Properties

Label 3648.2.a.q.1.1
Level $3648$
Weight $2$
Character 3648.1
Self dual yes
Analytic conductor $29.129$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3648,2,Mod(1,3648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3648, 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("3648.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.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: \(29.1294266574\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 228)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3648.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} +6.00000 q^{13} -3.00000 q^{15} -5.00000 q^{17} +1.00000 q^{19} +1.00000 q^{21} -4.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -6.00000 q^{31} +5.00000 q^{33} -3.00000 q^{35} +8.00000 q^{37} -6.00000 q^{39} -8.00000 q^{41} +9.00000 q^{43} +3.00000 q^{45} -1.00000 q^{47} -6.00000 q^{49} +5.00000 q^{51} -2.00000 q^{53} -15.0000 q^{55} -1.00000 q^{57} -8.00000 q^{59} -11.0000 q^{61} -1.00000 q^{63} +18.0000 q^{65} +4.00000 q^{69} +4.00000 q^{71} -11.0000 q^{73} -4.00000 q^{75} +5.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -15.0000 q^{85} +6.00000 q^{87} +10.0000 q^{89} -6.00000 q^{91} +6.00000 q^{93} +3.00000 q^{95} -10.0000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 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
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 9.00000 1.37249 0.686244 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) −1.00000 −0.145865 −0.0729325 0.997337i \(-0.523236\pi\)
−0.0729325 + 0.997337i \(0.523236\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 5.00000 0.700140
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −15.0000 −2.02260
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −11.0000 −1.40841 −0.704203 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 18.0000 2.23263
\(66\) 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
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) 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
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −15.0000 −1.62698
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 6.00000 0.622171
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) −12.0000 −1.11901
\(116\) 0 0
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) 0 0
\(129\) −9.00000 −0.792406
\(130\) 0 0
\(131\) 21.0000 1.83478 0.917389 0.397991i \(-0.130293\pi\)
0.917389 + 0.397991i \(0.130293\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) 0 0
\(141\) 1.00000 0.0842152
\(142\) 0 0
\(143\) −30.0000 −2.50873
\(144\) 0 0
\(145\) −18.0000 −1.49482
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) −11.0000 −0.901155 −0.450578 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) −18.0000 −1.44579
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 15.0000 1.16775
\(166\) 0 0
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −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
\(177\) 8.00000 0.601317
\(178\) 0 0
\(179\) −14.0000 −1.04641 −0.523205 0.852207i \(-0.675264\pi\)
−0.523205 + 0.852207i \(0.675264\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 11.0000 0.813143
\(184\) 0 0
\(185\) 24.0000 1.76452
\(186\) 0 0
\(187\) 25.0000 1.82818
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −7.00000 −0.506502 −0.253251 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) −18.0000 −1.28901
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −17.0000 −1.20510 −0.602549 0.798082i \(-0.705848\pi\)
−0.602549 + 0.798082i \(0.705848\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −24.0000 −1.67623
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) −4.00000 −0.274075
\(214\) 0 0
\(215\) 27.0000 1.84138
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) 11.0000 0.743311
\(220\) 0 0
\(221\) −30.0000 −2.01802
\(222\) 0 0
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) −10.0000 −0.663723 −0.331862 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) 0 0
\(229\) −13.0000 −0.859064 −0.429532 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −27.0000 −1.74648 −0.873242 0.487286i \(-0.837987\pi\)
−0.873242 + 0.487286i \(0.837987\pi\)
\(240\) 0 0
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −18.0000 −1.14998
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) 20.0000 1.25739
\(254\) 0 0
\(255\) 15.0000 0.939336
\(256\) 0 0
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 15.0000 0.924940 0.462470 0.886635i \(-0.346963\pi\)
0.462470 + 0.886635i \(0.346963\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 0 0
\(269\) 22.0000 1.34136 0.670682 0.741745i \(-0.266002\pi\)
0.670682 + 0.741745i \(0.266002\pi\)
\(270\) 0 0
\(271\) 4.00000 0.242983 0.121491 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) 0 0
\(273\) 6.00000 0.363137
\(274\) 0 0
\(275\) −20.0000 −1.20605
\(276\) 0 0
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) −11.0000 −0.653882 −0.326941 0.945045i \(-0.606018\pi\)
−0.326941 + 0.945045i \(0.606018\pi\)
\(284\) 0 0
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) −20.0000 −1.16841 −0.584206 0.811605i \(-0.698594\pi\)
−0.584206 + 0.811605i \(0.698594\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) −33.0000 −1.88957
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 11.0000 0.623753 0.311876 0.950123i \(-0.399043\pi\)
0.311876 + 0.950123i \(0.399043\pi\)
\(312\) 0 0
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) 30.0000 1.67968
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) 0 0
\(323\) −5.00000 −0.278207
\(324\) 0 0
\(325\) 24.0000 1.33128
\(326\) 0 0
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) 1.00000 0.0551318
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) 8.00000 0.438397
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 30.0000 1.62459
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) −11.0000 −0.590511 −0.295255 0.955418i \(-0.595405\pi\)
−0.295255 + 0.955418i \(0.595405\pi\)
\(348\) 0 0
\(349\) 35.0000 1.87351 0.936754 0.349990i \(-0.113815\pi\)
0.936754 + 0.349990i \(0.113815\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) −5.00000 −0.264628
\(358\) 0 0
\(359\) −3.00000 −0.158334 −0.0791670 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) −33.0000 −1.72730
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) −32.0000 −1.65690 −0.828449 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) 14.0000 0.719132 0.359566 0.933120i \(-0.382925\pi\)
0.359566 + 0.933120i \(0.382925\pi\)
\(380\) 0 0
\(381\) −14.0000 −0.717242
\(382\) 0 0
\(383\) −26.0000 −1.32854 −0.664269 0.747494i \(-0.731257\pi\)
−0.664269 + 0.747494i \(0.731257\pi\)
\(384\) 0 0
\(385\) 15.0000 0.764471
\(386\) 0 0
\(387\) 9.00000 0.457496
\(388\) 0 0
\(389\) 19.0000 0.963338 0.481669 0.876353i \(-0.340031\pi\)
0.481669 + 0.876353i \(0.340031\pi\)
\(390\) 0 0
\(391\) 20.0000 1.01144
\(392\) 0 0
\(393\) −21.0000 −1.05931
\(394\) 0 0
\(395\) 24.0000 1.20757
\(396\) 0 0
\(397\) 27.0000 1.35509 0.677546 0.735481i \(-0.263044\pi\)
0.677546 + 0.735481i \(0.263044\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) 0 0
\(403\) −36.0000 −1.79329
\(404\) 0 0
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) −40.0000 −1.98273
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) −3.00000 −0.147979
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 19.0000 0.930434
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) −1.00000 −0.0486217
\(424\) 0 0
\(425\) −20.0000 −0.970143
\(426\) 0 0
\(427\) 11.0000 0.532327
\(428\) 0 0
\(429\) 30.0000 1.44841
\(430\) 0 0
\(431\) −2.00000 −0.0963366 −0.0481683 0.998839i \(-0.515338\pi\)
−0.0481683 + 0.998839i \(0.515338\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) 18.0000 0.863034
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −15.0000 −0.712672 −0.356336 0.934358i \(-0.615974\pi\)
−0.356336 + 0.934358i \(0.615974\pi\)
\(444\) 0 0
\(445\) 30.0000 1.42214
\(446\) 0 0
\(447\) 11.0000 0.520282
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 0 0
\(453\) 16.0000 0.751746
\(454\) 0 0
\(455\) −18.0000 −0.843853
\(456\) 0 0
\(457\) 11.0000 0.514558 0.257279 0.966337i \(-0.417174\pi\)
0.257279 + 0.966337i \(0.417174\pi\)
\(458\) 0 0
\(459\) 5.00000 0.233380
\(460\) 0 0
\(461\) −3.00000 −0.139724 −0.0698620 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) −27.0000 −1.25480 −0.627398 0.778699i \(-0.715880\pi\)
−0.627398 + 0.778699i \(0.715880\pi\)
\(464\) 0 0
\(465\) 18.0000 0.834730
\(466\) 0 0
\(467\) 17.0000 0.786666 0.393333 0.919396i \(-0.371322\pi\)
0.393333 + 0.919396i \(0.371322\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) −45.0000 −2.06910
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 0 0
\(483\) −4.00000 −0.182006
\(484\) 0 0
\(485\) −30.0000 −1.36223
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 30.0000 1.35113
\(494\) 0 0
\(495\) −15.0000 −0.674200
\(496\) 0 0
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −13.0000 −0.581960 −0.290980 0.956729i \(-0.593981\pi\)
−0.290980 + 0.956729i \(0.593981\pi\)
\(500\) 0 0
\(501\) −18.0000 −0.804181
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 0 0
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 11.0000 0.486611
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 18.0000 0.793175
\(516\) 0 0
\(517\) 5.00000 0.219900
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 0 0
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) 30.0000 1.30682
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) −48.0000 −2.07911
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) 14.0000 0.604145
\(538\) 0 0
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) 25.0000 1.07483 0.537417 0.843317i \(-0.319400\pi\)
0.537417 + 0.843317i \(0.319400\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) 12.0000 0.514024
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) 0 0
\(549\) −11.0000 −0.469469
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) −24.0000 −1.01874
\(556\) 0 0
\(557\) 17.0000 0.720313 0.360157 0.932892i \(-0.382723\pi\)
0.360157 + 0.932892i \(0.382723\pi\)
\(558\) 0 0
\(559\) 54.0000 2.28396
\(560\) 0 0
\(561\) −25.0000 −1.05550
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −42.0000 −1.76695
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 7.00000 0.292429
\(574\) 0 0
\(575\) −16.0000 −0.667246
\(576\) 0 0
\(577\) −5.00000 −0.208153 −0.104076 0.994569i \(-0.533189\pi\)
−0.104076 + 0.994569i \(0.533189\pi\)
\(578\) 0 0
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 10.0000 0.414158
\(584\) 0 0
\(585\) 18.0000 0.744208
\(586\) 0 0
\(587\) 37.0000 1.52715 0.763577 0.645717i \(-0.223441\pi\)
0.763577 + 0.645717i \(0.223441\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 0 0
\(593\) 10.0000 0.410651 0.205325 0.978694i \(-0.434175\pi\)
0.205325 + 0.978694i \(0.434175\pi\)
\(594\) 0 0
\(595\) 15.0000 0.614940
\(596\) 0 0
\(597\) 17.0000 0.695764
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 42.0000 1.70754
\(606\) 0 0
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 19.0000 0.767403 0.383701 0.923457i \(-0.374649\pi\)
0.383701 + 0.923457i \(0.374649\pi\)
\(614\) 0 0
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) −33.0000 −1.32853 −0.664265 0.747497i \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 5.00000 0.199681
\(628\) 0 0
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) −5.00000 −0.199047 −0.0995234 0.995035i \(-0.531732\pi\)
−0.0995234 + 0.995035i \(0.531732\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) 42.0000 1.66672
\(636\) 0 0
\(637\) −36.0000 −1.42637
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −38.0000 −1.50091 −0.750455 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) 0 0
\(643\) 1.00000 0.0394362 0.0197181 0.999806i \(-0.493723\pi\)
0.0197181 + 0.999806i \(0.493723\pi\)
\(644\) 0 0
\(645\) −27.0000 −1.06312
\(646\) 0 0
\(647\) 9.00000 0.353827 0.176913 0.984226i \(-0.443389\pi\)
0.176913 + 0.984226i \(0.443389\pi\)
\(648\) 0 0
\(649\) 40.0000 1.57014
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) 21.0000 0.821794 0.410897 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(654\) 0 0
\(655\) 63.0000 2.46161
\(656\) 0 0
\(657\) −11.0000 −0.429151
\(658\) 0 0
\(659\) 22.0000 0.856998 0.428499 0.903542i \(-0.359042\pi\)
0.428499 + 0.903542i \(0.359042\pi\)
\(660\) 0 0
\(661\) 20.0000 0.777910 0.388955 0.921257i \(-0.372836\pi\)
0.388955 + 0.921257i \(0.372836\pi\)
\(662\) 0 0
\(663\) 30.0000 1.16510
\(664\) 0 0
\(665\) −3.00000 −0.116335
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 12.0000 0.463947
\(670\) 0 0
\(671\) 55.0000 2.12325
\(672\) 0 0
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 10.0000 0.383201
\(682\) 0 0
\(683\) 30.0000 1.14792 0.573959 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) 0 0
\(685\) 9.00000 0.343872
\(686\) 0 0
\(687\) 13.0000 0.495981
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 7.00000 0.266293 0.133146 0.991096i \(-0.457492\pi\)
0.133146 + 0.991096i \(0.457492\pi\)
\(692\) 0 0
\(693\) 5.00000 0.189934
\(694\) 0 0
\(695\) −57.0000 −2.16213
\(696\) 0 0
\(697\) 40.0000 1.51511
\(698\) 0 0
\(699\) −21.0000 −0.794293
\(700\) 0 0
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 3.00000 0.112987
\(706\) 0 0
\(707\) 2.00000 0.0752177
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) −90.0000 −3.36581
\(716\) 0 0
\(717\) 27.0000 1.00833
\(718\) 0 0
\(719\) 41.0000 1.52904 0.764521 0.644599i \(-0.222976\pi\)
0.764521 + 0.644599i \(0.222976\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) 0 0
\(723\) 20.0000 0.743808
\(724\) 0 0
\(725\) −24.0000 −0.891338
\(726\) 0 0
\(727\) 13.0000 0.482143 0.241072 0.970507i \(-0.422501\pi\)
0.241072 + 0.970507i \(0.422501\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −45.0000 −1.66439
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 0 0
\(735\) 18.0000 0.663940
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 37.0000 1.36107 0.680534 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −33.0000 −1.20903
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) −21.0000 −0.765283
\(754\) 0 0
\(755\) −48.0000 −1.74690
\(756\) 0 0
\(757\) −11.0000 −0.399802 −0.199901 0.979816i \(-0.564062\pi\)
−0.199901 + 0.979816i \(0.564062\pi\)
\(758\) 0 0
\(759\) −20.0000 −0.725954
\(760\) 0 0
\(761\) −13.0000 −0.471250 −0.235625 0.971844i \(-0.575714\pi\)
−0.235625 + 0.971844i \(0.575714\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 0 0
\(765\) −15.0000 −0.542326
\(766\) 0 0
\(767\) −48.0000 −1.73318
\(768\) 0 0
\(769\) 27.0000 0.973645 0.486822 0.873501i \(-0.338156\pi\)
0.486822 + 0.873501i \(0.338156\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) 0 0
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) 0 0
\(775\) −24.0000 −0.862105
\(776\) 0 0
\(777\) 8.00000 0.286998
\(778\) 0 0
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 54.0000 1.92734
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 0 0
\(789\) −15.0000 −0.534014
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) −66.0000 −2.34373
\(794\) 0 0
\(795\) 6.00000 0.212798
\(796\) 0 0
\(797\) −44.0000 −1.55856 −0.779280 0.626676i \(-0.784415\pi\)
−0.779280 + 0.626676i \(0.784415\pi\)
\(798\) 0 0
\(799\) 5.00000 0.176887
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) 55.0000 1.94091
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) −22.0000 −0.774437
\(808\) 0 0
\(809\) 45.0000 1.58212 0.791058 0.611741i \(-0.209531\pi\)
0.791058 + 0.611741i \(0.209531\pi\)
\(810\) 0 0
\(811\) 22.0000 0.772524 0.386262 0.922389i \(-0.373766\pi\)
0.386262 + 0.922389i \(0.373766\pi\)
\(812\) 0 0
\(813\) −4.00000 −0.140286
\(814\) 0 0
\(815\) −48.0000 −1.68137
\(816\) 0 0
\(817\) 9.00000 0.314870
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 21.0000 0.732905 0.366453 0.930437i \(-0.380572\pi\)
0.366453 + 0.930437i \(0.380572\pi\)
\(822\) 0 0
\(823\) −25.0000 −0.871445 −0.435723 0.900081i \(-0.643507\pi\)
−0.435723 + 0.900081i \(0.643507\pi\)
\(824\) 0 0
\(825\) 20.0000 0.696311
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) 1.00000 0.0346896
\(832\) 0 0
\(833\) 30.0000 1.03944
\(834\) 0 0
\(835\) 54.0000 1.86875
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 0 0
\(839\) 54.0000 1.86429 0.932144 0.362089i \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 14.0000 0.482186
\(844\) 0 0
\(845\) 69.0000 2.37367
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 0 0
\(849\) 11.0000 0.377519
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) 0 0
\(855\) 3.00000 0.102598
\(856\) 0 0
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) −11.0000 −0.375315 −0.187658 0.982235i \(-0.560090\pi\)
−0.187658 + 0.982235i \(0.560090\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) 28.0000 0.953131 0.476566 0.879139i \(-0.341881\pi\)
0.476566 + 0.879139i \(0.341881\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −40.0000 −1.35691
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 0 0
\(879\) 20.0000 0.674583
\(880\) 0 0
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 0 0
\(883\) −3.00000 −0.100958 −0.0504790 0.998725i \(-0.516075\pi\)
−0.0504790 + 0.998725i \(0.516075\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) −1.00000 −0.0334637
\(894\) 0 0
\(895\) −42.0000 −1.40391
\(896\) 0 0
\(897\) 24.0000 0.801337
\(898\) 0 0
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) 10.0000 0.333148
\(902\) 0 0
\(903\) 9.00000 0.299501
\(904\) 0 0
\(905\) 18.0000 0.598340
\(906\) 0 0
\(907\) −14.0000 −0.464862 −0.232431 0.972613i \(-0.574668\pi\)
−0.232431 + 0.972613i \(0.574668\pi\)
\(908\) 0 0
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 42.0000 1.39152 0.695761 0.718273i \(-0.255067\pi\)
0.695761 + 0.718273i \(0.255067\pi\)
\(912\) 0 0
\(913\) 20.0000 0.661903
\(914\) 0 0
\(915\) 33.0000 1.09095
\(916\) 0 0
\(917\) −21.0000 −0.693481
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 32.0000 1.05215
\(926\) 0 0
\(927\) 6.00000 0.197066
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) −11.0000 −0.360124
\(934\) 0 0
\(935\) 75.0000 2.45276
\(936\) 0 0
\(937\) 29.0000 0.947389 0.473694 0.880689i \(-0.342920\pi\)
0.473694 + 0.880689i \(0.342920\pi\)
\(938\) 0 0
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) −66.0000 −2.14245
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) −21.0000 −0.679544
\(956\) 0 0
\(957\) −30.0000 −0.969762
\(958\) 0 0
\(959\) −3.00000 −0.0968751
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 2.00000 0.0644491
\(964\) 0 0
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) 0 0
\(969\) 5.00000 0.160623
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 19.0000 0.609112
\(974\) 0 0
\(975\) −24.0000 −0.768615
\(976\) 0 0
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) −50.0000 −1.59801
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) −1.00000 −0.0318304
\(988\) 0 0
\(989\) −36.0000 −1.14473
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) −51.0000 −1.61681
\(996\) 0 0
\(997\) −29.0000 −0.918439 −0.459220 0.888323i \(-0.651871\pi\)
−0.459220 + 0.888323i \(0.651871\pi\)
\(998\) 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 3648.2.a.q.1.1 1
4.3 odd 2 3648.2.a.bi.1.1 1
8.3 odd 2 228.2.a.a.1.1 1
8.5 even 2 912.2.a.f.1.1 1
24.5 odd 2 2736.2.a.t.1.1 1
24.11 even 2 684.2.a.c.1.1 1
40.3 even 4 5700.2.f.a.3649.1 2
40.19 odd 2 5700.2.a.l.1.1 1
40.27 even 4 5700.2.f.a.3649.2 2
152.75 even 2 4332.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
228.2.a.a.1.1 1 8.3 odd 2
684.2.a.c.1.1 1 24.11 even 2
912.2.a.f.1.1 1 8.5 even 2
2736.2.a.t.1.1 1 24.5 odd 2
3648.2.a.q.1.1 1 1.1 even 1 trivial
3648.2.a.bi.1.1 1 4.3 odd 2
4332.2.a.b.1.1 1 152.75 even 2
5700.2.a.l.1.1 1 40.19 odd 2
5700.2.f.a.3649.1 2 40.3 even 4
5700.2.f.a.3649.2 2 40.27 even 4