↑

Properties

Label 6027.2.a.d.1.1
Level $6027$
Weight $2$
Character 6027.1
Self dual yes
Analytic conductor $48.126$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6027,2,Mod(1,6027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6027.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6027.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: \(48.1258372982\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 861)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6027.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} +8.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} +2.00000 q^{30} -4.00000 q^{31} -5.00000 q^{32} -2.00000 q^{34} -1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} -6.00000 q^{40} +1.00000 q^{41} +12.0000 q^{43} -2.00000 q^{45} -8.00000 q^{46} -1.00000 q^{48} +1.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} -14.0000 q^{61} +4.00000 q^{62} +7.00000 q^{64} +4.00000 q^{65} -4.00000 q^{67} -2.00000 q^{68} +8.00000 q^{69} +4.00000 q^{71} +3.00000 q^{72} -2.00000 q^{73} -6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +2.00000 q^{78} +8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -1.00000 q^{82} -4.00000 q^{83} -4.00000 q^{85} -12.0000 q^{86} -6.00000 q^{87} +10.0000 q^{89} +2.00000 q^{90} -8.00000 q^{92} -4.00000 q^{93} -8.00000 q^{95} -5.00000 q^{96} -6.00000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 3.00000 0.612372
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) −6.00000 −0.948683
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −8.00000 −1.17954
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 3.00000 0.353553
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −1.00000 −0.110432
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −12.0000 −1.29399
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) −5.00000 −0.510310
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −2.00000 −0.198030
\(103\) −12.0000 −1.18240 −0.591198 − 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −4.00000 −0.374634
\(115\) −16.0000 −1.49201
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) −6.00000 −0.547723
\(121\) −11.0000 −1.00000
\(122\) 14.0000 1.26750
\(123\) 1.00000 0.0901670
\(124\) 4.00000 0.359211
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 3.00000 0.265165
\(129\) 12.0000 1.05654
\(130\) −4.00000 −0.350823
\(131\) 20.0000 1.74741 0.873704 − 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) −2.00000 −0.172133
\(136\) 6.00000 0.514496
\(137\) −18.0000 −1.53784 −0.768922 − 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) −8.00000 −0.681005
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 12.0000 0.996546
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 1.00000 0.0816497
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 12.0000 0.973329
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 2.00000 0.160128
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) 10.0000 0.790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) −1.00000 −0.0780869
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 4.00000 0.306786
\(171\) 4.00000 0.305888
\(172\) −12.0000 −0.914991
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) −10.0000 −0.749532
\(179\) 8.00000 0.597948 0.298974 − 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) 2.00000 0.149071
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) 24.0000 1.76930
\(185\) −12.0000 −0.882258
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 20.0000 1.44715 0.723575 − 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 7.00000 0.505181
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 6.00000 0.430775
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −3.00000 −0.212132
\(201\) −4.00000 −0.282138
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) −2.00000 −0.139686
\(206\) 12.0000 0.836080
\(207\) 8.00000 0.556038
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 6.00000 0.412082
\(213\) 4.00000 0.274075
\(214\) −12.0000 −0.820303
\(215\) −24.0000 −1.63679
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) −6.00000 −0.402694
\(223\) 20.0000 1.33930 0.669650 − 0.742677i \(-0.266444\pi\)
0.669650 + 0.742677i \(0.266444\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 6.00000 0.399114
\(227\) 28.0000 1.85843 0.929213 − 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) −4.00000 −0.264906
\(229\) −2.00000 −0.132164 −0.0660819 − 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 16.0000 1.05501
\(231\) 0 0
\(232\) −18.0000 −1.18176
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 2.00000 0.129099
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) −1.00000 −0.0637577
\(247\) −8.00000 −0.509028
\(248\) −12.0000 −0.762001
\(249\) −4.00000 −0.253490
\(250\) −12.0000 −0.758947
\(251\) −4.00000 −0.252478 −0.126239 − 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) −4.00000 −0.250490
\(256\) −17.0000 −1.06250
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −12.0000 −0.747087
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) −6.00000 −0.371391
\(262\) −20.0000 −1.23560
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 2.00000 0.121716
\(271\) 28.0000 1.70088 0.850439 − 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) 6.00000 0.360505 0.180253 − 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 − 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) −4.00000 −0.237356
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) −12.0000 −0.704664
\(291\) −6.00000 −0.351726
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 18.0000 1.04623
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) −16.0000 −0.925304
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) 6.00000 0.344691
\(304\) −4.00000 −0.229416
\(305\) 28.0000 1.60328
\(306\) −2.00000 −0.114332
\(307\) −16.0000 −0.913168 −0.456584 − 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) −8.00000 −0.454369
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −6.00000 −0.339683
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) −14.0000 −0.782624
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) −1.00000 −0.0555556
\(325\) 2.00000 0.110940
\(326\) 20.0000 1.10770
\(327\) 14.0000 0.774202
\(328\) 3.00000 0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) −16.0000 −0.875481
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 36.0000 1.94099
\(345\) −16.0000 −0.861411
\(346\) 18.0000 0.967686
\(347\) −8.00000 −0.429463 −0.214731 − 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 6.00000 0.321634
\(349\) 26.0000 1.39175 0.695874 − 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 12.0000 0.637793
\(355\) −8.00000 −0.424596
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −8.00000 −0.422813
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −6.00000 −0.316228
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 14.0000 0.731792
\(367\) 4.00000 0.208798 0.104399 − 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) −8.00000 −0.417029
\(369\) 1.00000 0.0520579
\(370\) 12.0000 0.623850
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) 22.0000 1.13912 0.569558 − 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 8.00000 0.410391
\(381\) 8.00000 0.409852
\(382\) −20.0000 −1.02329
\(383\) −8.00000 −0.408781 −0.204390 − 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 12.0000 0.609994
\(388\) 6.00000 0.304604
\(389\) 22.0000 1.11544 0.557722 − 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) −4.00000 −0.202548
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) 20.0000 1.00887
\(394\) 2.00000 0.100759
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 38.0000 1.90717 0.953583 − 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 4.00000 0.199502
\(403\) 8.00000 0.398508
\(404\) −6.00000 −0.298511
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) 6.00000 0.296681 0.148340 − 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 2.00000 0.0987730
\(411\) −18.0000 −0.887875
\(412\) 12.0000 0.591198
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 8.00000 0.392705
\(416\) 10.0000 0.490290
\(417\) 0 0
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 − 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 − 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) −2.00000 −0.0970143
\(426\) −4.00000 −0.193801
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 24.0000 1.15738
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) −14.0000 −0.670478
\(437\) 32.0000 1.53077
\(438\) 2.00000 0.0955637
\(439\) −16.0000 −0.763638 −0.381819 − 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −6.00000 −0.284747
\(445\) −20.0000 −0.948091
\(446\) −20.0000 −0.947027
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 16.0000 0.751746
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) 2.00000 0.0935561 0.0467780 − 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 2.00000 0.0934539
\(459\) 2.00000 0.0933520
\(460\) 16.0000 0.746004
\(461\) −2.00000 −0.0931493 −0.0465746 − 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 − 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 6.00000 0.278543
\(465\) 8.00000 0.370991
\(466\) −6.00000 −0.277945
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) −36.0000 −1.65703
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −12.0000 −0.548867
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 10.0000 0.456435
\(481\) −12.0000 −0.547153
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 12.0000 0.544892
\(486\) −1.00000 −0.0453609
\(487\) −32.0000 −1.45006 −0.725029 − 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) −42.0000 −1.90125
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −4.00000 −0.180517 −0.0902587 − 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) −1.00000 −0.0450835
\(493\) −12.0000 −0.540453
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −12.0000 −0.536656
\(501\) 16.0000 0.714827
\(502\) 4.00000 0.178529
\(503\) 8.00000 0.356702 0.178351 − 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) −8.00000 −0.354943
\(509\) 14.0000 0.620539 0.310270 − 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 4.00000 0.176604
\(514\) −18.0000 −0.793946
\(515\) 24.0000 1.05757
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 12.0000 0.526235
\(521\) 10.0000 0.438108 0.219054 − 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 6.00000 0.262613
\(523\) 8.00000 0.349816 0.174908 − 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −12.0000 −0.521247
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −2.00000 −0.0866296
\(534\) −10.0000 −0.432742
\(535\) −24.0000 −1.03761
\(536\) −12.0000 −0.518321
\(537\) 8.00000 0.345225
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −34.0000 −1.46177 −0.730887 − 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −28.0000 −1.20270
\(543\) −2.00000 −0.0858282
\(544\) −10.0000 −0.428746
\(545\) −28.0000 −1.19939
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 18.0000 0.768922
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 24.0000 1.02151
\(553\) 0 0
\(554\) −6.00000 −0.254916
\(555\) −12.0000 −0.509372
\(556\) 0 0
\(557\) 2.00000 0.0847427 0.0423714 − 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 4.00000 0.169334
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) −16.0000 −0.672530
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) −30.0000 −1.25767 −0.628833 − 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 8.00000 0.335083
\(571\) 12.0000 0.502184 0.251092 − 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) 20.0000 0.835512
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 7.00000 0.291667
\(577\) 26.0000 1.08239 0.541197 − 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) 13.0000 0.540729
\(579\) −14.0000 −0.581820
\(580\) −12.0000 −0.498273
\(581\) 0 0
\(582\) 6.00000 0.248708
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 4.00000 0.165380
\(586\) −6.00000 −0.247858
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) −24.0000 −0.988064
\(591\) −2.00000 −0.0822690
\(592\) −6.00000 −0.246598
\(593\) 34.0000 1.39621 0.698106 − 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −8.00000 −0.327418
\(598\) 16.0000 0.654289
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −3.00000 −0.122474
\(601\) −14.0000 −0.571072 −0.285536 − 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −16.0000 −0.651031
\(605\) 22.0000 0.894427
\(606\) −6.00000 −0.243733
\(607\) −4.00000 −0.162355 −0.0811775 − 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) −20.0000 −0.811107
\(609\) 0 0
\(610\) −28.0000 −1.13369
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) −42.0000 −1.69636 −0.848182 − 0.529705i \(-0.822303\pi\)
−0.848182 + 0.529705i \(0.822303\pi\)
\(614\) 16.0000 0.645707
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 12.0000 0.482711
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) −8.00000 −0.321288
\(621\) 8.00000 0.321029
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) 14.0000 0.559553
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 24.0000 0.954669
\(633\) 4.00000 0.158986
\(634\) −18.0000 −0.714871
\(635\) −16.0000 −0.634941
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) −6.00000 −0.237171
\(641\) −42.0000 −1.65890 −0.829450 − 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) −12.0000 −0.473602
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) −8.00000 −0.314756
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 3.00000 0.117851
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −14.0000 −0.547862 −0.273931 − 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) −14.0000 −0.547443
\(655\) −40.0000 −1.56293
\(656\) −1.00000 −0.0390434
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 − 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) −4.00000 −0.155464
\(663\) −4.00000 −0.155347
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) −48.0000 −1.85857
\(668\) −16.0000 −0.619059
\(669\) 20.0000 0.773245
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 − 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 14.0000 0.539260
\(675\) −1.00000 −0.0384900
\(676\) 9.00000 0.346154
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) −12.0000 −0.460179
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) 8.00000 0.306111 0.153056 − 0.988218i \(-0.451089\pi\)
0.153056 + 0.988218i \(0.451089\pi\)
\(684\) −4.00000 −0.152944
\(685\) 36.0000 1.37549
\(686\) 0 0
\(687\) −2.00000 −0.0763048
\(688\) −12.0000 −0.457496
\(689\) 12.0000 0.457164
\(690\) 16.0000 0.609110
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 8.00000 0.303676
\(695\) 0 0
\(696\) −18.0000 −0.682288
\(697\) 2.00000 0.0757554
\(698\) −26.0000 −0.984115
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 2.00000 0.0754851
\(703\) 24.0000 0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 8.00000 0.300235
\(711\) 8.00000 0.300023
\(712\) 30.0000 1.12430
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) −8.00000 −0.298974
\(717\) 12.0000 0.448148
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 − 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −18.0000 −0.669427
\(724\) 2.00000 0.0743294
\(725\) 6.00000 0.222834
\(726\) 11.0000 0.408248
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) 24.0000 0.887672
\(732\) 14.0000 0.517455
\(733\) 42.0000 1.55131 0.775653 − 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) −40.0000 −1.47442
\(737\) 0 0
\(738\) −1.00000 −0.0368105
\(739\) 12.0000 0.441427 0.220714 − 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 12.0000 0.441129
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 − 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) −12.0000 −0.439941
\(745\) −36.0000 −1.31894
\(746\) −22.0000 −0.805477
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) −4.00000 −0.145768
\(754\) −12.0000 −0.437014
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) −24.0000 −0.870572
\(761\) −6.00000 −0.217500 −0.108750 − 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) −20.0000 −0.723575
\(765\) −4.00000 −0.144620
\(766\) 8.00000 0.289052
\(767\) 24.0000 0.866590
\(768\) −17.0000 −0.613435
\(769\) −18.0000 −0.649097 −0.324548 − 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 14.0000 0.503871
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −12.0000 −0.431331
\(775\) 4.00000 0.143684
\(776\) −18.0000 −0.646162
\(777\) 0 0
\(778\) −22.0000 −0.788738
\(779\) 4.00000 0.143315
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) −16.0000 −0.572159
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) −20.0000 −0.713376
\(787\) 32.0000 1.14068 0.570338 − 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 2.00000 0.0712470
\(789\) 12.0000 0.427211
\(790\) 16.0000 0.569254
\(791\) 0 0
\(792\) 0 0
\(793\) 28.0000 0.994309
\(794\) −38.0000 −1.34857
\(795\) 12.0000 0.425596
\(796\) 8.00000 0.283552
\(797\) −18.0000 −0.637593 −0.318796 − 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 10.0000 0.353333
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 6.00000 0.211210
\(808\) 18.0000 0.633238
\(809\) −26.0000 −0.914111 −0.457056 − 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 2.00000 0.0702728
\(811\) 8.00000 0.280918 0.140459 − 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 0 0
\(813\) 28.0000 0.982003
\(814\) 0 0
\(815\) 40.0000 1.40114
\(816\) −2.00000 −0.0700140
\(817\) 48.0000 1.67931
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 18.0000 0.627822
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −36.0000 −1.25412
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 − 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) −8.00000 −0.278019
\(829\) 50.0000 1.73657 0.868286 − 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) −8.00000 −0.277684
\(831\) 6.00000 0.208138
\(832\) −14.0000 −0.485363
\(833\) 0 0
\(834\) 0 0
\(835\) −32.0000 −1.10741
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) −28.0000 −0.967244
\(839\) 16.0000 0.552381 0.276191 − 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −30.0000 −1.03387
\(843\) −2.00000 −0.0688837
\(844\) −4.00000 −0.137686
\(845\) 18.0000 0.619219
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 16.0000 0.549119
\(850\) 2.00000 0.0685994
\(851\) 48.0000 1.64542
\(852\) −4.00000 −0.137038
\(853\) −54.0000 −1.84892 −0.924462 − 0.381273i \(-0.875486\pi\)
−0.924462 + 0.381273i \(0.875486\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 36.0000 1.23045
\(857\) 26.0000 0.888143 0.444072 − 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) 0 0
\(859\) 16.0000 0.545913 0.272956 − 0.962026i \(-0.411998\pi\)
0.272956 + 0.962026i \(0.411998\pi\)
\(860\) 24.0000 0.818393
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −5.00000 −0.170103
\(865\) 36.0000 1.22404
\(866\) 2.00000 0.0679628
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) −12.0000 −0.406838
\(871\) 8.00000 0.271070
\(872\) 42.0000 1.42230
\(873\) −6.00000 −0.203069
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −2.00000 −0.0675352 −0.0337676 − 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 16.0000 0.539974
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 − 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 4.00000 0.134535
\(885\) 24.0000 0.806751
\(886\) 4.00000 0.134383
\(887\) 32.0000 1.07445 0.537227 − 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 18.0000 0.604040
\(889\) 0 0
\(890\) 20.0000 0.670402
\(891\) 0 0
\(892\) −20.0000 −0.669650
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) −16.0000 −0.534821
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 6.00000 0.200223
\(899\) 24.0000 0.800445
\(900\) 1.00000 0.0333333
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 4.00000 0.132964
\(906\) −16.0000 −0.531564
\(907\) −4.00000 −0.132818 −0.0664089 − 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) −28.0000 −0.929213
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 56.0000 1.85536 0.927681 − 0.373373i \(-0.121799\pi\)
0.927681 + 0.373373i \(0.121799\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) 28.0000 0.925651
\(916\) 2.00000 0.0660819
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) −48.0000 −1.58251
\(921\) −16.0000 −0.527218
\(922\) 2.00000 0.0658665
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −40.0000 −1.31448
\(927\) −12.0000 −0.394132
\(928\) 30.0000 0.984798
\(929\) −30.0000 −0.984268 −0.492134 − 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) −8.00000 −0.262330
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 24.0000 0.785725
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) −22.0000 −0.718709 −0.359354 − 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 − 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 2.00000 0.0651635
\(943\) 8.00000 0.260516
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −8.00000 −0.259828
\(949\) 4.00000 0.129845
\(950\) 4.00000 0.129777
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 − 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 6.00000 0.194257
\(955\) −40.0000 −1.29437
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) −14.0000 −0.451848
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) 12.0000 0.386695
\(964\) 18.0000 0.579741
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) −56.0000 −1.80084 −0.900419 − 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) −33.0000 −1.06066
\(969\) 8.00000 0.256997
\(970\) −12.0000 −0.385297
\(971\) −20.0000 −0.641831 −0.320915 − 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 32.0000 1.02535
\(975\) 2.00000 0.0640513
\(976\) 14.0000 0.448129
\(977\) −58.0000 −1.85558 −0.927792 − 0.373097i \(-0.878296\pi\)
−0.927792 + 0.373097i \(0.878296\pi\)
\(978\) 20.0000 0.639529
\(979\) 0 0
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 4.00000 0.127645
\(983\) 16.0000 0.510321 0.255160 − 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 3.00000 0.0956365
\(985\) 4.00000 0.127451
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 96.0000 3.05262
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 − 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 20.0000 0.635001
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 4.00000 0.126745
\(997\) −2.00000 −0.0633406 −0.0316703 − 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) −4.00000 −0.126618
\(999\) 6.00000 0.189832
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 6027.2.a.d.1.1 1
7.6 odd 2 861.2.a.a.1.1 ✓ 1
21.20 even 2 2583.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
861.2.a.a.1.1 ✓ 1 7.6 odd 2
2583.2.a.e.1.1 1 21.20 even 2
6027.2.a.d.1.1 1 1.1 even 1 trivial