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Properties

Label 55.2.a.a.1.1
Level $55$
Weight $2$
Character 55.1
Self dual yes
Analytic conductor $0.439$
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] = [55,2,Mod(1,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("55.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 55.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: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 55.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{16} +6.00000 q^{17} -3.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +6.00000 q^{29} -8.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} +3.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} -3.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -3.00000 q^{45} +4.00000 q^{46} -12.0000 q^{47} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{55} +6.00000 q^{58} +4.00000 q^{59} -10.0000 q^{61} -8.00000 q^{62} +7.00000 q^{64} +2.00000 q^{65} -16.0000 q^{67} -6.00000 q^{68} +8.00000 q^{71} +9.00000 q^{72} +14.0000 q^{73} -2.00000 q^{74} +4.00000 q^{76} +8.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} +6.00000 q^{85} +4.00000 q^{86} +3.00000 q^{88} +10.0000 q^{89} -3.00000 q^{90} -4.00000 q^{92} -12.0000 q^{94} -4.00000 q^{95} +10.0000 q^{97} -7.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −3.00000 −1.06066
\(9\) −3.00000 −1.00000
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −3.00000 −0.707107
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) −3.00000 −0.447214
\(46\) 4.00000 0.589768
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −16.0000 −1.95471 −0.977356 − 0.211604i \(-0.932131\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 9.00000 1.06066
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 9.00000 1.00000
\(82\) 2.00000 0.220863
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −7.00000 −0.707107
\(99\) 3.00000 0.301511
\(100\) −1.00000 −0.100000
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 − 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) −6.00000 −0.557086
\(117\) −6.00000 −0.554700
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −16.0000 −1.38219
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −2.00000 −0.167248
\(144\) 3.00000 0.250000
\(145\) 6.00000 0.498273
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 12.0000 0.973329
\(153\) −18.0000 −1.45521
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 12.0000 0.917663
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 4.00000 0.298974 0.149487 − 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 3.00000 0.223607
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −26.0000 −1.87152 −0.935760 − 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) 2.00000 0.142494 0.0712470 − 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 3.00000 0.213201
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) −4.00000 −0.278693
\(207\) −12.0000 −0.834058
\(208\) −2.00000 −0.138675
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) −18.0000 −1.21911
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 − 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) −6.00000 −0.399114
\(227\) −20.0000 −1.32745 −0.663723 − 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) −18.0000 −1.18176
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −6.00000 −0.392232
\(235\) −12.0000 −0.782794
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 − 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) −7.00000 −0.447214
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 24.0000 1.52400
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) −18.0000 −1.11417
\(262\) −12.0000 −0.741362
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 0 0
\(268\) 16.0000 0.977356
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 12.0000 0.719712
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 0 0
\(288\) −15.0000 −0.883883
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) 10.0000 0.584206 0.292103 − 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −10.0000 −0.572598
\(306\) −18.0000 −1.02899
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 − 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 7.00000 0.391312
\(321\) 0 0
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) −9.00000 −0.500000
\(325\) 2.00000 0.110940
\(326\) 16.0000 0.886158
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) −8.00000 −0.437741
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 − 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) 8.00000 0.433224
\(342\) 12.0000 0.648886
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −5.00000 −0.266501
\(353\) 18.0000 0.958043 0.479022 − 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −32.0000 −1.68890 −0.844448 − 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 9.00000 0.474342
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 − 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) −4.00000 −0.208514
\(369\) −6.00000 −0.312348
\(370\) −2.00000 −0.103975
\(371\) 0 0
\(372\) 0 0
\(373\) 18.0000 0.932005 0.466002 − 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) 36.0000 1.85656
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) −12.0000 −0.613171 −0.306586 − 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −26.0000 −1.32337
\(387\) −12.0000 −0.609994
\(388\) −10.0000 −0.507673
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 21.0000 1.06066
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 8.00000 0.402524
\(396\) −3.00000 −0.150756
\(397\) 30.0000 1.50566 0.752828 − 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 2.00000 0.0998752 0.0499376 − 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) 10.0000 0.497519
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 2.00000 0.0987730
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) −12.0000 −0.589768
\(415\) −4.00000 −0.196352
\(416\) 10.0000 0.490290
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) −28.0000 −1.36789 −0.683945 − 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 − 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 4.00000 0.194717
\(423\) 36.0000 1.75038
\(424\) 6.00000 0.291386
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 − 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 3.00000 0.143019
\(441\) 21.0000 1.00000
\(442\) 12.0000 0.570782
\(443\) 8.00000 0.380091 0.190046 − 0.981775i \(-0.439136\pi\)
0.190046 + 0.981775i \(0.439136\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 − 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −3.00000 −0.141421
\(451\) −2.00000 −0.0941763
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 − 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) −34.0000 −1.58354 −0.791769 − 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) −36.0000 −1.67306 −0.836531 − 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) −12.0000 −0.553519
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 8.00000 0.365911
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 10.0000 0.454077
\(486\) 0 0
\(487\) 28.0000 1.26880 0.634401 − 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 30.0000 1.35804
\(489\) 0 0
\(490\) −7.00000 −0.316228
\(491\) −28.0000 −1.26362 −0.631811 − 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) −8.00000 −0.359937
\(495\) 3.00000 0.134840
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 − 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) 14.0000 0.620539 0.310270 − 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) −6.00000 −0.262865 −0.131432 − 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −18.0000 −0.787839
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −2.00000 −0.0868744
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 48.0000 2.07328
\(537\) 0 0
\(538\) −18.0000 −0.776035
\(539\) 7.00000 0.301511
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 − 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 30.0000 1.28624
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 − 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) −18.0000 −0.768922
\(549\) 30.0000 1.28037
\(550\) −1.00000 −0.0426401
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 10.0000 0.423714 0.211857 − 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) 24.0000 1.01600
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) −24.0000 −1.00702
\(569\) 26.0000 1.08998 0.544988 − 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 − 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) −21.0000 −0.875000
\(577\) −22.0000 −0.915872 −0.457936 − 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 0 0
\(582\) 0 0
\(583\) 2.00000 0.0828315
\(584\) −42.0000 −1.73797
\(585\) −6.00000 −0.248069
\(586\) 10.0000 0.413096
\(587\) −24.0000 −0.990586 −0.495293 − 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) 22.0000 0.903432 0.451716 − 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 48.0000 1.95471
\(604\) −8.00000 −0.325515
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 − 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) −20.0000 −0.811107
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) −24.0000 −0.970936
\(612\) 18.0000 0.727607
\(613\) 34.0000 1.37325 0.686624 − 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 − 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −24.0000 −0.954669
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) −14.0000 −0.554700
\(638\) −6.00000 −0.237542
\(639\) −24.0000 −0.949425
\(640\) −3.00000 −0.118585
\(641\) 34.0000 1.34292 0.671460 − 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 − 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 20.0000 0.786281 0.393141 − 0.919478i \(-0.371389\pi\)
0.393141 + 0.919478i \(0.371389\pi\)
\(648\) −27.0000 −1.06066
\(649\) −4.00000 −0.157014
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −10.0000 −0.391330 −0.195665 − 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) −2.00000 −0.0780869
\(657\) −42.0000 −1.63858
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 − 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 24.0000 0.929284
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) −16.0000 −0.618134
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 − 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) −38.0000 −1.46046 −0.730229 − 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −18.0000 −0.690268
\(681\) 0 0
\(682\) 8.00000 0.306336
\(683\) −16.0000 −0.612223 −0.306111 − 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) −12.0000 −0.458831
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) −10.0000 −0.378506
\(699\) 0 0
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 − 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 8.00000 0.300235
\(711\) −24.0000 −0.900070
\(712\) −30.0000 −1.12430
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −32.0000 −1.19423
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 3.00000 0.111803
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 52.0000 1.92857 0.964287 − 0.264861i \(-0.0853260\pi\)
0.964287 + 0.264861i \(0.0853260\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 14.0000 0.518163
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 42.0000 1.55131 0.775653 − 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 20.0000 0.737210
\(737\) 16.0000 0.589368
\(738\) −6.00000 −0.220863
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) 0 0
\(743\) 40.0000 1.46746 0.733729 − 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 18.0000 0.659027
\(747\) 12.0000 0.439057
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 − 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 − 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) 10.0000 0.362500 0.181250 − 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) −18.0000 −0.650791
\(766\) −12.0000 −0.433578
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 − 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 26.0000 0.935760
\(773\) 14.0000 0.503545 0.251773 − 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) −12.0000 −0.431331
\(775\) −8.00000 −0.287368
\(776\) −30.0000 −1.07694
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 24.0000 0.858238
\(783\) 0 0
\(784\) 7.00000 0.250000
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 − 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) 0 0
\(792\) −9.00000 −0.319801
\(793\) −20.0000 −0.710221
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) 0 0
\(797\) −2.00000 −0.0708436 −0.0354218 − 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 5.00000 0.176777
\(801\) −30.0000 −1.06000
\(802\) 2.00000 0.0706225
\(803\) −14.0000 −0.494049
\(804\) 0 0
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 0 0
\(808\) 30.0000 1.05540
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 9.00000 0.316228
\(811\) −12.0000 −0.421377 −0.210688 − 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 2.00000 0.0701000
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) −18.0000 −0.628204 −0.314102 − 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) −36.0000 −1.25488 −0.627441 − 0.778664i \(-0.715897\pi\)
−0.627441 + 0.778664i \(0.715897\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 12.0000 0.417029
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) 14.0000 0.485363
\(833\) −42.0000 −1.45521
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) −28.0000 −0.967244
\(839\) −32.0000 −1.10476 −0.552381 − 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 6.00000 0.206774
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) −9.00000 −0.309609
\(846\) 36.0000 1.23771
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) −36.0000 −1.23045
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 − 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 4.00000 0.136162 0.0680808 − 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) −22.0000 −0.747590
\(867\) 0 0
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) 54.0000 1.82867
\(873\) −30.0000 −1.01535
\(874\) −16.0000 −0.541208
\(875\) 0 0
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 − 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 21.0000 0.707107
\(883\) −16.0000 −0.538443 −0.269221 − 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 8.00000 0.268765
\(887\) 56.0000 1.88030 0.940148 − 0.340766i \(-0.110687\pi\)
0.940148 + 0.340766i \(0.110687\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 10.0000 0.335201
\(891\) −9.00000 −0.301511
\(892\) 4.00000 0.133930
\(893\) 48.0000 1.60626
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) 2.00000 0.0667409
\(899\) −48.0000 −1.60089
\(900\) 3.00000 0.100000
\(901\) −12.0000 −0.399778
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) −40.0000 −1.32818 −0.664089 − 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) 20.0000 0.663723
\(909\) 30.0000 0.995037
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 − 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 − 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) −12.0000 −0.395628
\(921\) 0 0
\(922\) −34.0000 −1.11973
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −36.0000 −1.18303
\(927\) 12.0000 0.394132
\(928\) 30.0000 0.984798
\(929\) −14.0000 −0.459325 −0.229663 − 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) 0 0
\(935\) −6.00000 −0.196221
\(936\) 18.0000 0.588348
\(937\) 6.00000 0.196011 0.0980057 − 0.995186i \(-0.468754\pi\)
0.0980057 + 0.995186i \(0.468754\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 12.0000 0.391397
\(941\) −2.00000 −0.0651981 −0.0325991 − 0.999469i \(-0.510378\pi\)
−0.0325991 + 0.999469i \(0.510378\pi\)
\(942\) 0 0
\(943\) 8.00000 0.260516
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 − 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 6.00000 0.194257
\(955\) 8.00000 0.258874
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −4.00000 −0.128965
\(963\) −36.0000 −1.16008
\(964\) −10.0000 −0.322078
\(965\) −26.0000 −0.836970
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −3.00000 −0.0964237
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 10.0000 0.319928 0.159964 − 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) 0 0
\(979\) −10.0000 −0.319601
\(980\) 7.00000 0.223607
\(981\) 54.0000 1.72409
\(982\) −28.0000 −0.893516
\(983\) −4.00000 −0.127580 −0.0637901 − 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 16.0000 0.508770
\(990\) 3.00000 0.0953463
\(991\) 8.00000 0.254128 0.127064 − 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −40.0000 −1.27000
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −46.0000 −1.45683 −0.728417 − 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −36.0000 −1.13956
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 55.2.a.a.1.1 ✓ 1
3.2 odd 2 495.2.a.a.1.1 1
4.3 odd 2 880.2.a.h.1.1 1
5.2 odd 4 275.2.b.b.199.2 2
5.3 odd 4 275.2.b.b.199.1 2
5.4 even 2 275.2.a.a.1.1 1
7.6 odd 2 2695.2.a.c.1.1 1
8.3 odd 2 3520.2.a.n.1.1 1
8.5 even 2 3520.2.a.p.1.1 1
11.2 odd 10 605.2.g.c.81.1 4
11.3 even 5 605.2.g.a.251.1 4
11.4 even 5 605.2.g.a.511.1 4
11.5 even 5 605.2.g.a.366.1 4
11.6 odd 10 605.2.g.c.366.1 4
11.7 odd 10 605.2.g.c.511.1 4
11.8 odd 10 605.2.g.c.251.1 4
11.9 even 5 605.2.g.a.81.1 4
11.10 odd 2 605.2.a.b.1.1 1
12.11 even 2 7920.2.a.i.1.1 1
13.12 even 2 9295.2.a.b.1.1 1
15.2 even 4 2475.2.c.f.199.1 2
15.8 even 4 2475.2.c.f.199.2 2
15.14 odd 2 2475.2.a.i.1.1 1
20.3 even 4 4400.2.b.n.4049.2 2
20.7 even 4 4400.2.b.n.4049.1 2
20.19 odd 2 4400.2.a.p.1.1 1
33.32 even 2 5445.2.a.i.1.1 1
44.43 even 2 9680.2.a.r.1.1 1
55.54 odd 2 3025.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
275.2.a.a.1.1 1 5.4 even 2
275.2.b.b.199.1 2 5.3 odd 4
275.2.b.b.199.2 2 5.2 odd 4
495.2.a.a.1.1 1 3.2 odd 2
605.2.a.b.1.1 1 11.10 odd 2
605.2.g.a.81.1 4 11.9 even 5
605.2.g.a.251.1 4 11.3 even 5
605.2.g.a.366.1 4 11.5 even 5
605.2.g.a.511.1 4 11.4 even 5
605.2.g.c.81.1 4 11.2 odd 10
605.2.g.c.251.1 4 11.8 odd 10
605.2.g.c.366.1 4 11.6 odd 10
605.2.g.c.511.1 4 11.7 odd 10
880.2.a.h.1.1 1 4.3 odd 2
2475.2.a.i.1.1 1 15.14 odd 2
2475.2.c.f.199.1 2 15.2 even 4
2475.2.c.f.199.2 2 15.8 even 4
2695.2.a.c.1.1 1 7.6 odd 2
3025.2.a.f.1.1 1 55.54 odd 2
3520.2.a.n.1.1 1 8.3 odd 2
3520.2.a.p.1.1 1 8.5 even 2
4400.2.a.p.1.1 1 20.19 odd 2
4400.2.b.n.4049.1 2 20.7 even 4
4400.2.b.n.4049.2 2 20.3 even 4
5445.2.a.i.1.1 1 33.32 even 2
7920.2.a.i.1.1 1 12.11 even 2
9295.2.a.b.1.1 1 13.12 even 2
9680.2.a.r.1.1 1 44.43 even 2