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Properties

Label 4998.2.a.t.1.1
Level $4998$
Weight $2$
Character 4998.1
Self dual yes
Analytic conductor $39.909$
Analytic rank $1$
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] = [4998,2,Mod(1,4998)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4998, 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("4998.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4998.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: \(39.9092309302\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 4998.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} +4.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -10.0000 q^{29} -2.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} -8.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} -6.00000 q^{46} -4.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} -1.00000 q^{51} +2.00000 q^{52} +14.0000 q^{53} -1.00000 q^{54} -8.00000 q^{55} -4.00000 q^{57} +10.0000 q^{58} +14.0000 q^{59} +2.00000 q^{60} -14.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -1.00000 q^{68} +6.00000 q^{69} -10.0000 q^{71} -1.00000 q^{72} -14.0000 q^{73} -1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} +4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +6.00000 q^{83} -2.00000 q^{85} +8.00000 q^{86} -10.0000 q^{87} +4.00000 q^{88} -2.00000 q^{89} -2.00000 q^{90} +6.00000 q^{92} -8.00000 q^{93} +4.00000 q^{94} -8.00000 q^{95} -1.00000 q^{96} -2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −2.00000 −0.365148
\(31\) −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
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) −2.00000 −0.316228
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) 2.00000 0.298142
\(46\) −6.00000 −0.884652
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −1.00000 −0.140028
\(52\) 2.00000 0.277350
\(53\) 14.0000 1.92305 0.961524 − 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) −1.00000 −0.136083
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 10.0000 1.31306
\(59\) 14.0000 1.82264 0.911322 − 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 2.00000 0.258199
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 4.00000 0.492366
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −1.00000 −0.121268
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 − 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 8.00000 0.862662
\(87\) −10.0000 −1.07211
\(88\) 4.00000 0.426401
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) −8.00000 −0.829561
\(94\) 4.00000 0.412568
\(95\) −8.00000 −0.820783
\(96\) −1.00000 −0.102062
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) −4.00000 −0.398015 −0.199007 − 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 1.00000 0.0990148
\(103\) 14.0000 1.37946 0.689730 − 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −14.0000 −1.35980
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) −8.00000 −0.766261 −0.383131 − 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 8.00000 0.762770
\(111\) 0 0
\(112\) 0 0
\(113\) −20.0000 −1.88144 −0.940721 − 0.339182i \(-0.889850\pi\)
−0.940721 + 0.339182i \(0.889850\pi\)
\(114\) 4.00000 0.374634
\(115\) 12.0000 1.11901
\(116\) −10.0000 −0.928477
\(117\) 2.00000 0.184900
\(118\) −14.0000 −1.28880
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 5.00000 0.454545
\(122\) 14.0000 1.26750
\(123\) −6.00000 −0.541002
\(124\) −8.00000 −0.718421
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) −4.00000 −0.350823
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 2.00000 0.172133
\(136\) 1.00000 0.0857493
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −6.00000 −0.510754
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 10.0000 0.839181
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) −20.0000 −1.66091
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 0 0
\(149\) 2.00000 0.163846 0.0819232 − 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 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
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) 2.00000 0.160128
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −4.00000 −0.318223
\(159\) 14.0000 1.11027
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 18.0000 1.40987 0.704934 − 0.709273i \(-0.250976\pi\)
0.704934 + 0.709273i \(0.250976\pi\)
\(164\) −6.00000 −0.468521
\(165\) −8.00000 −0.622799
\(166\) −6.00000 −0.465690
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) −4.00000 −0.305888
\(172\) −8.00000 −0.609994
\(173\) 22.0000 1.67263 0.836315 − 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 10.0000 0.758098
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 14.0000 1.05230
\(178\) 2.00000 0.149906
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 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
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 4.00000 0.292509
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 2.00000 0.143592
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 4.00000 0.284268
\(199\) 4.00000 0.283552 0.141776 − 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 4.00000 0.281439
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) −12.0000 −0.838116
\(206\) −14.0000 −0.975426
\(207\) 6.00000 0.417029
\(208\) 2.00000 0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 − 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 14.0000 0.961524
\(213\) −10.0000 −0.685189
\(214\) 8.00000 0.546869
\(215\) −16.0000 −1.09119
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 8.00000 0.541828
\(219\) −14.0000 −0.946032
\(220\) −8.00000 −0.539360
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 − 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 20.0000 1.33038
\(227\) 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 \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −12.0000 −0.791257
\(231\) 0 0
\(232\) 10.0000 0.656532
\(233\) 8.00000 0.524097 0.262049 − 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) −2.00000 −0.130744
\(235\) −8.00000 −0.521862
\(236\) 14.0000 0.911322
\(237\) 4.00000 0.259828
\(238\) 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
\(241\) −26.0000 −1.67481 −0.837404 − 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) −8.00000 −0.509028
\(248\) 8.00000 0.508001
\(249\) 6.00000 0.380235
\(250\) 12.0000 0.758947
\(251\) −10.0000 −0.631194 −0.315597 − 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) 0 0
\(253\) −24.0000 −1.50887
\(254\) −8.00000 −0.501965
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 − 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 8.00000 0.498058
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) −10.0000 −0.618984
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 − 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 4.00000 0.246183
\(265\) 28.0000 1.72003
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) 4.00000 0.244339
\(269\) −10.0000 −0.609711 −0.304855 − 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) −2.00000 −0.121716
\(271\) −10.0000 −0.607457 −0.303728 − 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 4.00000 0.241209
\(276\) 6.00000 0.361158
\(277\) −32.0000 −1.92269 −0.961347 − 0.275340i \(-0.911209\pi\)
−0.961347 + 0.275340i \(0.911209\pi\)
\(278\) 12.0000 0.719712
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 − 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 4.00000 0.238197
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −10.0000 −0.593391
\(285\) −8.00000 −0.473879
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 20.0000 1.17444
\(291\) −2.00000 −0.117242
\(292\) −14.0000 −0.819288
\(293\) 24.0000 1.40209 0.701047 − 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 28.0000 1.63022
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) −2.00000 −0.115857
\(299\) 12.0000 0.693978
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) −4.00000 −0.229794
\(304\) −4.00000 −0.229416
\(305\) −28.0000 −1.60328
\(306\) 1.00000 0.0571662
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 16.0000 0.908739
\(311\) 16.0000 0.907277 0.453638 − 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) −2.00000 −0.113228
\(313\) −18.0000 −1.01742 −0.508710 − 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −14.0000 −0.785081
\(319\) 40.0000 2.23957
\(320\) 2.00000 0.111803
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −18.0000 −0.996928
\(327\) −8.00000 −0.442401
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 8.00000 0.440386
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 9.00000 0.489535
\(339\) −20.0000 −1.08625
\(340\) −2.00000 −0.108465
\(341\) 32.0000 1.73290
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 8.00000 0.431331
\(345\) 12.0000 0.646058
\(346\) −22.0000 −1.18273
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −10.0000 −0.536056
\(349\) 6.00000 0.321173 0.160586 − 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 4.00000 0.213201
\(353\) 18.0000 0.958043 0.479022 − 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −14.0000 −0.744092
\(355\) −20.0000 −1.06149
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −20.0000 −1.05556 −0.527780 − 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −28.0000 −1.46559
\(366\) 14.0000 0.731792
\(367\) 28.0000 1.46159 0.730794 − 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 6.00000 0.312772
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −4.00000 −0.206835
\(375\) −12.0000 −0.619677
\(376\) 4.00000 0.206284
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) −18.0000 −0.924598 −0.462299 − 0.886724i \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) −8.00000 −0.410391
\(381\) 8.00000 0.409852
\(382\) −24.0000 −1.22795
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) −8.00000 −0.406663
\(388\) −2.00000 −0.101535
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −4.00000 −0.202548
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 8.00000 0.402524
\(396\) −4.00000 −0.201008
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −12.0000 −0.599251 −0.299626 − 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) −4.00000 −0.199502
\(403\) −16.0000 −0.797017
\(404\) −4.00000 −0.199007
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 1.00000 0.0495074
\(409\) 24.0000 1.18672 0.593362 − 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) 12.0000 0.592638
\(411\) 2.00000 0.0986527
\(412\) 14.0000 0.689730
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 12.0000 0.589057
\(416\) −2.00000 −0.0980581
\(417\) −12.0000 −0.587643
\(418\) −16.0000 −0.782586
\(419\) −16.0000 −0.781651 −0.390826 − 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 − 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) −2.00000 −0.0973585
\(423\) −4.00000 −0.194487
\(424\) −14.0000 −0.679900
\(425\) 1.00000 0.0485071
\(426\) 10.0000 0.484502
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) −8.00000 −0.386244
\(430\) 16.0000 0.771589
\(431\) 22.0000 1.05970 0.529851 − 0.848091i \(-0.322248\pi\)
0.529851 + 0.848091i \(0.322248\pi\)
\(432\) 1.00000 0.0481125
\(433\) −4.00000 −0.192228 −0.0961139 − 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) −20.0000 −0.958927
\(436\) −8.00000 −0.383131
\(437\) −24.0000 −1.14808
\(438\) 14.0000 0.668946
\(439\) −12.0000 −0.572729 −0.286364 − 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 8.00000 0.381385
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) 2.00000 0.0947027
\(447\) 2.00000 0.0945968
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 − 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 1.00000 0.0471405
\(451\) 24.0000 1.13012
\(452\) −20.0000 −0.940721
\(453\) −8.00000 −0.375873
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −2.00000 −0.0934539
\(459\) −1.00000 −0.0466760
\(460\) 12.0000 0.559503
\(461\) 24.0000 1.11779 0.558896 − 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −10.0000 −0.464238
\(465\) −16.0000 −0.741982
\(466\) −8.00000 −0.370593
\(467\) −26.0000 −1.20314 −0.601568 − 0.798821i \(-0.705457\pi\)
−0.601568 + 0.798821i \(0.705457\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 8.00000 0.369012
\(471\) 14.0000 0.645086
\(472\) −14.0000 −0.644402
\(473\) 32.0000 1.47136
\(474\) −4.00000 −0.183726
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 14.0000 0.641016
\(478\) −12.0000 −0.548867
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 0 0
\(482\) 26.0000 1.18427
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −4.00000 −0.181631
\(486\) −1.00000 −0.0453609
\(487\) 28.0000 1.26880 0.634401 − 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 14.0000 0.633750
\(489\) 18.0000 0.813988
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −6.00000 −0.270501
\(493\) 10.0000 0.450377
\(494\) 8.00000 0.359937
\(495\) −8.00000 −0.359573
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) 6.00000 0.268597 0.134298 − 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) −12.0000 −0.536656
\(501\) −8.00000 −0.357414
\(502\) 10.0000 0.446322
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 24.0000 1.06693
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) 16.0000 0.709188 0.354594 − 0.935020i \(-0.384619\pi\)
0.354594 + 0.935020i \(0.384619\pi\)
\(510\) 2.00000 0.0885615
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −22.0000 −0.970378
\(515\) 28.0000 1.23383
\(516\) −8.00000 −0.352180
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 22.0000 0.965693
\(520\) −4.00000 −0.175412
\(521\) 10.0000 0.438108 0.219054 − 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 10.0000 0.437688
\(523\) 8.00000 0.349816 0.174908 − 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 8.00000 0.348485
\(528\) −4.00000 −0.174078
\(529\) 13.0000 0.565217
\(530\) −28.0000 −1.21624
\(531\) 14.0000 0.607548
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 2.00000 0.0865485
\(535\) −16.0000 −0.691740
\(536\) −4.00000 −0.172774
\(537\) −12.0000 −0.517838
\(538\) 10.0000 0.431131
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −20.0000 −0.859867 −0.429934 − 0.902861i \(-0.641463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) 10.0000 0.429537
\(543\) −2.00000 −0.0858282
\(544\) 1.00000 0.0428746
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 − 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 2.00000 0.0854358
\(549\) −14.0000 −0.597505
\(550\) −4.00000 −0.170561
\(551\) 40.0000 1.70406
\(552\) −6.00000 −0.255377
\(553\) 0 0
\(554\) 32.0000 1.35955
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 8.00000 0.338667
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 30.0000 1.26547
\(563\) −14.0000 −0.590030 −0.295015 − 0.955493i \(-0.595325\pi\)
−0.295015 + 0.955493i \(0.595325\pi\)
\(564\) −4.00000 −0.168430
\(565\) −40.0000 −1.68281
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 10.0000 0.419591
\(569\) 42.0000 1.76073 0.880366 − 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 8.00000 0.335083
\(571\) −30.0000 −1.25546 −0.627730 − 0.778431i \(-0.716016\pi\)
−0.627730 + 0.778431i \(0.716016\pi\)
\(572\) −8.00000 −0.334497
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) 28.0000 1.16566 0.582828 − 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −10.0000 −0.415586
\(580\) −20.0000 −0.830455
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) −56.0000 −2.31928
\(584\) 14.0000 0.579324
\(585\) 4.00000 0.165380
\(586\) −24.0000 −0.991431
\(587\) −46.0000 −1.89862 −0.949312 − 0.314337i \(-0.898218\pi\)
−0.949312 + 0.314337i \(0.898218\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) −28.0000 −1.15274
\(591\) 18.0000 0.740421
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) 4.00000 0.163709
\(598\) −12.0000 −0.490716
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 1.00000 0.0408248
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) −8.00000 −0.325515
\(605\) 10.0000 0.406558
\(606\) 4.00000 0.162489
\(607\) 12.0000 0.487065 0.243532 − 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 28.0000 1.13369
\(611\) −8.00000 −0.323645
\(612\) −1.00000 −0.0404226
\(613\) −22.0000 −0.888572 −0.444286 − 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −16.0000 −0.645707
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −8.00000 −0.322068 −0.161034 − 0.986949i \(-0.551483\pi\)
−0.161034 + 0.986949i \(0.551483\pi\)
\(618\) −14.0000 −0.563163
\(619\) −44.0000 −1.76851 −0.884255 − 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) −16.0000 −0.642575
\(621\) 6.00000 0.240772
\(622\) −16.0000 −0.641542
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) 18.0000 0.719425
\(627\) 16.0000 0.638978
\(628\) 14.0000 0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −4.00000 −0.159111
\(633\) 2.00000 0.0794929
\(634\) 30.0000 1.19145
\(635\) 16.0000 0.634941
\(636\) 14.0000 0.555136
\(637\) 0 0
\(638\) −40.0000 −1.58362
\(639\) −10.0000 −0.395594
\(640\) −2.00000 −0.0790569
\(641\) −32.0000 −1.26392 −0.631962 − 0.774999i \(-0.717750\pi\)
−0.631962 + 0.774999i \(0.717750\pi\)
\(642\) 8.00000 0.315735
\(643\) −12.0000 −0.473234 −0.236617 − 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) −16.0000 −0.629999
\(646\) −4.00000 −0.157378
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −56.0000 −2.19819
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 18.0000 0.704934
\(653\) 26.0000 1.01746 0.508729 − 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 − 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) −8.00000 −0.311400
\(661\) −22.0000 −0.855701 −0.427850 − 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 4.00000 0.155464
\(663\) −2.00000 −0.0776736
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 0 0
\(667\) −60.0000 −2.32321
\(668\) −8.00000 −0.309529
\(669\) −2.00000 −0.0773245
\(670\) −8.00000 −0.309067
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 22.0000 0.847408
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 20.0000 0.768095
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) 28.0000 1.07296
\(682\) −32.0000 −1.22534
\(683\) −32.0000 −1.22445 −0.612223 − 0.790685i \(-0.709725\pi\)
−0.612223 + 0.790685i \(0.709725\pi\)
\(684\) −4.00000 −0.152944
\(685\) 4.00000 0.152832
\(686\) 0 0
\(687\) 2.00000 0.0763048
\(688\) −8.00000 −0.304997
\(689\) 28.0000 1.06672
\(690\) −12.0000 −0.456832
\(691\) −44.0000 −1.67384 −0.836919 − 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 22.0000 0.836315
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −24.0000 −0.910372
\(696\) 10.0000 0.379049
\(697\) 6.00000 0.227266
\(698\) −6.00000 −0.227103
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 − 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) −4.00000 −0.150756
\(705\) −8.00000 −0.301297
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 14.0000 0.526152
\(709\) 40.0000 1.50223 0.751116 − 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 20.0000 0.750587
\(711\) 4.00000 0.150012
\(712\) 2.00000 0.0749532
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) −12.0000 −0.448461
\(717\) 12.0000 0.448148
\(718\) 20.0000 0.746393
\(719\) 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
\(723\) −26.0000 −0.966950
\(724\) −2.00000 −0.0743294
\(725\) 10.0000 0.371391
\(726\) −5.00000 −0.185567
\(727\) 26.0000 0.964287 0.482143 − 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 28.0000 1.03633
\(731\) 8.00000 0.295891
\(732\) −14.0000 −0.517455
\(733\) 22.0000 0.812589 0.406294 − 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −16.0000 −0.589368
\(738\) 6.00000 0.220863
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 − 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 8.00000 0.293294
\(745\) 4.00000 0.146549
\(746\) −10.0000 −0.366126
\(747\) 6.00000 0.219529
\(748\) 4.00000 0.146254
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 12.0000 0.437886 0.218943 − 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) −4.00000 −0.145865
\(753\) −10.0000 −0.364420
\(754\) 20.0000 0.728357
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −14.0000 −0.508839 −0.254419 − 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) 18.0000 0.653789
\(759\) −24.0000 −0.871145
\(760\) 8.00000 0.290191
\(761\) 26.0000 0.942499 0.471250 − 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) 24.0000 0.868290
\(765\) −2.00000 −0.0723102
\(766\) −24.0000 −0.867155
\(767\) 28.0000 1.01102
\(768\) 1.00000 0.0360844
\(769\) 32.0000 1.15395 0.576975 − 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) −10.0000 −0.359908
\(773\) −24.0000 −0.863220 −0.431610 − 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 8.00000 0.287554
\(775\) 8.00000 0.287368
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) 24.0000 0.859889
\(780\) 4.00000 0.143223
\(781\) 40.0000 1.43131
\(782\) 6.00000 0.214560
\(783\) −10.0000 −0.357371
\(784\) 0 0
\(785\) 28.0000 0.999363
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 18.0000 0.641223
\(789\) 8.00000 0.284808
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −28.0000 −0.994309
\(794\) −2.00000 −0.0709773
\(795\) 28.0000 0.993058
\(796\) 4.00000 0.141776
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 1.00000 0.0353553
\(801\) −2.00000 −0.0706665
\(802\) 12.0000 0.423735
\(803\) 56.0000 1.97620
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) −10.0000 −0.352017
\(808\) 4.00000 0.140720
\(809\) 12.0000 0.421898 0.210949 − 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −4.00000 −0.140459 −0.0702295 − 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) −10.0000 −0.350715
\(814\) 0 0
\(815\) 36.0000 1.26102
\(816\) −1.00000 −0.0350070
\(817\) 32.0000 1.11954
\(818\) −24.0000 −0.839140
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) −22.0000 −0.767805 −0.383903 − 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 16.0000 0.557725 0.278862 − 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −14.0000 −0.487713
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 − 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 6.00000 0.208514
\(829\) 22.0000 0.764092 0.382046 − 0.924143i \(-0.375220\pi\)
0.382046 + 0.924143i \(0.375220\pi\)
\(830\) −12.0000 −0.416526
\(831\) −32.0000 −1.11007
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 12.0000 0.415526
\(835\) −16.0000 −0.553703
\(836\) 16.0000 0.553372
\(837\) −8.00000 −0.276520
\(838\) 16.0000 0.552711
\(839\) −24.0000 −0.828572 −0.414286 − 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 30.0000 1.03387
\(843\) −30.0000 −1.03325
\(844\) 2.00000 0.0688428
\(845\) −18.0000 −0.619219
\(846\) 4.00000 0.137523
\(847\) 0 0
\(848\) 14.0000 0.480762
\(849\) 4.00000 0.137280
\(850\) −1.00000 −0.0342997
\(851\) 0 0
\(852\) −10.0000 −0.342594
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 8.00000 0.273434
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 8.00000 0.273115
\(859\) −12.0000 −0.409435 −0.204717 − 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) −16.0000 −0.545595
\(861\) 0 0
\(862\) −22.0000 −0.749323
\(863\) −48.0000 −1.63394 −0.816970 − 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 44.0000 1.49604
\(866\) 4.00000 0.135926
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 20.0000 0.678064
\(871\) 8.00000 0.271070
\(872\) 8.00000 0.270914
\(873\) −2.00000 −0.0676897
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) −12.0000 −0.405211 −0.202606 − 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) 12.0000 0.404980
\(879\) 24.0000 0.809500
\(880\) −8.00000 −0.269680
\(881\) 2.00000 0.0673817 0.0336909 − 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 28.0000 0.941210
\(886\) 12.0000 0.403148
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 4.00000 0.134080
\(891\) −4.00000 −0.134005
\(892\) −2.00000 −0.0669650
\(893\) 16.0000 0.535420
\(894\) −2.00000 −0.0668900
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) −12.0000 −0.400445
\(899\) 80.0000 2.66815
\(900\) −1.00000 −0.0333333
\(901\) −14.0000 −0.466408
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 20.0000 0.665190
\(905\) −4.00000 −0.132964
\(906\) 8.00000 0.265782
\(907\) 46.0000 1.52740 0.763702 − 0.645568i \(-0.223379\pi\)
0.763702 + 0.645568i \(0.223379\pi\)
\(908\) 28.0000 0.929213
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) 34.0000 1.12647 0.563235 − 0.826297i \(-0.309557\pi\)
0.563235 + 0.826297i \(0.309557\pi\)
\(912\) −4.00000 −0.132453
\(913\) −24.0000 −0.794284
\(914\) 22.0000 0.727695
\(915\) −28.0000 −0.925651
\(916\) 2.00000 0.0660819
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) −48.0000 −1.58337 −0.791687 − 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) −12.0000 −0.395628
\(921\) 16.0000 0.527218
\(922\) −24.0000 −0.790398
\(923\) −20.0000 −0.658308
\(924\) 0 0
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 14.0000 0.459820
\(928\) 10.0000 0.328266
\(929\) 54.0000 1.77168 0.885841 − 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 16.0000 0.524661
\(931\) 0 0
\(932\) 8.00000 0.262049
\(933\) 16.0000 0.523816
\(934\) 26.0000 0.850746
\(935\) 8.00000 0.261628
\(936\) −2.00000 −0.0653720
\(937\) 36.0000 1.17607 0.588034 − 0.808836i \(-0.299902\pi\)
0.588034 + 0.808836i \(0.299902\pi\)
\(938\) 0 0
\(939\) −18.0000 −0.587408
\(940\) −8.00000 −0.260931
\(941\) 6.00000 0.195594 0.0977972 − 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −14.0000 −0.456145
\(943\) −36.0000 −1.17232
\(944\) 14.0000 0.455661
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) −4.00000 −0.129983 −0.0649913 − 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 4.00000 0.129914
\(949\) −28.0000 −0.908918
\(950\) −4.00000 −0.129777
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −14.0000 −0.453267
\(955\) 48.0000 1.55324
\(956\) 12.0000 0.388108
\(957\) 40.0000 1.29302
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −8.00000 −0.257796
\(964\) −26.0000 −0.837404
\(965\) −20.0000 −0.643823
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −5.00000 −0.160706
\(969\) 4.00000 0.128499
\(970\) 4.00000 0.128432
\(971\) −22.0000 −0.706014 −0.353007 − 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −28.0000 −0.897178
\(975\) −2.00000 −0.0640513
\(976\) −14.0000 −0.448129
\(977\) 2.00000 0.0639857 0.0319928 − 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) −18.0000 −0.575577
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) −8.00000 −0.255420
\(982\) −12.0000 −0.382935
\(983\) −16.0000 −0.510321 −0.255160 − 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 6.00000 0.191273
\(985\) 36.0000 1.14706
\(986\) −10.0000 −0.318465
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −48.0000 −1.52631
\(990\) 8.00000 0.254257
\(991\) −4.00000 −0.127064 −0.0635321 − 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 8.00000 0.254000
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 6.00000 0.190117
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −6.00000 −0.189927
\(999\) 0 0
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 4998.2.a.t.1.1 yes 1
7.6 odd 2 4998.2.a.b.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4998.2.a.b.1.1 ✓ 1 7.6 odd 2
4998.2.a.t.1.1 yes 1 1.1 even 1 trivial