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Properties

Label 6762.2.a.bg.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
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] = [6762,2,Mod(1,6762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6762, 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("6762.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.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: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6762.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{12} +2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} -2.00000 q^{20} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} -2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +8.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} -10.0000 q^{41} +8.00000 q^{43} -2.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +8.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +8.00000 q^{67} -2.00000 q^{68} -1.00000 q^{69} +1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +8.00000 q^{76} +2.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +16.0000 q^{83} +4.00000 q^{85} +8.00000 q^{86} -2.00000 q^{87} -18.0000 q^{89} -2.00000 q^{90} -1.00000 q^{92} +8.00000 q^{93} -8.00000 q^{94} -16.0000 q^{95} +1.00000 q^{96} -10.0000 q^{97} +O(q^{100})\)

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
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −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
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −2.00000 −0.365148
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 8.00000 1.29777
\(39\) 2.00000 0.320256
\(40\) −2.00000 −0.316228
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −1.00000 −0.147442
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −2.00000 −0.258199
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −2.00000 −0.242536
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 16.0000 1.75623 0.878114 − 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 8.00000 0.862662
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) −8.00000 −0.825137
\(95\) −16.0000 −1.64157
\(96\) 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
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 18.0000 1.79107 0.895533 − 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) −2.00000 −0.198030
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 8.00000 0.773389 0.386695 − 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 8.00000 0.749269
\(115\) 2.00000 0.186501
\(116\) −2.00000 −0.185695
\(117\) 2.00000 0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) −11.0000 −1.00000
\(122\) −2.00000 −0.181071
\(123\) −10.0000 −0.901670
\(124\) 8.00000 0.718421
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 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
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) −2.00000 −0.172133
\(136\) −2.00000 −0.171499
\(137\) 10.0000 0.854358 0.427179 − 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 2.00000 0.163846 0.0819232 − 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 8.00000 0.648886
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) 2.00000 0.160128
\(157\) 22.0000 1.75579 0.877896 − 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 8.00000 0.636446
\(159\) 2.00000 0.158610
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 4.00000 0.306786
\(171\) 8.00000 0.611775
\(172\) 8.00000 0.609994
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) −18.0000 −1.34916
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −2.00000 −0.149071
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −1.00000 −0.0737210
\(185\) −4.00000 −0.294086
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −16.0000 −1.16076
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −10.0000 −0.717958
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 − 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) 18.0000 1.26648
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 20.0000 1.39686
\(206\) −8.00000 −0.557386
\(207\) −1.00000 −0.0695048
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) −16.0000 −1.09119
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 2.00000 0.134231
\(223\) −24.0000 −1.60716 −0.803579 − 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −6.00000 −0.399114
\(227\) 8.00000 0.530979 0.265489 − 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 8.00000 0.529813
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −22.0000 −1.44127 −0.720634 − 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 2.00000 0.130744
\(235\) 16.0000 1.04372
\(236\) 4.00000 0.260378
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −2.00000 −0.129099
\(241\) 22.0000 1.41714 0.708572 − 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) 16.0000 1.01806
\(248\) 8.00000 0.508001
\(249\) 16.0000 1.01396
\(250\) 12.0000 0.758947
\(251\) −8.00000 −0.504956 −0.252478 − 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 8.00000 0.498058
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) −2.00000 −0.123797
\(262\) 12.0000 0.741362
\(263\) 8.00000 0.493301 0.246651 − 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 8.00000 0.488678
\(269\) 2.00000 0.121942 0.0609711 − 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) −2.00000 −0.121716
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 20.0000 1.19952
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 − 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) −8.00000 −0.476393
\(283\) −24.0000 −1.42665 −0.713326 − 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 0 0
\(285\) −16.0000 −0.947758
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 4.00000 0.234888
\(291\) −10.0000 −0.586210
\(292\) 6.00000 0.351123
\(293\) 30.0000 1.75262 0.876309 − 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) −2.00000 −0.115663
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 18.0000 1.03407
\(304\) 8.00000 0.458831
\(305\) 4.00000 0.229039
\(306\) −2.00000 −0.114332
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) −16.0000 −0.908739
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 2.00000 0.113228
\(313\) −2.00000 −0.113047 −0.0565233 − 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 14.0000 0.786318 0.393159 − 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 2.00000 0.112154
\(319\) 0 0
\(320\) −2.00000 −0.111803
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −4.00000 −0.221540
\(327\) 2.00000 0.110600
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 16.0000 0.878114
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 − 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) −9.00000 −0.489535
\(339\) −6.00000 −0.325875
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) 8.00000 0.432590
\(343\) 0 0
\(344\) 8.00000 0.431331
\(345\) 2.00000 0.107676
\(346\) −14.0000 −0.752645
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −2.00000 −0.107211
\(349\) −6.00000 −0.321173 −0.160586 − 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 − 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −2.00000 −0.105409
\(361\) 45.0000 2.36842
\(362\) −10.0000 −0.525588
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) −2.00000 −0.104542
\(367\) 24.0000 1.25279 0.626395 − 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −10.0000 −0.520579
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) 18.0000 0.932005 0.466002 − 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) −8.00000 −0.412568
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −16.0000 −0.820783
\(381\) 16.0000 0.819705
\(382\) 24.0000 1.22795
\(383\) −16.0000 −0.817562 −0.408781 − 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 8.00000 0.406663
\(388\) −10.0000 −0.507673
\(389\) −22.0000 −1.11544 −0.557722 − 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) −4.00000 −0.202548
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 22.0000 1.10834
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) −30.0000 −1.50566 −0.752828 − 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −14.0000 −0.699127 −0.349563 − 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 8.00000 0.399004
\(403\) 16.0000 0.797017
\(404\) 18.0000 0.895533
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) −2.00000 −0.0990148
\(409\) 22.0000 1.08783 0.543915 − 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 20.0000 0.987730
\(411\) 10.0000 0.493264
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −32.0000 −1.57082
\(416\) 2.00000 0.0980581
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 − 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) −12.0000 −0.584151
\(423\) −8.00000 −0.388973
\(424\) 2.00000 0.0971286
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000 0.0481125
\(433\) 22.0000 1.05725 0.528626 − 0.848855i \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) 2.00000 0.0957826
\(437\) −8.00000 −0.382692
\(438\) 6.00000 0.286691
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 2.00000 0.0949158
\(445\) 36.0000 1.70656
\(446\) −24.0000 −1.13643
\(447\) 2.00000 0.0945968
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 16.0000 0.751746
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) 18.0000 0.842004 0.421002 − 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 6.00000 0.280362
\(459\) −2.00000 −0.0933520
\(460\) 2.00000 0.0932505
\(461\) −30.0000 −1.39724 −0.698620 − 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −16.0000 −0.741982
\(466\) −22.0000 −1.01913
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 16.0000 0.738025
\(471\) 22.0000 1.01371
\(472\) 4.00000 0.184115
\(473\) 0 0
\(474\) 8.00000 0.367452
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 4.00000 0.182384
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 20.0000 0.908153
\(486\) 1.00000 0.0453609
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 4.00000 0.180517 0.0902587 − 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) −10.0000 −0.450835
\(493\) 4.00000 0.180151
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 16.0000 0.716977
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −8.00000 −0.357057
\(503\) 32.0000 1.42681 0.713405 − 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) 0 0
\(505\) −36.0000 −1.60198
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 16.0000 0.709885
\(509\) −30.0000 −1.32973 −0.664863 − 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) −18.0000 −0.793946
\(515\) 16.0000 0.705044
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) −4.00000 −0.175412
\(521\) −26.0000 −1.13908 −0.569540 − 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −4.00000 −0.173749
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) −18.0000 −0.778936
\(535\) −16.0000 −0.691740
\(536\) 8.00000 0.345547
\(537\) 12.0000 0.517838
\(538\) 2.00000 0.0862261
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 16.0000 0.687259
\(543\) −10.0000 −0.429141
\(544\) −2.00000 −0.0857493
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 − 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) 10.0000 0.427179
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −16.0000 −0.681623
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) −4.00000 −0.169791
\(556\) 20.0000 0.848189
\(557\) 2.00000 0.0847427 0.0423714 − 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 8.00000 0.338667
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −8.00000 −0.336861
\(565\) 12.0000 0.504844
\(566\) −24.0000 −1.00880
\(567\) 0 0
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 − 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) −16.0000 −0.670166
\(571\) 16.0000 0.669579 0.334790 − 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 − 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −13.0000 −0.540729
\(579\) −14.0000 −0.581820
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) −10.0000 −0.414513
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) −4.00000 −0.165380
\(586\) 30.0000 1.23929
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 64.0000 2.63707
\(590\) −8.00000 −0.329355
\(591\) 22.0000 0.904959
\(592\) 2.00000 0.0821995
\(593\) −34.0000 −1.39621 −0.698106 − 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) −16.0000 −0.654836
\(598\) −2.00000 −0.0817861
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −42.0000 −1.71322 −0.856608 − 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 16.0000 0.651031
\(605\) 22.0000 0.894427
\(606\) 18.0000 0.731200
\(607\) 24.0000 0.974130 0.487065 − 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) −16.0000 −0.647291
\(612\) −2.00000 −0.0808452
\(613\) 10.0000 0.403896 0.201948 − 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 4.00000 0.161427
\(615\) 20.0000 0.806478
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −8.00000 −0.321807
\(619\) 40.0000 1.60774 0.803868 − 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) −16.0000 −0.642575
\(621\) −1.00000 −0.0401286
\(622\) 8.00000 0.320771
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) −2.00000 −0.0799361
\(627\) 0 0
\(628\) 22.0000 0.877896
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 − 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 8.00000 0.318223
\(633\) −12.0000 −0.476957
\(634\) 14.0000 0.556011
\(635\) −32.0000 −1.26988
\(636\) 2.00000 0.0793052
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 8.00000 0.315735
\(643\) −16.0000 −0.630978 −0.315489 − 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) −16.0000 −0.629999
\(646\) −16.0000 −0.629512
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −10.0000 −0.391330 −0.195665 − 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 2.00000 0.0782062
\(655\) −24.0000 −0.937758
\(656\) −10.0000 −0.390434
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 32.0000 1.24654 0.623272 − 0.782006i \(-0.285803\pi\)
0.623272 + 0.782006i \(0.285803\pi\)
\(660\) 0 0
\(661\) −42.0000 −1.63361 −0.816805 − 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) −4.00000 −0.155464
\(663\) −4.00000 −0.155347
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 2.00000 0.0774403
\(668\) 0 0
\(669\) −24.0000 −0.927894
\(670\) −16.0000 −0.618134
\(671\) 0 0
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 − 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) −30.0000 −1.15556
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −2.00000 −0.0768662 −0.0384331 − 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 8.00000 0.305888
\(685\) −20.0000 −0.764161
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) 8.00000 0.304997
\(689\) 4.00000 0.152388
\(690\) 2.00000 0.0761387
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −40.0000 −1.51729
\(696\) −2.00000 −0.0758098
\(697\) 20.0000 0.757554
\(698\) −6.00000 −0.227103
\(699\) −22.0000 −0.832116
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 − 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 2.00000 0.0754851
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 16.0000 0.602595
\(706\) 14.0000 0.526897
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) −14.0000 −0.525781 −0.262891 − 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −18.0000 −0.674579
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) −48.0000 −1.79010 −0.895049 − 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 45.0000 1.67473
\(723\) 22.0000 0.818189
\(724\) −10.0000 −0.371647
\(725\) 2.00000 0.0742781
\(726\) −11.0000 −0.408248
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −12.0000 −0.444140
\(731\) −16.0000 −0.591781
\(732\) −2.00000 −0.0739221
\(733\) 6.00000 0.221615 0.110808 − 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 24.0000 0.885856
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) −10.0000 −0.368105
\(739\) −20.0000 −0.735712 −0.367856 − 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) −4.00000 −0.147043
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 8.00000 0.293294
\(745\) −4.00000 −0.146549
\(746\) 18.0000 0.659027
\(747\) 16.0000 0.585409
\(748\) 0 0
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 16.0000 0.583848 0.291924 − 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −8.00000 −0.291730
\(753\) −8.00000 −0.291536
\(754\) −4.00000 −0.145671
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 − 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) −16.0000 −0.580381
\(761\) −42.0000 −1.52250 −0.761249 − 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) 24.0000 0.868290
\(765\) 4.00000 0.144620
\(766\) −16.0000 −0.578103
\(767\) 8.00000 0.288863
\(768\) 1.00000 0.0360844
\(769\) −26.0000 −0.937584 −0.468792 − 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −14.0000 −0.503871
\(773\) 38.0000 1.36677 0.683383 − 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 8.00000 0.287554
\(775\) −8.00000 −0.287368
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) −22.0000 −0.788738
\(779\) −80.0000 −2.86630
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) 2.00000 0.0715199
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) −44.0000 −1.57043
\(786\) 12.0000 0.428026
\(787\) 32.0000 1.14068 0.570338 − 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 22.0000 0.783718
\(789\) 8.00000 0.284808
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) −30.0000 −1.06466
\(795\) −4.00000 −0.141865
\(796\) −16.0000 −0.567105
\(797\) −2.00000 −0.0708436 −0.0354218 − 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) −1.00000 −0.0353553
\(801\) −18.0000 −0.635999
\(802\) −14.0000 −0.494357
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 2.00000 0.0704033
\(808\) 18.0000 0.633238
\(809\) −38.0000 −1.33601 −0.668004 − 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 4.00000 0.140459 0.0702295 − 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) −2.00000 −0.0700140
\(817\) 64.0000 2.23908
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 20.0000 0.698430
\(821\) −50.0000 −1.74501 −0.872506 − 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 10.0000 0.348790
\(823\) −32.0000 −1.11545 −0.557725 − 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) 24.0000 0.834562 0.417281 − 0.908778i \(-0.362983\pi\)
0.417281 + 0.908778i \(0.362983\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −22.0000 −0.764092 −0.382046 − 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) −32.0000 −1.11074
\(831\) −2.00000 −0.0693792
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) 56.0000 1.93333 0.966667 − 0.256036i \(-0.0824164\pi\)
0.966667 + 0.256036i \(0.0824164\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −30.0000 −1.03387
\(843\) 2.00000 0.0688837
\(844\) −12.0000 −0.413057
\(845\) 18.0000 0.619219
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) −24.0000 −0.823678
\(850\) 2.00000 0.0685994
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) −16.0000 −0.547188
\(856\) 8.00000 0.273434
\(857\) 54.0000 1.84460 0.922302 − 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 − 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) −16.0000 −0.545595
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 32.0000 1.08929 0.544646 − 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 1.00000 0.0340207
\(865\) 28.0000 0.952029
\(866\) 22.0000 0.747590
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 4.00000 0.135613
\(871\) 16.0000 0.542139
\(872\) 2.00000 0.0677285
\(873\) −10.0000 −0.338449
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 8.00000 0.269987
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) 22.0000 0.741199 0.370599 − 0.928793i \(-0.379152\pi\)
0.370599 + 0.928793i \(0.379152\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −4.00000 −0.134535
\(885\) −8.00000 −0.268917
\(886\) 4.00000 0.134383
\(887\) 48.0000 1.61168 0.805841 − 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) 36.0000 1.20672
\(891\) 0 0
\(892\) −24.0000 −0.803579
\(893\) −64.0000 −2.14168
\(894\) 2.00000 0.0668900
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) −30.0000 −1.00111
\(899\) −16.0000 −0.533630
\(900\) −1.00000 −0.0333333
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 20.0000 0.664822
\(906\) 16.0000 0.531564
\(907\) 24.0000 0.796907 0.398453 − 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) 8.00000 0.265489
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 − 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) 18.0000 0.595387
\(915\) 4.00000 0.132236
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 2.00000 0.0659380
\(921\) 4.00000 0.131804
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) −2.00000 −0.0656532
\(929\) −34.0000 −1.11550 −0.557752 − 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) −16.0000 −0.524661
\(931\) 0 0
\(932\) −22.0000 −0.720634
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 22.0000 0.718709 0.359354 − 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) −2.00000 −0.0652675
\(940\) 16.0000 0.521862
\(941\) −58.0000 −1.89075 −0.945373 − 0.325991i \(-0.894302\pi\)
−0.945373 + 0.325991i \(0.894302\pi\)
\(942\) 22.0000 0.716799
\(943\) 10.0000 0.325645
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) 44.0000 1.42981 0.714904 − 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) 8.00000 0.259828
\(949\) 12.0000 0.389536
\(950\) −8.00000 −0.259554
\(951\) 14.0000 0.453981
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 − 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 2.00000 0.0647524
\(955\) −48.0000 −1.55324
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) 33.0000 1.06452
\(962\) 4.00000 0.128965
\(963\) 8.00000 0.257796
\(964\) 22.0000 0.708572
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 − 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −11.0000 −0.353553
\(969\) −16.0000 −0.513994
\(970\) 20.0000 0.642161
\(971\) 48.0000 1.54039 0.770197 − 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) −2.00000 −0.0640513
\(976\) −2.00000 −0.0640184
\(977\) −14.0000 −0.447900 −0.223950 − 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) −4.00000 −0.127906
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 4.00000 0.127645
\(983\) −48.0000 −1.53096 −0.765481 − 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) −10.0000 −0.318788
\(985\) −44.0000 −1.40196
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 − 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) 8.00000 0.254000
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 32.0000 1.01447
\(996\) 16.0000 0.506979
\(997\) −14.0000 −0.443384 −0.221692 − 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) −20.0000 −0.633089
\(999\) 2.00000 0.0632772
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 6762.2.a.bg.1.1 1
7.6 odd 2 138.2.a.c.1.1 ✓ 1
21.20 even 2 414.2.a.a.1.1 1
28.27 even 2 1104.2.a.g.1.1 1
35.13 even 4 3450.2.d.f.2899.1 2
35.27 even 4 3450.2.d.f.2899.2 2
35.34 odd 2 3450.2.a.k.1.1 1
56.13 odd 2 4416.2.a.s.1.1 1
56.27 even 2 4416.2.a.c.1.1 1
84.83 odd 2 3312.2.a.d.1.1 1
161.160 even 2 3174.2.a.e.1.1 1
483.482 odd 2 9522.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.c.1.1 ✓ 1 7.6 odd 2
414.2.a.a.1.1 1 21.20 even 2
1104.2.a.g.1.1 1 28.27 even 2
3174.2.a.e.1.1 1 161.160 even 2
3312.2.a.d.1.1 1 84.83 odd 2
3450.2.a.k.1.1 1 35.34 odd 2
3450.2.d.f.2899.1 2 35.13 even 4
3450.2.d.f.2899.2 2 35.27 even 4
4416.2.a.c.1.1 1 56.27 even 2
4416.2.a.s.1.1 1 56.13 odd 2
6762.2.a.bg.1.1 1 1.1 even 1 trivial
9522.2.a.d.1.1 1 483.482 odd 2