Properties

Label 2175.2.a.j.1.1
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +4.00000 q^{14} -4.00000 q^{16} -8.00000 q^{17} +2.00000 q^{18} -2.00000 q^{21} -6.00000 q^{22} +1.00000 q^{23} -8.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +1.00000 q^{29} -8.00000 q^{31} -8.00000 q^{32} +3.00000 q^{33} -16.0000 q^{34} +2.00000 q^{36} +7.00000 q^{37} +4.00000 q^{39} +7.00000 q^{41} -4.00000 q^{42} -9.00000 q^{43} -6.00000 q^{44} +2.00000 q^{46} +12.0000 q^{47} +4.00000 q^{48} -3.00000 q^{49} +8.00000 q^{51} -8.00000 q^{52} -9.00000 q^{53} -2.00000 q^{54} +2.00000 q^{58} +10.0000 q^{59} +2.00000 q^{61} -16.0000 q^{62} +2.00000 q^{63} -8.00000 q^{64} +6.00000 q^{66} -8.00000 q^{67} -16.0000 q^{68} -1.00000 q^{69} -8.00000 q^{71} +1.00000 q^{73} +14.0000 q^{74} -6.00000 q^{77} +8.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} +14.0000 q^{82} -9.00000 q^{83} -4.00000 q^{84} -18.0000 q^{86} -1.00000 q^{87} +10.0000 q^{89} -8.00000 q^{91} +2.00000 q^{92} +8.00000 q^{93} +24.0000 q^{94} +8.00000 q^{96} -13.0000 q^{97} -6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −2.00000 −0.577350
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −8.00000 −1.94029 −0.970143 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 2.00000 0.471405
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) −6.00000 −1.27920
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.00000 −1.56893
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −8.00000 −1.41421
\(33\) 3.00000 0.522233
\(34\) −16.0000 −2.74398
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) −4.00000 −0.617213
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 4.00000 0.577350
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) −8.00000 −1.10940
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −2.00000 −0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −16.0000 −2.03200
\(63\) 2.00000 0.251976
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −16.0000 −1.94029
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 14.0000 1.62747
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 8.00000 0.905822
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 14.0000 1.54604
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −18.0000 −1.94099
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 2.00000 0.208514
\(93\) 8.00000 0.829561
\(94\) 24.0000 2.47541
\(95\) 0 0
\(96\) 8.00000 0.816497
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) −6.00000 −0.606092
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 17.0000 1.69156 0.845782 0.533529i \(-0.179135\pi\)
0.845782 + 0.533529i \(0.179135\pi\)
\(102\) 16.0000 1.58424
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −18.0000 −1.74831
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −2.00000 −0.192450
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) −8.00000 −0.755929
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) −4.00000 −0.369800
\(118\) 20.0000 1.84115
\(119\) −16.0000 −1.46672
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 4.00000 0.362143
\(123\) −7.00000 −0.631169
\(124\) −16.0000 −1.43684
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) −3.00000 −0.266207 −0.133103 0.991102i \(-0.542494\pi\)
−0.133103 + 0.991102i \(0.542494\pi\)
\(128\) 0 0
\(129\) 9.00000 0.792406
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) −16.0000 −1.38219
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −2.00000 −0.170251
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) −16.0000 −1.34269
\(143\) 12.0000 1.00349
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 3.00000 0.247436
\(148\) 14.0000 1.15079
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −3.00000 −0.244137 −0.122068 0.992522i \(-0.538953\pi\)
−0.122068 + 0.992522i \(0.538953\pi\)
\(152\) 0 0
\(153\) −8.00000 −0.646762
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) 8.00000 0.640513
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −20.0000 −1.59111
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 2.00000 0.157135
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 14.0000 1.09322
\(165\) 0 0
\(166\) −18.0000 −1.39707
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) −18.0000 −1.37249
\(173\) −9.00000 −0.684257 −0.342129 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 12.0000 0.904534
\(177\) −10.0000 −0.751646
\(178\) 20.0000 1.49906
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) −16.0000 −1.18600
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 16.0000 1.17318
\(187\) 24.0000 1.75505
\(188\) 24.0000 1.75038
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 8.00000 0.577350
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) −26.0000 −1.86669
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −13.0000 −0.926212 −0.463106 0.886303i \(-0.653265\pi\)
−0.463106 + 0.886303i \(0.653265\pi\)
\(198\) −6.00000 −0.426401
\(199\) −25.0000 −1.77220 −0.886102 0.463491i \(-0.846597\pi\)
−0.886102 + 0.463491i \(0.846597\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 34.0000 2.39223
\(203\) 2.00000 0.140372
\(204\) 16.0000 1.12022
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 1.00000 0.0695048
\(208\) 16.0000 1.10940
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −18.0000 −1.23625
\(213\) 8.00000 0.548151
\(214\) 24.0000 1.64061
\(215\) 0 0
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) −10.0000 −0.677285
\(219\) −1.00000 −0.0675737
\(220\) 0 0
\(221\) 32.0000 2.15255
\(222\) −14.0000 −0.939618
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) −16.0000 −1.06904
\(225\) 0 0
\(226\) −28.0000 −1.86253
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) 1.00000 0.0655122 0.0327561 0.999463i \(-0.489572\pi\)
0.0327561 + 0.999463i \(0.489572\pi\)
\(234\) −8.00000 −0.522976
\(235\) 0 0
\(236\) 20.0000 1.30189
\(237\) 10.0000 0.649570
\(238\) −32.0000 −2.07425
\(239\) 30.0000 1.94054 0.970269 0.242028i \(-0.0778125\pi\)
0.970269 + 0.242028i \(0.0778125\pi\)
\(240\) 0 0
\(241\) −3.00000 −0.193247 −0.0966235 0.995321i \(-0.530804\pi\)
−0.0966235 + 0.995321i \(0.530804\pi\)
\(242\) −4.00000 −0.257130
\(243\) −1.00000 −0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) −14.0000 −0.892607
\(247\) 0 0
\(248\) 0 0
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 4.00000 0.251976
\(253\) −3.00000 −0.188608
\(254\) −6.00000 −0.376473
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −23.0000 −1.43470 −0.717350 0.696713i \(-0.754645\pi\)
−0.717350 + 0.696713i \(0.754645\pi\)
\(258\) 18.0000 1.12063
\(259\) 14.0000 0.869918
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 24.0000 1.48272
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −16.0000 −0.977356
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 32.0000 1.94029
\(273\) 8.00000 0.484182
\(274\) 4.00000 0.241649
\(275\) 0 0
\(276\) −2.00000 −0.120386
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 10.0000 0.599760
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −24.0000 −1.42918
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) 14.0000 0.826394
\(288\) −8.00000 −0.471405
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) 13.0000 0.762073
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −18.0000 −1.03750
\(302\) −6.00000 −0.345261
\(303\) −17.0000 −0.976624
\(304\) 0 0
\(305\) 0 0
\(306\) −16.0000 −0.914659
\(307\) 17.0000 0.970241 0.485121 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) −12.0000 −0.683763
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 17.0000 0.963982 0.481991 0.876176i \(-0.339914\pi\)
0.481991 + 0.876176i \(0.339914\pi\)
\(312\) 0 0
\(313\) −4.00000 −0.226093 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −20.0000 −1.12509
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 18.0000 1.00939
\(319\) −3.00000 −0.167968
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 4.00000 0.222911
\(323\) 0 0
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) 2.00000 0.110770
\(327\) 5.00000 0.276501
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) −18.0000 −0.987878
\(333\) 7.00000 0.383598
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) 8.00000 0.436436
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 6.00000 0.326357
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −3.00000 −0.161048 −0.0805242 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) −2.00000 −0.107211
\(349\) −5.00000 −0.267644 −0.133822 0.991005i \(-0.542725\pi\)
−0.133822 + 0.991005i \(0.542725\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 24.0000 1.27920
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) −20.0000 −1.06299
\(355\) 0 0
\(356\) 20.0000 1.06000
\(357\) 16.0000 0.846810
\(358\) 20.0000 1.05703
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 34.0000 1.78700
\(363\) 2.00000 0.104973
\(364\) −16.0000 −0.838628
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) −3.00000 −0.156599 −0.0782994 0.996930i \(-0.524949\pi\)
−0.0782994 + 0.996930i \(0.524949\pi\)
\(368\) −4.00000 −0.208514
\(369\) 7.00000 0.364405
\(370\) 0 0
\(371\) −18.0000 −0.934513
\(372\) 16.0000 0.829561
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 48.0000 2.48202
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) −4.00000 −0.205738
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) 3.00000 0.153695
\(382\) −6.00000 −0.306987
\(383\) −39.0000 −1.99281 −0.996403 0.0847358i \(-0.972995\pi\)
−0.996403 + 0.0847358i \(0.972995\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 12.0000 0.610784
\(387\) −9.00000 −0.457496
\(388\) −26.0000 −1.31995
\(389\) −25.0000 −1.26755 −0.633775 0.773517i \(-0.718496\pi\)
−0.633775 + 0.773517i \(0.718496\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) −26.0000 −1.30986
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) −50.0000 −2.50627
\(399\) 0 0
\(400\) 0 0
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 16.0000 0.798007
\(403\) 32.0000 1.59403
\(404\) 34.0000 1.69156
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) −21.0000 −1.04093
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) −8.00000 −0.394132
\(413\) 20.0000 0.984136
\(414\) 2.00000 0.0982946
\(415\) 0 0
\(416\) 32.0000 1.56893
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 24.0000 1.16830
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 4.00000 0.193574
\(428\) 24.0000 1.16008
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 4.00000 0.192450
\(433\) −19.0000 −0.913082 −0.456541 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) −32.0000 −1.53605
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 64.0000 3.04417
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) −14.0000 −0.664411
\(445\) 0 0
\(446\) 12.0000 0.568216
\(447\) 0 0
\(448\) −16.0000 −0.755929
\(449\) 5.00000 0.235965 0.117982 0.993016i \(-0.462357\pi\)
0.117982 + 0.993016i \(0.462357\pi\)
\(450\) 0 0
\(451\) −21.0000 −0.988851
\(452\) −28.0000 −1.31701
\(453\) 3.00000 0.140952
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 40.0000 1.86908
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) −13.0000 −0.605470 −0.302735 0.953075i \(-0.597900\pi\)
−0.302735 + 0.953075i \(0.597900\pi\)
\(462\) 12.0000 0.558291
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 2.00000 0.0926482
\(467\) 42.0000 1.94353 0.971764 0.235954i \(-0.0758216\pi\)
0.971764 + 0.235954i \(0.0758216\pi\)
\(468\) −8.00000 −0.369800
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 27.0000 1.24146
\(474\) 20.0000 0.918630
\(475\) 0 0
\(476\) −32.0000 −1.46672
\(477\) −9.00000 −0.412082
\(478\) 60.0000 2.74434
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −28.0000 −1.27669
\(482\) −6.00000 −0.273293
\(483\) −2.00000 −0.0910032
\(484\) −4.00000 −0.181818
\(485\) 0 0
\(486\) −2.00000 −0.0907218
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 0 0
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) −14.0000 −0.631169
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) 32.0000 1.43684
\(497\) −16.0000 −0.717698
\(498\) 18.0000 0.806599
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 24.0000 1.07117
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) −3.00000 −0.133235
\(508\) −6.00000 −0.266207
\(509\) −40.0000 −1.77297 −0.886484 0.462758i \(-0.846860\pi\)
−0.886484 + 0.462758i \(0.846860\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) −46.0000 −2.02897
\(515\) 0 0
\(516\) 18.0000 0.792406
\(517\) −36.0000 −1.58328
\(518\) 28.0000 1.23025
\(519\) 9.00000 0.395056
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 2.00000 0.0875376
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 24.0000 1.04844
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 64.0000 2.78788
\(528\) −12.0000 −0.522233
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 0 0
\(533\) −28.0000 −1.21281
\(534\) −20.0000 −0.865485
\(535\) 0 0
\(536\) 0 0
\(537\) −10.0000 −0.431532
\(538\) −20.0000 −0.862261
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 24.0000 1.03089
\(543\) −17.0000 −0.729540
\(544\) 64.0000 2.74398
\(545\) 0 0
\(546\) 16.0000 0.684737
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 4.00000 0.170872
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −20.0000 −0.850487
\(554\) −56.0000 −2.37921
\(555\) 0 0
\(556\) 10.0000 0.424094
\(557\) 47.0000 1.99145 0.995727 0.0923462i \(-0.0294367\pi\)
0.995727 + 0.0923462i \(0.0294367\pi\)
\(558\) −16.0000 −0.677334
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) −36.0000 −1.51857
\(563\) 46.0000 1.93867 0.969334 0.245745i \(-0.0790327\pi\)
0.969334 + 0.245745i \(0.0790327\pi\)
\(564\) −24.0000 −1.01058
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −23.0000 −0.962520 −0.481260 0.876578i \(-0.659821\pi\)
−0.481260 + 0.876578i \(0.659821\pi\)
\(572\) 24.0000 1.00349
\(573\) 3.00000 0.125327
\(574\) 28.0000 1.16870
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 94.0000 3.90988
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) 26.0000 1.07773
\(583\) 27.0000 1.11823
\(584\) 0 0
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 6.00000 0.247436
\(589\) 0 0
\(590\) 0 0
\(591\) 13.0000 0.534749
\(592\) −28.0000 −1.15079
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) 0 0
\(597\) 25.0000 1.02318
\(598\) −8.00000 −0.327144
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) −36.0000 −1.46725
\(603\) −8.00000 −0.325785
\(604\) −6.00000 −0.244137
\(605\) 0 0
\(606\) −34.0000 −1.38116
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) −16.0000 −0.646762
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 34.0000 1.37213
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 8.00000 0.321807
\(619\) −30.0000 −1.20580 −0.602901 0.797816i \(-0.705989\pi\)
−0.602901 + 0.797816i \(0.705989\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 34.0000 1.36328
\(623\) 20.0000 0.801283
\(624\) −16.0000 −0.640513
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) −56.0000 −2.23287
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 24.0000 0.953162
\(635\) 0 0
\(636\) 18.0000 0.713746
\(637\) 12.0000 0.475457
\(638\) −6.00000 −0.237542
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −43.0000 −1.69840 −0.849199 0.528073i \(-0.822915\pi\)
−0.849199 + 0.528073i \(0.822915\pi\)
\(642\) −24.0000 −0.947204
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) −33.0000 −1.29736 −0.648682 0.761060i \(-0.724679\pi\)
−0.648682 + 0.761060i \(0.724679\pi\)
\(648\) 0 0
\(649\) −30.0000 −1.17760
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 2.00000 0.0783260
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −28.0000 −1.09322
\(657\) 1.00000 0.0390137
\(658\) 48.0000 1.87123
\(659\) 25.0000 0.973862 0.486931 0.873441i \(-0.338116\pi\)
0.486931 + 0.873441i \(0.338116\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 64.0000 2.48743
\(663\) −32.0000 −1.24278
\(664\) 0 0
\(665\) 0 0
\(666\) 14.0000 0.542489
\(667\) 1.00000 0.0387202
\(668\) −16.0000 −0.619059
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) −6.00000 −0.231627
\(672\) 16.0000 0.617213
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 44.0000 1.69482
\(675\) 0 0
\(676\) 6.00000 0.230769
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 28.0000 1.07533
\(679\) −26.0000 −0.997788
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 48.0000 1.83801
\(683\) −39.0000 −1.49229 −0.746147 0.665782i \(-0.768098\pi\)
−0.746147 + 0.665782i \(0.768098\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −40.0000 −1.52721
\(687\) −20.0000 −0.763048
\(688\) 36.0000 1.37249
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −18.0000 −0.684257
\(693\) −6.00000 −0.227921
\(694\) −6.00000 −0.227757
\(695\) 0 0
\(696\) 0 0
\(697\) −56.0000 −2.12115
\(698\) −10.0000 −0.378506
\(699\) −1.00000 −0.0378235
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 8.00000 0.301941
\(703\) 0 0
\(704\) 24.0000 0.904534
\(705\) 0 0
\(706\) 52.0000 1.95705
\(707\) 34.0000 1.27870
\(708\) −20.0000 −0.751646
\(709\) −45.0000 −1.69001 −0.845005 0.534758i \(-0.820403\pi\)
−0.845005 + 0.534758i \(0.820403\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 32.0000 1.19757
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) −30.0000 −1.12037
\(718\) −30.0000 −1.11959
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) −38.0000 −1.41421
\(723\) 3.00000 0.111571
\(724\) 34.0000 1.26360
\(725\) 0 0
\(726\) 4.00000 0.148454
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 72.0000 2.66302
\(732\) −4.00000 −0.147844
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) −6.00000 −0.221464
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) 24.0000 0.884051
\(738\) 14.0000 0.515347
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −36.0000 −1.32160
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −8.00000 −0.292901
\(747\) −9.00000 −0.329293
\(748\) 48.0000 1.75505
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 22.0000 0.802791 0.401396 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) −48.0000 −1.75038
\(753\) −12.0000 −0.437304
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) −3.00000 −0.109037 −0.0545184 0.998513i \(-0.517362\pi\)
−0.0545184 + 0.998513i \(0.517362\pi\)
\(758\) −20.0000 −0.726433
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 6.00000 0.217357
\(763\) −10.0000 −0.362024
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) −78.0000 −2.81825
\(767\) −40.0000 −1.44432
\(768\) −16.0000 −0.577350
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 23.0000 0.828325
\(772\) 12.0000 0.431889
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) −18.0000 −0.646997
\(775\) 0 0
\(776\) 0 0
\(777\) −14.0000 −0.502247
\(778\) −50.0000 −1.79259
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) −16.0000 −0.572159
\(783\) −1.00000 −0.0357371
\(784\) 12.0000 0.428571
\(785\) 0 0
\(786\) −24.0000 −0.856052
\(787\) −38.0000 −1.35455 −0.677277 0.735728i \(-0.736840\pi\)
−0.677277 + 0.735728i \(0.736840\pi\)
\(788\) −26.0000 −0.926212
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) −28.0000 −0.995565
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) −16.0000 −0.567819
\(795\) 0 0
\(796\) −50.0000 −1.77220
\(797\) −8.00000 −0.283375 −0.141687 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) −96.0000 −3.39624
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) −56.0000 −1.97743
\(803\) −3.00000 −0.105868
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 64.0000 2.25430
\(807\) 10.0000 0.352017
\(808\) 0 0
\(809\) −45.0000 −1.58212 −0.791058 0.611741i \(-0.790469\pi\)
−0.791058 + 0.611741i \(0.790469\pi\)
\(810\) 0 0
\(811\) −13.0000 −0.456492 −0.228246 0.973604i \(-0.573299\pi\)
−0.228246 + 0.973604i \(0.573299\pi\)
\(812\) 4.00000 0.140372
\(813\) −12.0000 −0.420858
\(814\) −42.0000 −1.47210
\(815\) 0 0
\(816\) −32.0000 −1.12022
\(817\) 0 0
\(818\) −20.0000 −0.699284
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) −48.0000 −1.67521 −0.837606 0.546275i \(-0.816045\pi\)
−0.837606 + 0.546275i \(0.816045\pi\)
\(822\) −4.00000 −0.139516
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 40.0000 1.39178
\(827\) 22.0000 0.765015 0.382507 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) 2.00000 0.0695048
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 0 0
\(831\) 28.0000 0.971309
\(832\) 32.0000 1.10940
\(833\) 24.0000 0.831551
\(834\) −10.0000 −0.346272
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −56.0000 −1.92989
\(843\) 18.0000 0.619953
\(844\) 24.0000 0.826114
\(845\) 0 0
\(846\) 24.0000 0.825137
\(847\) −4.00000 −0.137442
\(848\) 36.0000 1.23625
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) 7.00000 0.239957
\(852\) 16.0000 0.548151
\(853\) −19.0000 −0.650548 −0.325274 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 0 0
\(857\) 7.00000 0.239115 0.119558 0.992827i \(-0.461852\pi\)
0.119558 + 0.992827i \(0.461852\pi\)
\(858\) −24.0000 −0.819346
\(859\) −30.0000 −1.02359 −0.511793 0.859109i \(-0.671019\pi\)
−0.511793 + 0.859109i \(0.671019\pi\)
\(860\) 0 0
\(861\) −14.0000 −0.477119
\(862\) 24.0000 0.817443
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 8.00000 0.272166
\(865\) 0 0
\(866\) −38.0000 −1.29129
\(867\) −47.0000 −1.59620
\(868\) −32.0000 −1.08615
\(869\) 30.0000 1.01768
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) 0 0
\(873\) −13.0000 −0.439983
\(874\) 0 0
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 52.0000 1.75592 0.877958 0.478738i \(-0.158906\pi\)
0.877958 + 0.478738i \(0.158906\pi\)
\(878\) 40.0000 1.34993
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 57.0000 1.92038 0.960189 0.279350i \(-0.0901189\pi\)
0.960189 + 0.279350i \(0.0901189\pi\)
\(882\) −6.00000 −0.202031
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 64.0000 2.15255
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 22.0000 0.738688 0.369344 0.929293i \(-0.379582\pi\)
0.369344 + 0.929293i \(0.379582\pi\)
\(888\) 0 0
\(889\) −6.00000 −0.201234
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 12.0000 0.401790
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 10.0000 0.333704
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 72.0000 2.39867
\(902\) −42.0000 −1.39845
\(903\) 18.0000 0.599002
\(904\) 0 0
\(905\) 0 0
\(906\) 6.00000 0.199337
\(907\) 37.0000 1.22856 0.614282 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) −6.00000 −0.199117
\(909\) 17.0000 0.563854
\(910\) 0 0
\(911\) 27.0000 0.894550 0.447275 0.894397i \(-0.352395\pi\)
0.447275 + 0.894397i \(0.352395\pi\)
\(912\) 0 0
\(913\) 27.0000 0.893570
\(914\) −56.0000 −1.85232
\(915\) 0 0
\(916\) 40.0000 1.32164
\(917\) 24.0000 0.792550
\(918\) 16.0000 0.528079
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −17.0000 −0.560169
\(922\) −26.0000 −0.856264
\(923\) 32.0000 1.05329
\(924\) 12.0000 0.394771
\(925\) 0 0
\(926\) 12.0000 0.394344
\(927\) −4.00000 −0.131377
\(928\) −8.00000 −0.262613
\(929\) 20.0000 0.656179 0.328089 0.944647i \(-0.393595\pi\)
0.328089 + 0.944647i \(0.393595\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.00000 0.0655122
\(933\) −17.0000 −0.556555
\(934\) 84.0000 2.74856
\(935\) 0 0
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) −32.0000 −1.04484
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) −4.00000 −0.130327
\(943\) 7.00000 0.227951
\(944\) −40.0000 −1.30189
\(945\) 0 0
\(946\) 54.0000 1.75569
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 20.0000 0.649570
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −18.0000 −0.582772
\(955\) 0 0
\(956\) 60.0000 1.94054
\(957\) 3.00000 0.0969762
\(958\) 0 0
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −56.0000 −1.80551
\(963\) 12.0000 0.386695
\(964\) −6.00000 −0.193247
\(965\) 0 0
\(966\) −4.00000 −0.128698
\(967\) −3.00000 −0.0964735 −0.0482367 0.998836i \(-0.515360\pi\)
−0.0482367 + 0.998836i \(0.515360\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.00000 −0.0962746 −0.0481373 0.998841i \(-0.515328\pi\)
−0.0481373 + 0.998841i \(0.515328\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 10.0000 0.320585
\(974\) 44.0000 1.40985
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −13.0000 −0.415907 −0.207953 0.978139i \(-0.566680\pi\)
−0.207953 + 0.978139i \(0.566680\pi\)
\(978\) −2.00000 −0.0639529
\(979\) −30.0000 −0.958804
\(980\) 0 0
\(981\) −5.00000 −0.159638
\(982\) −56.0000 −1.78703
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −16.0000 −0.509544
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) −9.00000 −0.286183
\(990\) 0 0
\(991\) −33.0000 −1.04828 −0.524140 0.851632i \(-0.675613\pi\)
−0.524140 + 0.851632i \(0.675613\pi\)
\(992\) 64.0000 2.03200
\(993\) −32.0000 −1.01549
\(994\) −32.0000 −1.01498
\(995\) 0 0
\(996\) 18.0000 0.570352
\(997\) 7.00000 0.221692 0.110846 0.993838i \(-0.464644\pi\)
0.110846 + 0.993838i \(0.464644\pi\)
\(998\) 40.0000 1.26618
\(999\) −7.00000 −0.221470
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 2175.2.a.j.1.1 1
3.2 odd 2 6525.2.a.b.1.1 1
5.2 odd 4 435.2.c.a.349.2 yes 2
5.3 odd 4 435.2.c.a.349.1 2
5.4 even 2 2175.2.a.a.1.1 1
15.2 even 4 1305.2.c.a.784.1 2
15.8 even 4 1305.2.c.a.784.2 2
15.14 odd 2 6525.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.a.349.1 2 5.3 odd 4
435.2.c.a.349.2 yes 2 5.2 odd 4
1305.2.c.a.784.1 2 15.2 even 4
1305.2.c.a.784.2 2 15.8 even 4
2175.2.a.a.1.1 1 5.4 even 2
2175.2.a.j.1.1 1 1.1 even 1 trivial
6525.2.a.b.1.1 1 3.2 odd 2
6525.2.a.l.1.1 1 15.14 odd 2