Properties

Label 8281.2.a.ck.1.5
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, 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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.332375\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.332375 q^{2} +1.45984 q^{3} -1.88953 q^{4} +1.44562 q^{5} +0.485214 q^{6} -1.29278 q^{8} -0.868875 q^{9} +O(q^{10})\) \(q+0.332375 q^{2} +1.45984 q^{3} -1.88953 q^{4} +1.44562 q^{5} +0.485214 q^{6} -1.29278 q^{8} -0.868875 q^{9} +0.480489 q^{10} -5.95516 q^{11} -2.75840 q^{12} +2.11037 q^{15} +3.34936 q^{16} +4.32871 q^{17} -0.288793 q^{18} +1.95753 q^{19} -2.73154 q^{20} -1.97935 q^{22} -0.540163 q^{23} -1.88725 q^{24} -2.91018 q^{25} -5.64793 q^{27} +7.15857 q^{29} +0.701436 q^{30} -6.10800 q^{31} +3.69881 q^{32} -8.69356 q^{33} +1.43876 q^{34} +1.64176 q^{36} +8.02881 q^{37} +0.650636 q^{38} -1.86887 q^{40} -7.55362 q^{41} +4.24839 q^{43} +11.2524 q^{44} -1.25606 q^{45} -0.179537 q^{46} +6.26084 q^{47} +4.88953 q^{48} -0.967272 q^{50} +6.31922 q^{51} -2.77905 q^{53} -1.87723 q^{54} -8.60891 q^{55} +2.85768 q^{57} +2.37933 q^{58} -0.851152 q^{59} -3.98760 q^{60} +6.77905 q^{61} -2.03015 q^{62} -5.46933 q^{64} -2.88953 q^{66} -0.987106 q^{67} -8.17922 q^{68} -0.788550 q^{69} +3.76223 q^{71} +1.12327 q^{72} -9.13519 q^{73} +2.66858 q^{74} -4.24839 q^{75} -3.69881 q^{76} -0.131125 q^{79} +4.84191 q^{80} -5.63843 q^{81} -2.51064 q^{82} +2.66812 q^{83} +6.25768 q^{85} +1.41206 q^{86} +10.4503 q^{87} +7.69873 q^{88} +9.71739 q^{89} -0.417485 q^{90} +1.02065 q^{92} -8.91668 q^{93} +2.08095 q^{94} +2.82985 q^{95} +5.39966 q^{96} +6.58319 q^{97} +5.17429 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9} - 6 q^{10} + 18 q^{12} - 2 q^{16} + 8 q^{17} - 18 q^{22} - 12 q^{23} + 16 q^{27} - 8 q^{29} + 38 q^{30} + 28 q^{36} + 34 q^{38} + 4 q^{40} - 8 q^{43} + 18 q^{48} + 16 q^{51} + 20 q^{53} + 12 q^{55} + 12 q^{61} - 22 q^{62} - 44 q^{64} - 2 q^{66} + 2 q^{68} + 28 q^{69} - 42 q^{74} + 8 q^{75} - 20 q^{79} + 24 q^{81} - 16 q^{82} + 68 q^{87} + 4 q^{88} + 108 q^{90} + 6 q^{92} + 26 q^{94} - 16 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.332375 0.235025 0.117512 0.993071i \(-0.462508\pi\)
0.117512 + 0.993071i \(0.462508\pi\)
\(3\) 1.45984 0.842837 0.421419 0.906866i \(-0.361532\pi\)
0.421419 + 0.906866i \(0.361532\pi\)
\(4\) −1.88953 −0.944763
\(5\) 1.44562 0.646502 0.323251 0.946313i \(-0.395224\pi\)
0.323251 + 0.946313i \(0.395224\pi\)
\(6\) 0.485214 0.198088
\(7\) 0 0
\(8\) −1.29278 −0.457068
\(9\) −0.868875 −0.289625
\(10\) 0.480489 0.151944
\(11\) −5.95516 −1.79555 −0.897774 0.440456i \(-0.854817\pi\)
−0.897774 + 0.440456i \(0.854817\pi\)
\(12\) −2.75840 −0.796282
\(13\) 0 0
\(14\) 0 0
\(15\) 2.11037 0.544896
\(16\) 3.34936 0.837341
\(17\) 4.32871 1.04987 0.524933 0.851143i \(-0.324090\pi\)
0.524933 + 0.851143i \(0.324090\pi\)
\(18\) −0.288793 −0.0680691
\(19\) 1.95753 0.449089 0.224545 0.974464i \(-0.427911\pi\)
0.224545 + 0.974464i \(0.427911\pi\)
\(20\) −2.73154 −0.610791
\(21\) 0 0
\(22\) −1.97935 −0.421998
\(23\) −0.540163 −0.112632 −0.0563158 0.998413i \(-0.517935\pi\)
−0.0563158 + 0.998413i \(0.517935\pi\)
\(24\) −1.88725 −0.385234
\(25\) −2.91018 −0.582036
\(26\) 0 0
\(27\) −5.64793 −1.08694
\(28\) 0 0
\(29\) 7.15857 1.32931 0.664656 0.747149i \(-0.268578\pi\)
0.664656 + 0.747149i \(0.268578\pi\)
\(30\) 0.701436 0.128064
\(31\) −6.10800 −1.09703 −0.548514 0.836141i \(-0.684806\pi\)
−0.548514 + 0.836141i \(0.684806\pi\)
\(32\) 3.69881 0.653864
\(33\) −8.69356 −1.51336
\(34\) 1.43876 0.246745
\(35\) 0 0
\(36\) 1.64176 0.273627
\(37\) 8.02881 1.31993 0.659964 0.751297i \(-0.270571\pi\)
0.659964 + 0.751297i \(0.270571\pi\)
\(38\) 0.650636 0.105547
\(39\) 0 0
\(40\) −1.86887 −0.295495
\(41\) −7.55362 −1.17968 −0.589839 0.807521i \(-0.700809\pi\)
−0.589839 + 0.807521i \(0.700809\pi\)
\(42\) 0 0
\(43\) 4.24839 0.647873 0.323936 0.946079i \(-0.394994\pi\)
0.323936 + 0.946079i \(0.394994\pi\)
\(44\) 11.2524 1.69637
\(45\) −1.25606 −0.187243
\(46\) −0.179537 −0.0264713
\(47\) 6.26084 0.913237 0.456618 0.889663i \(-0.349060\pi\)
0.456618 + 0.889663i \(0.349060\pi\)
\(48\) 4.88953 0.705742
\(49\) 0 0
\(50\) −0.967272 −0.136793
\(51\) 6.31922 0.884867
\(52\) 0 0
\(53\) −2.77905 −0.381732 −0.190866 0.981616i \(-0.561130\pi\)
−0.190866 + 0.981616i \(0.561130\pi\)
\(54\) −1.87723 −0.255459
\(55\) −8.60891 −1.16082
\(56\) 0 0
\(57\) 2.85768 0.378509
\(58\) 2.37933 0.312421
\(59\) −0.851152 −0.110811 −0.0554053 0.998464i \(-0.517645\pi\)
−0.0554053 + 0.998464i \(0.517645\pi\)
\(60\) −3.98760 −0.514798
\(61\) 6.77905 0.867969 0.433984 0.900920i \(-0.357107\pi\)
0.433984 + 0.900920i \(0.357107\pi\)
\(62\) −2.03015 −0.257829
\(63\) 0 0
\(64\) −5.46933 −0.683667
\(65\) 0 0
\(66\) −2.88953 −0.355676
\(67\) −0.987106 −0.120594 −0.0602971 0.998180i \(-0.519205\pi\)
−0.0602971 + 0.998180i \(0.519205\pi\)
\(68\) −8.17922 −0.991876
\(69\) −0.788550 −0.0949302
\(70\) 0 0
\(71\) 3.76223 0.446494 0.223247 0.974762i \(-0.428334\pi\)
0.223247 + 0.974762i \(0.428334\pi\)
\(72\) 1.12327 0.132378
\(73\) −9.13519 −1.06919 −0.534597 0.845107i \(-0.679537\pi\)
−0.534597 + 0.845107i \(0.679537\pi\)
\(74\) 2.66858 0.310216
\(75\) −4.24839 −0.490561
\(76\) −3.69881 −0.424283
\(77\) 0 0
\(78\) 0 0
\(79\) −0.131125 −0.0147527 −0.00737636 0.999973i \(-0.502348\pi\)
−0.00737636 + 0.999973i \(0.502348\pi\)
\(80\) 4.84191 0.541342
\(81\) −5.63843 −0.626492
\(82\) −2.51064 −0.277253
\(83\) 2.66812 0.292865 0.146432 0.989221i \(-0.453221\pi\)
0.146432 + 0.989221i \(0.453221\pi\)
\(84\) 0 0
\(85\) 6.25768 0.678741
\(86\) 1.41206 0.152266
\(87\) 10.4503 1.12039
\(88\) 7.69873 0.820687
\(89\) 9.71739 1.03004 0.515021 0.857178i \(-0.327784\pi\)
0.515021 + 0.857178i \(0.327784\pi\)
\(90\) −0.417485 −0.0440068
\(91\) 0 0
\(92\) 1.02065 0.106410
\(93\) −8.91668 −0.924617
\(94\) 2.08095 0.214633
\(95\) 2.82985 0.290337
\(96\) 5.39966 0.551101
\(97\) 6.58319 0.668422 0.334211 0.942498i \(-0.391530\pi\)
0.334211 + 0.942498i \(0.391530\pi\)
\(98\) 0 0
\(99\) 5.17429 0.520036
\(100\) 5.49886 0.549886
\(101\) −0.0708289 −0.00704774 −0.00352387 0.999994i \(-0.501122\pi\)
−0.00352387 + 0.999994i \(0.501122\pi\)
\(102\) 2.10035 0.207966
\(103\) 6.33821 0.624522 0.312261 0.949996i \(-0.398914\pi\)
0.312261 + 0.949996i \(0.398914\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −0.923689 −0.0897166
\(107\) 7.74953 0.749175 0.374588 0.927192i \(-0.377784\pi\)
0.374588 + 0.927192i \(0.377784\pi\)
\(108\) 10.6719 1.02691
\(109\) −0.0335623 −0.00321468 −0.00160734 0.999999i \(-0.500512\pi\)
−0.00160734 + 0.999999i \(0.500512\pi\)
\(110\) −2.86139 −0.272823
\(111\) 11.7208 1.11249
\(112\) 0 0
\(113\) −9.19987 −0.865451 −0.432725 0.901526i \(-0.642448\pi\)
−0.432725 + 0.901526i \(0.642448\pi\)
\(114\) 0.949823 0.0889591
\(115\) −0.780871 −0.0728166
\(116\) −13.5263 −1.25589
\(117\) 0 0
\(118\) −0.282902 −0.0260432
\(119\) 0 0
\(120\) −2.72825 −0.249054
\(121\) 24.4639 2.22399
\(122\) 2.25319 0.203994
\(123\) −11.0271 −0.994276
\(124\) 11.5412 1.03643
\(125\) −11.4351 −1.02279
\(126\) 0 0
\(127\) 14.3952 1.27737 0.638683 0.769470i \(-0.279480\pi\)
0.638683 + 0.769470i \(0.279480\pi\)
\(128\) −9.21550 −0.814542
\(129\) 6.20195 0.546052
\(130\) 0 0
\(131\) −9.46828 −0.827248 −0.413624 0.910448i \(-0.635737\pi\)
−0.413624 + 0.910448i \(0.635737\pi\)
\(132\) 16.4267 1.42976
\(133\) 0 0
\(134\) −0.328090 −0.0283426
\(135\) −8.16477 −0.702711
\(136\) −5.59609 −0.479860
\(137\) 16.6063 1.41877 0.709384 0.704822i \(-0.248973\pi\)
0.709384 + 0.704822i \(0.248973\pi\)
\(138\) −0.262094 −0.0223110
\(139\) 18.4778 1.56726 0.783632 0.621225i \(-0.213365\pi\)
0.783632 + 0.621225i \(0.213365\pi\)
\(140\) 0 0
\(141\) 9.13980 0.769710
\(142\) 1.25047 0.104937
\(143\) 0 0
\(144\) −2.91018 −0.242515
\(145\) 10.3486 0.859402
\(146\) −3.03631 −0.251287
\(147\) 0 0
\(148\) −15.1707 −1.24702
\(149\) −3.08080 −0.252389 −0.126195 0.992006i \(-0.540276\pi\)
−0.126195 + 0.992006i \(0.540276\pi\)
\(150\) −1.41206 −0.115294
\(151\) 2.54885 0.207422 0.103711 0.994607i \(-0.466928\pi\)
0.103711 + 0.994607i \(0.466928\pi\)
\(152\) −2.53067 −0.205264
\(153\) −3.76111 −0.304068
\(154\) 0 0
\(155\) −8.82985 −0.709231
\(156\) 0 0
\(157\) −9.40904 −0.750923 −0.375461 0.926838i \(-0.622516\pi\)
−0.375461 + 0.926838i \(0.622516\pi\)
\(158\) −0.0435828 −0.00346726
\(159\) −4.05697 −0.321738
\(160\) 5.34708 0.422724
\(161\) 0 0
\(162\) −1.87408 −0.147241
\(163\) 0.695157 0.0544489 0.0272244 0.999629i \(-0.491333\pi\)
0.0272244 + 0.999629i \(0.491333\pi\)
\(164\) 14.2728 1.11452
\(165\) −12.5676 −0.978387
\(166\) 0.886819 0.0688305
\(167\) 13.9840 1.08211 0.541056 0.840986i \(-0.318025\pi\)
0.541056 + 0.840986i \(0.318025\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 2.07990 0.159521
\(171\) −1.70085 −0.130067
\(172\) −8.02744 −0.612087
\(173\) −5.43648 −0.413328 −0.206664 0.978412i \(-0.566261\pi\)
−0.206664 + 0.978412i \(0.566261\pi\)
\(174\) 3.47344 0.263321
\(175\) 0 0
\(176\) −19.9460 −1.50349
\(177\) −1.24254 −0.0933953
\(178\) 3.22982 0.242085
\(179\) 5.35824 0.400493 0.200247 0.979745i \(-0.435826\pi\)
0.200247 + 0.979745i \(0.435826\pi\)
\(180\) 2.37337 0.176900
\(181\) 7.54016 0.560456 0.280228 0.959933i \(-0.409590\pi\)
0.280228 + 0.959933i \(0.409590\pi\)
\(182\) 0 0
\(183\) 9.89632 0.731557
\(184\) 0.698313 0.0514803
\(185\) 11.6066 0.853336
\(186\) −2.96369 −0.217308
\(187\) −25.7782 −1.88509
\(188\) −11.8300 −0.862793
\(189\) 0 0
\(190\) 0.940574 0.0682364
\(191\) 13.5463 0.980178 0.490089 0.871672i \(-0.336964\pi\)
0.490089 + 0.871672i \(0.336964\pi\)
\(192\) −7.98434 −0.576220
\(193\) 18.5562 1.33571 0.667853 0.744293i \(-0.267213\pi\)
0.667853 + 0.744293i \(0.267213\pi\)
\(194\) 2.18809 0.157096
\(195\) 0 0
\(196\) 0 0
\(197\) −2.66812 −0.190096 −0.0950480 0.995473i \(-0.530300\pi\)
−0.0950480 + 0.995473i \(0.530300\pi\)
\(198\) 1.71981 0.122221
\(199\) 20.1999 1.43193 0.715965 0.698136i \(-0.245987\pi\)
0.715965 + 0.698136i \(0.245987\pi\)
\(200\) 3.76223 0.266030
\(201\) −1.44101 −0.101641
\(202\) −0.0235418 −0.00165639
\(203\) 0 0
\(204\) −11.9403 −0.835990
\(205\) −10.9197 −0.762663
\(206\) 2.10666 0.146778
\(207\) 0.469334 0.0326210
\(208\) 0 0
\(209\) −11.6574 −0.806361
\(210\) 0 0
\(211\) 13.1268 0.903683 0.451842 0.892098i \(-0.350767\pi\)
0.451842 + 0.892098i \(0.350767\pi\)
\(212\) 5.25109 0.360647
\(213\) 5.49224 0.376322
\(214\) 2.57575 0.176075
\(215\) 6.14156 0.418851
\(216\) 7.30155 0.496807
\(217\) 0 0
\(218\) −0.0111553 −0.000755530 0
\(219\) −13.3359 −0.901157
\(220\) 16.2668 1.09670
\(221\) 0 0
\(222\) 3.89569 0.261462
\(223\) −2.22334 −0.148886 −0.0744428 0.997225i \(-0.523718\pi\)
−0.0744428 + 0.997225i \(0.523718\pi\)
\(224\) 0 0
\(225\) 2.52858 0.168572
\(226\) −3.05781 −0.203402
\(227\) 27.1045 1.79899 0.899495 0.436931i \(-0.143935\pi\)
0.899495 + 0.436931i \(0.143935\pi\)
\(228\) −5.39966 −0.357601
\(229\) −18.9887 −1.25481 −0.627406 0.778693i \(-0.715883\pi\)
−0.627406 + 0.778693i \(0.715883\pi\)
\(230\) −0.259542 −0.0171137
\(231\) 0 0
\(232\) −9.25447 −0.607586
\(233\) 21.7400 1.42424 0.712118 0.702059i \(-0.247736\pi\)
0.712118 + 0.702059i \(0.247736\pi\)
\(234\) 0 0
\(235\) 9.05080 0.590409
\(236\) 1.60827 0.104690
\(237\) −0.191421 −0.0124342
\(238\) 0 0
\(239\) 19.9695 1.29172 0.645861 0.763455i \(-0.276499\pi\)
0.645861 + 0.763455i \(0.276499\pi\)
\(240\) 7.06841 0.456264
\(241\) 3.23048 0.208094 0.104047 0.994572i \(-0.466821\pi\)
0.104047 + 0.994572i \(0.466821\pi\)
\(242\) 8.13120 0.522694
\(243\) 8.71259 0.558913
\(244\) −12.8092 −0.820025
\(245\) 0 0
\(246\) −3.66512 −0.233680
\(247\) 0 0
\(248\) 7.89632 0.501417
\(249\) 3.89503 0.246837
\(250\) −3.80075 −0.240381
\(251\) 12.4916 0.788466 0.394233 0.919011i \(-0.371010\pi\)
0.394233 + 0.919011i \(0.371010\pi\)
\(252\) 0 0
\(253\) 3.21675 0.202236
\(254\) 4.78460 0.300213
\(255\) 9.13519 0.572068
\(256\) 7.87566 0.492229
\(257\) 5.82757 0.363514 0.181757 0.983343i \(-0.441822\pi\)
0.181757 + 0.983343i \(0.441822\pi\)
\(258\) 2.06138 0.128336
\(259\) 0 0
\(260\) 0 0
\(261\) −6.21990 −0.385002
\(262\) −3.14702 −0.194424
\(263\) 17.5147 1.08000 0.540002 0.841664i \(-0.318424\pi\)
0.540002 + 0.841664i \(0.318424\pi\)
\(264\) 11.2389 0.691706
\(265\) −4.01746 −0.246791
\(266\) 0 0
\(267\) 14.1858 0.868157
\(268\) 1.86516 0.113933
\(269\) 22.3287 1.36141 0.680703 0.732560i \(-0.261675\pi\)
0.680703 + 0.732560i \(0.261675\pi\)
\(270\) −2.71377 −0.165155
\(271\) −26.3695 −1.60183 −0.800916 0.598777i \(-0.795654\pi\)
−0.800916 + 0.598777i \(0.795654\pi\)
\(272\) 14.4984 0.879097
\(273\) 0 0
\(274\) 5.51951 0.333446
\(275\) 17.3306 1.04507
\(276\) 1.48999 0.0896866
\(277\) −9.37618 −0.563360 −0.281680 0.959508i \(-0.590892\pi\)
−0.281680 + 0.959508i \(0.590892\pi\)
\(278\) 6.14156 0.368346
\(279\) 5.30709 0.317727
\(280\) 0 0
\(281\) −17.7754 −1.06039 −0.530195 0.847876i \(-0.677881\pi\)
−0.530195 + 0.847876i \(0.677881\pi\)
\(282\) 3.03785 0.180901
\(283\) −9.60662 −0.571055 −0.285527 0.958371i \(-0.592169\pi\)
−0.285527 + 0.958371i \(0.592169\pi\)
\(284\) −7.10883 −0.421832
\(285\) 4.13113 0.244707
\(286\) 0 0
\(287\) 0 0
\(288\) −3.21380 −0.189375
\(289\) 1.73775 0.102221
\(290\) 3.43961 0.201981
\(291\) 9.61039 0.563371
\(292\) 17.2612 1.01013
\(293\) −11.6338 −0.679654 −0.339827 0.940488i \(-0.610369\pi\)
−0.339827 + 0.940488i \(0.610369\pi\)
\(294\) 0 0
\(295\) −1.23044 −0.0716392
\(296\) −10.3795 −0.603297
\(297\) 33.6343 1.95166
\(298\) −1.02398 −0.0593177
\(299\) 0 0
\(300\) 8.02744 0.463464
\(301\) 0 0
\(302\) 0.847174 0.0487494
\(303\) −0.103399 −0.00594010
\(304\) 6.55649 0.376041
\(305\) 9.79995 0.561143
\(306\) −1.25010 −0.0714635
\(307\) 13.8280 0.789204 0.394602 0.918852i \(-0.370882\pi\)
0.394602 + 0.918852i \(0.370882\pi\)
\(308\) 0 0
\(309\) 9.25275 0.526371
\(310\) −2.93483 −0.166687
\(311\) 30.7144 1.74165 0.870827 0.491590i \(-0.163584\pi\)
0.870827 + 0.491590i \(0.163584\pi\)
\(312\) 0 0
\(313\) 11.0867 0.626657 0.313328 0.949645i \(-0.398556\pi\)
0.313328 + 0.949645i \(0.398556\pi\)
\(314\) −3.12733 −0.176486
\(315\) 0 0
\(316\) 0.247764 0.0139378
\(317\) −23.8834 −1.34142 −0.670712 0.741718i \(-0.734011\pi\)
−0.670712 + 0.741718i \(0.734011\pi\)
\(318\) −1.34844 −0.0756165
\(319\) −42.6304 −2.38684
\(320\) −7.90659 −0.441992
\(321\) 11.3130 0.631433
\(322\) 0 0
\(323\) 8.47360 0.471484
\(324\) 10.6540 0.591887
\(325\) 0 0
\(326\) 0.231053 0.0127968
\(327\) −0.0489954 −0.00270945
\(328\) 9.76519 0.539192
\(329\) 0 0
\(330\) −4.17716 −0.229945
\(331\) −18.3240 −1.00718 −0.503589 0.863943i \(-0.667987\pi\)
−0.503589 + 0.863943i \(0.667987\pi\)
\(332\) −5.04149 −0.276688
\(333\) −6.97603 −0.382284
\(334\) 4.64793 0.254323
\(335\) −1.42698 −0.0779643
\(336\) 0 0
\(337\) −7.21762 −0.393169 −0.196584 0.980487i \(-0.562985\pi\)
−0.196584 + 0.980487i \(0.562985\pi\)
\(338\) 0 0
\(339\) −13.4303 −0.729434
\(340\) −11.8241 −0.641249
\(341\) 36.3741 1.96977
\(342\) −0.565321 −0.0305691
\(343\) 0 0
\(344\) −5.49224 −0.296122
\(345\) −1.13994 −0.0613725
\(346\) −1.80695 −0.0971423
\(347\) −21.0782 −1.13154 −0.565770 0.824563i \(-0.691421\pi\)
−0.565770 + 0.824563i \(0.691421\pi\)
\(348\) −19.7462 −1.05851
\(349\) −30.7629 −1.64670 −0.823350 0.567534i \(-0.807898\pi\)
−0.823350 + 0.567534i \(0.807898\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −22.0270 −1.17404
\(353\) 6.12173 0.325827 0.162913 0.986640i \(-0.447911\pi\)
0.162913 + 0.986640i \(0.447911\pi\)
\(354\) −0.412991 −0.0219502
\(355\) 5.43876 0.288659
\(356\) −18.3613 −0.973145
\(357\) 0 0
\(358\) 1.78095 0.0941259
\(359\) −19.4287 −1.02541 −0.512703 0.858566i \(-0.671356\pi\)
−0.512703 + 0.858566i \(0.671356\pi\)
\(360\) 1.62382 0.0855827
\(361\) −15.1681 −0.798319
\(362\) 2.50616 0.131721
\(363\) 35.7133 1.87446
\(364\) 0 0
\(365\) −13.2060 −0.691235
\(366\) 3.28929 0.171934
\(367\) −5.40467 −0.282122 −0.141061 0.990001i \(-0.545051\pi\)
−0.141061 + 0.990001i \(0.545051\pi\)
\(368\) −1.80920 −0.0943111
\(369\) 6.56315 0.341664
\(370\) 3.85776 0.200555
\(371\) 0 0
\(372\) 16.8483 0.873544
\(373\) 16.2507 0.841428 0.420714 0.907193i \(-0.361780\pi\)
0.420714 + 0.907193i \(0.361780\pi\)
\(374\) −8.56803 −0.443042
\(375\) −16.6934 −0.862045
\(376\) −8.09390 −0.417411
\(377\) 0 0
\(378\) 0 0
\(379\) 25.1730 1.29305 0.646525 0.762893i \(-0.276222\pi\)
0.646525 + 0.762893i \(0.276222\pi\)
\(380\) −5.34708 −0.274300
\(381\) 21.0146 1.07661
\(382\) 4.50247 0.230366
\(383\) 3.81438 0.194906 0.0974529 0.995240i \(-0.468930\pi\)
0.0974529 + 0.995240i \(0.468930\pi\)
\(384\) −13.4531 −0.686527
\(385\) 0 0
\(386\) 6.16764 0.313924
\(387\) −3.69132 −0.187640
\(388\) −12.4391 −0.631500
\(389\) −2.87096 −0.145563 −0.0727817 0.997348i \(-0.523188\pi\)
−0.0727817 + 0.997348i \(0.523188\pi\)
\(390\) 0 0
\(391\) −2.33821 −0.118248
\(392\) 0 0
\(393\) −13.8222 −0.697236
\(394\) −0.886819 −0.0446773
\(395\) −0.189557 −0.00953766
\(396\) −9.77696 −0.491310
\(397\) −19.1184 −0.959524 −0.479762 0.877399i \(-0.659277\pi\)
−0.479762 + 0.877399i \(0.659277\pi\)
\(398\) 6.71394 0.336539
\(399\) 0 0
\(400\) −9.74725 −0.487362
\(401\) 2.99824 0.149725 0.0748625 0.997194i \(-0.476148\pi\)
0.0748625 + 0.997194i \(0.476148\pi\)
\(402\) −0.478958 −0.0238882
\(403\) 0 0
\(404\) 0.133833 0.00665844
\(405\) −8.15104 −0.405028
\(406\) 0 0
\(407\) −47.8129 −2.37000
\(408\) −8.16937 −0.404444
\(409\) −34.0805 −1.68517 −0.842587 0.538560i \(-0.818969\pi\)
−0.842587 + 0.538560i \(0.818969\pi\)
\(410\) −3.62943 −0.179245
\(411\) 24.2424 1.19579
\(412\) −11.9762 −0.590026
\(413\) 0 0
\(414\) 0.155995 0.00766674
\(415\) 3.85710 0.189337
\(416\) 0 0
\(417\) 26.9746 1.32095
\(418\) −3.87464 −0.189515
\(419\) 34.7759 1.69891 0.849457 0.527657i \(-0.176929\pi\)
0.849457 + 0.527657i \(0.176929\pi\)
\(420\) 0 0
\(421\) −24.1400 −1.17651 −0.588257 0.808674i \(-0.700186\pi\)
−0.588257 + 0.808674i \(0.700186\pi\)
\(422\) 4.36301 0.212388
\(423\) −5.43988 −0.264496
\(424\) 3.59271 0.174478
\(425\) −12.5973 −0.611060
\(426\) 1.82549 0.0884451
\(427\) 0 0
\(428\) −14.6429 −0.707793
\(429\) 0 0
\(430\) 2.04130 0.0984404
\(431\) −4.76477 −0.229511 −0.114755 0.993394i \(-0.536608\pi\)
−0.114755 + 0.993394i \(0.536608\pi\)
\(432\) −18.9170 −0.910143
\(433\) 22.0231 1.05836 0.529181 0.848509i \(-0.322499\pi\)
0.529181 + 0.848509i \(0.322499\pi\)
\(434\) 0 0
\(435\) 15.1072 0.724337
\(436\) 0.0634168 0.00303711
\(437\) −1.05739 −0.0505817
\(438\) −4.43252 −0.211794
\(439\) 3.43240 0.163819 0.0819097 0.996640i \(-0.473898\pi\)
0.0819097 + 0.996640i \(0.473898\pi\)
\(440\) 11.1294 0.530576
\(441\) 0 0
\(442\) 0 0
\(443\) −8.70594 −0.413632 −0.206816 0.978380i \(-0.566310\pi\)
−0.206816 + 0.978380i \(0.566310\pi\)
\(444\) −22.1467 −1.05104
\(445\) 14.0477 0.665923
\(446\) −0.738982 −0.0349918
\(447\) −4.49747 −0.212723
\(448\) 0 0
\(449\) −17.6120 −0.831159 −0.415580 0.909557i \(-0.636421\pi\)
−0.415580 + 0.909557i \(0.636421\pi\)
\(450\) 0.840438 0.0396186
\(451\) 44.9830 2.11817
\(452\) 17.3834 0.817646
\(453\) 3.72090 0.174823
\(454\) 9.00887 0.422807
\(455\) 0 0
\(456\) −3.69436 −0.173004
\(457\) 9.07268 0.424402 0.212201 0.977226i \(-0.431937\pi\)
0.212201 + 0.977226i \(0.431937\pi\)
\(458\) −6.31139 −0.294912
\(459\) −24.4483 −1.14115
\(460\) 1.47548 0.0687944
\(461\) −6.58319 −0.306610 −0.153305 0.988179i \(-0.548992\pi\)
−0.153305 + 0.988179i \(0.548992\pi\)
\(462\) 0 0
\(463\) −3.47344 −0.161424 −0.0807121 0.996737i \(-0.525719\pi\)
−0.0807121 + 0.996737i \(0.525719\pi\)
\(464\) 23.9766 1.11309
\(465\) −12.8901 −0.597766
\(466\) 7.22585 0.334731
\(467\) 29.7854 1.37830 0.689152 0.724617i \(-0.257983\pi\)
0.689152 + 0.724617i \(0.257983\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 3.00826 0.138761
\(471\) −13.7357 −0.632906
\(472\) 1.10035 0.0506479
\(473\) −25.2998 −1.16329
\(474\) −0.0636237 −0.00292233
\(475\) −5.69677 −0.261386
\(476\) 0 0
\(477\) 2.41465 0.110559
\(478\) 6.63738 0.303587
\(479\) −35.1855 −1.60767 −0.803833 0.594855i \(-0.797209\pi\)
−0.803833 + 0.594855i \(0.797209\pi\)
\(480\) 7.80587 0.356288
\(481\) 0 0
\(482\) 1.07373 0.0489072
\(483\) 0 0
\(484\) −46.2252 −2.10115
\(485\) 9.51680 0.432136
\(486\) 2.89585 0.131358
\(487\) 1.80154 0.0816355 0.0408178 0.999167i \(-0.487004\pi\)
0.0408178 + 0.999167i \(0.487004\pi\)
\(488\) −8.76384 −0.396721
\(489\) 1.01482 0.0458916
\(490\) 0 0
\(491\) −8.19322 −0.369755 −0.184877 0.982762i \(-0.559189\pi\)
−0.184877 + 0.982762i \(0.559189\pi\)
\(492\) 20.8359 0.939356
\(493\) 30.9874 1.39560
\(494\) 0 0
\(495\) 7.48006 0.336204
\(496\) −20.4579 −0.918587
\(497\) 0 0
\(498\) 1.29461 0.0580129
\(499\) 36.5164 1.63470 0.817350 0.576141i \(-0.195442\pi\)
0.817350 + 0.576141i \(0.195442\pi\)
\(500\) 21.6070 0.966293
\(501\) 20.4143 0.912045
\(502\) 4.15192 0.185309
\(503\) −3.02972 −0.135089 −0.0675443 0.997716i \(-0.521516\pi\)
−0.0675443 + 0.997716i \(0.521516\pi\)
\(504\) 0 0
\(505\) −0.102392 −0.00455637
\(506\) 1.06917 0.0475304
\(507\) 0 0
\(508\) −27.2001 −1.20681
\(509\) −29.3447 −1.30068 −0.650341 0.759642i \(-0.725374\pi\)
−0.650341 + 0.759642i \(0.725374\pi\)
\(510\) 3.03631 0.134450
\(511\) 0 0
\(512\) 21.0487 0.930229
\(513\) −11.0560 −0.488135
\(514\) 1.93694 0.0854348
\(515\) 9.16265 0.403755
\(516\) −11.7188 −0.515890
\(517\) −37.2843 −1.63976
\(518\) 0 0
\(519\) −7.93637 −0.348368
\(520\) 0 0
\(521\) −29.6838 −1.30047 −0.650236 0.759732i \(-0.725330\pi\)
−0.650236 + 0.759732i \(0.725330\pi\)
\(522\) −2.06734 −0.0904851
\(523\) 20.5727 0.899583 0.449791 0.893134i \(-0.351498\pi\)
0.449791 + 0.893134i \(0.351498\pi\)
\(524\) 17.8906 0.781553
\(525\) 0 0
\(526\) 5.82146 0.253828
\(527\) −26.4398 −1.15173
\(528\) −29.1179 −1.26719
\(529\) −22.7082 −0.987314
\(530\) −1.33530 −0.0580019
\(531\) 0.739544 0.0320935
\(532\) 0 0
\(533\) 0 0
\(534\) 4.71501 0.204039
\(535\) 11.2029 0.484343
\(536\) 1.27611 0.0551197
\(537\) 7.82216 0.337551
\(538\) 7.42151 0.319964
\(539\) 0 0
\(540\) 15.4275 0.663896
\(541\) 34.0668 1.46465 0.732324 0.680957i \(-0.238436\pi\)
0.732324 + 0.680957i \(0.238436\pi\)
\(542\) −8.76457 −0.376470
\(543\) 11.0074 0.472373
\(544\) 16.0111 0.686470
\(545\) −0.0485183 −0.00207830
\(546\) 0 0
\(547\) −0.850931 −0.0363832 −0.0181916 0.999835i \(-0.505791\pi\)
−0.0181916 + 0.999835i \(0.505791\pi\)
\(548\) −31.3780 −1.34040
\(549\) −5.89015 −0.251385
\(550\) 5.76026 0.245618
\(551\) 14.0131 0.596980
\(552\) 1.01942 0.0433895
\(553\) 0 0
\(554\) −3.11641 −0.132404
\(555\) 16.9438 0.719224
\(556\) −34.9143 −1.48069
\(557\) −17.7281 −0.751162 −0.375581 0.926789i \(-0.622557\pi\)
−0.375581 + 0.926789i \(0.622557\pi\)
\(558\) 1.76394 0.0746737
\(559\) 0 0
\(560\) 0 0
\(561\) −37.6319 −1.58882
\(562\) −5.90809 −0.249218
\(563\) −24.1806 −1.01909 −0.509545 0.860444i \(-0.670186\pi\)
−0.509545 + 0.860444i \(0.670186\pi\)
\(564\) −17.2699 −0.727194
\(565\) −13.2995 −0.559515
\(566\) −3.19301 −0.134212
\(567\) 0 0
\(568\) −4.86375 −0.204078
\(569\) −42.7749 −1.79322 −0.896608 0.442825i \(-0.853976\pi\)
−0.896608 + 0.442825i \(0.853976\pi\)
\(570\) 1.37308 0.0575122
\(571\) 7.36280 0.308124 0.154062 0.988061i \(-0.450765\pi\)
0.154062 + 0.988061i \(0.450765\pi\)
\(572\) 0 0
\(573\) 19.7754 0.826131
\(574\) 0 0
\(575\) 1.57197 0.0655557
\(576\) 4.75217 0.198007
\(577\) −8.19393 −0.341118 −0.170559 0.985347i \(-0.554557\pi\)
−0.170559 + 0.985347i \(0.554557\pi\)
\(578\) 0.577585 0.0240244
\(579\) 27.0891 1.12578
\(580\) −19.5539 −0.811932
\(581\) 0 0
\(582\) 3.19426 0.132406
\(583\) 16.5497 0.685419
\(584\) 11.8098 0.488694
\(585\) 0 0
\(586\) −3.86679 −0.159736
\(587\) 39.1141 1.61441 0.807205 0.590271i \(-0.200979\pi\)
0.807205 + 0.590271i \(0.200979\pi\)
\(588\) 0 0
\(589\) −11.9566 −0.492664
\(590\) −0.408969 −0.0168370
\(591\) −3.89503 −0.160220
\(592\) 26.8914 1.10523
\(593\) −1.21338 −0.0498276 −0.0249138 0.999690i \(-0.507931\pi\)
−0.0249138 + 0.999690i \(0.507931\pi\)
\(594\) 11.1792 0.458689
\(595\) 0 0
\(596\) 5.82125 0.238448
\(597\) 29.4885 1.20688
\(598\) 0 0
\(599\) 32.6638 1.33461 0.667303 0.744786i \(-0.267449\pi\)
0.667303 + 0.744786i \(0.267449\pi\)
\(600\) 5.49224 0.224220
\(601\) 2.50114 0.102024 0.0510118 0.998698i \(-0.483755\pi\)
0.0510118 + 0.998698i \(0.483755\pi\)
\(602\) 0 0
\(603\) 0.857671 0.0349271
\(604\) −4.81611 −0.195965
\(605\) 35.3656 1.43781
\(606\) −0.0343672 −0.00139607
\(607\) 12.6456 0.513271 0.256635 0.966508i \(-0.417386\pi\)
0.256635 + 0.966508i \(0.417386\pi\)
\(608\) 7.24055 0.293643
\(609\) 0 0
\(610\) 3.25726 0.131883
\(611\) 0 0
\(612\) 7.10672 0.287272
\(613\) 20.0280 0.808923 0.404462 0.914555i \(-0.367459\pi\)
0.404462 + 0.914555i \(0.367459\pi\)
\(614\) 4.59608 0.185483
\(615\) −15.9409 −0.642801
\(616\) 0 0
\(617\) −45.2926 −1.82341 −0.911705 0.410846i \(-0.865233\pi\)
−0.911705 + 0.410846i \(0.865233\pi\)
\(618\) 3.07539 0.123710
\(619\) −4.43315 −0.178183 −0.0890917 0.996023i \(-0.528396\pi\)
−0.0890917 + 0.996023i \(0.528396\pi\)
\(620\) 16.6842 0.670055
\(621\) 3.05080 0.122424
\(622\) 10.2087 0.409332
\(623\) 0 0
\(624\) 0 0
\(625\) −1.97997 −0.0791988
\(626\) 3.68494 0.147280
\(627\) −17.0179 −0.679631
\(628\) 17.7786 0.709444
\(629\) 34.7544 1.38575
\(630\) 0 0
\(631\) 19.7358 0.785672 0.392836 0.919609i \(-0.371494\pi\)
0.392836 + 0.919609i \(0.371494\pi\)
\(632\) 0.169516 0.00674300
\(633\) 19.1629 0.761658
\(634\) −7.93824 −0.315268
\(635\) 20.8100 0.825819
\(636\) 7.66574 0.303967
\(637\) 0 0
\(638\) −14.1693 −0.560968
\(639\) −3.26891 −0.129316
\(640\) −13.3221 −0.526603
\(641\) 39.6425 1.56579 0.782893 0.622157i \(-0.213743\pi\)
0.782893 + 0.622157i \(0.213743\pi\)
\(642\) 3.76018 0.148402
\(643\) −20.8300 −0.821453 −0.410727 0.911759i \(-0.634725\pi\)
−0.410727 + 0.911759i \(0.634725\pi\)
\(644\) 0 0
\(645\) 8.96568 0.353023
\(646\) 2.81642 0.110810
\(647\) −15.7441 −0.618965 −0.309482 0.950905i \(-0.600156\pi\)
−0.309482 + 0.950905i \(0.600156\pi\)
\(648\) 7.28927 0.286350
\(649\) 5.06874 0.198966
\(650\) 0 0
\(651\) 0 0
\(652\) −1.31352 −0.0514413
\(653\) −27.0264 −1.05762 −0.528812 0.848739i \(-0.677362\pi\)
−0.528812 + 0.848739i \(0.677362\pi\)
\(654\) −0.0162849 −0.000636789 0
\(655\) −13.6876 −0.534817
\(656\) −25.2998 −0.987792
\(657\) 7.93734 0.309665
\(658\) 0 0
\(659\) −6.79491 −0.264692 −0.132346 0.991204i \(-0.542251\pi\)
−0.132346 + 0.991204i \(0.542251\pi\)
\(660\) 23.7468 0.924344
\(661\) −7.20526 −0.280252 −0.140126 0.990134i \(-0.544751\pi\)
−0.140126 + 0.990134i \(0.544751\pi\)
\(662\) −6.09044 −0.236712
\(663\) 0 0
\(664\) −3.44930 −0.133859
\(665\) 0 0
\(666\) −2.31866 −0.0898463
\(667\) −3.86679 −0.149723
\(668\) −26.4231 −1.02234
\(669\) −3.24571 −0.125486
\(670\) −0.474293 −0.0183236
\(671\) −40.3703 −1.55848
\(672\) 0 0
\(673\) 8.32130 0.320763 0.160381 0.987055i \(-0.448728\pi\)
0.160381 + 0.987055i \(0.448728\pi\)
\(674\) −2.39896 −0.0924044
\(675\) 16.4365 0.632640
\(676\) 0 0
\(677\) −29.9956 −1.15283 −0.576413 0.817159i \(-0.695548\pi\)
−0.576413 + 0.817159i \(0.695548\pi\)
\(678\) −4.46391 −0.171435
\(679\) 0 0
\(680\) −8.08982 −0.310231
\(681\) 39.5682 1.51626
\(682\) 12.0899 0.462944
\(683\) 36.0839 1.38071 0.690356 0.723469i \(-0.257454\pi\)
0.690356 + 0.723469i \(0.257454\pi\)
\(684\) 3.21380 0.122883
\(685\) 24.0064 0.917236
\(686\) 0 0
\(687\) −27.7205 −1.05760
\(688\) 14.2294 0.542491
\(689\) 0 0
\(690\) −0.378889 −0.0144241
\(691\) 25.7677 0.980248 0.490124 0.871653i \(-0.336951\pi\)
0.490124 + 0.871653i \(0.336951\pi\)
\(692\) 10.2724 0.390497
\(693\) 0 0
\(694\) −7.00589 −0.265940
\(695\) 26.7119 1.01324
\(696\) −13.5100 −0.512096
\(697\) −32.6974 −1.23850
\(698\) −10.2248 −0.387016
\(699\) 31.7369 1.20040
\(700\) 0 0
\(701\) 41.7872 1.57828 0.789141 0.614213i \(-0.210526\pi\)
0.789141 + 0.614213i \(0.210526\pi\)
\(702\) 0 0
\(703\) 15.7167 0.592765
\(704\) 32.5708 1.22756
\(705\) 13.2127 0.497619
\(706\) 2.03471 0.0765774
\(707\) 0 0
\(708\) 2.34782 0.0882364
\(709\) −0.343847 −0.0129135 −0.00645673 0.999979i \(-0.502055\pi\)
−0.00645673 + 0.999979i \(0.502055\pi\)
\(710\) 1.80771 0.0678421
\(711\) 0.113931 0.00427276
\(712\) −12.5625 −0.470799
\(713\) 3.29931 0.123560
\(714\) 0 0
\(715\) 0 0
\(716\) −10.1245 −0.378372
\(717\) 29.1523 1.08871
\(718\) −6.45761 −0.240996
\(719\) 8.78010 0.327443 0.163721 0.986507i \(-0.447650\pi\)
0.163721 + 0.986507i \(0.447650\pi\)
\(720\) −4.20702 −0.156786
\(721\) 0 0
\(722\) −5.04149 −0.187625
\(723\) 4.71598 0.175389
\(724\) −14.2473 −0.529498
\(725\) −20.8327 −0.773707
\(726\) 11.8702 0.440546
\(727\) −17.3658 −0.644064 −0.322032 0.946729i \(-0.604366\pi\)
−0.322032 + 0.946729i \(0.604366\pi\)
\(728\) 0 0
\(729\) 29.6343 1.09757
\(730\) −4.38936 −0.162458
\(731\) 18.3900 0.680180
\(732\) −18.6994 −0.691148
\(733\) 9.05895 0.334600 0.167300 0.985906i \(-0.446495\pi\)
0.167300 + 0.985906i \(0.446495\pi\)
\(734\) −1.79638 −0.0663056
\(735\) 0 0
\(736\) −1.99796 −0.0736458
\(737\) 5.87837 0.216533
\(738\) 2.18143 0.0802995
\(739\) 7.07843 0.260384 0.130192 0.991489i \(-0.458441\pi\)
0.130192 + 0.991489i \(0.458441\pi\)
\(740\) −21.9310 −0.806201
\(741\) 0 0
\(742\) 0 0
\(743\) 14.6779 0.538479 0.269240 0.963073i \(-0.413228\pi\)
0.269240 + 0.963073i \(0.413228\pi\)
\(744\) 11.5273 0.422613
\(745\) −4.45367 −0.163170
\(746\) 5.40132 0.197756
\(747\) −2.31827 −0.0848209
\(748\) 48.7085 1.78096
\(749\) 0 0
\(750\) −5.54848 −0.202602
\(751\) −31.7113 −1.15716 −0.578580 0.815626i \(-0.696393\pi\)
−0.578580 + 0.815626i \(0.696393\pi\)
\(752\) 20.9698 0.764691
\(753\) 18.2358 0.664548
\(754\) 0 0
\(755\) 3.68467 0.134099
\(756\) 0 0
\(757\) 15.5317 0.564510 0.282255 0.959339i \(-0.408918\pi\)
0.282255 + 0.959339i \(0.408918\pi\)
\(758\) 8.36688 0.303899
\(759\) 4.69594 0.170452
\(760\) −3.65839 −0.132704
\(761\) −0.250369 −0.00907587 −0.00453794 0.999990i \(-0.501444\pi\)
−0.00453794 + 0.999990i \(0.501444\pi\)
\(762\) 6.98474 0.253030
\(763\) 0 0
\(764\) −25.5961 −0.926036
\(765\) −5.43714 −0.196580
\(766\) 1.26781 0.0458077
\(767\) 0 0
\(768\) 11.4972 0.414869
\(769\) 24.0146 0.865988 0.432994 0.901397i \(-0.357457\pi\)
0.432994 + 0.901397i \(0.357457\pi\)
\(770\) 0 0
\(771\) 8.50731 0.306383
\(772\) −35.0625 −1.26193
\(773\) −30.5062 −1.09723 −0.548616 0.836074i \(-0.684845\pi\)
−0.548616 + 0.836074i \(0.684845\pi\)
\(774\) −1.22690 −0.0441001
\(775\) 17.7754 0.638510
\(776\) −8.51064 −0.305514
\(777\) 0 0
\(778\) −0.954237 −0.0342110
\(779\) −14.7865 −0.529780
\(780\) 0 0
\(781\) −22.4047 −0.801702
\(782\) −0.777163 −0.0277913
\(783\) −40.4311 −1.44489
\(784\) 0 0
\(785\) −13.6019 −0.485473
\(786\) −4.59414 −0.163868
\(787\) −19.8492 −0.707548 −0.353774 0.935331i \(-0.615102\pi\)
−0.353774 + 0.935331i \(0.615102\pi\)
\(788\) 5.04149 0.179596
\(789\) 25.5686 0.910268
\(790\) −0.0630042 −0.00224159
\(791\) 0 0
\(792\) −6.68923 −0.237691
\(793\) 0 0
\(794\) −6.35448 −0.225512
\(795\) −5.86484 −0.208004
\(796\) −38.1682 −1.35284
\(797\) 52.2894 1.85219 0.926093 0.377296i \(-0.123146\pi\)
0.926093 + 0.377296i \(0.123146\pi\)
\(798\) 0 0
\(799\) 27.1014 0.958777
\(800\) −10.7642 −0.380572
\(801\) −8.44319 −0.298326
\(802\) 0.996542 0.0351891
\(803\) 54.4015 1.91979
\(804\) 2.72283 0.0960269
\(805\) 0 0
\(806\) 0 0
\(807\) 32.5963 1.14744
\(808\) 0.0915664 0.00322129
\(809\) −2.36460 −0.0831349 −0.0415674 0.999136i \(-0.513235\pi\)
−0.0415674 + 0.999136i \(0.513235\pi\)
\(810\) −2.70920 −0.0951917
\(811\) −23.6646 −0.830978 −0.415489 0.909598i \(-0.636390\pi\)
−0.415489 + 0.909598i \(0.636390\pi\)
\(812\) 0 0
\(813\) −38.4952 −1.35008
\(814\) −15.8918 −0.557008
\(815\) 1.00493 0.0352013
\(816\) 21.1654 0.740936
\(817\) 8.31636 0.290953
\(818\) −11.3275 −0.396058
\(819\) 0 0
\(820\) 20.6330 0.720536
\(821\) 3.57753 0.124857 0.0624284 0.998049i \(-0.480115\pi\)
0.0624284 + 0.998049i \(0.480115\pi\)
\(822\) 8.05759 0.281041
\(823\) −29.9422 −1.04372 −0.521859 0.853032i \(-0.674761\pi\)
−0.521859 + 0.853032i \(0.674761\pi\)
\(824\) −8.19393 −0.285449
\(825\) 25.2998 0.880827
\(826\) 0 0
\(827\) 9.32620 0.324304 0.162152 0.986766i \(-0.448157\pi\)
0.162152 + 0.986766i \(0.448157\pi\)
\(828\) −0.886819 −0.0308191
\(829\) 38.2268 1.32767 0.663836 0.747878i \(-0.268927\pi\)
0.663836 + 0.747878i \(0.268927\pi\)
\(830\) 1.28200 0.0444990
\(831\) −13.6877 −0.474821
\(832\) 0 0
\(833\) 0 0
\(834\) 8.96568 0.310456
\(835\) 20.2155 0.699587
\(836\) 22.0270 0.761820
\(837\) 34.4975 1.19241
\(838\) 11.5587 0.399287
\(839\) 23.4981 0.811244 0.405622 0.914041i \(-0.367055\pi\)
0.405622 + 0.914041i \(0.367055\pi\)
\(840\) 0 0
\(841\) 22.2451 0.767071
\(842\) −8.02356 −0.276510
\(843\) −25.9491 −0.893736
\(844\) −24.8034 −0.853767
\(845\) 0 0
\(846\) −1.80808 −0.0621632
\(847\) 0 0
\(848\) −9.30806 −0.319640
\(849\) −14.0241 −0.481306
\(850\) −4.18704 −0.143614
\(851\) −4.33686 −0.148666
\(852\) −10.3777 −0.355535
\(853\) 40.9295 1.40140 0.700700 0.713456i \(-0.252871\pi\)
0.700700 + 0.713456i \(0.252871\pi\)
\(854\) 0 0
\(855\) −2.45879 −0.0840888
\(856\) −10.0185 −0.342424
\(857\) −11.6620 −0.398366 −0.199183 0.979962i \(-0.563829\pi\)
−0.199183 + 0.979962i \(0.563829\pi\)
\(858\) 0 0
\(859\) 28.2776 0.964820 0.482410 0.875945i \(-0.339762\pi\)
0.482410 + 0.875945i \(0.339762\pi\)
\(860\) −11.6046 −0.395715
\(861\) 0 0
\(862\) −1.58369 −0.0539407
\(863\) 11.4442 0.389567 0.194783 0.980846i \(-0.437600\pi\)
0.194783 + 0.980846i \(0.437600\pi\)
\(864\) −20.8906 −0.710713
\(865\) −7.85909 −0.267217
\(866\) 7.31993 0.248741
\(867\) 2.53683 0.0861553
\(868\) 0 0
\(869\) 0.780871 0.0264892
\(870\) 5.02127 0.170237
\(871\) 0 0
\(872\) 0.0433887 0.00146933
\(873\) −5.71997 −0.193592
\(874\) −0.351449 −0.0118879
\(875\) 0 0
\(876\) 25.1985 0.851380
\(877\) 56.3486 1.90276 0.951378 0.308026i \(-0.0996684\pi\)
0.951378 + 0.308026i \(0.0996684\pi\)
\(878\) 1.14084 0.0385016
\(879\) −16.9835 −0.572838
\(880\) −28.8344 −0.972006
\(881\) −1.16418 −0.0392221 −0.0196111 0.999808i \(-0.506243\pi\)
−0.0196111 + 0.999808i \(0.506243\pi\)
\(882\) 0 0
\(883\) −12.1881 −0.410162 −0.205081 0.978745i \(-0.565746\pi\)
−0.205081 + 0.978745i \(0.565746\pi\)
\(884\) 0 0
\(885\) −1.79625 −0.0603802
\(886\) −2.89364 −0.0972138
\(887\) 30.6641 1.02960 0.514799 0.857311i \(-0.327866\pi\)
0.514799 + 0.857311i \(0.327866\pi\)
\(888\) −15.1524 −0.508481
\(889\) 0 0
\(890\) 4.66910 0.156509
\(891\) 33.5778 1.12490
\(892\) 4.20105 0.140662
\(893\) 12.2558 0.410125
\(894\) −1.49485 −0.0499952
\(895\) 7.74598 0.258920
\(896\) 0 0
\(897\) 0 0
\(898\) −5.85378 −0.195343
\(899\) −43.7245 −1.45829
\(900\) −4.77782 −0.159261
\(901\) −12.0297 −0.400768
\(902\) 14.9512 0.497822
\(903\) 0 0
\(904\) 11.8934 0.395570
\(905\) 10.9002 0.362336
\(906\) 1.23674 0.0410878
\(907\) −11.6479 −0.386763 −0.193382 0.981124i \(-0.561946\pi\)
−0.193382 + 0.981124i \(0.561946\pi\)
\(908\) −51.2147 −1.69962
\(909\) 0.0615414 0.00204120
\(910\) 0 0
\(911\) −26.5833 −0.880743 −0.440371 0.897816i \(-0.645153\pi\)
−0.440371 + 0.897816i \(0.645153\pi\)
\(912\) 9.57141 0.316941
\(913\) −15.8891 −0.525852
\(914\) 3.01553 0.0997450
\(915\) 14.3063 0.472953
\(916\) 35.8797 1.18550
\(917\) 0 0
\(918\) −8.12600 −0.268198
\(919\) −45.7079 −1.50777 −0.753883 0.657009i \(-0.771821\pi\)
−0.753883 + 0.657009i \(0.771821\pi\)
\(920\) 1.00950 0.0332821
\(921\) 20.1866 0.665171
\(922\) −2.18809 −0.0720609
\(923\) 0 0
\(924\) 0 0
\(925\) −23.3653 −0.768246
\(926\) −1.15448 −0.0379387
\(927\) −5.50711 −0.180877
\(928\) 26.4782 0.869189
\(929\) 11.0651 0.363035 0.181518 0.983388i \(-0.441899\pi\)
0.181518 + 0.983388i \(0.441899\pi\)
\(930\) −4.28437 −0.140490
\(931\) 0 0
\(932\) −41.0784 −1.34557
\(933\) 44.8380 1.46793
\(934\) 9.89994 0.323936
\(935\) −37.2655 −1.21871
\(936\) 0 0
\(937\) −57.6584 −1.88362 −0.941808 0.336150i \(-0.890875\pi\)
−0.941808 + 0.336150i \(0.890875\pi\)
\(938\) 0 0
\(939\) 16.1848 0.528170
\(940\) −17.1017 −0.557797
\(941\) 16.9745 0.553351 0.276676 0.960963i \(-0.410767\pi\)
0.276676 + 0.960963i \(0.410767\pi\)
\(942\) −4.56540 −0.148749
\(943\) 4.08018 0.132869
\(944\) −2.85082 −0.0927862
\(945\) 0 0
\(946\) −8.40904 −0.273401
\(947\) 16.6962 0.542552 0.271276 0.962502i \(-0.412554\pi\)
0.271276 + 0.962502i \(0.412554\pi\)
\(948\) 0.361696 0.0117473
\(949\) 0 0
\(950\) −1.89347 −0.0614322
\(951\) −34.8658 −1.13060
\(952\) 0 0
\(953\) −18.2473 −0.591089 −0.295545 0.955329i \(-0.595501\pi\)
−0.295545 + 0.955329i \(0.595501\pi\)
\(954\) 0.802570 0.0259842
\(955\) 19.5829 0.633687
\(956\) −37.7330 −1.22037
\(957\) −62.2334 −2.01172
\(958\) −11.6948 −0.377841
\(959\) 0 0
\(960\) −11.5423 −0.372527
\(961\) 6.30763 0.203472
\(962\) 0 0
\(963\) −6.73337 −0.216980
\(964\) −6.10408 −0.196599
\(965\) 26.8253 0.863537
\(966\) 0 0
\(967\) 29.5845 0.951374 0.475687 0.879615i \(-0.342199\pi\)
0.475687 + 0.879615i \(0.342199\pi\)
\(968\) −31.6265 −1.01652
\(969\) 12.3701 0.397384
\(970\) 3.16315 0.101563
\(971\) 15.1301 0.485548 0.242774 0.970083i \(-0.421943\pi\)
0.242774 + 0.970083i \(0.421943\pi\)
\(972\) −16.4627 −0.528040
\(973\) 0 0
\(974\) 0.598787 0.0191864
\(975\) 0 0
\(976\) 22.7055 0.726786
\(977\) −32.4636 −1.03860 −0.519301 0.854591i \(-0.673808\pi\)
−0.519301 + 0.854591i \(0.673808\pi\)
\(978\) 0.337300 0.0107857
\(979\) −57.8686 −1.84949
\(980\) 0 0
\(981\) 0.0291614 0.000931052 0
\(982\) −2.72322 −0.0869016
\(983\) 44.2945 1.41278 0.706388 0.707825i \(-0.250323\pi\)
0.706388 + 0.707825i \(0.250323\pi\)
\(984\) 14.2556 0.454452
\(985\) −3.85710 −0.122897
\(986\) 10.2994 0.328001
\(987\) 0 0
\(988\) 0 0
\(989\) −2.29482 −0.0729710
\(990\) 2.48619 0.0790163
\(991\) −8.52117 −0.270684 −0.135342 0.990799i \(-0.543213\pi\)
−0.135342 + 0.990799i \(0.543213\pi\)
\(992\) −22.5923 −0.717307
\(993\) −26.7500 −0.848887
\(994\) 0 0
\(995\) 29.2014 0.925746
\(996\) −7.35976 −0.233203
\(997\) −6.77905 −0.214695 −0.107347 0.994222i \(-0.534236\pi\)
−0.107347 + 0.994222i \(0.534236\pi\)
\(998\) 12.1372 0.384195
\(999\) −45.3462 −1.43469
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 8281.2.a.ck.1.5 8
7.2 even 3 1183.2.e.i.508.4 16
7.4 even 3 1183.2.e.i.170.4 16
7.6 odd 2 8281.2.a.cj.1.5 8
13.5 odd 4 637.2.c.f.246.4 8
13.8 odd 4 637.2.c.f.246.5 8
13.12 even 2 inner 8281.2.a.ck.1.4 8
91.5 even 12 637.2.r.f.116.4 16
91.18 odd 12 91.2.r.a.51.5 yes 16
91.25 even 6 1183.2.e.i.170.5 16
91.31 even 12 637.2.r.f.324.5 16
91.34 even 4 637.2.c.e.246.5 8
91.44 odd 12 91.2.r.a.25.4 16
91.47 even 12 637.2.r.f.116.5 16
91.51 even 6 1183.2.e.i.508.5 16
91.60 odd 12 91.2.r.a.51.4 yes 16
91.73 even 12 637.2.r.f.324.4 16
91.83 even 4 637.2.c.e.246.4 8
91.86 odd 12 91.2.r.a.25.5 yes 16
91.90 odd 2 8281.2.a.cj.1.4 8
273.44 even 12 819.2.dl.e.298.5 16
273.86 even 12 819.2.dl.e.298.4 16
273.200 even 12 819.2.dl.e.415.4 16
273.242 even 12 819.2.dl.e.415.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.r.a.25.4 16 91.44 odd 12
91.2.r.a.25.5 yes 16 91.86 odd 12
91.2.r.a.51.4 yes 16 91.60 odd 12
91.2.r.a.51.5 yes 16 91.18 odd 12
637.2.c.e.246.4 8 91.83 even 4
637.2.c.e.246.5 8 91.34 even 4
637.2.c.f.246.4 8 13.5 odd 4
637.2.c.f.246.5 8 13.8 odd 4
637.2.r.f.116.4 16 91.5 even 12
637.2.r.f.116.5 16 91.47 even 12
637.2.r.f.324.4 16 91.73 even 12
637.2.r.f.324.5 16 91.31 even 12
819.2.dl.e.298.4 16 273.86 even 12
819.2.dl.e.298.5 16 273.44 even 12
819.2.dl.e.415.4 16 273.200 even 12
819.2.dl.e.415.5 16 273.242 even 12
1183.2.e.i.170.4 16 7.4 even 3
1183.2.e.i.170.5 16 91.25 even 6
1183.2.e.i.508.4 16 7.2 even 3
1183.2.e.i.508.5 16 91.51 even 6
8281.2.a.cj.1.4 8 91.90 odd 2
8281.2.a.cj.1.5 8 7.6 odd 2
8281.2.a.ck.1.4 8 13.12 even 2 inner
8281.2.a.ck.1.5 8 1.1 even 1 trivial