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Properties

Label 7098.2.a.ba.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
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] = [7098,2,Mod(1,7098)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7098, 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("7098.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.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: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7098.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -5.00000 q^{19} -1.00000 q^{21} +3.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} -9.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} -5.00000 q^{38} -3.00000 q^{41} -1.00000 q^{42} +8.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -5.00000 q^{50} -3.00000 q^{51} -3.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -5.00000 q^{57} -9.00000 q^{58} +6.00000 q^{59} -7.00000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -2.00000 q^{67} -3.00000 q^{68} -6.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} +16.0000 q^{73} -8.00000 q^{74} -5.00000 q^{75} -5.00000 q^{76} -3.00000 q^{77} -13.0000 q^{79} +1.00000 q^{81} -3.00000 q^{82} -18.0000 q^{83} -1.00000 q^{84} +8.00000 q^{86} -9.00000 q^{87} +3.00000 q^{88} +15.0000 q^{89} -6.00000 q^{92} -8.00000 q^{93} -3.00000 q^{94} +1.00000 q^{96} +16.0000 q^{97} +1.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 1.00000 0.235702
\(19\) −5.00000 −1.14708 −0.573539 − 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 3.00000 0.639602
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.00000 0.522233
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −5.00000 −0.811107
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) −1.00000 −0.154303
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −5.00000 −0.707107
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 − 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −5.00000 −0.662266
\(58\) −9.00000 −1.18176
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −8.00000 −1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −3.00000 −0.363803
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) −8.00000 −0.929981
\(75\) −5.00000 −0.577350
\(76\) −5.00000 −0.573539
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 − 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) −18.0000 −1.97576 −0.987878 − 0.155230i \(-0.950388\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −9.00000 −0.964901
\(88\) 3.00000 0.319801
\(89\) 15.0000 1.59000 0.794998 − 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −8.00000 −0.829561
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 16.0000 1.62455 0.812277 − 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 1.00000 0.101015
\(99\) 3.00000 0.301511
\(100\) −5.00000 −0.500000
\(101\) 18.0000 1.79107 0.895533 − 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) −3.00000 −0.297044
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) 9.00000 0.870063 0.435031 − 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) −1.00000 −0.0944911
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) −5.00000 −0.468293
\(115\) 0 0
\(116\) −9.00000 −0.835629
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −7.00000 −0.633750
\(123\) −3.00000 −0.270501
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 3.00000 0.261116
\(133\) 5.00000 0.433555
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −6.00000 −0.510754
\(139\) 11.0000 0.933008 0.466504 − 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 16.0000 1.32417
\(147\) 1.00000 0.0824786
\(148\) −8.00000 −0.657596
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −5.00000 −0.408248
\(151\) 19.0000 1.54620 0.773099 − 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) −5.00000 −0.405554
\(153\) −3.00000 −0.242536
\(154\) −3.00000 −0.241747
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −13.0000 −1.03422
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) −18.0000 −1.39707
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) 0 0
\(171\) −5.00000 −0.382360
\(172\) 8.00000 0.609994
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −9.00000 −0.682288
\(175\) 5.00000 0.377964
\(176\) 3.00000 0.226134
\(177\) 6.00000 0.450988
\(178\) 15.0000 1.12430
\(179\) −24.0000 −1.79384 −0.896922 − 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 − 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) −7.00000 −0.517455
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) −9.00000 −0.658145
\(188\) −3.00000 −0.218797
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 − 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 1.00000 0.0721688
\(193\) −5.00000 −0.359908 −0.179954 − 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 3.00000 0.213741 0.106871 − 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 3.00000 0.213201
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −5.00000 −0.353553
\(201\) −2.00000 −0.141069
\(202\) 18.0000 1.26648
\(203\) 9.00000 0.631676
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 − 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) −3.00000 −0.206041
\(213\) 6.00000 0.411113
\(214\) 9.00000 0.615227
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 8.00000 0.543075
\(218\) −2.00000 −0.135457
\(219\) 16.0000 1.08118
\(220\) 0 0
\(221\) 0 0
\(222\) −8.00000 −0.536925
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −5.00000 −0.333333
\(226\) −12.0000 −0.798228
\(227\) 6.00000 0.398234 0.199117 − 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) −5.00000 −0.331133
\(229\) −5.00000 −0.330409 −0.165205 − 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) −9.00000 −0.590879
\(233\) −24.0000 −1.57229 −0.786146 − 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −13.0000 −0.844441
\(238\) 3.00000 0.194461
\(239\) 6.00000 0.388108 0.194054 − 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 − 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) −7.00000 −0.448129
\(245\) 0 0
\(246\) −3.00000 −0.191273
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 − 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −18.0000 −1.13165
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −15.0000 −0.935674 −0.467837 − 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) 8.00000 0.498058
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) 12.0000 0.741362
\(263\) 6.00000 0.369976 0.184988 − 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) 5.00000 0.306570
\(267\) 15.0000 0.917985
\(268\) −2.00000 −0.122169
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 − 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −15.0000 −0.904534
\(276\) −6.00000 −0.361158
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 11.0000 0.659736
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) −3.00000 −0.178647
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 3.00000 0.177084
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) 16.0000 0.936329
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 3.00000 0.174078
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) −5.00000 −0.288675
\(301\) −8.00000 −0.461112
\(302\) 19.0000 1.09333
\(303\) 18.0000 1.03407
\(304\) −5.00000 −0.286770
\(305\) 0 0
\(306\) −3.00000 −0.171499
\(307\) 13.0000 0.741949 0.370975 − 0.928643i \(-0.379024\pi\)
0.370975 + 0.928643i \(0.379024\pi\)
\(308\) −3.00000 −0.170941
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −21.0000 −1.19080 −0.595400 − 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 − 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −13.0000 −0.731307
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −3.00000 −0.168232
\(319\) −27.0000 −1.51171
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 6.00000 0.334367
\(323\) 15.0000 0.834622
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) −2.00000 −0.110600
\(328\) −3.00000 −0.165647
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −18.0000 −0.987878
\(333\) −8.00000 −0.438397
\(334\) 0 0
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 23.0000 1.25289 0.626445 − 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) −24.0000 −1.29967
\(342\) −5.00000 −0.270369
\(343\) −1.00000 −0.0539949
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 3.00000 0.161048 0.0805242 − 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) −9.00000 −0.482451
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 5.00000 0.267261
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) 15.0000 0.794998
\(357\) 3.00000 0.158777
\(358\) −24.0000 −1.26844
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 5.00000 0.262794
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) −7.00000 −0.365896
\(367\) −10.0000 −0.521996 −0.260998 − 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) −6.00000 −0.312772
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) −8.00000 −0.414781
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −9.00000 −0.465379
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) 10.0000 0.513665 0.256833 − 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 6.00000 0.306987
\(383\) 21.0000 1.07305 0.536525 − 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −5.00000 −0.254493
\(387\) 8.00000 0.406663
\(388\) 16.0000 0.812277
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 1.00000 0.0505076
\(393\) 12.0000 0.605320
\(394\) 3.00000 0.151138
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) 25.0000 1.25471 0.627357 − 0.778732i \(-0.284137\pi\)
0.627357 + 0.778732i \(0.284137\pi\)
\(398\) −4.00000 −0.200502
\(399\) 5.00000 0.250313
\(400\) −5.00000 −0.250000
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) −24.0000 −1.18964
\(408\) −3.00000 −0.148522
\(409\) 22.0000 1.08783 0.543915 − 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) −4.00000 −0.197066
\(413\) −6.00000 −0.295241
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 0 0
\(417\) 11.0000 0.538672
\(418\) −15.0000 −0.733674
\(419\) −6.00000 −0.293119 −0.146560 − 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −22.0000 −1.07094
\(423\) −3.00000 −0.145865
\(424\) −3.00000 −0.145693
\(425\) 15.0000 0.727607
\(426\) 6.00000 0.290701
\(427\) 7.00000 0.338754
\(428\) 9.00000 0.435031
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 − 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 30.0000 1.43509
\(438\) 16.0000 0.764510
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 33.0000 1.56788 0.783939 − 0.620838i \(-0.213208\pi\)
0.783939 + 0.620838i \(0.213208\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 6.00000 0.283790
\(448\) −1.00000 −0.0472456
\(449\) −24.0000 −1.13263 −0.566315 − 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) −5.00000 −0.235702
\(451\) −9.00000 −0.423793
\(452\) −12.0000 −0.564433
\(453\) 19.0000 0.892698
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) −5.00000 −0.234146
\(457\) −38.0000 −1.77757 −0.888783 − 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) −5.00000 −0.233635
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) −3.00000 −0.139573
\(463\) 31.0000 1.44069 0.720346 − 0.693615i \(-0.243983\pi\)
0.720346 + 0.693615i \(0.243983\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) −24.0000 −1.11059 −0.555294 − 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 6.00000 0.276172
\(473\) 24.0000 1.10352
\(474\) −13.0000 −0.597110
\(475\) 25.0000 1.14708
\(476\) 3.00000 0.137505
\(477\) −3.00000 −0.137361
\(478\) 6.00000 0.274434
\(479\) −9.00000 −0.411220 −0.205610 − 0.978634i \(-0.565918\pi\)
−0.205610 + 0.978634i \(0.565918\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −8.00000 −0.364390
\(483\) 6.00000 0.273009
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 13.0000 0.589086 0.294543 − 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) −7.00000 −0.316875
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −3.00000 −0.135250
\(493\) 27.0000 1.21602
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −6.00000 −0.269137
\(498\) −18.0000 −0.806599
\(499\) −32.0000 −1.43252 −0.716258 − 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.0000 0.803379
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −18.0000 −0.800198
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) −30.0000 −1.32973 −0.664863 − 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 1.00000 0.0441942
\(513\) −5.00000 −0.220755
\(514\) −15.0000 −0.661622
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) −9.00000 −0.395820
\(518\) 8.00000 0.351500
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 21.0000 0.920027 0.460013 − 0.887912i \(-0.347845\pi\)
0.460013 + 0.887912i \(0.347845\pi\)
\(522\) −9.00000 −0.393919
\(523\) −19.0000 −0.830812 −0.415406 − 0.909636i \(-0.636360\pi\)
−0.415406 + 0.909636i \(0.636360\pi\)
\(524\) 12.0000 0.524222
\(525\) 5.00000 0.218218
\(526\) 6.00000 0.261612
\(527\) 24.0000 1.04546
\(528\) 3.00000 0.130558
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 5.00000 0.216777
\(533\) 0 0
\(534\) 15.0000 0.649113
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) −24.0000 −1.03568
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 34.0000 1.46177 0.730887 − 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) −20.0000 −0.859074
\(543\) 5.00000 0.214571
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) −34.0000 −1.45374 −0.726868 − 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) 6.00000 0.256307
\(549\) −7.00000 −0.298753
\(550\) −15.0000 −0.639602
\(551\) 45.0000 1.91706
\(552\) −6.00000 −0.255377
\(553\) 13.0000 0.552816
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 11.0000 0.466504
\(557\) −3.00000 −0.127114 −0.0635570 − 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) −8.00000 −0.338667
\(559\) 0 0
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 18.0000 0.759284
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) −3.00000 −0.126323
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) −1.00000 −0.0419961
\(568\) 6.00000 0.251754
\(569\) −18.0000 −0.754599 −0.377300 − 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 26.0000 1.08807 0.544033 − 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) 0 0
\(573\) 6.00000 0.250654
\(574\) 3.00000 0.125218
\(575\) 30.0000 1.25109
\(576\) 1.00000 0.0416667
\(577\) 10.0000 0.416305 0.208153 − 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −8.00000 −0.332756
\(579\) −5.00000 −0.207793
\(580\) 0 0
\(581\) 18.0000 0.746766
\(582\) 16.0000 0.663221
\(583\) −9.00000 −0.372742
\(584\) 16.0000 0.662085
\(585\) 0 0
\(586\) 0 0
\(587\) −24.0000 −0.990586 −0.495293 − 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 1.00000 0.0412393
\(589\) 40.0000 1.64817
\(590\) 0 0
\(591\) 3.00000 0.123404
\(592\) −8.00000 −0.328798
\(593\) −15.0000 −0.615976 −0.307988 − 0.951390i \(-0.599656\pi\)
−0.307988 + 0.951390i \(0.599656\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 − 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) −5.00000 −0.204124
\(601\) 8.00000 0.326327 0.163163 − 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) −8.00000 −0.326056
\(603\) −2.00000 −0.0814463
\(604\) 19.0000 0.773099
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −5.00000 −0.202777
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) 0 0
\(612\) −3.00000 −0.121268
\(613\) 40.0000 1.61558 0.807792 − 0.589467i \(-0.200662\pi\)
0.807792 + 0.589467i \(0.200662\pi\)
\(614\) 13.0000 0.524637
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) 42.0000 1.69086 0.845428 − 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) −4.00000 −0.160904
\(619\) 1.00000 0.0401934 0.0200967 − 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) −21.0000 −0.842023
\(623\) −15.0000 −0.600962
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 26.0000 1.03917
\(627\) −15.0000 −0.599042
\(628\) 2.00000 0.0798087
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 13.0000 0.517522 0.258761 − 0.965941i \(-0.416686\pi\)
0.258761 + 0.965941i \(0.416686\pi\)
\(632\) −13.0000 −0.517112
\(633\) −22.0000 −0.874421
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 0 0
\(638\) −27.0000 −1.06894
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 9.00000 0.355202
\(643\) 31.0000 1.22252 0.611260 − 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 15.0000 0.590167
\(647\) 33.0000 1.29736 0.648682 − 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(648\) 1.00000 0.0392837
\(649\) 18.0000 0.706562
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 16.0000 0.626608
\(653\) 21.0000 0.821794 0.410897 − 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) 16.0000 0.624219
\(658\) 3.00000 0.116952
\(659\) −33.0000 −1.28550 −0.642749 − 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 − 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) −18.0000 −0.698535
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 54.0000 2.09089
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) −21.0000 −0.810696
\(672\) −1.00000 −0.0385758
\(673\) −1.00000 −0.0385472 −0.0192736 − 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 23.0000 0.885927
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −12.0000 −0.460857
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) −24.0000 −0.919007
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −5.00000 −0.191180
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) −5.00000 −0.190762
\(688\) 8.00000 0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 − 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −6.00000 −0.228086
\(693\) −3.00000 −0.113961
\(694\) 3.00000 0.113878
\(695\) 0 0
\(696\) −9.00000 −0.341144
\(697\) 9.00000 0.340899
\(698\) −2.00000 −0.0757011
\(699\) −24.0000 −0.907763
\(700\) 5.00000 0.188982
\(701\) −21.0000 −0.793159 −0.396580 − 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −18.0000 −0.676960
\(708\) 6.00000 0.225494
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) −13.0000 −0.487538
\(712\) 15.0000 0.562149
\(713\) 48.0000 1.79761
\(714\) 3.00000 0.112272
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 6.00000 0.224074
\(718\) −24.0000 −0.895672
\(719\) 9.00000 0.335643 0.167822 − 0.985817i \(-0.446327\pi\)
0.167822 + 0.985817i \(0.446327\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 6.00000 0.223297
\(723\) −8.00000 −0.297523
\(724\) 5.00000 0.185824
\(725\) 45.0000 1.67126
\(726\) −2.00000 −0.0742270
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) −7.00000 −0.258727
\(733\) 37.0000 1.36663 0.683313 − 0.730125i \(-0.260538\pi\)
0.683313 + 0.730125i \(0.260538\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −6.00000 −0.221013
\(738\) −3.00000 −0.110432
\(739\) −2.00000 −0.0735712 −0.0367856 − 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 3.00000 0.110133
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) −18.0000 −0.658586
\(748\) −9.00000 −0.329073
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) −1.00000 −0.0364905 −0.0182453 − 0.999834i \(-0.505808\pi\)
−0.0182453 + 0.999834i \(0.505808\pi\)
\(752\) −3.00000 −0.109399
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −34.0000 −1.23575 −0.617876 − 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 10.0000 0.363216
\(759\) −18.0000 −0.653359
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 − 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) −16.0000 −0.579619
\(763\) 2.00000 0.0724049
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) 21.0000 0.758761
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −8.00000 −0.288487 −0.144244 − 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 0 0
\(771\) −15.0000 −0.540212
\(772\) −5.00000 −0.179954
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 8.00000 0.287554
\(775\) 40.0000 1.43684
\(776\) 16.0000 0.574367
\(777\) 8.00000 0.286998
\(778\) −18.0000 −0.645331
\(779\) 15.0000 0.537431
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 18.0000 0.643679
\(783\) −9.00000 −0.321634
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) −35.0000 −1.24762 −0.623808 − 0.781578i \(-0.714415\pi\)
−0.623808 + 0.781578i \(0.714415\pi\)
\(788\) 3.00000 0.106871
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 3.00000 0.106600
\(793\) 0 0
\(794\) 25.0000 0.887217
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −54.0000 −1.91278 −0.956389 − 0.292096i \(-0.905647\pi\)
−0.956389 + 0.292096i \(0.905647\pi\)
\(798\) 5.00000 0.176998
\(799\) 9.00000 0.318397
\(800\) −5.00000 −0.176777
\(801\) 15.0000 0.529999
\(802\) −6.00000 −0.211867
\(803\) 48.0000 1.69388
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 18.0000 0.633238
\(809\) −54.0000 −1.89854 −0.949269 − 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 − 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 9.00000 0.315838
\(813\) −20.0000 −0.701431
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) −3.00000 −0.105021
\(817\) −40.0000 −1.39942
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 0 0
\(821\) 15.0000 0.523504 0.261752 − 0.965135i \(-0.415700\pi\)
0.261752 + 0.965135i \(0.415700\pi\)
\(822\) 6.00000 0.209274
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −4.00000 −0.139347
\(825\) −15.0000 −0.522233
\(826\) −6.00000 −0.208767
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −6.00000 −0.208514
\(829\) 11.0000 0.382046 0.191023 − 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) −3.00000 −0.103944
\(834\) 11.0000 0.380899
\(835\) 0 0
\(836\) −15.0000 −0.518786
\(837\) −8.00000 −0.276520
\(838\) −6.00000 −0.207267
\(839\) 12.0000 0.414286 0.207143 − 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −26.0000 −0.896019
\(843\) 18.0000 0.619953
\(844\) −22.0000 −0.757271
\(845\) 0 0
\(846\) −3.00000 −0.103142
\(847\) 2.00000 0.0687208
\(848\) −3.00000 −0.103020
\(849\) −4.00000 −0.137280
\(850\) 15.0000 0.514496
\(851\) 48.0000 1.64542
\(852\) 6.00000 0.205557
\(853\) 1.00000 0.0342393 0.0171197 − 0.999853i \(-0.494550\pi\)
0.0171197 + 0.999853i \(0.494550\pi\)
\(854\) 7.00000 0.239535
\(855\) 0 0
\(856\) 9.00000 0.307614
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 5.00000 0.170598 0.0852989 − 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) 0 0
\(861\) 3.00000 0.102240
\(862\) 18.0000 0.613082
\(863\) −30.0000 −1.02121 −0.510606 − 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) −8.00000 −0.271694
\(868\) 8.00000 0.271538
\(869\) −39.0000 −1.32298
\(870\) 0 0
\(871\) 0 0
\(872\) −2.00000 −0.0677285
\(873\) 16.0000 0.541518
\(874\) 30.0000 1.01477
\(875\) 0 0
\(876\) 16.0000 0.540590
\(877\) −50.0000 −1.68838 −0.844190 − 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 32.0000 1.07995
\(879\) 0 0
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 − 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 1.00000 0.0336718
\(883\) −16.0000 −0.538443 −0.269221 − 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 33.0000 1.10866
\(887\) 57.0000 1.91387 0.956936 − 0.290298i \(-0.0937544\pi\)
0.956936 + 0.290298i \(0.0937544\pi\)
\(888\) −8.00000 −0.268462
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) −8.00000 −0.267860
\(893\) 15.0000 0.501956
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −24.0000 −0.800890
\(899\) 72.0000 2.40133
\(900\) −5.00000 −0.166667
\(901\) 9.00000 0.299833
\(902\) −9.00000 −0.299667
\(903\) −8.00000 −0.266223
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) 19.0000 0.631233
\(907\) −22.0000 −0.730498 −0.365249 − 0.930910i \(-0.619016\pi\)
−0.365249 + 0.930910i \(0.619016\pi\)
\(908\) 6.00000 0.199117
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −42.0000 −1.39152 −0.695761 − 0.718273i \(-0.744933\pi\)
−0.695761 + 0.718273i \(0.744933\pi\)
\(912\) −5.00000 −0.165567
\(913\) −54.0000 −1.78714
\(914\) −38.0000 −1.25693
\(915\) 0 0
\(916\) −5.00000 −0.165205
\(917\) −12.0000 −0.396275
\(918\) −3.00000 −0.0990148
\(919\) 29.0000 0.956622 0.478311 − 0.878191i \(-0.341249\pi\)
0.478311 + 0.878191i \(0.341249\pi\)
\(920\) 0 0
\(921\) 13.0000 0.428365
\(922\) 0 0
\(923\) 0 0
\(924\) −3.00000 −0.0986928
\(925\) 40.0000 1.31519
\(926\) 31.0000 1.01872
\(927\) −4.00000 −0.131377
\(928\) −9.00000 −0.295439
\(929\) −39.0000 −1.27955 −0.639774 − 0.768563i \(-0.720972\pi\)
−0.639774 + 0.768563i \(0.720972\pi\)
\(930\) 0 0
\(931\) −5.00000 −0.163868
\(932\) −24.0000 −0.786146
\(933\) −21.0000 −0.687509
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) 0 0
\(937\) −28.0000 −0.914720 −0.457360 − 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 2.00000 0.0653023
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 2.00000 0.0651635
\(943\) 18.0000 0.586161
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 24.0000 0.780307
\(947\) −3.00000 −0.0974869 −0.0487435 − 0.998811i \(-0.515522\pi\)
−0.0487435 + 0.998811i \(0.515522\pi\)
\(948\) −13.0000 −0.422220
\(949\) 0 0
\(950\) 25.0000 0.811107
\(951\) −30.0000 −0.972817
\(952\) 3.00000 0.0972306
\(953\) −36.0000 −1.16615 −0.583077 − 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) −3.00000 −0.0971286
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) −27.0000 −0.872786
\(958\) −9.00000 −0.290777
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 9.00000 0.290021
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) 6.00000 0.193047
\(967\) 16.0000 0.514525 0.257263 − 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 15.0000 0.481869
\(970\) 0 0
\(971\) 6.00000 0.192549 0.0962746 − 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 1.00000 0.0320750
\(973\) −11.0000 −0.352644
\(974\) 13.0000 0.416547
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 16.0000 0.511624
\(979\) 45.0000 1.43821
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) −12.0000 −0.382935
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 0 0
\(986\) 27.0000 0.859855
\(987\) 3.00000 0.0954911
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 17.0000 0.540023 0.270011 − 0.962857i \(-0.412973\pi\)
0.270011 + 0.962857i \(0.412973\pi\)
\(992\) −8.00000 −0.254000
\(993\) 10.0000 0.317340
\(994\) −6.00000 −0.190308
\(995\) 0 0
\(996\) −18.0000 −0.570352
\(997\) −19.0000 −0.601736 −0.300868 − 0.953666i \(-0.597276\pi\)
−0.300868 + 0.953666i \(0.597276\pi\)
\(998\) −32.0000 −1.01294
\(999\) −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 7098.2.a.ba.1.1 1
13.4 even 6 546.2.l.e.211.1 ✓ 2
13.10 even 6 546.2.l.e.295.1 yes 2
13.12 even 2 7098.2.a.m.1.1 1
39.17 odd 6 1638.2.r.f.757.1 2
39.23 odd 6 1638.2.r.f.1387.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.e.211.1 ✓ 2 13.4 even 6
546.2.l.e.295.1 yes 2 13.10 even 6
1638.2.r.f.757.1 2 39.17 odd 6
1638.2.r.f.1387.1 2 39.23 odd 6
7098.2.a.m.1.1 1 13.12 even 2
7098.2.a.ba.1.1 1 1.1 even 1 trivial