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Properties

Label 1610.2.a.g.1.1
Level $1610$
Weight $2$
Character 1610.1
Self dual yes
Analytic conductor $12.856$
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] = [1610,2,Mod(1,1610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1610, 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("1610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1610 = 2 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1610.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: \(12.8559147254\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1610.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -3.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} -1.00000 q^{35} -3.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} +1.00000 q^{40} -6.00000 q^{41} -8.00000 q^{43} -4.00000 q^{44} -3.00000 q^{45} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} +14.0000 q^{53} -4.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} +12.0000 q^{59} +6.00000 q^{61} +4.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -8.00000 q^{67} -6.00000 q^{68} -1.00000 q^{70} -3.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} -4.00000 q^{76} +4.00000 q^{77} -4.00000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} -6.00000 q^{85} -8.00000 q^{86} -4.00000 q^{88} -6.00000 q^{89} -3.00000 q^{90} +2.00000 q^{91} -1.00000 q^{92} +8.00000 q^{94} -4.00000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +12.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 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
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −3.00000 −1.00000
\(10\) 1.00000 0.316228
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −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
\(22\) −4.00000 −0.852803
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) −1.00000 −0.169031
\(36\) −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
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) −3.00000 −0.447214
\(46\) −1.00000 −0.147442
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 14.0000 1.92305 0.961524 − 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 4.00000 0.508001
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −3.00000 −0.353553
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −3.00000 −0.316228
\(91\) 2.00000 0.209657
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 12.0000 1.20605
\(100\) 1.00000 0.100000
\(101\) −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 \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 14.0000 1.35980
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −10.0000 −0.940721 −0.470360 − 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) −2.00000 −0.185695
\(117\) 6.00000 0.554700
\(118\) 12.0000 1.10469
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) 3.00000 0.267261
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −18.0000 −1.53784 −0.768922 − 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 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
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) −3.00000 −0.250000
\(145\) −2.00000 −0.166091
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −4.00000 −0.324443
\(153\) 18.0000 1.45521
\(154\) 4.00000 0.322329
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 1.00000 0.0788110
\(162\) 9.00000 0.707107
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) 12.0000 0.917663
\(172\) −8.00000 −0.609994
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −3.00000 −0.223607
\(181\) 6.00000 0.445976 0.222988 − 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 − 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 12.0000 0.852803
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 16.0000 1.11477
\(207\) 3.00000 0.208514
\(208\) −2.00000 −0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 14.0000 0.961524
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 − 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −3.00000 −0.200000
\(226\) −10.0000 −0.665190
\(227\) 4.00000 0.265489 0.132745 − 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −14.0000 −0.917170 −0.458585 − 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 6.00000 0.392232
\(235\) 8.00000 0.521862
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 3.00000 0.188982
\(253\) 4.00000 0.251478
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) −2.00000 −0.124035
\(261\) 6.00000 0.371391
\(262\) 4.00000 0.247121
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 14.0000 0.860013
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 10.0000 0.609711 0.304855 − 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 − 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) −20.0000 −1.19952
\(279\) −12.0000 −0.718421
\(280\) −1.00000 −0.0597614
\(281\) −30.0000 −1.78965 −0.894825 − 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 6.00000 0.354169
\(288\) −3.00000 −0.176777
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 6.00000 0.343559
\(306\) 18.0000 1.02899
\(307\) −8.00000 −0.456584 −0.228292 − 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 34.0000 1.92179 0.960897 − 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) −2.00000 −0.112867
\(315\) 3.00000 0.169031
\(316\) −4.00000 −0.225018
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) 24.0000 1.33540
\(324\) 9.00000 0.500000
\(325\) −2.00000 −0.110940
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 − 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) −16.0000 −0.866449
\(342\) 12.0000 0.648886
\(343\) −1.00000 −0.0539949
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) 30.0000 1.59674 0.798369 − 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −20.0000 −1.05556 −0.527780 − 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) −3.00000 −0.158114
\(361\) −3.00000 −0.157895
\(362\) 6.00000 0.315353
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 18.0000 0.937043
\(370\) −2.00000 −0.103975
\(371\) −14.0000 −0.726844
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 − 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −36.0000 −1.84920 −0.924598 − 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 10.0000 0.508987
\(387\) 24.0000 1.21999
\(388\) −14.0000 −0.710742
\(389\) 26.0000 1.31825 0.659126 − 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −4.00000 −0.201262
\(396\) 12.0000 0.603023
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 10.0000 0.499376 0.249688 − 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) −14.0000 −0.696526
\(405\) 9.00000 0.447214
\(406\) 2.00000 0.0992583
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 − 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) −12.0000 −0.590481
\(414\) 3.00000 0.147442
\(415\) 4.00000 0.196352
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 16.0000 0.782586
\(419\) −4.00000 −0.195413 −0.0977064 − 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 − 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −20.0000 −0.973585
\(423\) −24.0000 −1.16692
\(424\) 14.0000 0.679900
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) 0 0
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 36.0000 1.73406 0.867029 − 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) −12.0000 −0.572729 −0.286364 − 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) −4.00000 −0.190693
\(441\) −3.00000 −0.142857
\(442\) 12.0000 0.570782
\(443\) 20.0000 0.950229 0.475114 − 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 24.0000 1.13643
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 2.00000 0.0943858 0.0471929 − 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −3.00000 −0.141421
\(451\) 24.0000 1.13012
\(452\) −10.0000 −0.470360
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 − 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) −22.0000 −1.02464 −0.512321 − 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 6.00000 0.277350
\(469\) 8.00000 0.369406
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) 32.0000 1.47136
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 6.00000 0.275010
\(477\) −42.0000 −1.92305
\(478\) −8.00000 −0.365911
\(479\) −16.0000 −0.731059 −0.365529 − 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 − 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 8.00000 0.359937
\(495\) 12.0000 0.539360
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 − 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 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 \(0.616099\pi\)
\(504\) 3.00000 0.133631
\(505\) −14.0000 −0.622992
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −14.0000 −0.620539 −0.310270 − 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) −32.0000 −1.40736
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) −22.0000 −0.963837 −0.481919 − 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 6.00000 0.262613
\(523\) −12.0000 −0.524723 −0.262362 − 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 14.0000 0.608121
\(531\) −36.0000 −1.56227
\(532\) 4.00000 0.173422
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 − 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) 2.00000 0.0856706
\(546\) 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
\(549\) −18.0000 −0.768221
\(550\) −4.00000 −0.170561
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 46.0000 1.94908 0.974541 − 0.224208i \(-0.0719796\pi\)
0.974541 + 0.224208i \(0.0719796\pi\)
\(558\) −12.0000 −0.508001
\(559\) 16.0000 0.676728
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −30.0000 −1.26547
\(563\) −20.0000 −0.842900 −0.421450 − 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) −10.0000 −0.420703
\(566\) 20.0000 0.840663
\(567\) −9.00000 −0.377964
\(568\) 0 0
\(569\) −46.0000 −1.92842 −0.964210 − 0.265139i \(-0.914582\pi\)
−0.964210 + 0.265139i \(0.914582\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 − 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 8.00000 0.334497
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) −1.00000 −0.0417029
\(576\) −3.00000 −0.125000
\(577\) −42.0000 −1.74848 −0.874241 − 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) −2.00000 −0.0830455
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −56.0000 −2.31928
\(584\) −2.00000 −0.0827606
\(585\) 6.00000 0.248069
\(586\) 6.00000 0.247858
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 12.0000 0.494032
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 14.0000 0.574911 0.287456 − 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 2.00000 0.0817861
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 − 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 8.00000 0.326056
\(603\) 24.0000 0.977356
\(604\) −8.00000 −0.325515
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) 16.0000 0.649420 0.324710 − 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 6.00000 0.242933
\(611\) −16.0000 −0.647291
\(612\) 18.0000 0.727607
\(613\) −2.00000 −0.0807792 −0.0403896 − 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 30.0000 1.20775 0.603877 − 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 34.0000 1.35891
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 12.0000 0.478471
\(630\) 3.00000 0.119523
\(631\) 12.0000 0.477712 0.238856 − 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −14.0000 −0.552967 −0.276483 − 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 − 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 9.00000 0.353553
\(649\) −48.0000 −1.88416
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 46.0000 1.80012 0.900060 − 0.435767i \(-0.143523\pi\)
0.900060 + 0.435767i \(0.143523\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) −6.00000 −0.234261
\(657\) 6.00000 0.234082
\(658\) −8.00000 −0.311872
\(659\) 20.0000 0.779089 0.389545 − 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 − 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 4.00000 0.155113
\(666\) 6.00000 0.232495
\(667\) 2.00000 0.0774403
\(668\) 0 0
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) 20.0000 0.765279 0.382639 − 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 12.0000 0.458831
\(685\) −18.0000 −0.687745
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) −28.0000 −1.06672
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 6.00000 0.228086
\(693\) −12.0000 −0.455842
\(694\) −12.0000 −0.455514
\(695\) −20.0000 −0.758643
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) −14.0000 −0.529908
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −6.00000 −0.226617 −0.113308 − 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 14.0000 0.526524
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 − 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) −6.00000 −0.224860
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −20.0000 −0.746393
\(719\) −44.0000 −1.64092 −0.820462 − 0.571702i \(-0.806283\pi\)
−0.820462 + 0.571702i \(0.806283\pi\)
\(720\) −3.00000 −0.111803
\(721\) −16.0000 −0.595871
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) 6.00000 0.222988
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 2.00000 0.0741249
\(729\) −27.0000 −1.00000
\(730\) −2.00000 −0.0740233
\(731\) 48.0000 1.77534
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 − 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 32.0000 1.17874
\(738\) 18.0000 0.662589
\(739\) 28.0000 1.03000 0.514998 − 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −14.0000 −0.513956
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −26.0000 −0.951928
\(747\) −12.0000 −0.439057
\(748\) 24.0000 0.877527
\(749\) 0 0
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 − 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 − 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −36.0000 −1.30758
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) 34.0000 1.23250 0.616250 − 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) −12.0000 −0.434145
\(765\) 18.0000 0.650791
\(766\) 24.0000 0.867155
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) −46.0000 −1.65880 −0.829401 − 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 4.00000 0.144150
\(771\) 0 0
\(772\) 10.0000 0.359908
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 24.0000 0.862662
\(775\) 4.00000 0.143684
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 6.00000 0.214560
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 − 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −4.00000 −0.142314
\(791\) 10.0000 0.355559
\(792\) 12.0000 0.426401
\(793\) −12.0000 −0.426132
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 0 0
\(797\) −34.0000 −1.20434 −0.602171 − 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 1.00000 0.0353553
\(801\) 18.0000 0.635999
\(802\) 10.0000 0.353112
\(803\) 8.00000 0.282314
\(804\) 0 0
\(805\) 1.00000 0.0352454
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) −14.0000 −0.492518
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 9.00000 0.316228
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 2.00000 0.0699284
\(819\) −6.00000 −0.209657
\(820\) −6.00000 −0.209529
\(821\) −42.0000 −1.46581 −0.732905 − 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 − 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 32.0000 1.11275 0.556375 − 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 3.00000 0.104257
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) −56.0000 −1.93333 −0.966667 − 0.256036i \(-0.917584\pi\)
−0.966667 + 0.256036i \(0.917584\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 2.00000 0.0689246
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) −9.00000 −0.309609
\(846\) −24.0000 −0.825137
\(847\) −5.00000 −0.171802
\(848\) 14.0000 0.480762
\(849\) 0 0
\(850\) −6.00000 −0.205798
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 − 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) −6.00000 −0.205316
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 52.0000 1.77422 0.887109 − 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 36.0000 1.22616
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) −14.0000 −0.475739
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 2.00000 0.0677285
\(873\) 42.0000 1.42148
\(874\) 4.00000 0.135302
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 − 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) −12.0000 −0.404980
\(879\) 0 0
\(880\) −4.00000 −0.134840
\(881\) −30.0000 −1.01073 −0.505363 − 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −3.00000 −0.101015
\(883\) −28.0000 −0.942275 −0.471138 − 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) −40.0000 −1.34307 −0.671534 − 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) −6.00000 −0.201120
\(891\) −36.0000 −1.20605
\(892\) 24.0000 0.803579
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 2.00000 0.0667409
\(899\) −8.00000 −0.266815
\(900\) −3.00000 −0.100000
\(901\) −84.0000 −2.79845
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) 32.0000 1.06254 0.531271 − 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) 4.00000 0.132745
\(909\) 42.0000 1.39305
\(910\) 2.00000 0.0662994
\(911\) −60.0000 −1.98789 −0.993944 − 0.109885i \(-0.964952\pi\)
−0.993944 + 0.109885i \(0.964952\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) 44.0000 1.45143 0.725713 − 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) −22.0000 −0.724531
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −24.0000 −0.788689
\(927\) −48.0000 −1.57653
\(928\) −2.00000 −0.0656532
\(929\) 42.0000 1.37798 0.688988 − 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) −14.0000 −0.458585
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 24.0000 0.784884
\(936\) 6.00000 0.196116
\(937\) 10.0000 0.326686 0.163343 − 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) 14.0000 0.456387 0.228193 − 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 32.0000 1.04041
\(947\) 36.0000 1.16984 0.584921 − 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 6.00000 0.194461
\(953\) 30.0000 0.971795 0.485898 − 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) −42.0000 −1.35980
\(955\) −12.0000 −0.388311
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 4.00000 0.128965
\(963\) 0 0
\(964\) −14.0000 −0.450910
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) −20.0000 −0.641831 −0.320915 − 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) −34.0000 −1.08776 −0.543878 − 0.839164i \(-0.683045\pi\)
−0.543878 + 0.839164i \(0.683045\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 1.00000 0.0319438
\(981\) −6.00000 −0.191565
\(982\) 20.0000 0.638226
\(983\) 24.0000 0.765481 0.382741 − 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 8.00000 0.254385
\(990\) 12.0000 0.381385
\(991\) −24.0000 −0.762385 −0.381193 − 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 54.0000 1.71020 0.855099 − 0.518465i \(-0.173497\pi\)
0.855099 + 0.518465i \(0.173497\pi\)
\(998\) −28.0000 −0.886325
\(999\) 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 1610.2.a.g.1.1 ✓ 1
5.4 even 2 8050.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1610.2.a.g.1.1 ✓ 1 1.1 even 1 trivial
8050.2.a.g.1.1 1 5.4 even 2