Properties

Label 99.2.a.b.1.1
Level $99$
Weight $2$
Character 99.1
Self dual yes
Analytic conductor $0.791$
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] = [99,2,Mod(1,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, 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("99.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.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: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 99.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} +4.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} +4.00000 q^{7} +3.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} -2.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} -2.00000 q^{20} +1.00000 q^{22} -8.00000 q^{23} -1.00000 q^{25} +2.00000 q^{26} -4.00000 q^{28} +6.00000 q^{29} -8.00000 q^{31} -5.00000 q^{32} -2.00000 q^{34} +8.00000 q^{35} +6.00000 q^{37} +6.00000 q^{40} +2.00000 q^{41} +1.00000 q^{44} +8.00000 q^{46} -8.00000 q^{47} +9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} -2.00000 q^{55} +12.0000 q^{56} -6.00000 q^{58} +4.00000 q^{59} +6.00000 q^{61} +8.00000 q^{62} +7.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} -2.00000 q^{68} -8.00000 q^{70} -14.0000 q^{73} -6.00000 q^{74} -4.00000 q^{77} -4.00000 q^{79} -2.00000 q^{80} -2.00000 q^{82} -12.0000 q^{83} +4.00000 q^{85} -3.00000 q^{88} +6.00000 q^{89} -8.00000 q^{91} +8.00000 q^{92} +8.00000 q^{94} +2.00000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) 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
\(34\) −2.00000 −0.342997
\(35\) 8.00000 1.35225
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 6.00000 0.948683
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 12.0000 1.60357
\(57\) 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
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −16.0000 −1.49201
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −6.00000 −0.543214
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) −8.00000 −0.676123
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) −16.0000 −1.28515
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) −10.0000 −0.790569
\(161\) −32.0000 −2.52195
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 8.00000 0.592999
\(183\) 0 0
\(184\) −24.0000 −1.76930
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) −32.0000 −2.17230
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −20.0000 −1.33631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 16.0000 1.05501
\(231\) 0 0
\(232\) 18.0000 1.18176
\(233\) −30.0000 −1.96537 −0.982683 0.185296i \(-0.940675\pi\)
−0.982683 + 0.185296i \(0.940675\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) −8.00000 −0.518563
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 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
\(244\) −6.00000 −0.384111
\(245\) 18.0000 1.14998
\(246\) 0 0
\(247\) 0 0
\(248\) −24.0000 −1.52400
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 8.00000 0.479808
\(279\) 0 0
\(280\) 24.0000 1.43427
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −12.0000 −0.704664
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 18.0000 1.04623
\(297\) 0 0
\(298\) −22.0000 −1.27443
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 0 0
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 32.0000 1.82634 0.913168 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) 16.0000 0.908739
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 14.0000 0.782624
\(321\) 0 0
\(322\) 32.0000 1.78329
\(323\) 0 0
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 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
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) 8.00000 0.419314
\(365\) −28.0000 −1.46559
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) −12.0000 −0.623850
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 27.0000 1.36371
\(393\) 0 0
\(394\) −14.0000 −0.705310
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −26.0000 −1.29838 −0.649189 0.760627i \(-0.724892\pi\)
−0.649189 + 0.760627i \(0.724892\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) −24.0000 −1.19110
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 16.0000 0.787309
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 10.0000 0.490290
\(417\) 0 0
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 24.0000 1.16144
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) −6.00000 −0.286039
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) 28.0000 1.32288
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) −16.0000 −0.750092
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 16.0000 0.746004
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 30.0000 1.38972
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 16.0000 0.738025
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) 24.0000 1.09773
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 4.00000 0.181631
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 18.0000 0.814822
\(489\) 0 0
\(490\) −18.0000 −0.813157
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) 32.0000 1.42681 0.713405 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) −56.0000 −2.47729
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) −24.0000 −1.05450
\(519\) 0 0
\(520\) −12.0000 −0.526235
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) −2.00000 −0.0862261
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) 46.0000 1.97769 0.988847 0.148933i \(-0.0475840\pi\)
0.988847 + 0.148933i \(0.0475840\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) −10.0000 −0.428746
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 2.00000 0.0854358
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −8.00000 −0.338062
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) 44.0000 1.85438 0.927189 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) −16.0000 −0.672530
\(567\) 0 0
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) −8.00000 −0.333914
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) −12.0000 −0.498273
\(581\) −48.0000 −1.99138
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) −42.0000 −1.73797
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) −38.0000 −1.56047 −0.780236 0.625485i \(-0.784901\pi\)
−0.780236 + 0.625485i \(0.784901\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) −22.0000 −0.901155
\(597\) 0 0
\(598\) −16.0000 −0.654289
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −20.0000 −0.813788
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −32.0000 −1.29141
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 16.0000 0.642575
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −12.0000 −0.477334
\(633\) 0 0
\(634\) 22.0000 0.873732
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) 6.00000 0.237171
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 32.0000 1.26098
\(645\) 0 0
\(646\) 0 0
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 2.00000 0.0782660 0.0391330 0.999234i \(-0.487540\pi\)
0.0391330 + 0.999234i \(0.487540\pi\)
\(654\) 0 0
\(655\) 24.0000 0.937758
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 32.0000 1.24749
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) −36.0000 −1.39707
\(665\) 0 0
\(666\) 0 0
\(667\) −48.0000 −1.85857
\(668\) 0 0
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) −6.00000 −0.231627
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) 12.0000 0.460179
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) −8.00000 −0.305441
\(687\) 0 0
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 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
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) −6.00000 −0.227103
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) −8.00000 −0.300871
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 18.0000 0.674579
\(713\) 64.0000 2.39682
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) −24.0000 −0.889499
\(729\) 0 0
\(730\) 28.0000 1.03633
\(731\) 0 0
\(732\) 0 0
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) 40.0000 1.47442
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) 44.0000 1.61204
\(746\) 2.00000 0.0732252
\(747\) 0 0
\(748\) 2.00000 0.0731272
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 40.0000 1.45575
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 16.0000 0.572159
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) 28.0000 0.999363
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) −14.0000 −0.498729
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) 0 0
\(797\) 10.0000 0.354218 0.177109 0.984191i \(-0.443325\pi\)
0.177109 + 0.984191i \(0.443325\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) 26.0000 0.918092
\(803\) 14.0000 0.494049
\(804\) 0 0
\(805\) −64.0000 −2.25570
\(806\) −16.0000 −0.563576
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) −56.0000 −1.96643 −0.983213 0.182462i \(-0.941593\pi\)
−0.983213 + 0.182462i \(0.941593\pi\)
\(812\) −24.0000 −0.842235
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) 0 0
\(818\) −18.0000 −0.629355
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 14.0000 0.488603 0.244302 0.969699i \(-0.421441\pi\)
0.244302 + 0.969699i \(0.421441\pi\)
\(822\) 0 0
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) −14.0000 −0.485363
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) 56.0000 1.93333 0.966667 0.256036i \(-0.0824164\pi\)
0.966667 + 0.256036i \(0.0824164\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) 0 0
\(845\) −18.0000 −0.619219
\(846\) 0 0
\(847\) 4.00000 0.137442
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) −48.0000 −1.64542
\(852\) 0 0
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) −24.0000 −0.821263
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) 32.0000 1.08615
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −6.00000 −0.203186
\(873\) 0 0
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) 0 0
\(877\) 6.00000 0.202606 0.101303 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) −12.0000 −0.402241
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 12.0000 0.400892
\(897\) 0 0
\(898\) 2.00000 0.0667409
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 44.0000 1.46261
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 16.0000 0.530395
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 48.0000 1.58510
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) −48.0000 −1.58251
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) −30.0000 −0.984798
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 30.0000 0.982683
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) 16.0000 0.521862
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) 0 0
\(952\) 24.0000 0.777844
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 8.00000 0.258468
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 12.0000 0.386896
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −28.0000 −0.901352
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 3.00000 0.0964237
\(969\) 0 0
\(970\) −4.00000 −0.128432
\(971\) 52.0000 1.66876 0.834380 0.551190i \(-0.185826\pi\)
0.834380 + 0.551190i \(0.185826\pi\)
\(972\) 0 0
\(973\) −32.0000 −1.02587
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) −18.0000 −0.574989
\(981\) 0 0
\(982\) 4.00000 0.127645
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 28.0000 0.892154
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 40.0000 1.27000
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 4.00000 0.126618
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 99.2.a.b.1.1 1
3.2 odd 2 33.2.a.a.1.1 1
4.3 odd 2 1584.2.a.o.1.1 1
5.2 odd 4 2475.2.c.d.199.1 2
5.3 odd 4 2475.2.c.d.199.2 2
5.4 even 2 2475.2.a.g.1.1 1
7.6 odd 2 4851.2.a.b.1.1 1
8.3 odd 2 6336.2.a.n.1.1 1
8.5 even 2 6336.2.a.x.1.1 1
9.2 odd 6 891.2.e.e.595.1 2
9.4 even 3 891.2.e.g.298.1 2
9.5 odd 6 891.2.e.e.298.1 2
9.7 even 3 891.2.e.g.595.1 2
11.10 odd 2 1089.2.a.j.1.1 1
12.11 even 2 528.2.a.g.1.1 1
15.2 even 4 825.2.c.a.199.2 2
15.8 even 4 825.2.c.a.199.1 2
15.14 odd 2 825.2.a.a.1.1 1
21.20 even 2 1617.2.a.j.1.1 1
24.5 odd 2 2112.2.a.bb.1.1 1
24.11 even 2 2112.2.a.j.1.1 1
33.2 even 10 363.2.e.g.202.1 4
33.5 odd 10 363.2.e.e.124.1 4
33.8 even 10 363.2.e.g.130.1 4
33.14 odd 10 363.2.e.e.130.1 4
33.17 even 10 363.2.e.g.124.1 4
33.20 odd 10 363.2.e.e.202.1 4
33.26 odd 10 363.2.e.e.148.1 4
33.29 even 10 363.2.e.g.148.1 4
33.32 even 2 363.2.a.b.1.1 1
39.38 odd 2 5577.2.a.a.1.1 1
51.50 odd 2 9537.2.a.m.1.1 1
132.131 odd 2 5808.2.a.t.1.1 1
165.164 even 2 9075.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.a.a.1.1 1 3.2 odd 2
99.2.a.b.1.1 1 1.1 even 1 trivial
363.2.a.b.1.1 1 33.32 even 2
363.2.e.e.124.1 4 33.5 odd 10
363.2.e.e.130.1 4 33.14 odd 10
363.2.e.e.148.1 4 33.26 odd 10
363.2.e.e.202.1 4 33.20 odd 10
363.2.e.g.124.1 4 33.17 even 10
363.2.e.g.130.1 4 33.8 even 10
363.2.e.g.148.1 4 33.29 even 10
363.2.e.g.202.1 4 33.2 even 10
528.2.a.g.1.1 1 12.11 even 2
825.2.a.a.1.1 1 15.14 odd 2
825.2.c.a.199.1 2 15.8 even 4
825.2.c.a.199.2 2 15.2 even 4
891.2.e.e.298.1 2 9.5 odd 6
891.2.e.e.595.1 2 9.2 odd 6
891.2.e.g.298.1 2 9.4 even 3
891.2.e.g.595.1 2 9.7 even 3
1089.2.a.j.1.1 1 11.10 odd 2
1584.2.a.o.1.1 1 4.3 odd 2
1617.2.a.j.1.1 1 21.20 even 2
2112.2.a.j.1.1 1 24.11 even 2
2112.2.a.bb.1.1 1 24.5 odd 2
2475.2.a.g.1.1 1 5.4 even 2
2475.2.c.d.199.1 2 5.2 odd 4
2475.2.c.d.199.2 2 5.3 odd 4
4851.2.a.b.1.1 1 7.6 odd 2
5577.2.a.a.1.1 1 39.38 odd 2
5808.2.a.t.1.1 1 132.131 odd 2
6336.2.a.n.1.1 1 8.3 odd 2
6336.2.a.x.1.1 1 8.5 even 2
9075.2.a.q.1.1 1 165.164 even 2
9537.2.a.m.1.1 1 51.50 odd 2