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Properties

Label 8002.2.a.c.1.1
Level $8002$
Weight $2$
Character 8002.1
Self dual yes
Analytic conductor $63.896$
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] = [8002,2,Mod(1,8002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8002, 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("8002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8002.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: \(63.8962916974\)
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 \(\chi\) \(=\) 8002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -6.00000 q^{11} +2.00000 q^{12} +2.00000 q^{13} -4.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} -2.00000 q^{20} -6.00000 q^{22} +4.00000 q^{23} +2.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -4.00000 q^{27} -6.00000 q^{29} -4.00000 q^{30} +1.00000 q^{32} -12.0000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +8.00000 q^{38} +4.00000 q^{39} -2.00000 q^{40} -2.00000 q^{41} -6.00000 q^{43} -6.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} +2.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -4.00000 q^{51} +2.00000 q^{52} -4.00000 q^{53} -4.00000 q^{54} +12.0000 q^{55} +16.0000 q^{57} -6.00000 q^{58} -4.00000 q^{60} -6.00000 q^{61} +1.00000 q^{64} -4.00000 q^{65} -12.0000 q^{66} +2.00000 q^{67} -2.00000 q^{68} +8.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -2.00000 q^{75} +8.00000 q^{76} +4.00000 q^{78} -16.0000 q^{79} -2.00000 q^{80} -11.0000 q^{81} -2.00000 q^{82} -12.0000 q^{83} +4.00000 q^{85} -6.00000 q^{86} -12.0000 q^{87} -6.00000 q^{88} -2.00000 q^{89} -2.00000 q^{90} +4.00000 q^{92} -16.0000 q^{95} +2.00000 q^{96} +2.00000 q^{97} -7.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 2.00000 0.816497
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 2.00000 0.577350
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 2.00000 0.408248
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −4.00000 −0.730297
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −12.0000 −2.08893
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 8.00000 1.29777
\(39\) 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 \(0.549916\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −6.00000 −0.904534
\(45\) −2.00000 −0.298142
\(46\) 4.00000 0.589768
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 2.00000 0.288675
\(49\) −7.00000 −1.00000
\(50\) −1.00000 −0.141421
\(51\) −4.00000 −0.560112
\(52\) 2.00000 0.277350
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −4.00000 −0.544331
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 16.0000 2.11925
\(58\) −6.00000 −0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −4.00000 −0.516398
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −12.0000 −1.47710
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −2.00000 −0.242536
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −2.00000 −0.223607
\(81\) −11.0000 −1.22222
\(82\) −2.00000 −0.220863
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) −6.00000 −0.646997
\(87\) −12.0000 −1.28654
\(88\) −6.00000 −0.639602
\(89\) −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
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 0 0
\(95\) −16.0000 −1.64157
\(96\) 2.00000 0.204124
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −7.00000 −0.707107
\(99\) −6.00000 −0.603023
\(100\) −1.00000 −0.100000
\(101\) −12.0000 −1.19404 −0.597022 − 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) −4.00000 −0.396059
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −4.00000 −0.384900
\(109\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 12.0000 1.14416
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 16.0000 1.49854
\(115\) −8.00000 −0.746004
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) −4.00000 −0.365148
\(121\) 25.0000 2.27273
\(122\) −6.00000 −0.543214
\(123\) −4.00000 −0.360668
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.0000 −1.05654
\(130\) −4.00000 −0.350823
\(131\) 14.0000 1.22319 0.611593 − 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) −12.0000 −1.04447
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 8.00000 0.688530
\(136\) −2.00000 −0.171499
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 8.00000 0.681005
\(139\) −6.00000 −0.508913 −0.254457 − 0.967084i \(-0.581897\pi\)
−0.254457 + 0.967084i \(0.581897\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 12.0000 0.996546
\(146\) 10.0000 0.827606
\(147\) −14.0000 −1.15470
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −2.00000 −0.163299
\(151\) −24.0000 −1.95309 −0.976546 − 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) 8.00000 0.648886
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) −22.0000 −1.75579 −0.877896 − 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −16.0000 −1.27289
\(159\) −8.00000 −0.634441
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) 14.0000 1.09656 0.548282 − 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) −2.00000 −0.156174
\(165\) 24.0000 1.86840
\(166\) −12.0000 −0.931381
\(167\) −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
\(170\) 4.00000 0.306786
\(171\) 8.00000 0.611775
\(172\) −6.00000 −0.457496
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −12.0000 −0.909718
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) −18.0000 −1.34538 −0.672692 − 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) −2.00000 −0.149071
\(181\) 8.00000 0.594635 0.297318 − 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) 0 0
\(190\) −16.0000 −1.16076
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 2.00000 0.144338
\(193\) 6.00000 0.431889 0.215945 − 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 2.00000 0.143592
\(195\) −8.00000 −0.572892
\(196\) −7.00000 −0.500000
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −6.00000 −0.426401
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 4.00000 0.279372
\(206\) 16.0000 1.11477
\(207\) 4.00000 0.278019
\(208\) 2.00000 0.138675
\(209\) −48.0000 −3.32023
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 − 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −4.00000 −0.274721
\(213\) −16.0000 −1.09630
\(214\) −12.0000 −0.820303
\(215\) 12.0000 0.818393
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) 0 0
\(219\) 20.0000 1.35147
\(220\) 12.0000 0.809040
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 − 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 2.00000 0.133038
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 16.0000 1.05963
\(229\) −6.00000 −0.396491 −0.198246 − 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −14.0000 −0.917170 −0.458585 − 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) −32.0000 −2.07862
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −4.00000 −0.258199
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 25.0000 1.60706
\(243\) −10.0000 −0.641500
\(244\) −6.00000 −0.384111
\(245\) 14.0000 0.894427
\(246\) −4.00000 −0.255031
\(247\) 16.0000 1.01806
\(248\) 0 0
\(249\) −24.0000 −1.52094
\(250\) 12.0000 0.758947
\(251\) −2.00000 −0.126239 −0.0631194 − 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) −24.0000 −1.50887
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −12.0000 −0.747087
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) −6.00000 −0.371391
\(262\) 14.0000 0.864923
\(263\) 20.0000 1.23325 0.616626 − 0.787256i \(-0.288499\pi\)
0.616626 + 0.787256i \(0.288499\pi\)
\(264\) −12.0000 −0.738549
\(265\) 8.00000 0.491436
\(266\) 0 0
\(267\) −4.00000 −0.244796
\(268\) 2.00000 0.122169
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 8.00000 0.486864
\(271\) 28.0000 1.70088 0.850439 − 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 6.00000 0.361814
\(276\) 8.00000 0.481543
\(277\) −14.0000 −0.841178 −0.420589 − 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) −6.00000 −0.359856
\(279\) 0 0
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 − 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 − 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) −8.00000 −0.474713
\(285\) −32.0000 −1.89552
\(286\) −12.0000 −0.709575
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 12.0000 0.704664
\(291\) 4.00000 0.234484
\(292\) 10.0000 0.585206
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −14.0000 −0.816497
\(295\) 0 0
\(296\) 0 0
\(297\) 24.0000 1.39262
\(298\) 6.00000 0.347571
\(299\) 8.00000 0.462652
\(300\) −2.00000 −0.115470
\(301\) 0 0
\(302\) −24.0000 −1.38104
\(303\) −24.0000 −1.37876
\(304\) 8.00000 0.458831
\(305\) 12.0000 0.687118
\(306\) −2.00000 −0.114332
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 32.0000 1.82042
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 − 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 4.00000 0.226455
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 12.0000 0.673987 0.336994 − 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) −8.00000 −0.448618
\(319\) 36.0000 2.01561
\(320\) −2.00000 −0.111803
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) −11.0000 −0.611111
\(325\) −2.00000 −0.110940
\(326\) 14.0000 0.775388
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 24.0000 1.32116
\(331\) −22.0000 −1.20923 −0.604615 − 0.796518i \(-0.706673\pi\)
−0.604615 + 0.796518i \(0.706673\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −9.00000 −0.489535
\(339\) 4.00000 0.217250
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) 8.00000 0.432590
\(343\) 0 0
\(344\) −6.00000 −0.323498
\(345\) −16.0000 −0.861411
\(346\) 6.00000 0.322562
\(347\) −16.0000 −0.858925 −0.429463 − 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) −12.0000 −0.643268
\(349\) 4.00000 0.214115 0.107058 − 0.994253i \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(350\) 0 0
\(351\) −8.00000 −0.427008
\(352\) −6.00000 −0.319801
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) −18.0000 −0.951330
\(359\) −12.0000 −0.633336 −0.316668 − 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) −2.00000 −0.105409
\(361\) 45.0000 2.36842
\(362\) 8.00000 0.420471
\(363\) 50.0000 2.62432
\(364\) 0 0
\(365\) −20.0000 −1.04685
\(366\) −12.0000 −0.627250
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 4.00000 0.208514
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 32.0000 1.65690 0.828449 − 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 12.0000 0.620505
\(375\) 24.0000 1.23935
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 − 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) −16.0000 −0.820783
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) −16.0000 −0.817562 −0.408781 − 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) −6.00000 −0.304997
\(388\) 2.00000 0.101535
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) −8.00000 −0.405096
\(391\) −8.00000 −0.404577
\(392\) −7.00000 −0.353553
\(393\) 28.0000 1.41241
\(394\) −18.0000 −0.906827
\(395\) 32.0000 1.61009
\(396\) −6.00000 −0.301511
\(397\) 30.0000 1.50566 0.752828 − 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) −20.0000 −1.00251
\(399\) 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
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) 22.0000 1.09319
\(406\) 0 0
\(407\) 0 0
\(408\) −4.00000 −0.198030
\(409\) 38.0000 1.87898 0.939490 − 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 4.00000 0.197546
\(411\) 36.0000 1.77575
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 24.0000 1.17811
\(416\) 2.00000 0.0980581
\(417\) −12.0000 −0.587643
\(418\) −48.0000 −2.34776
\(419\) 26.0000 1.27018 0.635092 − 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 − 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) −16.0000 −0.778868
\(423\) 0 0
\(424\) −4.00000 −0.194257
\(425\) 2.00000 0.0970143
\(426\) −16.0000 −0.775203
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −24.0000 −1.15873
\(430\) 12.0000 0.578691
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) −4.00000 −0.192450
\(433\) 34.0000 1.63394 0.816968 − 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 24.0000 1.15071
\(436\) 0 0
\(437\) 32.0000 1.53077
\(438\) 20.0000 0.955637
\(439\) 16.0000 0.763638 0.381819 − 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 12.0000 0.572078
\(441\) −7.00000 −0.333333
\(442\) −4.00000 −0.190261
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) 4.00000 0.189618
\(446\) 24.0000 1.13643
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 − 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 12.0000 0.565058
\(452\) 2.00000 0.0940721
\(453\) −48.0000 −2.25524
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) 16.0000 0.749269
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −6.00000 −0.280362
\(459\) 8.00000 0.373408
\(460\) −8.00000 −0.373002
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) −6.00000 −0.277647 −0.138823 − 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −44.0000 −2.02741
\(472\) 0 0
\(473\) 36.0000 1.65528
\(474\) −32.0000 −1.46981
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) −4.00000 −0.183147
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −4.00000 −0.182574
\(481\) 0 0
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) −4.00000 −0.181631
\(486\) −10.0000 −0.453609
\(487\) −4.00000 −0.181257 −0.0906287 − 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) −6.00000 −0.271607
\(489\) 28.0000 1.26620
\(490\) 14.0000 0.632456
\(491\) −30.0000 −1.35388 −0.676941 − 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) −4.00000 −0.180334
\(493\) 12.0000 0.540453
\(494\) 16.0000 0.719874
\(495\) 12.0000 0.539360
\(496\) 0 0
\(497\) 0 0
\(498\) −24.0000 −1.07547
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 12.0000 0.536656
\(501\) −16.0000 −0.714827
\(502\) −2.00000 −0.0892644
\(503\) 8.00000 0.356702 0.178351 − 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) −24.0000 −1.06693
\(507\) −18.0000 −0.799408
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 8.00000 0.354246
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −32.0000 −1.41283
\(514\) −14.0000 −0.617514
\(515\) −32.0000 −1.41009
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) −4.00000 −0.175412
\(521\) 6.00000 0.262865 0.131432 − 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −6.00000 −0.262613
\(523\) 34.0000 1.48672 0.743358 − 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) 14.0000 0.611593
\(525\) 0 0
\(526\) 20.0000 0.872041
\(527\) 0 0
\(528\) −12.0000 −0.522233
\(529\) −7.00000 −0.304348
\(530\) 8.00000 0.347498
\(531\) 0 0
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) −4.00000 −0.173097
\(535\) 24.0000 1.03761
\(536\) 2.00000 0.0863868
\(537\) −36.0000 −1.55351
\(538\) 6.00000 0.258678
\(539\) 42.0000 1.80907
\(540\) 8.00000 0.344265
\(541\) 22.0000 0.945854 0.472927 − 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 28.0000 1.20270
\(543\) 16.0000 0.686626
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −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
\(550\) 6.00000 0.255841
\(551\) −48.0000 −2.04487
\(552\) 8.00000 0.340503
\(553\) 0 0
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) −6.00000 −0.254457
\(557\) −38.0000 −1.61011 −0.805056 − 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 2.00000 0.0843649
\(563\) 18.0000 0.758610 0.379305 − 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) 6.00000 0.252199
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −32.0000 −1.34033
\(571\) 12.0000 0.502184 0.251092 − 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −12.0000 −0.501745
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 − 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −13.0000 −0.540729
\(579\) 12.0000 0.498703
\(580\) 12.0000 0.498273
\(581\) 0 0
\(582\) 4.00000 0.165805
\(583\) 24.0000 0.993978
\(584\) 10.0000 0.413803
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) 38.0000 1.56843 0.784214 − 0.620491i \(-0.213066\pi\)
0.784214 + 0.620491i \(0.213066\pi\)
\(588\) −14.0000 −0.577350
\(589\) 0 0
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 24.0000 0.984732
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −40.0000 −1.63709
\(598\) 8.00000 0.327144
\(599\) −12.0000 −0.490307 −0.245153 − 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −2.00000 −0.0816497
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) −24.0000 −0.976546
\(605\) −50.0000 −2.03279
\(606\) −24.0000 −0.974933
\(607\) 40.0000 1.62355 0.811775 − 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) 22.0000 0.888572 0.444286 − 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) −28.0000 −1.12999
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 32.0000 1.28723
\(619\) 16.0000 0.643094 0.321547 − 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) 0 0
\(621\) −16.0000 −0.642058
\(622\) 28.0000 1.12270
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) −19.0000 −0.760000
\(626\) 14.0000 0.559553
\(627\) −96.0000 −3.83387
\(628\) −22.0000 −0.877896
\(629\) 0 0
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −16.0000 −0.636446
\(633\) −32.0000 −1.27189
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) −8.00000 −0.317221
\(637\) −14.0000 −0.554700
\(638\) 36.0000 1.42525
\(639\) −8.00000 −0.316475
\(640\) −2.00000 −0.0790569
\(641\) −2.00000 −0.0789953 −0.0394976 − 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) −24.0000 −0.947204
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 24.0000 0.944999
\(646\) −16.0000 −0.629512
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 14.0000 0.548282
\(653\) 32.0000 1.25226 0.626128 − 0.779720i \(-0.284639\pi\)
0.626128 + 0.779720i \(0.284639\pi\)
\(654\) 0 0
\(655\) −28.0000 −1.09405
\(656\) −2.00000 −0.0780869
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) −10.0000 −0.389545 −0.194772 − 0.980848i \(-0.562397\pi\)
−0.194772 + 0.980848i \(0.562397\pi\)
\(660\) 24.0000 0.934199
\(661\) 24.0000 0.933492 0.466746 − 0.884391i \(-0.345426\pi\)
0.466746 + 0.884391i \(0.345426\pi\)
\(662\) −22.0000 −0.855054
\(663\) −8.00000 −0.310694
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) −8.00000 −0.309529
\(669\) 48.0000 1.85579
\(670\) −4.00000 −0.154533
\(671\) 36.0000 1.38976
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 − 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 14.0000 0.539260
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) −4.00000 −0.153732 −0.0768662 − 0.997041i \(-0.524491\pi\)
−0.0768662 + 0.997041i \(0.524491\pi\)
\(678\) 4.00000 0.153619
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) 36.0000 1.37952
\(682\) 0 0
\(683\) 14.0000 0.535695 0.267848 − 0.963461i \(-0.413688\pi\)
0.267848 + 0.963461i \(0.413688\pi\)
\(684\) 8.00000 0.305888
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) −12.0000 −0.457829
\(688\) −6.00000 −0.228748
\(689\) −8.00000 −0.304776
\(690\) −16.0000 −0.609110
\(691\) −10.0000 −0.380418 −0.190209 − 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −16.0000 −0.607352
\(695\) 12.0000 0.455186
\(696\) −12.0000 −0.454859
\(697\) 4.00000 0.151511
\(698\) 4.00000 0.151402
\(699\) −28.0000 −1.05906
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 − 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) −8.00000 −0.301941
\(703\) 0 0
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) 24.0000 0.901339 0.450669 − 0.892691i \(-0.351185\pi\)
0.450669 + 0.892691i \(0.351185\pi\)
\(710\) 16.0000 0.600469
\(711\) −16.0000 −0.600047
\(712\) −2.00000 −0.0749532
\(713\) 0 0
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) −18.0000 −0.672692
\(717\) 0 0
\(718\) −12.0000 −0.447836
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 45.0000 1.67473
\(723\) 4.00000 0.148762
\(724\) 8.00000 0.297318
\(725\) 6.00000 0.222834
\(726\) 50.0000 1.85567
\(727\) −4.00000 −0.148352 −0.0741759 − 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) −20.0000 −0.740233
\(731\) 12.0000 0.443836
\(732\) −12.0000 −0.443533
\(733\) 44.0000 1.62518 0.812589 − 0.582838i \(-0.198058\pi\)
0.812589 + 0.582838i \(0.198058\pi\)
\(734\) 0 0
\(735\) 28.0000 1.03280
\(736\) 4.00000 0.147442
\(737\) −12.0000 −0.442026
\(738\) −2.00000 −0.0736210
\(739\) −6.00000 −0.220714 −0.110357 − 0.993892i \(-0.535199\pi\)
−0.110357 + 0.993892i \(0.535199\pi\)
\(740\) 0 0
\(741\) 32.0000 1.17555
\(742\) 0 0
\(743\) −20.0000 −0.733729 −0.366864 − 0.930274i \(-0.619569\pi\)
−0.366864 + 0.930274i \(0.619569\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 32.0000 1.17160
\(747\) −12.0000 −0.439057
\(748\) 12.0000 0.438763
\(749\) 0 0
\(750\) 24.0000 0.876356
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) −4.00000 −0.145768
\(754\) −12.0000 −0.437014
\(755\) 48.0000 1.74690
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 − 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 8.00000 0.290573
\(759\) −48.0000 −1.74229
\(760\) −16.0000 −0.580381
\(761\) 6.00000 0.217500 0.108750 − 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 4.00000 0.144620
\(766\) −16.0000 −0.578103
\(767\) 0 0
\(768\) 2.00000 0.0721688
\(769\) 30.0000 1.08183 0.540914 − 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) −28.0000 −1.00840
\(772\) 6.00000 0.215945
\(773\) −16.0000 −0.575480 −0.287740 − 0.957709i \(-0.592904\pi\)
−0.287740 + 0.957709i \(0.592904\pi\)
\(774\) −6.00000 −0.215666
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) −16.0000 −0.573259
\(780\) −8.00000 −0.286446
\(781\) 48.0000 1.71758
\(782\) −8.00000 −0.286079
\(783\) 24.0000 0.857690
\(784\) −7.00000 −0.250000
\(785\) 44.0000 1.57043
\(786\) 28.0000 0.998727
\(787\) −36.0000 −1.28326 −0.641631 − 0.767014i \(-0.721742\pi\)
−0.641631 + 0.767014i \(0.721742\pi\)
\(788\) −18.0000 −0.641223
\(789\) 40.0000 1.42404
\(790\) 32.0000 1.13851
\(791\) 0 0
\(792\) −6.00000 −0.213201
\(793\) −12.0000 −0.426132
\(794\) 30.0000 1.06466
\(795\) 16.0000 0.567462
\(796\) −20.0000 −0.708881
\(797\) −42.0000 −1.48772 −0.743858 − 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −2.00000 −0.0706665
\(802\) −18.0000 −0.635602
\(803\) −60.0000 −2.11735
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) −12.0000 −0.422159
\(809\) −2.00000 −0.0703163 −0.0351581 − 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 22.0000 0.773001
\(811\) −42.0000 −1.47482 −0.737410 − 0.675446i \(-0.763951\pi\)
−0.737410 + 0.675446i \(0.763951\pi\)
\(812\) 0 0
\(813\) 56.0000 1.96401
\(814\) 0 0
\(815\) −28.0000 −0.980797
\(816\) −4.00000 −0.140028
\(817\) −48.0000 −1.67931
\(818\) 38.0000 1.32864
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 36.0000 1.25564
\(823\) −24.0000 −0.836587 −0.418294 − 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 16.0000 0.557386
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) −10.0000 −0.347734 −0.173867 − 0.984769i \(-0.555626\pi\)
−0.173867 + 0.984769i \(0.555626\pi\)
\(828\) 4.00000 0.139010
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 24.0000 0.833052
\(831\) −28.0000 −0.971309
\(832\) 2.00000 0.0693375
\(833\) 14.0000 0.485071
\(834\) −12.0000 −0.415526
\(835\) 16.0000 0.553703
\(836\) −48.0000 −1.66011
\(837\) 0 0
\(838\) 26.0000 0.898155
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −8.00000 −0.275698
\(843\) 4.00000 0.137767
\(844\) −16.0000 −0.550743
\(845\) 18.0000 0.619219
\(846\) 0 0
\(847\) 0 0
\(848\) −4.00000 −0.137361
\(849\) 12.0000 0.411839
\(850\) 2.00000 0.0685994
\(851\) 0 0
\(852\) −16.0000 −0.548151
\(853\) −30.0000 −1.02718 −0.513590 − 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 0 0
\(855\) −16.0000 −0.547188
\(856\) −12.0000 −0.410152
\(857\) 22.0000 0.751506 0.375753 − 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) −24.0000 −0.819346
\(859\) 30.0000 1.02359 0.511793 − 0.859109i \(-0.328981\pi\)
0.511793 + 0.859109i \(0.328981\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 8.00000 0.272323 0.136162 − 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) −4.00000 −0.136083
\(865\) −12.0000 −0.408012
\(866\) 34.0000 1.15537
\(867\) −26.0000 −0.883006
\(868\) 0 0
\(869\) 96.0000 3.25658
\(870\) 24.0000 0.813676
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) 20.0000 0.675737
\(877\) 8.00000 0.270141 0.135070 − 0.990836i \(-0.456874\pi\)
0.135070 + 0.990836i \(0.456874\pi\)
\(878\) 16.0000 0.539974
\(879\) 0 0
\(880\) 12.0000 0.404520
\(881\) 26.0000 0.875962 0.437981 − 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) −7.00000 −0.235702
\(883\) −18.0000 −0.605748 −0.302874 − 0.953031i \(-0.597946\pi\)
−0.302874 + 0.953031i \(0.597946\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) −32.0000 −1.07445 −0.537227 − 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 4.00000 0.134080
\(891\) 66.0000 2.21108
\(892\) 24.0000 0.803579
\(893\) 0 0
\(894\) 12.0000 0.401340
\(895\) 36.0000 1.20335
\(896\) 0 0
\(897\) 16.0000 0.534224
\(898\) −2.00000 −0.0667409
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) 8.00000 0.266519
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) −16.0000 −0.531858
\(906\) −48.0000 −1.59469
\(907\) 34.0000 1.12895 0.564476 − 0.825450i \(-0.309078\pi\)
0.564476 + 0.825450i \(0.309078\pi\)
\(908\) 18.0000 0.597351
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 − 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 16.0000 0.529813
\(913\) 72.0000 2.38285
\(914\) −22.0000 −0.727695
\(915\) 24.0000 0.793416
\(916\) −6.00000 −0.198246
\(917\) 0 0
\(918\) 8.00000 0.264039
\(919\) 32.0000 1.05558 0.527791 − 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) −8.00000 −0.263752
\(921\) −56.0000 −1.84526
\(922\) 0 0
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 16.0000 0.525509
\(928\) −6.00000 −0.196960
\(929\) −18.0000 −0.590561 −0.295280 − 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −56.0000 −1.83533
\(932\) −14.0000 −0.458585
\(933\) 56.0000 1.83336
\(934\) −6.00000 −0.196326
\(935\) −24.0000 −0.784884
\(936\) 2.00000 0.0653720
\(937\) −42.0000 −1.37208 −0.686040 − 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) 28.0000 0.913745
\(940\) 0 0
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) −44.0000 −1.43360
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 0 0
\(946\) 36.0000 1.17046
\(947\) −18.0000 −0.584921 −0.292461 − 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) −32.0000 −1.03931
\(949\) 20.0000 0.649227
\(950\) −8.00000 −0.259554
\(951\) 24.0000 0.778253
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −4.00000 −0.129505
\(955\) 24.0000 0.776622
\(956\) 0 0
\(957\) 72.0000 2.32743
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) −4.00000 −0.129099
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 2.00000 0.0644157
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 − 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 25.0000 0.803530
\(969\) −32.0000 −1.02799
\(970\) −4.00000 −0.128432
\(971\) −16.0000 −0.513464 −0.256732 − 0.966483i \(-0.582646\pi\)
−0.256732 + 0.966483i \(0.582646\pi\)
\(972\) −10.0000 −0.320750
\(973\) 0 0
\(974\) −4.00000 −0.128168
\(975\) −4.00000 −0.128103
\(976\) −6.00000 −0.192055
\(977\) −42.0000 −1.34370 −0.671850 − 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 28.0000 0.895341
\(979\) 12.0000 0.383522
\(980\) 14.0000 0.447214
\(981\) 0 0
\(982\) −30.0000 −0.957338
\(983\) −48.0000 −1.53096 −0.765481 − 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) −4.00000 −0.127515
\(985\) 36.0000 1.14706
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) −24.0000 −0.763156
\(990\) 12.0000 0.381385
\(991\) 20.0000 0.635321 0.317660 − 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 0 0
\(993\) −44.0000 −1.39630
\(994\) 0 0
\(995\) 40.0000 1.26809
\(996\) −24.0000 −0.760469
\(997\) 22.0000 0.696747 0.348373 − 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 8002.2.a.c.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8002.2.a.c.1.1 ✓ 1 1.1 even 1 trivial