.918659 0.459329 0.888266i \(-0.348090\pi\)
0.459329 + 0.888266i \(0.348090\pi\)
\(878\) −0.754874 −0.0254758
\(879\) 7.16686 0.241732
\(880\) 0 0
\(881\) 10.3570 0.348935 0.174467 0.984663i \(-0.444180\pi\)
0.174467 + 0.984663i \(0.444180\pi\)
\(882\) −21.4776 −0.723188
\(883\) 7.39489 0.248858 0.124429 0.992229i \(-0.460290\pi\)
0.124429 + 0.992229i \(0.460290\pi\)
\(884\) −12.0422 −0.405025
\(885\) 8.28370 0.278453
\(886\) −11.8560 −0.398309
\(887\) 18.0590 0.606363 0.303181 0.952933i \(-0.401951\pi\)
0.303181 + 0.952933i \(0.401951\pi\)
\(888\) 42.7065 1.43314
\(889\) −12.2837 −0.411982
\(890\) −6.02000 −0.201791
\(891\) 0 0
\(892\) 23.3563 0.782027
\(893\) 44.8084 1.49946
\(894\) −7.84864 −0.262498
\(895\) −1.46463 −0.0489572
\(896\) 10.3673 0.346348
\(897\) 23.8148 0.795154
\(898\) 26.4386 0.882267
\(899\) −21.9607 −0.732431
\(900\) −7.14094 −0.238031
\(901\) 28.4141 0.946612
\(902\) 0 0
\(903\) 22.1817 0.738160
\(904\) −0.777253 −0.0258511
\(905\) −9.28900 −0.308777
\(906\) −50.5137 −1.67821
\(907\) 26.2971 0.873179 0.436590 0.899661i \(-0.356186\pi\)
0.436590 + 0.899661i \(0.356186\pi\)
\(908\) −35.1687 −1.16711
\(909\) 36.9030 1.22399
\(910\) −2.62395 −0.0869830
\(911\) 28.6489 0.949179 0.474589 0.880207i \(-0.342597\pi\)
0.474589 + 0.880207i \(0.342597\pi\)
\(912\) −22.2379 −0.736371
\(913\) 0 0
\(914\) 18.5535 0.613694
\(915\) −6.97371 −0.230544
\(916\) 21.8591 0.722245
\(917\) −11.5385 −0.381035
\(918\) 9.47430 0.312698
\(919\) −38.8568 −1.28177 −0.640884 0.767638i \(-0.721432\pi\)
−0.640884 + 0.767638i \(0.721432\pi\)
\(920\) 6.26055 0.206404
\(921\) 25.2836 0.833123
\(922\) −4.90996 −0.161701
\(923\) −6.97781 −0.229677
\(924\) 0 0
\(925\) 5.95858 0.195917
\(926\) 28.6006 0.939873
\(927\) −46.5096 −1.52757
\(928\) 34.8875 1.14524
\(929\) 7.42132 0.243486 0.121743 0.992562i \(-0.461152\pi\)
0.121743 + 0.992562i \(0.461152\pi\)
\(930\) −7.64904 −0.250822
\(931\) −45.5240 −1.49199
\(932\) 16.6349 0.544893
\(933\) 56.5775 1.85227
\(934\) 16.6334 0.544260
\(935\) 0 0
\(936\) 43.1245 1.40957
\(937\) 14.9360 0.487937 0.243968 0.969783i \(-0.421551\pi\)
0.243968 + 0.969783i \(0.421551\pi\)
\(938\) 4.67657 0.152696
\(939\) −19.9757 −0.651881
\(940\) 8.50666 0.277457
\(941\) 52.3409 1.70626 0.853132 0.521695i \(-0.174700\pi\)
0.853132 + 0.521695i \(0.174700\pi\)
\(942\) −14.9988 −0.488687
\(943\) −9.66700 −0.314801
\(944\) 3.03875 0.0989030
\(945\) −5.52373 −0.179687
\(946\) 0 0
\(947\) −3.69553 −0.120088 −0.0600442 0.998196i \(-0.519124\pi\)
−0.0600442 + 0.998196i \(0.519124\pi\)
\(948\) 49.4398 1.60573
\(949\) −2.84841 −0.0924631
\(950\) 5.65682 0.183532
\(951\) −6.44756 −0.209076
\(952\) −6.30538 −0.204358
\(953\) −43.1526 −1.39785 −0.698925 0.715195i \(-0.746338\pi\)
−0.698925 + 0.715195i \(0.746338\pi\)
\(954\) −42.8666 −1.38786
\(955\) 4.47296 0.144742
\(956\) 39.9011 1.29049
\(957\) 0 0
\(958\) 1.20373 0.0388907
\(959\) 4.41559 0.142587
\(960\) 6.35192 0.205007
\(961\) −17.3972 −0.561199
\(962\) −15.1593 −0.488755
\(963\) 22.7919 0.734458
\(964\) 15.9255 0.512927
\(965\) −22.6660 −0.729645
\(966\) 5.25314 0.169017
\(967\) 29.2144 0.939471 0.469736 0.882807i \(-0.344349\pi\)
0.469736 + 0.882807i \(0.344349\pi\)
\(968\) 0 0
\(969\) 51.7087 1.66112
\(970\) −1.80189 −0.0578554
\(971\) −26.4046 −0.847365 −0.423683 0.905811i \(-0.639263\pi\)
−0.423683 + 0.905811i \(0.639263\pi\)
\(972\) −21.9870 −0.705234
\(973\) −6.14001 −0.196840
\(974\) 0.774743 0.0248244
\(975\) 9.69704 0.310554
\(976\) −2.55820 −0.0818861
\(977\) −15.8434 −0.506875 −0.253437 0.967352i \(-0.581561\pi\)
−0.253437 + 0.967352i \(0.581561\pi\)
\(978\) −38.7598 −1.23940
\(979\) 0 0
\(980\) −8.64252 −0.276075
\(981\) 26.1345 0.834410
\(982\) 10.2636 0.327525
\(983\) 35.1039 1.11964 0.559821 0.828614i \(-0.310870\pi\)
0.559821 + 0.828614i \(0.310870\pi\)
\(984\) −28.2121 −0.899368
\(985\) −11.2080 −0.357116
\(986\) −10.5333 −0.335449
\(987\) 16.9433 0.539311
\(988\) 38.5077 1.22509
\(989\) 18.7861 0.597363
\(990\) 0 0
\(991\) 18.9700 0.602600 0.301300 0.953529i \(-0.402579\pi\)
0.301300 + 0.953529i \(0.402579\pi\)
\(992\) −21.6099 −0.686114
\(993\) −43.2843 −1.37359
\(994\) −1.53918 −0.0488200
\(995\) 7.81979 0.247904
\(996\) −6.59031 −0.208822
\(997\) −30.1347 −0.954375 −0.477188 0.878801i \(-0.658344\pi\)
−0.477188 + 0.878801i \(0.658344\pi\)
\(998\) 12.8108 0.405519
\(999\) −31.9122 −1.00966
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 605.2.a.k.1.3 4
3.2 odd 2 5445.2.a.bi.1.2 4
4.3 odd 2 9680.2.a.cm.1.4 4
5.4 even 2 3025.2.a.w.1.2 4
11.2 odd 10 55.2.g.b.26.2 8
11.3 even 5 605.2.g.e.251.2 8
11.4 even 5 605.2.g.e.511.2 8
11.5 even 5 605.2.g.k.366.1 8
11.6 odd 10 55.2.g.b.36.2 yes 8
11.7 odd 10 605.2.g.m.511.1 8
11.8 odd 10 605.2.g.m.251.1 8
11.9 even 5 605.2.g.k.81.1 8
11.10 odd 2 605.2.a.j.1.2 4
33.2 even 10 495.2.n.e.136.1 8
33.17 even 10 495.2.n.e.91.1 8
33.32 even 2 5445.2.a.bp.1.3 4
44.35 even 10 880.2.bo.h.81.2 8
44.39 even 10 880.2.bo.h.641.2 8
44.43 even 2 9680.2.a.cn.1.4 4
55.2 even 20 275.2.z.a.224.2 16
55.13 even 20 275.2.z.a.224.3 16
55.17 even 20 275.2.z.a.124.3 16
55.24 odd 10 275.2.h.a.26.1 8
55.28 even 20 275.2.z.a.124.2 16
55.39 odd 10 275.2.h.a.201.1 8
55.54 odd 2 3025.2.a.bd.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.g.b.26.2 8 11.2 odd 10
55.2.g.b.36.2 yes 8 11.6 odd 10
275.2.h.a.26.1 8 55.24 odd 10
275.2.h.a.201.1 8 55.39 odd 10
275.2.z.a.124.2 16 55.28 even 20
275.2.z.a.124.3 16 55.17 even 20
275.2.z.a.224.2 16 55.2 even 20
275.2.z.a.224.3 16 55.13 even 20
495.2.n.e.91.1 8 33.17 even 10
495.2.n.e.136.1 8 33.2 even 10
605.2.a.j.1.2 4 11.10 odd 2
605.2.a.k.1.3 4 1.1 even 1 trivial
605.2.g.e.251.2 8 11.3 even 5
605.2.g.e.511.2 8 11.4 even 5
605.2.g.k.81.1 8 11.9 even 5
605.2.g.k.366.1 8 11.5 even 5
605.2.g.m.251.1 8 11.8 odd 10
605.2.g.m.511.1 8 11.7 odd 10
880.2.bo.h.81.2 8 44.35 even 10
880.2.bo.h.641.2 8 44.39 even 10
3025.2.a.w.1.2 4 5.4 even 2
3025.2.a.bd.1.3 4 55.54 odd 2
5445.2.a.bi.1.2 4 3.2 odd 2
5445.2.a.bp.1.3 4 33.32 even 2
9680.2.a.cm.1.4 4 4.3 odd 2
9680.2.a.cn.1.4 4 44.43 even 2