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Properties

Label 4998.2.a.c.1.1
Level $4998$
Weight $2$
Character 4998.1
Self dual yes
Analytic conductor $39.909$
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] = [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: \(0\)
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 \(\chi\) \(=\) 4998.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} -2.00000 q^{20} -2.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +8.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} +2.00000 q^{44} -2.00000 q^{45} -6.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{50} +1.00000 q^{51} +4.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} -4.00000 q^{55} +2.00000 q^{57} -8.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -8.00000 q^{65} +2.00000 q^{66} +4.00000 q^{67} -1.00000 q^{68} -6.00000 q^{69} +2.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} -2.00000 q^{76} +4.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} +2.00000 q^{85} -4.00000 q^{86} -8.00000 q^{87} -2.00000 q^{88} +2.00000 q^{89} +2.00000 q^{90} +6.00000 q^{92} +4.00000 q^{93} -4.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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 \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 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
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) −2.00000 −0.365148
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 2.00000 0.324443
\(39\) −4.00000 −0.640513
\(40\) 2.00000 0.316228
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.00000 0.301511
\(45\) −2.00000 −0.298142
\(46\) −6.00000 −0.884652
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 1.00000 0.140028
\(52\) 4.00000 0.554700
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) −8.00000 −1.05045
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 2.00000 0.258199
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) 2.00000 0.246183
\(67\) 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
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) −8.00000 −0.857690
\(88\) −2.00000 −0.213201
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 4.00000 0.414781
\(94\) −4.00000 −0.412568
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) −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
\(103\) −14.0000 −1.37946 −0.689730 − 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −2.00000 −0.193347 −0.0966736 − 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 4.00000 0.381385
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 16.0000 1.50515 0.752577 − 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) −2.00000 −0.187317
\(115\) −12.0000 −1.11901
\(116\) 8.00000 0.742781
\(117\) 4.00000 0.369800
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) −7.00000 −0.636364
\(122\) 10.0000 0.905357
\(123\) −6.00000 −0.541002
\(124\) −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 \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 8.00000 0.701646
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −2.00000 −0.174078
\(133\) 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
\(142\) −2.00000 −0.167836
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) −16.0000 −1.32873
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 2.00000 0.162221
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) −4.00000 −0.320256
\(157\) 4.00000 0.319235 0.159617 − 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 8.00000 0.636446
\(159\) −2.00000 −0.158610
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 6.00000 0.468521
\(165\) 4.00000 0.311400
\(166\) 12.0000 0.931381
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −2.00000 −0.153393
\(171\) −2.00000 −0.152944
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) −4.00000 −0.300658
\(178\) −2.00000 −0.149906
\(179\) −24.0000 −1.79384 −0.896922 − 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) −2.00000 −0.149071
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) −6.00000 −0.442326
\(185\) 12.0000 0.882258
\(186\) −4.00000 −0.293294
\(187\) −2.00000 −0.146254
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) −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
\(195\) 8.00000 0.572892
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 − 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) −2.00000 −0.142134
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) 1.00000 0.0700140
\(205\) −12.0000 −0.838116
\(206\) 14.0000 0.975426
\(207\) 6.00000 0.417029
\(208\) 4.00000 0.277350
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 2.00000 0.137361
\(213\) −2.00000 −0.137038
\(214\) 2.00000 0.136717
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) −2.00000 −0.135147
\(220\) −4.00000 −0.269680
\(221\) −4.00000 −0.269069
\(222\) −6.00000 −0.402694
\(223\) 2.00000 0.133930 0.0669650 − 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −16.0000 −1.06430
\(227\) −4.00000 −0.265489 −0.132745 − 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 2.00000 0.132453
\(229\) 16.0000 1.05731 0.528655 − 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 12.0000 0.791257
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) 20.0000 1.31024 0.655122 − 0.755523i \(-0.272617\pi\)
0.655122 + 0.755523i \(0.272617\pi\)
\(234\) −4.00000 −0.261488
\(235\) −8.00000 −0.521862
\(236\) 4.00000 0.260378
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 2.00000 0.129099
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) −8.00000 −0.509028
\(248\) 4.00000 0.254000
\(249\) 12.0000 0.760469
\(250\) −12.0000 −0.758947
\(251\) 16.0000 1.00991 0.504956 − 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) −8.00000 −0.501965
\(255\) −2.00000 −0.125245
\(256\) 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
\(259\) 0 0
\(260\) −8.00000 −0.496139
\(261\) 8.00000 0.495188
\(262\) −12.0000 −0.741362
\(263\) −4.00000 −0.246651 −0.123325 − 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 2.00000 0.123091
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) 4.00000 0.244339
\(269\) −14.0000 −0.853595 −0.426798 − 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) −2.00000 −0.121716
\(271\) 10.0000 0.607457 0.303728 − 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) −2.00000 −0.120605
\(276\) −6.00000 −0.361158
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 12.0000 0.719712
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 4.00000 0.238197
\(283\) 8.00000 0.475551 0.237775 − 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 2.00000 0.118678
\(285\) −4.00000 −0.236940
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 16.0000 0.939552
\(291\) 10.0000 0.586210
\(292\) 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
\(295\) −8.00000 −0.465778
\(296\) 6.00000 0.348743
\(297\) −2.00000 −0.116052
\(298\) 10.0000 0.579284
\(299\) 24.0000 1.38796
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) 2.00000 0.114897
\(304\) −2.00000 −0.114708
\(305\) 20.0000 1.14520
\(306\) 1.00000 0.0571662
\(307\) 2.00000 0.114146 0.0570730 − 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) −8.00000 −0.454369
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 4.00000 0.226455
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 12.0000 0.673987 0.336994 − 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 2.00000 0.112154
\(319\) 16.0000 0.895828
\(320\) −2.00000 −0.111803
\(321\) 2.00000 0.111629
\(322\) 0 0
\(323\) 2.00000 0.111283
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) −12.0000 −0.664619
\(327\) 2.00000 0.110600
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −12.0000 −0.658586
\(333\) −6.00000 −0.328798
\(334\) 16.0000 0.875481
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −3.00000 −0.163178
\(339\) −16.0000 −0.869001
\(340\) 2.00000 0.108465
\(341\) −8.00000 −0.433224
\(342\) 2.00000 0.108148
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 12.0000 0.646058
\(346\) −14.0000 −0.752645
\(347\) 6.00000 0.322097 0.161048 − 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) −8.00000 −0.428845
\(349\) 24.0000 1.28469 0.642345 − 0.766415i \(-0.277962\pi\)
0.642345 + 0.766415i \(0.277962\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) −2.00000 −0.106600
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 4.00000 0.212598
\(355\) −4.00000 −0.212298
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) −8.00000 −0.422224 −0.211112 − 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 2.00000 0.105409
\(361\) −15.0000 −0.789474
\(362\) −2.00000 −0.105118
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) −10.0000 −0.522708
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 6.00000 0.312772
\(369\) 6.00000 0.312348
\(370\) −12.0000 −0.623850
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 2.00000 0.103418
\(375\) −12.0000 −0.619677
\(376\) −4.00000 −0.206284
\(377\) 32.0000 1.64808
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 − 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 4.00000 0.205196
\(381\) −8.00000 −0.409852
\(382\) 24.0000 1.22795
\(383\) 36.0000 1.83951 0.919757 − 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −26.0000 −1.32337
\(387\) 4.00000 0.203331
\(388\) −10.0000 −0.507673
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) −8.00000 −0.405096
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) −12.0000 −0.604551
\(395\) 16.0000 0.805047
\(396\) 2.00000 0.100504
\(397\) −38.0000 −1.90717 −0.953583 − 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) −20.0000 −1.00251
\(399\) 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
\(404\) −2.00000 −0.0995037
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) −1.00000 −0.0495074
\(409\) −12.0000 −0.593362 −0.296681 − 0.954977i \(-0.595880\pi\)
−0.296681 + 0.954977i \(0.595880\pi\)
\(410\) 12.0000 0.592638
\(411\) −2.00000 −0.0986527
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 24.0000 1.17811
\(416\) −4.00000 −0.196116
\(417\) 12.0000 0.587643
\(418\) 4.00000 0.195646
\(419\) 4.00000 0.195413 0.0977064 − 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 − 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −20.0000 −0.973585
\(423\) 4.00000 0.194487
\(424\) −2.00000 −0.0971286
\(425\) 1.00000 0.0485071
\(426\) 2.00000 0.0969003
\(427\) 0 0
\(428\) −2.00000 −0.0966736
\(429\) −8.00000 −0.386244
\(430\) 8.00000 0.385794
\(431\) −2.00000 −0.0963366 −0.0481683 − 0.998839i \(-0.515338\pi\)
−0.0481683 + 0.998839i \(0.515338\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 16.0000 0.768911 0.384455 − 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 16.0000 0.767141
\(436\) −2.00000 −0.0957826
\(437\) −12.0000 −0.574038
\(438\) 2.00000 0.0955637
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 6.00000 0.284747
\(445\) −4.00000 −0.189618
\(446\) −2.00000 −0.0947027
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) 36.0000 1.69895 0.849473 − 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 1.00000 0.0471405
\(451\) 12.0000 0.565058
\(452\) 16.0000 0.752577
\(453\) −16.0000 −0.751746
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −16.0000 −0.747631
\(459\) 1.00000 0.0466760
\(460\) −12.0000 −0.559503
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 8.00000 0.371391
\(465\) −8.00000 −0.370991
\(466\) −20.0000 −0.926482
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) 8.00000 0.369012
\(471\) −4.00000 −0.184310
\(472\) −4.00000 −0.184115
\(473\) 8.00000 0.367840
\(474\) −8.00000 −0.367452
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) −24.0000 −1.09773
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −24.0000 −1.09431
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 20.0000 0.908153
\(486\) 1.00000 0.0453609
\(487\) 4.00000 0.181257 0.0906287 − 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 10.0000 0.452679
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 − 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) −6.00000 −0.270501
\(493\) −8.00000 −0.360302
\(494\) 8.00000 0.359937
\(495\) −4.00000 −0.179787
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 24.0000 1.07439 0.537194 − 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 12.0000 0.536656
\(501\) 16.0000 0.714827
\(502\) −16.0000 −0.714115
\(503\) −40.0000 −1.78351 −0.891756 − 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) −12.0000 −0.533465
\(507\) −3.00000 −0.133235
\(508\) 8.00000 0.354943
\(509\) 26.0000 1.15243 0.576215 − 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 2.00000 0.0885615
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) −2.00000 −0.0882162
\(515\) 28.0000 1.23383
\(516\) −4.00000 −0.176090
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 8.00000 0.350823
\(521\) 26.0000 1.13908 0.569540 − 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) −8.00000 −0.350150
\(523\) −26.0000 −1.13690 −0.568450 − 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) 4.00000 0.174243
\(528\) −2.00000 −0.0870388
\(529\) 13.0000 0.565217
\(530\) 4.00000 0.173749
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 2.00000 0.0865485
\(535\) 4.00000 0.172935
\(536\) −4.00000 −0.172774
\(537\) 24.0000 1.03568
\(538\) 14.0000 0.603583
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) 34.0000 1.46177 0.730887 − 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) −10.0000 −0.429537
\(543\) −2.00000 −0.0858282
\(544\) 1.00000 0.0428746
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) 40.0000 1.71028 0.855138 − 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) 2.00000 0.0854358
\(549\) −10.0000 −0.426790
\(550\) 2.00000 0.0852803
\(551\) −16.0000 −0.681623
\(552\) 6.00000 0.255377
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) −12.0000 −0.509372
\(556\) −12.0000 −0.508913
\(557\) 42.0000 1.77960 0.889799 − 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 4.00000 0.169334
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) 6.00000 0.253095
\(563\) 44.0000 1.85438 0.927189 − 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) −4.00000 −0.168430
\(565\) −32.0000 −1.34625
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) −2.00000 −0.0839181
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 4.00000 0.167542
\(571\) −12.0000 −0.502184 −0.251092 − 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 8.00000 0.334497
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) 20.0000 0.832611 0.416305 − 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −26.0000 −1.08052
\(580\) −16.0000 −0.664364
\(581\) 0 0
\(582\) −10.0000 −0.414513
\(583\) 4.00000 0.165663
\(584\) −2.00000 −0.0827606
\(585\) −8.00000 −0.330759
\(586\) −6.00000 −0.247858
\(587\) 16.0000 0.660391 0.330195 − 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 8.00000 0.329355
\(591\) −12.0000 −0.493614
\(592\) −6.00000 −0.246598
\(593\) 30.0000 1.23195 0.615976 − 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −20.0000 −0.818546
\(598\) −24.0000 −0.981433
\(599\) −36.0000 −1.47092 −0.735460 − 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 10.0000 0.407909 0.203954 − 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 16.0000 0.651031
\(605\) 14.0000 0.569181
\(606\) −2.00000 −0.0812444
\(607\) −36.0000 −1.46119 −0.730597 − 0.682808i \(-0.760758\pi\)
−0.730597 + 0.682808i \(0.760758\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) −20.0000 −0.809776
\(611\) 16.0000 0.647291
\(612\) −1.00000 −0.0404226
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 − 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) −14.0000 −0.563163
\(619\) 32.0000 1.28619 0.643094 − 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 8.00000 0.321288
\(621\) −6.00000 −0.240772
\(622\) −8.00000 −0.320771
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) −19.0000 −0.760000
\(626\) −6.00000 −0.239808
\(627\) 4.00000 0.159745
\(628\) 4.00000 0.159617
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −24.0000 −0.955425 −0.477712 − 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) 8.00000 0.318223
\(633\) −20.0000 −0.794929
\(634\) −12.0000 −0.476581
\(635\) −16.0000 −0.634941
\(636\) −2.00000 −0.0793052
\(637\) 0 0
\(638\) −16.0000 −0.633446
\(639\) 2.00000 0.0791188
\(640\) 2.00000 0.0790569
\(641\) −8.00000 −0.315981 −0.157991 − 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) −2.00000 −0.0789337
\(643\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) −2.00000 −0.0786889
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 8.00000 0.314027
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) 20.0000 0.782660 0.391330 − 0.920250i \(-0.372015\pi\)
0.391330 + 0.920250i \(0.372015\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −24.0000 −0.937758
\(656\) 6.00000 0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −44.0000 −1.71400 −0.856998 − 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 4.00000 0.155700
\(661\) −32.0000 −1.24466 −0.622328 − 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) −20.0000 −0.777322
\(663\) 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
\(669\) −2.00000 −0.0773245
\(670\) 8.00000 0.309067
\(671\) −20.0000 −0.772091
\(672\) 0 0
\(673\) 38.0000 1.46479 0.732396 − 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) 22.0000 0.847408
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 16.0000 0.614476
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 4.00000 0.153280
\(682\) 8.00000 0.306336
\(683\) −14.0000 −0.535695 −0.267848 − 0.963461i \(-0.586312\pi\)
−0.267848 + 0.963461i \(0.586312\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −4.00000 −0.152832
\(686\) 0 0
\(687\) −16.0000 −0.610438
\(688\) 4.00000 0.152499
\(689\) 8.00000 0.304776
\(690\) −12.0000 −0.456832
\(691\) 32.0000 1.21734 0.608669 − 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) −6.00000 −0.227757
\(695\) 24.0000 0.910372
\(696\) 8.00000 0.303239
\(697\) −6.00000 −0.227266
\(698\) −24.0000 −0.908413
\(699\) −20.0000 −0.756469
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 − 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 4.00000 0.150970
\(703\) 12.0000 0.452589
\(704\) 2.00000 0.0753778
\(705\) 8.00000 0.301297
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) 22.0000 0.826227 0.413114 − 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 4.00000 0.150117
\(711\) −8.00000 −0.300023
\(712\) −2.00000 −0.0749532
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) −24.0000 −0.896922
\(717\) −24.0000 −0.896296
\(718\) 8.00000 0.298557
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) 10.0000 0.371904
\(724\) 2.00000 0.0743294
\(725\) −8.00000 −0.297113
\(726\) −7.00000 −0.259794
\(727\) 10.0000 0.370879 0.185440 − 0.982656i \(-0.440629\pi\)
0.185440 + 0.982656i \(0.440629\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) −4.00000 −0.147945
\(732\) 10.0000 0.369611
\(733\) −4.00000 −0.147743 −0.0738717 − 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 8.00000 0.294684
\(738\) −6.00000 −0.220863
\(739\) −44.0000 −1.61857 −0.809283 − 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 12.0000 0.441129
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 6.00000 0.220119 0.110059 − 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) −4.00000 −0.146647
\(745\) 20.0000 0.732743
\(746\) −10.0000 −0.366126
\(747\) −12.0000 −0.439057
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) −12.0000 −0.437886 −0.218943 − 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 4.00000 0.145865
\(753\) −16.0000 −0.583072
\(754\) −32.0000 −1.16537
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 − 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 12.0000 0.435860
\(759\) −12.0000 −0.435572
\(760\) −4.00000 −0.145095
\(761\) 10.0000 0.362500 0.181250 − 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) −24.0000 −0.868290
\(765\) 2.00000 0.0723102
\(766\) −36.0000 −1.30073
\(767\) 16.0000 0.577727
\(768\) −1.00000 −0.0360844
\(769\) −44.0000 −1.58668 −0.793340 − 0.608778i \(-0.791660\pi\)
−0.793340 + 0.608778i \(0.791660\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 26.0000 0.935760
\(773\) −18.0000 −0.647415 −0.323708 − 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) −4.00000 −0.143777
\(775\) 4.00000 0.143684
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −12.0000 −0.429945
\(780\) 8.00000 0.286446
\(781\) 4.00000 0.143131
\(782\) 6.00000 0.214560
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) −8.00000 −0.285532
\(786\) 12.0000 0.428026
\(787\) 32.0000 1.14068 0.570338 − 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 12.0000 0.427482
\(789\) 4.00000 0.142404
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) −40.0000 −1.42044
\(794\) 38.0000 1.34857
\(795\) 4.00000 0.141865
\(796\) 20.0000 0.708881
\(797\) −18.0000 −0.637593 −0.318796 − 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 1.00000 0.0353553
\(801\) 2.00000 0.0706665
\(802\) 12.0000 0.423735
\(803\) 4.00000 0.141157
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 14.0000 0.492823
\(808\) 2.00000 0.0703598
\(809\) 12.0000 0.421898 0.210949 − 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 2.00000 0.0702728
\(811\) −44.0000 −1.54505 −0.772524 − 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) −10.0000 −0.350715
\(814\) 12.0000 0.420600
\(815\) −24.0000 −0.840683
\(816\) 1.00000 0.0350070
\(817\) −8.00000 −0.279885
\(818\) 12.0000 0.419570
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) −28.0000 −0.977207 −0.488603 − 0.872506i \(-0.662493\pi\)
−0.488603 + 0.872506i \(0.662493\pi\)
\(822\) 2.00000 0.0697580
\(823\) −20.0000 −0.697156 −0.348578 − 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 14.0000 0.487713
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) −18.0000 −0.625921 −0.312961 − 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 6.00000 0.208514
\(829\) −40.0000 −1.38926 −0.694629 − 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(830\) −24.0000 −0.833052
\(831\) −10.0000 −0.346896
\(832\) 4.00000 0.138675
\(833\) 0 0
\(834\) −12.0000 −0.415526
\(835\) 32.0000 1.10741
\(836\) −4.00000 −0.138343
\(837\) 4.00000 0.138260
\(838\) −4.00000 −0.138178
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −30.0000 −1.03387
\(843\) 6.00000 0.206651
\(844\) 20.0000 0.688428
\(845\) −6.00000 −0.206406
\(846\) −4.00000 −0.137523
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) −8.00000 −0.274559
\(850\) −1.00000 −0.0342997
\(851\) −36.0000 −1.23406
\(852\) −2.00000 −0.0685189
\(853\) 46.0000 1.57501 0.787505 − 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 2.00000 0.0683586
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 8.00000 0.273115
\(859\) 6.00000 0.204717 0.102359 − 0.994748i \(-0.467361\pi\)
0.102359 + 0.994748i \(0.467361\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 2.00000 0.0681203
\(863\) 12.0000 0.408485 0.204242 − 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 1.00000 0.0340207
\(865\) −28.0000 −0.952029
\(866\) −16.0000 −0.543702
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) −16.0000 −0.542451
\(871\) 16.0000 0.542139
\(872\) 2.00000 0.0677285
\(873\) −10.0000 −0.338449
\(874\) 12.0000 0.405906
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) −6.00000 −0.202606 −0.101303 − 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) −4.00000 −0.134840
\(881\) −50.0000 −1.68454 −0.842271 − 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) 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
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) 4.00000 0.134080
\(891\) 2.00000 0.0670025
\(892\) 2.00000 0.0669650
\(893\) −8.00000 −0.267710
\(894\) −10.0000 −0.334450
\(895\) 48.0000 1.60446
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) −36.0000 −1.20134
\(899\) −32.0000 −1.06726
\(900\) −1.00000 −0.0333333
\(901\) −2.00000 −0.0666297
\(902\) −12.0000 −0.399556
\(903\) 0 0
\(904\) −16.0000 −0.532152
\(905\) −4.00000 −0.132964
\(906\) 16.0000 0.531564
\(907\) −8.00000 −0.265636 −0.132818 − 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) −4.00000 −0.132745
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 22.0000 0.728893 0.364446 − 0.931224i \(-0.381258\pi\)
0.364446 + 0.931224i \(0.381258\pi\)
\(912\) 2.00000 0.0662266
\(913\) −24.0000 −0.794284
\(914\) 10.0000 0.330771
\(915\) −20.0000 −0.661180
\(916\) 16.0000 0.528655
\(917\) 0 0
\(918\) −1.00000 −0.0330049
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 12.0000 0.395628
\(921\) −2.00000 −0.0659022
\(922\) 18.0000 0.592798
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) −8.00000 −0.262896
\(927\) −14.0000 −0.459820
\(928\) −8.00000 −0.262613
\(929\) 54.0000 1.77168 0.885841 − 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 8.00000 0.262330
\(931\) 0 0
\(932\) 20.0000 0.655122
\(933\) −8.00000 −0.261908
\(934\) −8.00000 −0.261768
\(935\) 4.00000 0.130814
\(936\) −4.00000 −0.130744
\(937\) −12.0000 −0.392023 −0.196011 − 0.980602i \(-0.562799\pi\)
−0.196011 + 0.980602i \(0.562799\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) −8.00000 −0.260931
\(941\) −30.0000 −0.977972 −0.488986 − 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 4.00000 0.130327
\(943\) 36.0000 1.17232
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 38.0000 1.23483 0.617417 − 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(948\) 8.00000 0.259828
\(949\) 8.00000 0.259691
\(950\) −2.00000 −0.0648886
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 48.0000 1.55324
\(956\) 24.0000 0.776215
\(957\) −16.0000 −0.517207
\(958\) 0 0
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) −15.0000 −0.483871
\(962\) 24.0000 0.773791
\(963\) −2.00000 −0.0644491
\(964\) −10.0000 −0.322078
\(965\) −52.0000 −1.67394
\(966\) 0 0
\(967\) −56.0000 −1.80084 −0.900419 − 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 7.00000 0.224989
\(969\) −2.00000 −0.0642493
\(970\) −20.0000 −0.642161
\(971\) 40.0000 1.28366 0.641831 − 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −4.00000 −0.128168
\(975\) 4.00000 0.128103
\(976\) −10.0000 −0.320092
\(977\) −46.0000 −1.47167 −0.735835 − 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) 12.0000 0.383718
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 24.0000 0.765871
\(983\) 40.0000 1.27580 0.637901 − 0.770118i \(-0.279803\pi\)
0.637901 + 0.770118i \(0.279803\pi\)
\(984\) 6.00000 0.191273
\(985\) −24.0000 −0.764704
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 24.0000 0.763156
\(990\) 4.00000 0.127128
\(991\) −40.0000 −1.27064 −0.635321 − 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 4.00000 0.127000
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −40.0000 −1.26809
\(996\) 12.0000 0.380235
\(997\) 46.0000 1.45683 0.728417 − 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 4998.2.a.c.1.1 ✓ 1
7.6 odd 2 4998.2.a.v.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4998.2.a.c.1.1 ✓ 1 1.1 even 1 trivial
4998.2.a.v.1.1 yes 1 7.6 odd 2