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Properties

Label 1502.2.a.a.1.1
Level $1502$
Weight $2$
Character 1502.1
Self dual yes
Analytic conductor $11.994$
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] = [1502,2,Mod(1,1502)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1502, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1502.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1502 = 2 \cdot 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1502.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: \(11.9935303836\)
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 \(\chi\) \(=\) 1502.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +2.00000 q^{11} -2.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -8.00000 q^{21} +2.00000 q^{22} -2.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} +4.00000 q^{28} -4.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{39} +2.00000 q^{40} -2.00000 q^{41} -8.00000 q^{42} -4.00000 q^{43} +2.00000 q^{44} +2.00000 q^{45} -8.00000 q^{47} -2.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -12.0000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +4.00000 q^{54} +4.00000 q^{55} +4.00000 q^{56} +12.0000 q^{59} -4.00000 q^{60} +6.00000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{66} +2.00000 q^{67} +6.00000 q^{68} +8.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} -14.0000 q^{73} -2.00000 q^{74} +2.00000 q^{75} +8.00000 q^{77} +4.00000 q^{78} +12.0000 q^{79} +2.00000 q^{80} -11.0000 q^{81} -2.00000 q^{82} +6.00000 q^{83} -8.00000 q^{84} +12.0000 q^{85} -4.00000 q^{86} +2.00000 q^{88} -2.00000 q^{89} +2.00000 q^{90} -8.00000 q^{91} +8.00000 q^{93} -8.00000 q^{94} -2.00000 q^{96} +10.0000 q^{97} +9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −2.00000 −0.816497
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 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 \(0.402509\pi\)
\(12\) −2.00000 −0.577350
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) −4.00000 −1.03280
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) −8.00000 −1.74574
\(22\) 2.00000 0.426401
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −2.00000 −0.408248
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 4.00000 0.769800
\(28\) 4.00000 0.755929
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −4.00000 −0.730297
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 6.00000 1.02899
\(35\) 8.00000 1.35225
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 2.00000 0.316228
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −8.00000 −1.23443
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000 0.301511
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −2.00000 −0.288675
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −12.0000 −1.68034
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 4.00000 0.544331
\(55\) 4.00000 0.539360
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −4.00000 −0.516398
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −4.00000 −0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −4.00000 −0.492366
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 8.00000 0.956183
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −2.00000 −0.232495
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 4.00000 0.452911
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 2.00000 0.223607
\(81\) −11.0000 −1.22222
\(82\) −2.00000 −0.220863
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −8.00000 −0.872872
\(85\) 12.0000 1.30158
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 2.00000 0.210819
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 9.00000 0.909137
\(99\) 2.00000 0.201008
\(100\) −1.00000 −0.100000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −12.0000 −1.18818
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) −16.0000 −1.56144
\(106\) 6.00000 0.582772
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 4.00000 0.384900
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 4.00000 0.381385
\(111\) 4.00000 0.379663
\(112\) 4.00000 0.377964
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 12.0000 1.10469
\(119\) 24.0000 2.20008
\(120\) −4.00000 −0.365148
\(121\) −7.00000 −0.636364
\(122\) 6.00000 0.543214
\(123\) 4.00000 0.360668
\(124\) −4.00000 −0.359211
\(125\) −12.0000 −1.07331
\(126\) 4.00000 0.356348
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) −4.00000 −0.350823
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 8.00000 0.688530
\(136\) 6.00000 0.514496
\(137\) −18.0000 −1.53784 −0.768922 − 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 8.00000 0.676123
\(141\) 16.0000 1.34744
\(142\) 8.00000 0.671345
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) −18.0000 −1.48461
\(148\) −2.00000 −0.164399
\(149\) −14.0000 −1.14692 −0.573462 − 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 2.00000 0.163299
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 8.00000 0.644658
\(155\) −8.00000 −0.642575
\(156\) 4.00000 0.320256
\(157\) −8.00000 −0.638470 −0.319235 − 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 12.0000 0.954669
\(159\) −12.0000 −0.951662
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −2.00000 −0.156174
\(165\) −8.00000 −0.622799
\(166\) 6.00000 0.465690
\(167\) −24.0000 −1.85718 −0.928588 − 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) −8.00000 −0.617213
\(169\) −9.00000 −0.692308
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 8.00000 0.608229 0.304114 − 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 2.00000 0.150756
\(177\) −24.0000 −1.80395
\(178\) −2.00000 −0.149906
\(179\) 8.00000 0.597948 0.298974 − 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) 2.00000 0.149071
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −8.00000 −0.592999
\(183\) −12.0000 −0.887066
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 8.00000 0.586588
\(187\) 12.0000 0.877527
\(188\) −8.00000 −0.583460
\(189\) 16.0000 1.16383
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) −2.00000 −0.144338
\(193\) −18.0000 −1.29567 −0.647834 − 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 10.0000 0.717958
\(195\) 8.00000 0.572892
\(196\) 9.00000 0.642857
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 2.00000 0.142134
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 0 0
\(204\) −12.0000 −0.840168
\(205\) −4.00000 −0.279372
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) −16.0000 −1.10410
\(211\) 12.0000 0.826114 0.413057 − 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 6.00000 0.412082
\(213\) −16.0000 −1.09630
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 4.00000 0.272166
\(217\) −16.0000 −1.08615
\(218\) 2.00000 0.135457
\(219\) 28.0000 1.89206
\(220\) 4.00000 0.269680
\(221\) −12.0000 −0.807207
\(222\) 4.00000 0.268462
\(223\) 12.0000 0.803579 0.401790 − 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 4.00000 0.267261
\(225\) −1.00000 −0.0666667
\(226\) −6.00000 −0.399114
\(227\) −14.0000 −0.929213 −0.464606 − 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) −16.0000 −1.04372
\(236\) 12.0000 0.781133
\(237\) −24.0000 −1.55897
\(238\) 24.0000 1.55569
\(239\) −24.0000 −1.55243 −0.776215 − 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) −4.00000 −0.258199
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −7.00000 −0.449977
\(243\) 10.0000 0.641500
\(244\) 6.00000 0.384111
\(245\) 18.0000 1.14998
\(246\) 4.00000 0.255031
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) −12.0000 −0.760469
\(250\) −12.0000 −0.758947
\(251\) −6.00000 −0.378717 −0.189358 − 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) −24.0000 −1.50294
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 8.00000 0.498058
\(259\) −8.00000 −0.497096
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) 4.00000 0.246651 0.123325 − 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) −4.00000 −0.246183
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 4.00000 0.244796
\(268\) 2.00000 0.122169
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 8.00000 0.486864
\(271\) −8.00000 −0.485965 −0.242983 − 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 6.00000 0.363803
\(273\) 16.0000 0.968364
\(274\) −18.0000 −1.08742
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) −20.0000 −1.20168 −0.600842 − 0.799368i \(-0.705168\pi\)
−0.600842 + 0.799368i \(0.705168\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 8.00000 0.478091
\(281\) −14.0000 −0.835170 −0.417585 − 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 16.0000 0.952786
\(283\) 26.0000 1.54554 0.772770 − 0.634686i \(-0.218871\pi\)
0.772770 + 0.634686i \(0.218871\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) −8.00000 −0.472225
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −20.0000 −1.17242
\(292\) −14.0000 −0.819288
\(293\) −4.00000 −0.233682 −0.116841 − 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) −18.0000 −1.04978
\(295\) 24.0000 1.39733
\(296\) −2.00000 −0.116248
\(297\) 8.00000 0.464207
\(298\) −14.0000 −0.810998
\(299\) 0 0
\(300\) 2.00000 0.115470
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 6.00000 0.342997
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 8.00000 0.455842
\(309\) −16.0000 −0.910208
\(310\) −8.00000 −0.454369
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 4.00000 0.226455
\(313\) −26.0000 −1.46961 −0.734803 − 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −8.00000 −0.451466
\(315\) 8.00000 0.450749
\(316\) 12.0000 0.675053
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 2.00000 0.110940
\(326\) 4.00000 0.221540
\(327\) −4.00000 −0.221201
\(328\) −2.00000 −0.110432
\(329\) −32.0000 −1.76422
\(330\) −8.00000 −0.440386
\(331\) −6.00000 −0.329790 −0.164895 − 0.986311i \(-0.552728\pi\)
−0.164895 + 0.986311i \(0.552728\pi\)
\(332\) 6.00000 0.329293
\(333\) −2.00000 −0.109599
\(334\) −24.0000 −1.31322
\(335\) 4.00000 0.218543
\(336\) −8.00000 −0.436436
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 12.0000 0.651751
\(340\) 12.0000 0.650791
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 8.00000 0.430083
\(347\) 20.0000 1.07366 0.536828 − 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 − 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) −4.00000 −0.213809
\(351\) −8.00000 −0.427008
\(352\) 2.00000 0.106600
\(353\) 18.0000 0.958043 0.479022 − 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −24.0000 −1.27559
\(355\) 16.0000 0.849192
\(356\) −2.00000 −0.106000
\(357\) −48.0000 −2.54043
\(358\) 8.00000 0.422813
\(359\) 16.0000 0.844448 0.422224 − 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 2.00000 0.105409
\(361\) −19.0000 −1.00000
\(362\) 2.00000 0.105118
\(363\) 14.0000 0.734809
\(364\) −8.00000 −0.419314
\(365\) −28.0000 −1.46559
\(366\) −12.0000 −0.627250
\(367\) 32.0000 1.67039 0.835193 − 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) −4.00000 −0.207950
\(371\) 24.0000 1.24602
\(372\) 8.00000 0.414781
\(373\) −20.0000 −1.03556 −0.517780 − 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) 12.0000 0.620505
\(375\) 24.0000 1.23935
\(376\) −8.00000 −0.412568
\(377\) 0 0
\(378\) 16.0000 0.822951
\(379\) −34.0000 −1.74646 −0.873231 − 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 0 0
\(381\) −32.0000 −1.63941
\(382\) −24.0000 −1.22795
\(383\) 4.00000 0.204390 0.102195 − 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) −2.00000 −0.102062
\(385\) 16.0000 0.815436
\(386\) −18.0000 −0.916176
\(387\) −4.00000 −0.203331
\(388\) 10.0000 0.507673
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 8.00000 0.405096
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) −8.00000 −0.403547
\(394\) −2.00000 −0.100759
\(395\) 24.0000 1.20757
\(396\) 2.00000 0.100504
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −4.00000 −0.199502
\(403\) 8.00000 0.398508
\(404\) 0 0
\(405\) −22.0000 −1.09319
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) −12.0000 −0.594089
\(409\) 22.0000 1.08783 0.543915 − 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) −4.00000 −0.197546
\(411\) 36.0000 1.77575
\(412\) 8.00000 0.394132
\(413\) 48.0000 2.36193
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 0 0
\(419\) 16.0000 0.781651 0.390826 − 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) −16.0000 −0.780720
\(421\) 8.00000 0.389896 0.194948 − 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 12.0000 0.584151
\(423\) −8.00000 −0.388973
\(424\) 6.00000 0.291386
\(425\) −6.00000 −0.291043
\(426\) −16.0000 −0.775203
\(427\) 24.0000 1.16144
\(428\) 0 0
\(429\) 8.00000 0.386244
\(430\) −8.00000 −0.385794
\(431\) 12.0000 0.578020 0.289010 − 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 4.00000 0.192450
\(433\) 18.0000 0.865025 0.432512 − 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 28.0000 1.33789
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 4.00000 0.190693
\(441\) 9.00000 0.428571
\(442\) −12.0000 −0.570782
\(443\) −10.0000 −0.475114 −0.237557 − 0.971374i \(-0.576347\pi\)
−0.237557 + 0.971374i \(0.576347\pi\)
\(444\) 4.00000 0.189832
\(445\) −4.00000 −0.189618
\(446\) 12.0000 0.568216
\(447\) 28.0000 1.32435
\(448\) 4.00000 0.188982
\(449\) 22.0000 1.03824 0.519122 − 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −4.00000 −0.188353
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) −14.0000 −0.657053
\(455\) −16.0000 −0.750092
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 − 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 6.00000 0.280362
\(459\) 24.0000 1.12022
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 − 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) −16.0000 −0.744387
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 16.0000 0.741982
\(466\) −6.00000 −0.277945
\(467\) −2.00000 −0.0925490 −0.0462745 − 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 8.00000 0.369406
\(470\) −16.0000 −0.738025
\(471\) 16.0000 0.737241
\(472\) 12.0000 0.552345
\(473\) −8.00000 −0.367840
\(474\) −24.0000 −1.10236
\(475\) 0 0
\(476\) 24.0000 1.10004
\(477\) 6.00000 0.274721
\(478\) −24.0000 −1.09773
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −4.00000 −0.182574
\(481\) 4.00000 0.182384
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 20.0000 0.908153
\(486\) 10.0000 0.453609
\(487\) −12.0000 −0.543772 −0.271886 − 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 6.00000 0.271607
\(489\) −8.00000 −0.361773
\(490\) 18.0000 0.813157
\(491\) −34.0000 −1.53440 −0.767199 − 0.641409i \(-0.778350\pi\)
−0.767199 + 0.641409i \(0.778350\pi\)
\(492\) 4.00000 0.180334
\(493\) 0 0
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) −4.00000 −0.179605
\(497\) 32.0000 1.43540
\(498\) −12.0000 −0.537733
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −12.0000 −0.536656
\(501\) 48.0000 2.14448
\(502\) −6.00000 −0.267793
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) 18.0000 0.799408
\(508\) 16.0000 0.709885
\(509\) 36.0000 1.59567 0.797836 − 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) −24.0000 −1.06274
\(511\) −56.0000 −2.47729
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 16.0000 0.705044
\(516\) 8.00000 0.352180
\(517\) −16.0000 −0.703679
\(518\) −8.00000 −0.351500
\(519\) −16.0000 −0.702322
\(520\) −4.00000 −0.175412
\(521\) −34.0000 −1.48957 −0.744784 − 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 4.00000 0.174741
\(525\) 8.00000 0.349149
\(526\) 4.00000 0.174408
\(527\) −24.0000 −1.04546
\(528\) −4.00000 −0.174078
\(529\) −23.0000 −1.00000
\(530\) 12.0000 0.521247
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 4.00000 0.173097
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) −16.0000 −0.690451
\(538\) −6.00000 −0.258678
\(539\) 18.0000 0.775315
\(540\) 8.00000 0.344265
\(541\) −40.0000 −1.71973 −0.859867 − 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) −8.00000 −0.343629
\(543\) −4.00000 −0.171656
\(544\) 6.00000 0.257248
\(545\) 4.00000 0.171341
\(546\) 16.0000 0.684737
\(547\) −22.0000 −0.940652 −0.470326 − 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −18.0000 −0.768922
\(549\) 6.00000 0.256074
\(550\) −2.00000 −0.0852803
\(551\) 0 0
\(552\) 0 0
\(553\) 48.0000 2.04117
\(554\) −20.0000 −0.849719
\(555\) 8.00000 0.339581
\(556\) 0 0
\(557\) 40.0000 1.69485 0.847427 − 0.530912i \(-0.178150\pi\)
0.847427 + 0.530912i \(0.178150\pi\)
\(558\) −4.00000 −0.169334
\(559\) 8.00000 0.338364
\(560\) 8.00000 0.338062
\(561\) −24.0000 −1.01328
\(562\) −14.0000 −0.590554
\(563\) 10.0000 0.421450 0.210725 − 0.977545i \(-0.432418\pi\)
0.210725 + 0.977545i \(0.432418\pi\)
\(564\) 16.0000 0.673722
\(565\) −12.0000 −0.504844
\(566\) 26.0000 1.09286
\(567\) −44.0000 −1.84783
\(568\) 8.00000 0.335673
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −2.00000 −0.0836974 −0.0418487 − 0.999124i \(-0.513325\pi\)
−0.0418487 + 0.999124i \(0.513325\pi\)
\(572\) −4.00000 −0.167248
\(573\) 48.0000 2.00523
\(574\) −8.00000 −0.333914
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 26.0000 1.08239 0.541197 − 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) 19.0000 0.790296
\(579\) 36.0000 1.49611
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) −20.0000 −0.829027
\(583\) 12.0000 0.496989
\(584\) −14.0000 −0.579324
\(585\) −4.00000 −0.165380
\(586\) −4.00000 −0.165238
\(587\) −8.00000 −0.330195 −0.165098 − 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) −18.0000 −0.742307
\(589\) 0 0
\(590\) 24.0000 0.988064
\(591\) 4.00000 0.164538
\(592\) −2.00000 −0.0821995
\(593\) 34.0000 1.39621 0.698106 − 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 8.00000 0.328244
\(595\) 48.0000 1.96781
\(596\) −14.0000 −0.573462
\(597\) 0 0
\(598\) 0 0
\(599\) −32.0000 −1.30748 −0.653742 − 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 2.00000 0.0816497
\(601\) −6.00000 −0.244745 −0.122373 − 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) −16.0000 −0.652111
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) 12.0000 0.487065 0.243532 − 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) 16.0000 0.647291
\(612\) 6.00000 0.242536
\(613\) 34.0000 1.37325 0.686624 − 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 20.0000 0.807134
\(615\) 8.00000 0.322591
\(616\) 8.00000 0.322329
\(617\) −2.00000 −0.0805170 −0.0402585 − 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) −16.0000 −0.643614
\(619\) −38.0000 −1.52735 −0.763674 − 0.645601i \(-0.776607\pi\)
−0.763674 + 0.645601i \(0.776607\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) −8.00000 −0.320513
\(624\) 4.00000 0.160128
\(625\) −19.0000 −0.760000
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) −8.00000 −0.319235
\(629\) −12.0000 −0.478471
\(630\) 8.00000 0.318728
\(631\) 24.0000 0.955425 0.477712 − 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 12.0000 0.477334
\(633\) −24.0000 −0.953914
\(634\) 0 0
\(635\) 32.0000 1.26988
\(636\) −12.0000 −0.475831
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 2.00000 0.0790569
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 − 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 − 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) −11.0000 −0.432121
\(649\) 24.0000 0.942082
\(650\) 2.00000 0.0784465
\(651\) 32.0000 1.25418
\(652\) 4.00000 0.156652
\(653\) −36.0000 −1.40879 −0.704394 − 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) −4.00000 −0.156412
\(655\) 8.00000 0.312586
\(656\) −2.00000 −0.0780869
\(657\) −14.0000 −0.546192
\(658\) −32.0000 −1.24749
\(659\) −14.0000 −0.545363 −0.272681 − 0.962104i \(-0.587910\pi\)
−0.272681 + 0.962104i \(0.587910\pi\)
\(660\) −8.00000 −0.311400
\(661\) −8.00000 −0.311164 −0.155582 − 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) −6.00000 −0.233197
\(663\) 24.0000 0.932083
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) −24.0000 −0.928588
\(669\) −24.0000 −0.927894
\(670\) 4.00000 0.154533
\(671\) 12.0000 0.463255
\(672\) −8.00000 −0.308607
\(673\) −2.00000 −0.0770943 −0.0385472 − 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −22.0000 −0.847408
\(675\) −4.00000 −0.153960
\(676\) −9.00000 −0.346154
\(677\) −48.0000 −1.84479 −0.922395 − 0.386248i \(-0.873771\pi\)
−0.922395 + 0.386248i \(0.873771\pi\)
\(678\) 12.0000 0.460857
\(679\) 40.0000 1.53506
\(680\) 12.0000 0.460179
\(681\) 28.0000 1.07296
\(682\) −8.00000 −0.306336
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −36.0000 −1.37549
\(686\) 8.00000 0.305441
\(687\) −12.0000 −0.457829
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 8.00000 0.304114
\(693\) 8.00000 0.303895
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) −8.00000 −0.302804
\(699\) 12.0000 0.453882
\(700\) −4.00000 −0.151186
\(701\) −4.00000 −0.151078 −0.0755390 − 0.997143i \(-0.524068\pi\)
−0.0755390 + 0.997143i \(0.524068\pi\)
\(702\) −8.00000 −0.301941
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 32.0000 1.20519
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) −24.0000 −0.901975
\(709\) 28.0000 1.05156 0.525781 − 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) 16.0000 0.600469
\(711\) 12.0000 0.450035
\(712\) −2.00000 −0.0749532
\(713\) 0 0
\(714\) −48.0000 −1.79635
\(715\) −8.00000 −0.299183
\(716\) 8.00000 0.298974
\(717\) 48.0000 1.79259
\(718\) 16.0000 0.597115
\(719\) −24.0000 −0.895049 −0.447524 − 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 2.00000 0.0745356
\(721\) 32.0000 1.19174
\(722\) −19.0000 −0.707107
\(723\) 20.0000 0.743808
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) −8.00000 −0.296500
\(729\) 13.0000 0.481481
\(730\) −28.0000 −1.03633
\(731\) −24.0000 −0.887672
\(732\) −12.0000 −0.443533
\(733\) 28.0000 1.03420 0.517102 − 0.855924i \(-0.327011\pi\)
0.517102 + 0.855924i \(0.327011\pi\)
\(734\) 32.0000 1.18114
\(735\) −36.0000 −1.32788
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) −2.00000 −0.0736210
\(739\) −16.0000 −0.588570 −0.294285 − 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) −8.00000 −0.293492 −0.146746 − 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 8.00000 0.293294
\(745\) −28.0000 −1.02584
\(746\) −20.0000 −0.732252
\(747\) 6.00000 0.219529
\(748\) 12.0000 0.438763
\(749\) 0 0
\(750\) 24.0000 0.876356
\(751\) −1.00000 −0.0364905
\(752\) −8.00000 −0.291730
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 16.0000 0.581914
\(757\) −32.0000 −1.16306 −0.581530 − 0.813525i \(-0.697546\pi\)
−0.581530 + 0.813525i \(0.697546\pi\)
\(758\) −34.0000 −1.23494
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 − 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −32.0000 −1.15924
\(763\) 8.00000 0.289619
\(764\) −24.0000 −0.868290
\(765\) 12.0000 0.433861
\(766\) 4.00000 0.144526
\(767\) −24.0000 −0.866590
\(768\) −2.00000 −0.0721688
\(769\) 10.0000 0.360609 0.180305 − 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 16.0000 0.576600
\(771\) −36.0000 −1.29651
\(772\) −18.0000 −0.647834
\(773\) −12.0000 −0.431610 −0.215805 − 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) −4.00000 −0.143777
\(775\) 4.00000 0.143684
\(776\) 10.0000 0.358979
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) 0 0
\(780\) 8.00000 0.286446
\(781\) 16.0000 0.572525
\(782\) 0 0
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) −16.0000 −0.571064
\(786\) −8.00000 −0.285351
\(787\) 36.0000 1.28326 0.641631 − 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −8.00000 −0.284808
\(790\) 24.0000 0.853882
\(791\) −24.0000 −0.853342
\(792\) 2.00000 0.0710669
\(793\) −12.0000 −0.426132
\(794\) 18.0000 0.638796
\(795\) −24.0000 −0.851192
\(796\) 0 0
\(797\) −22.0000 −0.779280 −0.389640 − 0.920967i \(-0.627401\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) −1.00000 −0.0353553
\(801\) −2.00000 −0.0706665
\(802\) −18.0000 −0.635602
\(803\) −28.0000 −0.988099
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 12.0000 0.422420
\(808\) 0 0
\(809\) −2.00000 −0.0703163 −0.0351581 − 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) −22.0000 −0.773001
\(811\) 30.0000 1.05344 0.526721 − 0.850038i \(-0.323421\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) −4.00000 −0.140200
\(815\) 8.00000 0.280228
\(816\) −12.0000 −0.420084
\(817\) 0 0
\(818\) 22.0000 0.769212
\(819\) −8.00000 −0.279543
\(820\) −4.00000 −0.139686
\(821\) −48.0000 −1.67521 −0.837606 − 0.546275i \(-0.816045\pi\)
−0.837606 + 0.546275i \(0.816045\pi\)
\(822\) 36.0000 1.25564
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 8.00000 0.278693
\(825\) 4.00000 0.139262
\(826\) 48.0000 1.67013
\(827\) −56.0000 −1.94731 −0.973655 − 0.228024i \(-0.926773\pi\)
−0.973655 + 0.228024i \(0.926773\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 − 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 12.0000 0.416526
\(831\) 40.0000 1.38758
\(832\) −2.00000 −0.0693375
\(833\) 54.0000 1.87099
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) 16.0000 0.552711
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) −16.0000 −0.552052
\(841\) −29.0000 −1.00000
\(842\) 8.00000 0.275698
\(843\) 28.0000 0.964371
\(844\) 12.0000 0.413057
\(845\) −18.0000 −0.619219
\(846\) −8.00000 −0.275046
\(847\) −28.0000 −0.962091
\(848\) 6.00000 0.206041
\(849\) −52.0000 −1.78464
\(850\) −6.00000 −0.205798
\(851\) 0 0
\(852\) −16.0000 −0.548151
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 24.0000 0.821263
\(855\) 0 0
\(856\) 0 0
\(857\) −10.0000 −0.341593 −0.170797 − 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 8.00000 0.273115
\(859\) 30.0000 1.02359 0.511793 − 0.859109i \(-0.328981\pi\)
0.511793 + 0.859109i \(0.328981\pi\)
\(860\) −8.00000 −0.272798
\(861\) 16.0000 0.545279
\(862\) 12.0000 0.408722
\(863\) 48.0000 1.63394 0.816970 − 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 4.00000 0.136083
\(865\) 16.0000 0.544016
\(866\) 18.0000 0.611665
\(867\) −38.0000 −1.29055
\(868\) −16.0000 −0.543075
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 2.00000 0.0677285
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) 28.0000 0.946032
\(877\) −16.0000 −0.540282 −0.270141 − 0.962821i \(-0.587070\pi\)
−0.270141 + 0.962821i \(0.587070\pi\)
\(878\) −8.00000 −0.269987
\(879\) 8.00000 0.269833
\(880\) 4.00000 0.134840
\(881\) −26.0000 −0.875962 −0.437981 − 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 9.00000 0.303046
\(883\) −12.0000 −0.403832 −0.201916 − 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) −12.0000 −0.403604
\(885\) −48.0000 −1.61350
\(886\) −10.0000 −0.335957
\(887\) −8.00000 −0.268614 −0.134307 − 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 4.00000 0.134231
\(889\) 64.0000 2.14649
\(890\) −4.00000 −0.134080
\(891\) −22.0000 −0.737028
\(892\) 12.0000 0.401790
\(893\) 0 0
\(894\) 28.0000 0.936460
\(895\) 16.0000 0.534821
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 22.0000 0.734150
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) 36.0000 1.19933
\(902\) −4.00000 −0.133185
\(903\) 32.0000 1.06489
\(904\) −6.00000 −0.199557
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) 54.0000 1.79304 0.896520 − 0.443003i \(-0.146087\pi\)
0.896520 + 0.443003i \(0.146087\pi\)
\(908\) −14.0000 −0.464606
\(909\) 0 0
\(910\) −16.0000 −0.530395
\(911\) −32.0000 −1.06021 −0.530104 − 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 38.0000 1.25693
\(915\) −24.0000 −0.793416
\(916\) 6.00000 0.198246
\(917\) 16.0000 0.528367
\(918\) 24.0000 0.792118
\(919\) 8.00000 0.263896 0.131948 − 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) −40.0000 −1.31804
\(922\) −14.0000 −0.461065
\(923\) −16.0000 −0.526646
\(924\) −16.0000 −0.526361
\(925\) 2.00000 0.0657596
\(926\) −8.00000 −0.262896
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 − 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 16.0000 0.524661
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 48.0000 1.57145
\(934\) −2.00000 −0.0654420
\(935\) 24.0000 0.784884
\(936\) −2.00000 −0.0653720
\(937\) 10.0000 0.326686 0.163343 − 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 8.00000 0.261209
\(939\) 52.0000 1.69696
\(940\) −16.0000 −0.521862
\(941\) 42.0000 1.36916 0.684580 − 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 16.0000 0.521308
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 32.0000 1.04096
\(946\) −8.00000 −0.260102
\(947\) 28.0000 0.909878 0.454939 − 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) −24.0000 −0.779484
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) 0 0
\(952\) 24.0000 0.777844
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 6.00000 0.194257
\(955\) −48.0000 −1.55324
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 0 0
\(959\) −72.0000 −2.32500
\(960\) −4.00000 −0.129099
\(961\) −15.0000 −0.483871
\(962\) 4.00000 0.128965
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −36.0000 −1.15888
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 20.0000 0.642161
\(971\) 34.0000 1.09111 0.545556 − 0.838074i \(-0.316319\pi\)
0.545556 + 0.838074i \(0.316319\pi\)
\(972\) 10.0000 0.320750
\(973\) 0 0
\(974\) −12.0000 −0.384505
\(975\) −4.00000 −0.128103
\(976\) 6.00000 0.192055
\(977\) 38.0000 1.21573 0.607864 − 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) −8.00000 −0.255812
\(979\) −4.00000 −0.127841
\(980\) 18.0000 0.574989
\(981\) 2.00000 0.0638551
\(982\) −34.0000 −1.08498
\(983\) −4.00000 −0.127580 −0.0637901 − 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 4.00000 0.127515
\(985\) −4.00000 −0.127451
\(986\) 0 0
\(987\) 64.0000 2.03714
\(988\) 0 0
\(989\) 0 0
\(990\) 4.00000 0.127128
\(991\) 36.0000 1.14358 0.571789 − 0.820401i \(-0.306250\pi\)
0.571789 + 0.820401i \(0.306250\pi\)
\(992\) −4.00000 −0.127000
\(993\) 12.0000 0.380808
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 22.0000 0.696747 0.348373 − 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) −4.00000 −0.126618
\(999\) −8.00000 −0.253109
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 1502.2.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1502.2.a.a.1.1 ✓ 1 1.1 even 1 trivial