Properties

Label 7623.2.a.e.1.1
Level $7623$
Weight $2$
Character 7623.1
Self dual yes
Analytic conductor $60.870$
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] = [7623,2,Mod(1,7623)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, 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("7623.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.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: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7623.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{10} -5.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} -7.00000 q^{17} +6.00000 q^{19} +1.00000 q^{20} +4.00000 q^{23} -4.00000 q^{25} +5.00000 q^{26} +1.00000 q^{28} +9.00000 q^{29} -2.00000 q^{31} -5.00000 q^{32} +7.00000 q^{34} +1.00000 q^{35} +9.00000 q^{37} -6.00000 q^{38} -3.00000 q^{40} -7.00000 q^{41} +6.00000 q^{43} -4.00000 q^{46} -2.00000 q^{47} +1.00000 q^{49} +4.00000 q^{50} +5.00000 q^{52} -3.00000 q^{53} -3.00000 q^{56} -9.00000 q^{58} -2.00000 q^{59} -6.00000 q^{61} +2.00000 q^{62} +7.00000 q^{64} +5.00000 q^{65} +8.00000 q^{67} +7.00000 q^{68} -1.00000 q^{70} -6.00000 q^{71} +10.0000 q^{73} -9.00000 q^{74} -6.00000 q^{76} +14.0000 q^{79} +1.00000 q^{80} +7.00000 q^{82} +7.00000 q^{85} -6.00000 q^{86} +9.00000 q^{89} +5.00000 q^{91} -4.00000 q^{92} +2.00000 q^{94} -6.00000 q^{95} +17.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 5.00000 0.980581
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 7.00000 1.20049
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 9.00000 1.47959 0.739795 0.672832i \(-0.234922\pi\)
0.739795 + 0.672832i \(0.234922\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) −9.00000 −1.18176
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 5.00000 0.620174
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 7.00000 0.848875
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −9.00000 −1.04623
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 7.00000 0.773021
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 7.00000 0.759257
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −15.0000 −1.47087
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −9.00000 −0.835629
\(117\) 0 0
\(118\) 2.00000 0.184115
\(119\) 7.00000 0.641689
\(120\) 0 0
\(121\) 0 0
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) −5.00000 −0.438529
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −21.0000 −1.80074
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −9.00000 −0.739795
\(149\) 13.0000 1.06500 0.532501 0.846430i \(-0.321252\pi\)
0.532501 + 0.846430i \(0.321252\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 18.0000 1.45999
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −14.0000 −1.11378
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 7.00000 0.546608
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) −7.00000 −0.536875
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) −9.00000 −0.674579
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) −5.00000 −0.370625
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) −9.00000 −0.661693
\(186\) 0 0
\(187\) 0 0
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 0 0
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) −17.0000 −1.22053
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) −7.00000 −0.498729 −0.249365 0.968410i \(-0.580222\pi\)
−0.249365 + 0.968410i \(0.580222\pi\)
\(198\) 0 0
\(199\) 6.00000 0.425329 0.212664 0.977125i \(-0.431786\pi\)
0.212664 + 0.977125i \(0.431786\pi\)
\(200\) −12.0000 −0.848528
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) −9.00000 −0.631676
\(204\) 0 0
\(205\) 7.00000 0.488901
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) 5.00000 0.346688
\(209\) 0 0
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) 3.00000 0.206041
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) −1.00000 −0.0677285
\(219\) 0 0
\(220\) 0 0
\(221\) 35.0000 2.35435
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 5.00000 0.334077
\(225\) 0 0
\(226\) −15.0000 −0.997785
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 0 0
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 27.0000 1.77264
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 0 0
\(235\) 2.00000 0.130466
\(236\) 2.00000 0.130189
\(237\) 0 0
\(238\) −7.00000 −0.453743
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −30.0000 −1.90885
\(248\) −6.00000 −0.381000
\(249\) 0 0
\(250\) −9.00000 −0.569210
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 17.0000 1.06043 0.530215 0.847863i \(-0.322111\pi\)
0.530215 + 0.847863i \(0.322111\pi\)
\(258\) 0 0
\(259\) −9.00000 −0.559233
\(260\) −5.00000 −0.310087
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 3.00000 0.184289
\(266\) 6.00000 0.367884
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 11.0000 0.670682 0.335341 0.942097i \(-0.391148\pi\)
0.335341 + 0.942097i \(0.391148\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 7.00000 0.424437
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 0 0
\(277\) −11.0000 −0.660926 −0.330463 0.943819i \(-0.607205\pi\)
−0.330463 + 0.943819i \(0.607205\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 7.00000 0.413197
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 9.00000 0.528498
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 2.00000 0.116445
\(296\) 27.0000 1.56934
\(297\) 0 0
\(298\) −13.0000 −0.753070
\(299\) −20.0000 −1.15663
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) −2.00000 −0.115087
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.00000 −0.113592
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −14.0000 −0.787562
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −7.00000 −0.391312
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) −42.0000 −2.33694
\(324\) 0 0
\(325\) 20.0000 1.10940
\(326\) −10.0000 −0.553849
\(327\) 0 0
\(328\) −21.0000 −1.15953
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 34.0000 1.86881 0.934405 0.356214i \(-0.115932\pi\)
0.934405 + 0.356214i \(0.115932\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) −7.00000 −0.379628
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 18.0000 0.970495
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −10.0000 −0.536828 −0.268414 0.963304i \(-0.586500\pi\)
−0.268414 + 0.963304i \(0.586500\pi\)
\(348\) 0 0
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) 33.0000 1.75641 0.878206 0.478282i \(-0.158740\pi\)
0.878206 + 0.478282i \(0.158740\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) −9.00000 −0.476999
\(357\) 0 0
\(358\) 6.00000 0.317110
\(359\) −28.0000 −1.47778 −0.738892 0.673824i \(-0.764651\pi\)
−0.738892 + 0.673824i \(0.764651\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 5.00000 0.262794
\(363\) 0 0
\(364\) −5.00000 −0.262071
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 9.00000 0.467888
\(371\) 3.00000 0.155752
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −45.0000 −2.31762
\(378\) 0 0
\(379\) −32.0000 −1.64373 −0.821865 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) −10.0000 −0.511645
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5.00000 0.254493
\(387\) 0 0
\(388\) −17.0000 −0.863044
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) −28.0000 −1.41602
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) 7.00000 0.352655
\(395\) −14.0000 −0.704416
\(396\) 0 0
\(397\) −29.0000 −1.45547 −0.727734 0.685859i \(-0.759427\pi\)
−0.727734 + 0.685859i \(0.759427\pi\)
\(398\) −6.00000 −0.300753
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −17.0000 −0.848939 −0.424470 0.905442i \(-0.639539\pi\)
−0.424470 + 0.905442i \(0.639539\pi\)
\(402\) 0 0
\(403\) 10.0000 0.498135
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) 0 0
\(408\) 0 0
\(409\) 9.00000 0.445021 0.222511 0.974930i \(-0.428575\pi\)
0.222511 + 0.974930i \(0.428575\pi\)
\(410\) −7.00000 −0.345705
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) 2.00000 0.0984136
\(414\) 0 0
\(415\) 0 0
\(416\) 25.0000 1.22573
\(417\) 0 0
\(418\) 0 0
\(419\) −32.0000 −1.56330 −0.781651 0.623716i \(-0.785622\pi\)
−0.781651 + 0.623716i \(0.785622\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) −24.0000 −1.16830
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 28.0000 1.35820
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −1.00000 −0.0478913
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −35.0000 −1.66478
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) 8.00000 0.378811
\(447\) 0 0
\(448\) −7.00000 −0.330719
\(449\) 19.0000 0.896665 0.448333 0.893867i \(-0.352018\pi\)
0.448333 + 0.893867i \(0.352018\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −15.0000 −0.705541
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) −5.00000 −0.234404
\(456\) 0 0
\(457\) −33.0000 −1.54367 −0.771837 0.635820i \(-0.780662\pi\)
−0.771837 + 0.635820i \(0.780662\pi\)
\(458\) −15.0000 −0.700904
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −21.0000 −0.978068 −0.489034 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) 9.00000 0.416917
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) −2.00000 −0.0922531
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) 0 0
\(475\) −24.0000 −1.10120
\(476\) −7.00000 −0.320844
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) −14.0000 −0.639676 −0.319838 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 0 0
\(481\) −45.0000 −2.05182
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 0 0
\(485\) −17.0000 −0.771930
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) −18.0000 −0.814822
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) −63.0000 −2.83738
\(494\) 30.0000 1.34976
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) −9.00000 −0.402492
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −17.0000 −0.749838
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) 9.00000 0.395437
\(519\) 0 0
\(520\) 15.0000 0.657794
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) 14.0000 0.609850
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −3.00000 −0.130312
\(531\) 0 0
\(532\) 6.00000 0.260133
\(533\) 35.0000 1.51602
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 24.0000 1.03664
\(537\) 0 0
\(538\) −11.0000 −0.474244
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) 35.0000 1.50061
\(545\) −1.00000 −0.0428353
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 2.00000 0.0854358
\(549\) 0 0
\(550\) 0 0
\(551\) 54.0000 2.30048
\(552\) 0 0
\(553\) −14.0000 −0.595341
\(554\) 11.0000 0.467345
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 38.0000 1.61011 0.805056 0.593199i \(-0.202135\pi\)
0.805056 + 0.593199i \(0.202135\pi\)
\(558\) 0 0
\(559\) −30.0000 −1.26886
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) −22.0000 −0.927189 −0.463595 0.886047i \(-0.653441\pi\)
−0.463595 + 0.886047i \(0.653441\pi\)
\(564\) 0 0
\(565\) −15.0000 −0.631055
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) −18.0000 −0.755263
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −7.00000 −0.292174
\(575\) −16.0000 −0.667246
\(576\) 0 0
\(577\) −15.0000 −0.624458 −0.312229 0.950007i \(-0.601076\pi\)
−0.312229 + 0.950007i \(0.601076\pi\)
\(578\) −32.0000 −1.33102
\(579\) 0 0
\(580\) 9.00000 0.373705
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 30.0000 1.24141
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) −2.00000 −0.0823387
\(591\) 0 0
\(592\) −9.00000 −0.369898
\(593\) 37.0000 1.51941 0.759704 0.650269i \(-0.225344\pi\)
0.759704 + 0.650269i \(0.225344\pi\)
\(594\) 0 0
\(595\) −7.00000 −0.286972
\(596\) −13.0000 −0.532501
\(597\) 0 0
\(598\) 20.0000 0.817861
\(599\) −10.0000 −0.408589 −0.204294 0.978909i \(-0.565490\pi\)
−0.204294 + 0.978909i \(0.565490\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 6.00000 0.244542
\(603\) 0 0
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) −30.0000 −1.21666
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) 10.0000 0.404557
\(612\) 0 0
\(613\) −31.0000 −1.25208 −0.626039 0.779792i \(-0.715325\pi\)
−0.626039 + 0.779792i \(0.715325\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) −45.0000 −1.81163 −0.905816 0.423672i \(-0.860741\pi\)
−0.905816 + 0.423672i \(0.860741\pi\)
\(618\) 0 0
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) −9.00000 −0.360577
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 19.0000 0.759393
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −63.0000 −2.51197
\(630\) 0 0
\(631\) 46.0000 1.83123 0.915616 0.402055i \(-0.131704\pi\)
0.915616 + 0.402055i \(0.131704\pi\)
\(632\) 42.0000 1.67067
\(633\) 0 0
\(634\) −14.0000 −0.556011
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −5.00000 −0.198107
\(638\) 0 0
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 42.0000 1.65247
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −20.0000 −0.784465
\(651\) 0 0
\(652\) −10.0000 −0.391630
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 7.00000 0.273304
\(657\) 0 0
\(658\) −2.00000 −0.0779681
\(659\) −22.0000 −0.856998 −0.428499 0.903542i \(-0.640958\pi\)
−0.428499 + 0.903542i \(0.640958\pi\)
\(660\) 0 0
\(661\) −17.0000 −0.661223 −0.330612 0.943767i \(-0.607255\pi\)
−0.330612 + 0.943767i \(0.607255\pi\)
\(662\) −34.0000 −1.32145
\(663\) 0 0
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) 0 0
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 13.0000 0.500741
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 51.0000 1.96009 0.980045 0.198778i \(-0.0636972\pi\)
0.980045 + 0.198778i \(0.0636972\pi\)
\(678\) 0 0
\(679\) −17.0000 −0.652400
\(680\) 21.0000 0.805313
\(681\) 0 0
\(682\) 0 0
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −6.00000 −0.228748
\(689\) 15.0000 0.571454
\(690\) 0 0
\(691\) −34.0000 −1.29342 −0.646710 0.762736i \(-0.723856\pi\)
−0.646710 + 0.762736i \(0.723856\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 10.0000 0.379595
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) 49.0000 1.85601
\(698\) −15.0000 −0.567758
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) −35.0000 −1.32193 −0.660966 0.750416i \(-0.729853\pi\)
−0.660966 + 0.750416i \(0.729853\pi\)
\(702\) 0 0
\(703\) 54.0000 2.03665
\(704\) 0 0
\(705\) 0 0
\(706\) −33.0000 −1.24197
\(707\) 14.0000 0.526524
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) −6.00000 −0.225176
\(711\) 0 0
\(712\) 27.0000 1.01187
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) 0 0
\(718\) 28.0000 1.04495
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) 5.00000 0.185824
\(725\) −36.0000 −1.33701
\(726\) 0 0
\(727\) −6.00000 −0.222528 −0.111264 0.993791i \(-0.535490\pi\)
−0.111264 + 0.993791i \(0.535490\pi\)
\(728\) 15.0000 0.555937
\(729\) 0 0
\(730\) 10.0000 0.370117
\(731\) −42.0000 −1.55343
\(732\) 0 0
\(733\) 3.00000 0.110808 0.0554038 0.998464i \(-0.482355\pi\)
0.0554038 + 0.998464i \(0.482355\pi\)
\(734\) 14.0000 0.516749
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) 0 0
\(738\) 0 0
\(739\) 2.00000 0.0735712 0.0367856 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) 9.00000 0.330847
\(741\) 0 0
\(742\) −3.00000 −0.110133
\(743\) −10.0000 −0.366864 −0.183432 0.983032i \(-0.558721\pi\)
−0.183432 + 0.983032i \(0.558721\pi\)
\(744\) 0 0
\(745\) −13.0000 −0.476283
\(746\) 26.0000 0.951928
\(747\) 0 0
\(748\) 0 0
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) 46.0000 1.67856 0.839282 0.543696i \(-0.182976\pi\)
0.839282 + 0.543696i \(0.182976\pi\)
\(752\) 2.00000 0.0729325
\(753\) 0 0
\(754\) 45.0000 1.63880
\(755\) −2.00000 −0.0727875
\(756\) 0 0
\(757\) 17.0000 0.617876 0.308938 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) 32.0000 1.16229
\(759\) 0 0
\(760\) −18.0000 −0.652929
\(761\) 45.0000 1.63125 0.815624 0.578582i \(-0.196394\pi\)
0.815624 + 0.578582i \(0.196394\pi\)
\(762\) 0 0
\(763\) −1.00000 −0.0362024
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 10.0000 0.361079
\(768\) 0 0
\(769\) −19.0000 −0.685158 −0.342579 0.939489i \(-0.611300\pi\)
−0.342579 + 0.939489i \(0.611300\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.00000 0.179954
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 51.0000 1.83079
\(777\) 0 0
\(778\) −9.00000 −0.322666
\(779\) −42.0000 −1.50481
\(780\) 0 0
\(781\) 0 0
\(782\) 28.0000 1.00128
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 40.0000 1.42585 0.712923 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 7.00000 0.249365
\(789\) 0 0
\(790\) 14.0000 0.498098
\(791\) −15.0000 −0.533339
\(792\) 0 0
\(793\) 30.0000 1.06533
\(794\) 29.0000 1.02917
\(795\) 0 0
\(796\) −6.00000 −0.212664
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 14.0000 0.495284
\(800\) 20.0000 0.707107
\(801\) 0 0
\(802\) 17.0000 0.600291
\(803\) 0 0
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) −10.0000 −0.352235
\(807\) 0 0
\(808\) −42.0000 −1.47755
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 9.00000 0.315838
\(813\) 0 0
\(814\) 0 0
\(815\) −10.0000 −0.350285
\(816\) 0 0
\(817\) 36.0000 1.25948
\(818\) −9.00000 −0.314678
\(819\) 0 0
\(820\) −7.00000 −0.244451
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) 12.0000 0.418294 0.209147 0.977884i \(-0.432931\pi\)
0.209147 + 0.977884i \(0.432931\pi\)
\(824\) −48.0000 −1.67216
\(825\) 0 0
\(826\) −2.00000 −0.0695889
\(827\) 34.0000 1.18230 0.591148 0.806563i \(-0.298675\pi\)
0.591148 + 0.806563i \(0.298675\pi\)
\(828\) 0 0
\(829\) −29.0000 −1.00721 −0.503606 0.863934i \(-0.667994\pi\)
−0.503606 + 0.863934i \(0.667994\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −35.0000 −1.21341
\(833\) −7.00000 −0.242536
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) 0 0
\(838\) 32.0000 1.10542
\(839\) 26.0000 0.897620 0.448810 0.893627i \(-0.351848\pi\)
0.448810 + 0.893627i \(0.351848\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −1.00000 −0.0344623
\(843\) 0 0
\(844\) −24.0000 −0.826114
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) 3.00000 0.103020
\(849\) 0 0
\(850\) −28.0000 −0.960392
\(851\) 36.0000 1.23406
\(852\) 0 0
\(853\) −25.0000 −0.855984 −0.427992 0.903783i \(-0.640779\pi\)
−0.427992 + 0.903783i \(0.640779\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) −54.0000 −1.84568
\(857\) −38.0000 −1.29806 −0.649028 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 6.00000 0.204598
\(861\) 0 0
\(862\) −30.0000 −1.02180
\(863\) 30.0000 1.02121 0.510606 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 11.0000 0.373795
\(867\) 0 0
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) −40.0000 −1.35535
\(872\) 3.00000 0.101593
\(873\) 0 0
\(874\) −24.0000 −0.811812
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) −27.0000 −0.911725 −0.455863 0.890050i \(-0.650669\pi\)
−0.455863 + 0.890050i \(0.650669\pi\)
\(878\) −4.00000 −0.134993
\(879\) 0 0
\(880\) 0 0
\(881\) 49.0000 1.65085 0.825426 0.564510i \(-0.190935\pi\)
0.825426 + 0.564510i \(0.190935\pi\)
\(882\) 0 0
\(883\) 22.0000 0.740359 0.370179 0.928960i \(-0.379296\pi\)
0.370179 + 0.928960i \(0.379296\pi\)
\(884\) −35.0000 −1.17718
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 9.00000 0.301681
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) −19.0000 −0.634038
\(899\) −18.0000 −0.600334
\(900\) 0 0
\(901\) 21.0000 0.699611
\(902\) 0 0
\(903\) 0 0
\(904\) 45.0000 1.49668
\(905\) 5.00000 0.166206
\(906\) 0 0
\(907\) −30.0000 −0.996134 −0.498067 0.867139i \(-0.665957\pi\)
−0.498067 + 0.867139i \(0.665957\pi\)
\(908\) −6.00000 −0.199117
\(909\) 0 0
\(910\) 5.00000 0.165748
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 33.0000 1.09154
\(915\) 0 0
\(916\) −15.0000 −0.495614
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −14.0000 −0.461817 −0.230909 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) −12.0000 −0.395628
\(921\) 0 0
\(922\) 21.0000 0.691598
\(923\) 30.0000 0.987462
\(924\) 0 0
\(925\) −36.0000 −1.18367
\(926\) 26.0000 0.854413
\(927\) 0 0
\(928\) −45.0000 −1.47720
\(929\) 33.0000 1.08269 0.541347 0.840799i \(-0.317914\pi\)
0.541347 + 0.840799i \(0.317914\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 9.00000 0.294805
\(933\) 0 0
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) 0 0
\(937\) 29.0000 0.947389 0.473694 0.880689i \(-0.342920\pi\)
0.473694 + 0.880689i \(0.342920\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) −2.00000 −0.0652328
\(941\) 3.00000 0.0977972 0.0488986 0.998804i \(-0.484429\pi\)
0.0488986 + 0.998804i \(0.484429\pi\)
\(942\) 0 0
\(943\) −28.0000 −0.911805
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 0 0
\(947\) −42.0000 −1.36482 −0.682408 0.730971i \(-0.739067\pi\)
−0.682408 + 0.730971i \(0.739067\pi\)
\(948\) 0 0
\(949\) −50.0000 −1.62307
\(950\) 24.0000 0.778663
\(951\) 0 0
\(952\) 21.0000 0.680614
\(953\) −37.0000 −1.19855 −0.599274 0.800544i \(-0.704544\pi\)
−0.599274 + 0.800544i \(0.704544\pi\)
\(954\) 0 0
\(955\) −10.0000 −0.323592
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 14.0000 0.452319
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 45.0000 1.45086
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) 5.00000 0.160956
\(966\) 0 0
\(967\) −50.0000 −1.60789 −0.803946 0.594703i \(-0.797270\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 17.0000 0.545837
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) −57.0000 −1.82359 −0.911796 0.410644i \(-0.865304\pi\)
−0.911796 + 0.410644i \(0.865304\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 28.0000 0.893516
\(983\) 30.0000 0.956851 0.478426 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) 7.00000 0.223039
\(986\) 63.0000 2.00633
\(987\) 0 0
\(988\) 30.0000 0.954427
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −14.0000 −0.444725 −0.222362 0.974964i \(-0.571377\pi\)
−0.222362 + 0.974964i \(0.571377\pi\)
\(992\) 10.0000 0.317500
\(993\) 0 0
\(994\) −6.00000 −0.190308
\(995\) −6.00000 −0.190213
\(996\) 0 0
\(997\) 47.0000 1.48850 0.744252 0.667898i \(-0.232806\pi\)
0.744252 + 0.667898i \(0.232806\pi\)
\(998\) 16.0000 0.506471
\(999\) 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 7623.2.a.e.1.1 1
3.2 odd 2 2541.2.a.i.1.1 yes 1
11.10 odd 2 7623.2.a.m.1.1 1
33.32 even 2 2541.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.d.1.1 1 33.32 even 2
2541.2.a.i.1.1 yes 1 3.2 odd 2
7623.2.a.e.1.1 1 1.1 even 1 trivial
7623.2.a.m.1.1 1 11.10 odd 2