Properties

Label 3570.2.a.t.1.1
Level $3570$
Weight $2$
Character 3570.1
Self dual yes
Analytic conductor $28.507$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3570,2,Mod(1,3570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3570, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3570.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3570.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: \(28.5065935216\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3570.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} +4.00000 q^{22} -1.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +1.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} +4.00000 q^{38} -6.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} +6.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -1.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} -2.00000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -4.00000 q^{66} +12.0000 q^{67} +1.00000 q^{68} -1.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -4.00000 q^{77} -6.00000 q^{78} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} +1.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} +4.00000 q^{88} +10.0000 q^{89} +1.00000 q^{90} -6.00000 q^{91} +8.00000 q^{93} +4.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.00000 0.218218
\(22\) 4.00000 0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.00000 −0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 1.00000 0.171499
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 4.00000 0.648886
\(39\) −6.00000 −0.960769
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −1.00000 −0.140028
\(52\) 6.00000 0.832050
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.00000 0.539360
\(56\) −1.00000 −0.133631
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −8.00000 −1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) −4.00000 −0.492366
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −10.0000 −1.16248
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) −4.00000 −0.455842
\(78\) −6.00000 −0.679366
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 0.109109
\(85\) 1.00000 0.108465
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) 4.00000 0.426401
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 1.00000 0.105409
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 1.00000 0.101015
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 6.00000 0.588348
\(105\) 1.00000 0.0975900
\(106\) 6.00000 0.582772
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 4.00000 0.381385
\(111\) 10.0000 0.949158
\(112\) −1.00000 −0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 6.00000 0.554700
\(118\) −12.0000 −1.10469
\(119\) −1.00000 −0.0916698
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) −2.00000 −0.180334
\(124\) −8.00000 −0.718421
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 6.00000 0.526235
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −4.00000 −0.348155
\(133\) −4.00000 −0.346844
\(134\) 12.0000 1.03664
\(135\) −1.00000 −0.0860663
\(136\) 1.00000 0.0857493
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 24.0000 2.00698
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 2.00000 0.165521
\(147\) −1.00000 −0.0824786
\(148\) −10.0000 −0.821995
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 4.00000 0.324443
\(153\) 1.00000 0.0808452
\(154\) −4.00000 −0.322329
\(155\) −8.00000 −0.642575
\(156\) −6.00000 −0.480384
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 2.00000 0.156174
\(165\) −4.00000 −0.311400
\(166\) 4.00000 0.310460
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 1.00000 0.0771517
\(169\) 23.0000 1.76923
\(170\) 1.00000 0.0766965
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 2.00000 0.151620
\(175\) −1.00000 −0.0755929
\(176\) 4.00000 0.301511
\(177\) 12.0000 0.901975
\(178\) 10.0000 0.749532
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −6.00000 −0.444750
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −10.0000 −0.735215
\(186\) 8.00000 0.586588
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 4.00000 0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) 10.0000 0.717958
\(195\) −6.00000 −0.429669
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 4.00000 0.284268
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000 0.0707107
\(201\) −12.0000 −0.846415
\(202\) −2.00000 −0.140720
\(203\) 2.00000 0.140372
\(204\) −1.00000 −0.0700140
\(205\) 2.00000 0.139686
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) 16.0000 1.10674
\(210\) 1.00000 0.0690066
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 6.00000 0.412082
\(213\) 8.00000 0.548151
\(214\) 4.00000 0.273434
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 8.00000 0.543075
\(218\) 14.0000 0.948200
\(219\) −2.00000 −0.135147
\(220\) 4.00000 0.269680
\(221\) 6.00000 0.403604
\(222\) 10.0000 0.671156
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 18.0000 1.19734
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) −4.00000 −0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) −2.00000 −0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) −8.00000 −0.519656
\(238\) −1.00000 −0.0648204
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 1.00000 0.0638877
\(246\) −2.00000 −0.127515
\(247\) 24.0000 1.52708
\(248\) −8.00000 −0.508001
\(249\) −4.00000 −0.253490
\(250\) 1.00000 0.0632456
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) −1.00000 −0.0626224
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) −4.00000 −0.249029
\(259\) 10.0000 0.621370
\(260\) 6.00000 0.372104
\(261\) −2.00000 −0.123797
\(262\) 12.0000 0.741362
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −4.00000 −0.246183
\(265\) 6.00000 0.368577
\(266\) −4.00000 −0.245256
\(267\) −10.0000 −0.611990
\(268\) 12.0000 0.733017
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 1.00000 0.0606339
\(273\) 6.00000 0.363137
\(274\) 10.0000 0.604122
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −12.0000 −0.719712
\(279\) −8.00000 −0.478947
\(280\) −1.00000 −0.0597614
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −8.00000 −0.474713
\(285\) −4.00000 −0.236940
\(286\) 24.0000 1.41915
\(287\) −2.00000 −0.118056
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) −2.00000 −0.117444
\(291\) −10.0000 −0.586210
\(292\) 2.00000 0.117041
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −12.0000 −0.698667
\(296\) −10.0000 −0.581238
\(297\) −4.00000 −0.232104
\(298\) −2.00000 −0.115857
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −4.00000 −0.230556
\(302\) −8.00000 −0.460348
\(303\) 2.00000 0.114897
\(304\) 4.00000 0.229416
\(305\) −2.00000 −0.114520
\(306\) 1.00000 0.0571662
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) −4.00000 −0.227921
\(309\) 8.00000 0.455104
\(310\) −8.00000 −0.454369
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) −6.00000 −0.339683
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −10.0000 −0.564333
\(315\) −1.00000 −0.0563436
\(316\) 8.00000 0.450035
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −6.00000 −0.336463
\(319\) −8.00000 −0.447914
\(320\) 1.00000 0.0559017
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 6.00000 0.332820
\(326\) −20.0000 −1.10770
\(327\) −14.0000 −0.774202
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 4.00000 0.219529
\(333\) −10.0000 −0.547997
\(334\) −16.0000 −0.875481
\(335\) 12.0000 0.655630
\(336\) 1.00000 0.0545545
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 23.0000 1.25104
\(339\) −18.0000 −0.977626
\(340\) 1.00000 0.0542326
\(341\) −32.0000 −1.73290
\(342\) 4.00000 0.216295
\(343\) −1.00000 −0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 36.0000 1.93258 0.966291 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) 2.00000 0.107211
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −6.00000 −0.320256
\(352\) 4.00000 0.213201
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 12.0000 0.637793
\(355\) −8.00000 −0.424596
\(356\) 10.0000 0.529999
\(357\) 1.00000 0.0529256
\(358\) −4.00000 −0.211407
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) −5.00000 −0.262432
\(364\) −6.00000 −0.314485
\(365\) 2.00000 0.104685
\(366\) 2.00000 0.104542
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) −10.0000 −0.519875
\(371\) −6.00000 −0.311504
\(372\) 8.00000 0.414781
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 4.00000 0.206835
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 1.00000 0.0514344
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 4.00000 0.205196
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −4.00000 −0.203859
\(386\) 26.0000 1.32337
\(387\) 4.00000 0.203331
\(388\) 10.0000 0.507673
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) −6.00000 −0.303822
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) −12.0000 −0.605320
\(394\) 6.00000 0.302276
\(395\) 8.00000 0.402524
\(396\) 4.00000 0.201008
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) −12.0000 −0.598506
\(403\) −48.0000 −2.39105
\(404\) −2.00000 −0.0995037
\(405\) 1.00000 0.0496904
\(406\) 2.00000 0.0992583
\(407\) −40.0000 −1.98273
\(408\) −1.00000 −0.0495074
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 2.00000 0.0987730
\(411\) −10.0000 −0.493264
\(412\) −8.00000 −0.394132
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 6.00000 0.294174
\(417\) 12.0000 0.587643
\(418\) 16.0000 0.782586
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 1.00000 0.0487950
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 1.00000 0.0485071
\(426\) 8.00000 0.387601
\(427\) 2.00000 0.0967868
\(428\) 4.00000 0.193347
\(429\) −24.0000 −1.15873
\(430\) 4.00000 0.192897
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 8.00000 0.384012
\(435\) 2.00000 0.0958927
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 4.00000 0.190693
\(441\) 1.00000 0.0476190
\(442\) 6.00000 0.285391
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 10.0000 0.474579
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) 2.00000 0.0945968
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 1.00000 0.0471405
\(451\) 8.00000 0.376705
\(452\) 18.0000 0.846649
\(453\) 8.00000 0.375873
\(454\) −4.00000 −0.187729
\(455\) −6.00000 −0.281284
\(456\) −4.00000 −0.187317
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −10.0000 −0.467269
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 4.00000 0.186097
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 8.00000 0.370991
\(466\) −6.00000 −0.277945
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 6.00000 0.277350
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −12.0000 −0.552345
\(473\) 16.0000 0.735681
\(474\) −8.00000 −0.367452
\(475\) 4.00000 0.183533
\(476\) −1.00000 −0.0458349
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −60.0000 −2.73576
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 10.0000 0.454077
\(486\) −1.00000 −0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 20.0000 0.904431
\(490\) 1.00000 0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −2.00000 −0.0900755
\(494\) 24.0000 1.07981
\(495\) 4.00000 0.179787
\(496\) −8.00000 −0.359211
\(497\) 8.00000 0.358849
\(498\) −4.00000 −0.179244
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 1.00000 0.0447214
\(501\) 16.0000 0.714827
\(502\) 4.00000 0.178529
\(503\) −32.0000 −1.42681 −0.713405 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 16.0000 0.709885
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) −1.00000 −0.0442807
\(511\) −2.00000 −0.0884748
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 2.00000 0.0882162
\(515\) −8.00000 −0.352522
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 10.0000 0.439375
\(519\) 18.0000 0.790112
\(520\) 6.00000 0.263117
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) −24.0000 −1.04645
\(527\) −8.00000 −0.348485
\(528\) −4.00000 −0.174078
\(529\) −23.0000 −1.00000
\(530\) 6.00000 0.260623
\(531\) −12.0000 −0.520756
\(532\) −4.00000 −0.173422
\(533\) 12.0000 0.519778
\(534\) −10.0000 −0.432742
\(535\) 4.00000 0.172935
\(536\) 12.0000 0.518321
\(537\) 4.00000 0.172613
\(538\) −18.0000 −0.776035
\(539\) 4.00000 0.172292
\(540\) −1.00000 −0.0430331
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −16.0000 −0.687259
\(543\) 10.0000 0.429141
\(544\) 1.00000 0.0428746
\(545\) 14.0000 0.599694
\(546\) 6.00000 0.256776
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 10.0000 0.427179
\(549\) −2.00000 −0.0853579
\(550\) 4.00000 0.170561
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 22.0000 0.934690
\(555\) 10.0000 0.424476
\(556\) −12.0000 −0.508913
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) −8.00000 −0.338667
\(559\) 24.0000 1.01509
\(560\) −1.00000 −0.0422577
\(561\) −4.00000 −0.168880
\(562\) −22.0000 −0.928014
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 18.0000 0.757266
\(566\) 4.00000 0.168133
\(567\) −1.00000 −0.0419961
\(568\) −8.00000 −0.335673
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −4.00000 −0.167542
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 24.0000 1.00349
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 1.00000 0.0415945
\(579\) −26.0000 −1.08052
\(580\) −2.00000 −0.0830455
\(581\) −4.00000 −0.165948
\(582\) −10.0000 −0.414513
\(583\) 24.0000 0.993978
\(584\) 2.00000 0.0827606
\(585\) 6.00000 0.248069
\(586\) −10.0000 −0.413096
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −32.0000 −1.31854
\(590\) −12.0000 −0.494032
\(591\) −6.00000 −0.246807
\(592\) −10.0000 −0.410997
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) −4.00000 −0.164122
\(595\) −1.00000 −0.0409960
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −4.00000 −0.163028
\(603\) 12.0000 0.488678
\(604\) −8.00000 −0.325515
\(605\) 5.00000 0.203279
\(606\) 2.00000 0.0812444
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 4.00000 0.162221
\(609\) −2.00000 −0.0810441
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 1.00000 0.0404226
\(613\) 30.0000 1.21169 0.605844 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) −4.00000 −0.161427
\(615\) −2.00000 −0.0806478
\(616\) −4.00000 −0.161165
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 8.00000 0.321807
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) −10.0000 −0.400642
\(624\) −6.00000 −0.240192
\(625\) 1.00000 0.0400000
\(626\) 2.00000 0.0799361
\(627\) −16.0000 −0.638978
\(628\) −10.0000 −0.399043
\(629\) −10.0000 −0.398726
\(630\) −1.00000 −0.0398410
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 8.00000 0.318223
\(633\) 20.0000 0.794929
\(634\) −2.00000 −0.0794301
\(635\) 16.0000 0.634941
\(636\) −6.00000 −0.237915
\(637\) 6.00000 0.237729
\(638\) −8.00000 −0.316723
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) −38.0000 −1.50091 −0.750455 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) −4.00000 −0.157867
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 4.00000 0.157378
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) −48.0000 −1.88416
\(650\) 6.00000 0.235339
\(651\) −8.00000 −0.313545
\(652\) −20.0000 −0.783260
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −14.0000 −0.547443
\(655\) 12.0000 0.468879
\(656\) 2.00000 0.0780869
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) −4.00000 −0.155700
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −20.0000 −0.777322
\(663\) −6.00000 −0.233021
\(664\) 4.00000 0.155230
\(665\) −4.00000 −0.155113
\(666\) −10.0000 −0.387492
\(667\) 0 0
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) −8.00000 −0.308837
\(672\) 1.00000 0.0385758
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 26.0000 1.00148
\(675\) −1.00000 −0.0384900
\(676\) 23.0000 0.884615
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) −18.0000 −0.691286
\(679\) −10.0000 −0.383765
\(680\) 1.00000 0.0383482
\(681\) 4.00000 0.153280
\(682\) −32.0000 −1.22534
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 4.00000 0.152944
\(685\) 10.0000 0.382080
\(686\) −1.00000 −0.0381802
\(687\) 10.0000 0.381524
\(688\) 4.00000 0.152499
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) −18.0000 −0.684257
\(693\) −4.00000 −0.151947
\(694\) 36.0000 1.36654
\(695\) −12.0000 −0.455186
\(696\) 2.00000 0.0758098
\(697\) 2.00000 0.0757554
\(698\) 30.0000 1.13552
\(699\) 6.00000 0.226941
\(700\) −1.00000 −0.0377964
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) −6.00000 −0.226455
\(703\) −40.0000 −1.50863
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 2.00000 0.0752177
\(708\) 12.0000 0.450988
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −8.00000 −0.300235
\(711\) 8.00000 0.300023
\(712\) 10.0000 0.374766
\(713\) 0 0
\(714\) 1.00000 0.0374241
\(715\) 24.0000 0.897549
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000 0.0372678
\(721\) 8.00000 0.297936
\(722\) −3.00000 −0.111648
\(723\) −2.00000 −0.0743808
\(724\) −10.0000 −0.371647
\(725\) −2.00000 −0.0742781
\(726\) −5.00000 −0.185567
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) 4.00000 0.147945
\(732\) 2.00000 0.0739221
\(733\) 38.0000 1.40356 0.701781 0.712393i \(-0.252388\pi\)
0.701781 + 0.712393i \(0.252388\pi\)
\(734\) −16.0000 −0.590571
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 2.00000 0.0736210
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) −10.0000 −0.367607
\(741\) −24.0000 −0.881662
\(742\) −6.00000 −0.220267
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 8.00000 0.293294
\(745\) −2.00000 −0.0732743
\(746\) 14.0000 0.512576
\(747\) 4.00000 0.146352
\(748\) 4.00000 0.146254
\(749\) −4.00000 −0.146157
\(750\) −1.00000 −0.0365148
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) −4.00000 −0.145768
\(754\) −12.0000 −0.437014
\(755\) −8.00000 −0.291150
\(756\) 1.00000 0.0363696
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −16.0000 −0.579619
\(763\) −14.0000 −0.506834
\(764\) 0 0
\(765\) 1.00000 0.0361551
\(766\) 16.0000 0.578103
\(767\) −72.0000 −2.59977
\(768\) −1.00000 −0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) −4.00000 −0.144150
\(771\) −2.00000 −0.0720282
\(772\) 26.0000 0.935760
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 4.00000 0.143777
\(775\) −8.00000 −0.287368
\(776\) 10.0000 0.358979
\(777\) −10.0000 −0.358748
\(778\) 14.0000 0.501924
\(779\) 8.00000 0.286630
\(780\) −6.00000 −0.214834
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 1.00000 0.0357143
\(785\) −10.0000 −0.356915
\(786\) −12.0000 −0.428026
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 6.00000 0.213741
\(789\) 24.0000 0.854423
\(790\) 8.00000 0.284627
\(791\) −18.0000 −0.640006
\(792\) 4.00000 0.142134
\(793\) −12.0000 −0.426132
\(794\) −2.00000 −0.0709773
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 4.00000 0.141598
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 10.0000 0.353333
\(802\) −22.0000 −0.776847
\(803\) 8.00000 0.282314
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) 18.0000 0.633630
\(808\) −2.00000 −0.0703598
\(809\) −46.0000 −1.61727 −0.808637 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) 1.00000 0.0351364
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) 2.00000 0.0701862
\(813\) 16.0000 0.561144
\(814\) −40.0000 −1.40200
\(815\) −20.0000 −0.700569
\(816\) −1.00000 −0.0350070
\(817\) 16.0000 0.559769
\(818\) 26.0000 0.909069
\(819\) −6.00000 −0.209657
\(820\) 2.00000 0.0698430
\(821\) −26.0000 −0.907406 −0.453703 0.891153i \(-0.649897\pi\)
−0.453703 + 0.891153i \(0.649897\pi\)
\(822\) −10.0000 −0.348790
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −8.00000 −0.278693
\(825\) −4.00000 −0.139262
\(826\) 12.0000 0.417533
\(827\) −44.0000 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 4.00000 0.138842
\(831\) −22.0000 −0.763172
\(832\) 6.00000 0.208013
\(833\) 1.00000 0.0346479
\(834\) 12.0000 0.415526
\(835\) −16.0000 −0.553703
\(836\) 16.0000 0.553372
\(837\) 8.00000 0.276520
\(838\) 28.0000 0.967244
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 1.00000 0.0345033
\(841\) −25.0000 −0.862069
\(842\) 6.00000 0.206774
\(843\) 22.0000 0.757720
\(844\) −20.0000 −0.688428
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 6.00000 0.206041
\(849\) −4.00000 −0.137280
\(850\) 1.00000 0.0342997
\(851\) 0 0
\(852\) 8.00000 0.274075
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 2.00000 0.0684386
\(855\) 4.00000 0.136797
\(856\) 4.00000 0.136717
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) −24.0000 −0.819346
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 4.00000 0.136399
\(861\) 2.00000 0.0681598
\(862\) −32.0000 −1.08992
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) 34.0000 1.15537
\(867\) −1.00000 −0.0339618
\(868\) 8.00000 0.271538
\(869\) 32.0000 1.08553
\(870\) 2.00000 0.0678064
\(871\) 72.0000 2.43963
\(872\) 14.0000 0.474100
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) −2.00000 −0.0675737
\(877\) 46.0000 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) 0 0
\(879\) 10.0000 0.337292
\(880\) 4.00000 0.134840
\(881\) −38.0000 −1.28025 −0.640126 0.768270i \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(882\) 1.00000 0.0336718
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 6.00000 0.201802
\(885\) 12.0000 0.403376
\(886\) −36.0000 −1.20944
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 10.0000 0.335578
\(889\) −16.0000 −0.536623
\(890\) 10.0000 0.335201
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) 0 0
\(894\) 2.00000 0.0668900
\(895\) −4.00000 −0.133705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 16.0000 0.533630
\(900\) 1.00000 0.0333333
\(901\) 6.00000 0.199889
\(902\) 8.00000 0.266371
\(903\) 4.00000 0.133112
\(904\) 18.0000 0.598671
\(905\) −10.0000 −0.332411
\(906\) 8.00000 0.265782
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) −4.00000 −0.132745
\(909\) −2.00000 −0.0663358
\(910\) −6.00000 −0.198898
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) −4.00000 −0.132453
\(913\) 16.0000 0.529523
\(914\) 26.0000 0.860004
\(915\) 2.00000 0.0661180
\(916\) −10.0000 −0.330409
\(917\) −12.0000 −0.396275
\(918\) −1.00000 −0.0330049
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) −10.0000 −0.329332
\(923\) −48.0000 −1.57994
\(924\) 4.00000 0.131590
\(925\) −10.0000 −0.328798
\(926\) 16.0000 0.525793
\(927\) −8.00000 −0.262754
\(928\) −2.00000 −0.0656532
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 8.00000 0.262330
\(931\) 4.00000 0.131095
\(932\) −6.00000 −0.196537
\(933\) 8.00000 0.261908
\(934\) −12.0000 −0.392652
\(935\) 4.00000 0.130814
\(936\) 6.00000 0.196116
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) −12.0000 −0.391814
\(939\) −2.00000 −0.0652675
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 10.0000 0.325818
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 1.00000 0.0325300
\(946\) 16.0000 0.520205
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −8.00000 −0.259828
\(949\) 12.0000 0.389536
\(950\) 4.00000 0.129777
\(951\) 2.00000 0.0648544
\(952\) −1.00000 −0.0324102
\(953\) 58.0000 1.87880 0.939402 0.342817i \(-0.111381\pi\)
0.939402 + 0.342817i \(0.111381\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) 8.00000 0.258603
\(958\) 16.0000 0.516937
\(959\) −10.0000 −0.322917
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) −60.0000 −1.93448
\(963\) 4.00000 0.128898
\(964\) 2.00000 0.0644157
\(965\) 26.0000 0.836970
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 5.00000 0.160706
\(969\) −4.00000 −0.128499
\(970\) 10.0000 0.321081
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 12.0000 0.384702
\(974\) −8.00000 −0.256337
\(975\) −6.00000 −0.192154
\(976\) −2.00000 −0.0640184
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 20.0000 0.639529
\(979\) 40.0000 1.27841
\(980\) 1.00000 0.0319438
\(981\) 14.0000 0.446986
\(982\) −12.0000 −0.382935
\(983\) 32.0000 1.02064 0.510321 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 6.00000 0.191176
\(986\) −2.00000 −0.0636930
\(987\) 0 0
\(988\) 24.0000 0.763542
\(989\) 0 0
\(990\) 4.00000 0.127128
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) −8.00000 −0.254000
\(993\) 20.0000 0.634681
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) −20.0000 −0.633089
\(999\) 10.0000 0.316386
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 3570.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3570.2.a.t.1.1 1 1.1 even 1 trivial