Properties

Label 6288.2.a.i.1.1
Level $6288$
Weight $2$
Character 6288.1
Self dual yes
Analytic conductor $50.210$
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] = [6288,2,Mod(1,6288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6288.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6288 = 2^{4} \cdot 3 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6288.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: \(50.2099327910\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 786)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6288.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} +3.00000 q^{13} -2.00000 q^{15} -5.00000 q^{17} -1.00000 q^{19} -2.00000 q^{21} -4.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} +9.00000 q^{29} +5.00000 q^{31} +3.00000 q^{33} +4.00000 q^{35} -8.00000 q^{37} +3.00000 q^{39} -12.0000 q^{41} +6.00000 q^{43} -2.00000 q^{45} +8.00000 q^{47} -3.00000 q^{49} -5.00000 q^{51} +12.0000 q^{53} -6.00000 q^{55} -1.00000 q^{57} +5.00000 q^{59} -3.00000 q^{61} -2.00000 q^{63} -6.00000 q^{65} -4.00000 q^{69} +8.00000 q^{71} -2.00000 q^{73} -1.00000 q^{75} -6.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +14.0000 q^{83} +10.0000 q^{85} +9.00000 q^{87} -14.0000 q^{89} -6.00000 q^{91} +5.00000 q^{93} +2.00000 q^{95} +12.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.0000 1.53670 0.768350 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 0 0
\(85\) 10.0000 1.08465
\(86\) 0 0
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −13.0000 −1.24517 −0.622587 0.782551i \(-0.713918\pi\)
−0.622587 + 0.782551i \(0.713918\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 0 0
\(117\) 3.00000 0.277350
\(118\) 0 0
\(119\) 10.0000 0.916698
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −15.0000 −1.33103 −0.665517 0.746382i \(-0.731789\pi\)
−0.665517 + 0.746382i \(0.731789\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 1.00000 0.0873704
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) −7.00000 −0.598050 −0.299025 0.954245i \(-0.596661\pi\)
−0.299025 + 0.954245i \(0.596661\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 9.00000 0.752618
\(144\) 0 0
\(145\) −18.0000 −1.49482
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 0 0
\(149\) 9.00000 0.737309 0.368654 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 0 0
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) −21.0000 −1.62503 −0.812514 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −11.0000 −0.836315 −0.418157 0.908375i \(-0.637324\pi\)
−0.418157 + 0.908375i \(0.637324\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 5.00000 0.375823
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) −3.00000 −0.221766
\(184\) 0 0
\(185\) 16.0000 1.17634
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 5.00000 0.361787 0.180894 0.983503i \(-0.442101\pi\)
0.180894 + 0.983503i \(0.442101\pi\)
\(192\) 0 0
\(193\) 19.0000 1.36765 0.683825 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) 0 0
\(195\) −6.00000 −0.429669
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 17.0000 1.20510 0.602549 0.798082i \(-0.294152\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −18.0000 −1.26335
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) −12.0000 −0.818393
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −15.0000 −1.00901
\(222\) 0 0
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) 20.0000 1.31024 0.655122 0.755523i \(-0.272617\pi\)
0.655122 + 0.755523i \(0.272617\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) −3.00000 −0.190885
\(248\) 0 0
\(249\) 14.0000 0.887214
\(250\) 0 0
\(251\) 22.0000 1.38863 0.694314 0.719672i \(-0.255708\pi\)
0.694314 + 0.719672i \(0.255708\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 0 0
\(255\) 10.0000 0.626224
\(256\) 0 0
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 0 0
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) −24.0000 −1.47431
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 0 0
\(269\) −8.00000 −0.487769 −0.243884 0.969804i \(-0.578422\pi\)
−0.243884 + 0.969804i \(0.578422\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) −6.00000 −0.363137
\(274\) 0 0
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) 3.00000 0.180253 0.0901263 0.995930i \(-0.471273\pi\)
0.0901263 + 0.995930i \(0.471273\pi\)
\(278\) 0 0
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 0 0
\(293\) 29.0000 1.69420 0.847099 0.531435i \(-0.178347\pi\)
0.847099 + 0.531435i \(0.178347\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) −10.0000 −0.570730 −0.285365 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 33.0000 1.87126 0.935629 0.352985i \(-0.114833\pi\)
0.935629 + 0.352985i \(0.114833\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 27.0000 1.51171
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 5.00000 0.278207
\(324\) 0 0
\(325\) −3.00000 −0.166410
\(326\) 0 0
\(327\) −13.0000 −0.718902
\(328\) 0 0
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 0 0
\(333\) −8.00000 −0.438397
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.00000 0.0544735 0.0272367 0.999629i \(-0.491329\pi\)
0.0272367 + 0.999629i \(0.491329\pi\)
\(338\) 0 0
\(339\) 4.00000 0.217250
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 8.00000 0.430706
\(346\) 0 0
\(347\) 30.0000 1.61048 0.805242 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) 0 0
\(353\) 28.0000 1.49029 0.745145 0.666903i \(-0.232380\pi\)
0.745145 + 0.666903i \(0.232380\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) 0 0
\(357\) 10.0000 0.529256
\(358\) 0 0
\(359\) 14.0000 0.738892 0.369446 0.929252i \(-0.379548\pi\)
0.369446 + 0.929252i \(0.379548\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 27.0000 1.39057
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) −15.0000 −0.768473
\(382\) 0 0
\(383\) 9.00000 0.459879 0.229939 0.973205i \(-0.426147\pi\)
0.229939 + 0.973205i \(0.426147\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) 6.00000 0.304997
\(388\) 0 0
\(389\) −21.0000 −1.06474 −0.532371 0.846511i \(-0.678699\pi\)
−0.532371 + 0.846511i \(0.678699\pi\)
\(390\) 0 0
\(391\) 20.0000 1.01144
\(392\) 0 0
\(393\) 1.00000 0.0504433
\(394\) 0 0
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) −17.0000 −0.848939 −0.424470 0.905442i \(-0.639539\pi\)
−0.424470 + 0.905442i \(0.639539\pi\)
\(402\) 0 0
\(403\) 15.0000 0.747203
\(404\) 0 0
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −7.00000 −0.345285
\(412\) 0 0
\(413\) −10.0000 −0.492068
\(414\) 0 0
\(415\) −28.0000 −1.37447
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −27.0000 −1.31590 −0.657950 0.753062i \(-0.728576\pi\)
−0.657950 + 0.753062i \(0.728576\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) 5.00000 0.242536
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 0 0
\(429\) 9.00000 0.434524
\(430\) 0 0
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 0 0
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) 0 0
\(435\) −18.0000 −0.863034
\(436\) 0 0
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) 28.0000 1.32733
\(446\) 0 0
\(447\) 9.00000 0.425685
\(448\) 0 0
\(449\) 23.0000 1.08544 0.542719 0.839915i \(-0.317395\pi\)
0.542719 + 0.839915i \(0.317395\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 0 0
\(453\) 12.0000 0.563809
\(454\) 0 0
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) −5.00000 −0.233380
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −13.0000 −0.604161 −0.302081 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(464\) 0 0
\(465\) −10.0000 −0.463739
\(466\) 0 0
\(467\) −25.0000 −1.15686 −0.578431 0.815731i \(-0.696335\pi\)
−0.578431 + 0.815731i \(0.696335\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 18.0000 0.827641
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 0 0
\(483\) 8.00000 0.364013
\(484\) 0 0
\(485\) −24.0000 −1.08978
\(486\) 0 0
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 0 0
\(489\) 1.00000 0.0452216
\(490\) 0 0
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) 0 0
\(493\) −45.0000 −2.02670
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 0 0
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 0 0
\(501\) −21.0000 −0.938211
\(502\) 0 0
\(503\) 2.00000 0.0891756 0.0445878 0.999005i \(-0.485803\pi\)
0.0445878 + 0.999005i \(0.485803\pi\)
\(504\) 0 0
\(505\) −28.0000 −1.24598
\(506\) 0 0
\(507\) −4.00000 −0.177646
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −11.0000 −0.482846
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) 0 0
\(523\) 23.0000 1.00572 0.502860 0.864368i \(-0.332281\pi\)
0.502860 + 0.864368i \(0.332281\pi\)
\(524\) 0 0
\(525\) 2.00000 0.0872872
\(526\) 0 0
\(527\) −25.0000 −1.08902
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 5.00000 0.216982
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) 4.00000 0.172613
\(538\) 0 0
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) 26.0000 1.11372
\(546\) 0 0
\(547\) 1.00000 0.0427569 0.0213785 0.999771i \(-0.493195\pi\)
0.0213785 + 0.999771i \(0.493195\pi\)
\(548\) 0 0
\(549\) −3.00000 −0.128037
\(550\) 0 0
\(551\) −9.00000 −0.383413
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) 16.0000 0.679162
\(556\) 0 0
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 18.0000 0.761319
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −8.00000 −0.336563
\(566\) 0 0
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −23.0000 −0.962520 −0.481260 0.876578i \(-0.659821\pi\)
−0.481260 + 0.876578i \(0.659821\pi\)
\(572\) 0 0
\(573\) 5.00000 0.208878
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 5.00000 0.208153 0.104076 0.994569i \(-0.466811\pi\)
0.104076 + 0.994569i \(0.466811\pi\)
\(578\) 0 0
\(579\) 19.0000 0.789613
\(580\) 0 0
\(581\) −28.0000 −1.16164
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) 33.0000 1.36206 0.681028 0.732257i \(-0.261533\pi\)
0.681028 + 0.732257i \(0.261533\pi\)
\(588\) 0 0
\(589\) −5.00000 −0.206021
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 19.0000 0.780236 0.390118 0.920765i \(-0.372434\pi\)
0.390118 + 0.920765i \(0.372434\pi\)
\(594\) 0 0
\(595\) −20.0000 −0.819920
\(596\) 0 0
\(597\) 17.0000 0.695764
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.00000 0.162623
\(606\) 0 0
\(607\) −44.0000 −1.78590 −0.892952 0.450151i \(-0.851370\pi\)
−0.892952 + 0.450151i \(0.851370\pi\)
\(608\) 0 0
\(609\) −18.0000 −0.729397
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) −3.00000 −0.121169 −0.0605844 0.998163i \(-0.519296\pi\)
−0.0605844 + 0.998163i \(0.519296\pi\)
\(614\) 0 0
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) 31.0000 1.24600 0.622998 0.782224i \(-0.285915\pi\)
0.622998 + 0.782224i \(0.285915\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 28.0000 1.12180
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −3.00000 −0.119808
\(628\) 0 0
\(629\) 40.0000 1.59490
\(630\) 0 0
\(631\) −26.0000 −1.03504 −0.517522 0.855670i \(-0.673145\pi\)
−0.517522 + 0.855670i \(0.673145\pi\)
\(632\) 0 0
\(633\) −2.00000 −0.0794929
\(634\) 0 0
\(635\) 30.0000 1.19051
\(636\) 0 0
\(637\) −9.00000 −0.356593
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −21.0000 −0.828159 −0.414080 0.910241i \(-0.635896\pi\)
−0.414080 + 0.910241i \(0.635896\pi\)
\(644\) 0 0
\(645\) −12.0000 −0.472500
\(646\) 0 0
\(647\) 31.0000 1.21874 0.609368 0.792888i \(-0.291423\pi\)
0.609368 + 0.792888i \(0.291423\pi\)
\(648\) 0 0
\(649\) 15.0000 0.588802
\(650\) 0 0
\(651\) −10.0000 −0.391931
\(652\) 0 0
\(653\) −38.0000 −1.48705 −0.743527 0.668705i \(-0.766849\pi\)
−0.743527 + 0.668705i \(0.766849\pi\)
\(654\) 0 0
\(655\) −2.00000 −0.0781465
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −13.0000 −0.506408 −0.253204 0.967413i \(-0.581484\pi\)
−0.253204 + 0.967413i \(0.581484\pi\)
\(660\) 0 0
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) 0 0
\(663\) −15.0000 −0.582552
\(664\) 0 0
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) −36.0000 −1.39393
\(668\) 0 0
\(669\) 19.0000 0.734582
\(670\) 0 0
\(671\) −9.00000 −0.347441
\(672\) 0 0
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 0 0
\(683\) −41.0000 −1.56882 −0.784411 0.620242i \(-0.787034\pi\)
−0.784411 + 0.620242i \(0.787034\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) 0 0
\(687\) 20.0000 0.763048
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) −6.00000 −0.227921
\(694\) 0 0
\(695\) 24.0000 0.910372
\(696\) 0 0
\(697\) 60.0000 2.27266
\(698\) 0 0
\(699\) 20.0000 0.756469
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) −16.0000 −0.602595
\(706\) 0 0
\(707\) −28.0000 −1.05305
\(708\) 0 0
\(709\) 18.0000 0.676004 0.338002 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) −20.0000 −0.749006
\(714\) 0 0
\(715\) −18.0000 −0.673162
\(716\) 0 0
\(717\) −8.00000 −0.298765
\(718\) 0 0
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 8.00000 0.297523
\(724\) 0 0
\(725\) −9.00000 −0.334252
\(726\) 0 0
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −30.0000 −1.10959
\(732\) 0 0
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 10.0000 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) 0 0
\(741\) −3.00000 −0.110208
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) 14.0000 0.512233
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −37.0000 −1.35015 −0.675075 0.737749i \(-0.735889\pi\)
−0.675075 + 0.737749i \(0.735889\pi\)
\(752\) 0 0
\(753\) 22.0000 0.801725
\(754\) 0 0
\(755\) −24.0000 −0.873449
\(756\) 0 0
\(757\) 19.0000 0.690567 0.345283 0.938498i \(-0.387783\pi\)
0.345283 + 0.938498i \(0.387783\pi\)
\(758\) 0 0
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −13.0000 −0.471250 −0.235625 0.971844i \(-0.575714\pi\)
−0.235625 + 0.971844i \(0.575714\pi\)
\(762\) 0 0
\(763\) 26.0000 0.941263
\(764\) 0 0
\(765\) 10.0000 0.361551
\(766\) 0 0
\(767\) 15.0000 0.541619
\(768\) 0 0
\(769\) −17.0000 −0.613036 −0.306518 0.951865i \(-0.599164\pi\)
−0.306518 + 0.951865i \(0.599164\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) 0 0
\(773\) −53.0000 −1.90628 −0.953139 0.302534i \(-0.902168\pi\)
−0.953139 + 0.302534i \(0.902168\pi\)
\(774\) 0 0
\(775\) −5.00000 −0.179605
\(776\) 0 0
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) 21.0000 0.747620
\(790\) 0 0
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) −9.00000 −0.319599
\(794\) 0 0
\(795\) −24.0000 −0.851192
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) 0 0
\(807\) −8.00000 −0.281613
\(808\) 0 0
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) 18.0000 0.632065 0.316033 0.948748i \(-0.397649\pi\)
0.316033 + 0.948748i \(0.397649\pi\)
\(812\) 0 0
\(813\) −4.00000 −0.140286
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) −6.00000 −0.209913
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 20.0000 0.698005 0.349002 0.937122i \(-0.386521\pi\)
0.349002 + 0.937122i \(0.386521\pi\)
\(822\) 0 0
\(823\) −47.0000 −1.63832 −0.819159 0.573567i \(-0.805559\pi\)
−0.819159 + 0.573567i \(0.805559\pi\)
\(824\) 0 0
\(825\) −3.00000 −0.104447
\(826\) 0 0
\(827\) −33.0000 −1.14752 −0.573761 0.819023i \(-0.694516\pi\)
−0.573761 + 0.819023i \(0.694516\pi\)
\(828\) 0 0
\(829\) −15.0000 −0.520972 −0.260486 0.965478i \(-0.583883\pi\)
−0.260486 + 0.965478i \(0.583883\pi\)
\(830\) 0 0
\(831\) 3.00000 0.104069
\(832\) 0 0
\(833\) 15.0000 0.519719
\(834\) 0 0
\(835\) 42.0000 1.45347
\(836\) 0 0
\(837\) 5.00000 0.172825
\(838\) 0 0
\(839\) 29.0000 1.00119 0.500596 0.865681i \(-0.333114\pi\)
0.500596 + 0.865681i \(0.333114\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) −10.0000 −0.344418
\(844\) 0 0
\(845\) 8.00000 0.275208
\(846\) 0 0
\(847\) 4.00000 0.137442
\(848\) 0 0
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) −44.0000 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) −45.0000 −1.53717 −0.768585 0.639747i \(-0.779039\pi\)
−0.768585 + 0.639747i \(0.779039\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) −5.00000 −0.170202 −0.0851010 0.996372i \(-0.527121\pi\)
−0.0851010 + 0.996372i \(0.527121\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) 0 0
\(879\) 29.0000 0.978146
\(880\) 0 0
\(881\) −50.0000 −1.68454 −0.842271 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) 0 0
\(883\) 17.0000 0.572096 0.286048 0.958215i \(-0.407658\pi\)
0.286048 + 0.958215i \(0.407658\pi\)
\(884\) 0 0
\(885\) −10.0000 −0.336146
\(886\) 0 0
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 30.0000 1.00617
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 0 0
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) 0 0
\(899\) 45.0000 1.50083
\(900\) 0 0
\(901\) −60.0000 −1.99889
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) 0 0
\(905\) −40.0000 −1.32964
\(906\) 0 0
\(907\) 34.0000 1.12895 0.564476 0.825450i \(-0.309078\pi\)
0.564476 + 0.825450i \(0.309078\pi\)
\(908\) 0 0
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −31.0000 −1.02708 −0.513538 0.858067i \(-0.671665\pi\)
−0.513538 + 0.858067i \(0.671665\pi\)
\(912\) 0 0
\(913\) 42.0000 1.39000
\(914\) 0 0
\(915\) 6.00000 0.198354
\(916\) 0 0
\(917\) −2.00000 −0.0660458
\(918\) 0 0
\(919\) −25.0000 −0.824674 −0.412337 0.911031i \(-0.635287\pi\)
−0.412337 + 0.911031i \(0.635287\pi\)
\(920\) 0 0
\(921\) −10.0000 −0.329511
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 0 0
\(933\) 33.0000 1.08037
\(934\) 0 0
\(935\) 30.0000 0.981105
\(936\) 0 0
\(937\) 17.0000 0.555366 0.277683 0.960673i \(-0.410434\pi\)
0.277683 + 0.960673i \(0.410434\pi\)
\(938\) 0 0
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) −49.0000 −1.59735 −0.798677 0.601760i \(-0.794466\pi\)
−0.798677 + 0.601760i \(0.794466\pi\)
\(942\) 0 0
\(943\) 48.0000 1.56310
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) 40.0000 1.29983 0.649913 0.760009i \(-0.274805\pi\)
0.649913 + 0.760009i \(0.274805\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 8.00000 0.259145 0.129573 0.991570i \(-0.458639\pi\)
0.129573 + 0.991570i \(0.458639\pi\)
\(954\) 0 0
\(955\) −10.0000 −0.323592
\(956\) 0 0
\(957\) 27.0000 0.872786
\(958\) 0 0
\(959\) 14.0000 0.452084
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) −3.00000 −0.0966736
\(964\) 0 0
\(965\) −38.0000 −1.22326
\(966\) 0 0
\(967\) −33.0000 −1.06121 −0.530604 0.847620i \(-0.678035\pi\)
−0.530604 + 0.847620i \(0.678035\pi\)
\(968\) 0 0
\(969\) 5.00000 0.160623
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 24.0000 0.769405
\(974\) 0 0
\(975\) −3.00000 −0.0960769
\(976\) 0 0
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) −42.0000 −1.34233
\(980\) 0 0
\(981\) −13.0000 −0.415058
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 46.0000 1.46124 0.730619 0.682785i \(-0.239232\pi\)
0.730619 + 0.682785i \(0.239232\pi\)
\(992\) 0 0
\(993\) 17.0000 0.539479
\(994\) 0 0
\(995\) −34.0000 −1.07787
\(996\) 0 0
\(997\) 35.0000 1.10846 0.554231 0.832363i \(-0.313013\pi\)
0.554231 + 0.832363i \(0.313013\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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 6288.2.a.i.1.1 1
4.3 odd 2 786.2.a.a.1.1 1
12.11 even 2 2358.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
786.2.a.a.1.1 1 4.3 odd 2
2358.2.a.w.1.1 1 12.11 even 2
6288.2.a.i.1.1 1 1.1 even 1 trivial