Properties

Label 2100.2.k.d.1849.1
Level $2100$
Weight $2$
Character 2100.1849
Analytic conductor $16.769$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1849,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.k (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1849
Dual form 2100.2.k.d.1849.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{13} -6.00000 q^{19} +1.00000 q^{21} +1.00000i q^{23} +1.00000i q^{27} -1.00000 q^{29} -2.00000 q^{31} +1.00000i q^{33} +7.00000i q^{37} -2.00000 q^{39} -8.00000 q^{41} -1.00000i q^{43} +2.00000i q^{47} -1.00000 q^{49} -14.0000i q^{53} +6.00000i q^{57} -10.0000 q^{59} -1.00000i q^{63} +3.00000i q^{67} +1.00000 q^{69} -9.00000 q^{71} -1.00000i q^{77} -1.00000 q^{79} +1.00000 q^{81} +2.00000i q^{83} +1.00000i q^{87} -2.00000 q^{89} +2.00000 q^{91} +2.00000i q^{93} +10.0000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} - 2 q^{11} - 12 q^{19} + 2 q^{21} - 2 q^{29} - 4 q^{31} - 4 q^{39} - 16 q^{41} - 2 q^{49} - 20 q^{59} + 2 q^{69} - 18 q^{71} - 2 q^{79} + 2 q^{81} - 4 q^{89} + 4 q^{91} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000i 0.208514i 0.994550 + 0.104257i \(0.0332465\pi\)
−0.994550 + 0.104257i \(0.966753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 1.00000i 0.174078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.00000i 1.15079i 0.817875 + 0.575396i \(0.195152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 14.0000i − 1.92305i −0.274721 0.961524i \(-0.588586\pi\)
0.274721 0.961524i \(-0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.00000i 0.366508i 0.983066 + 0.183254i \(0.0586631\pi\)
−0.983066 + 0.183254i \(0.941337\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −9.00000 −1.06810 −0.534052 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.00000i − 0.113961i
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000i 0.107211i
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 2.00000i 0.207390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 2.00000i 0.197066i 0.995134 + 0.0985329i \(0.0314150\pi\)
−0.995134 + 0.0985329i \(0.968585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) 0 0
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) − 9.00000i − 0.846649i −0.905978 0.423324i \(-0.860863\pi\)
0.905978 0.423324i \(-0.139137\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 8.00000i 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.0000i 0.976092i 0.872818 + 0.488046i \(0.162290\pi\)
−0.872818 + 0.488046i \(0.837710\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) − 6.00000i − 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 0 0
\(143\) 2.00000i 0.167248i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) 21.0000 1.72039 0.860194 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 0 0
\(151\) 3.00000 0.244137 0.122068 0.992522i \(-0.461047\pi\)
0.122068 + 0.992522i \(0.461047\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.00000i 0.319235i 0.987179 + 0.159617i \(0.0510260\pi\)
−0.987179 + 0.159617i \(0.948974\pi\)
\(158\) 0 0
\(159\) −14.0000 −1.11027
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 0 0
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 10.0000i 0.751646i
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) − 5.00000i − 0.359908i −0.983675 0.179954i \(-0.942405\pi\)
0.983675 0.179954i \(-0.0575949\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 11.0000i − 0.783718i −0.920025 0.391859i \(-0.871832\pi\)
0.920025 0.391859i \(-0.128168\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) 0 0
\(203\) − 1.00000i − 0.0701862i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.00000i − 0.0695048i
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 9.00000i 0.616670i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 2.00000i − 0.135769i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 12.0000i 0.803579i 0.915732 + 0.401790i \(0.131612\pi\)
−0.915732 + 0.401790i \(0.868388\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 26.0000i − 1.72568i −0.505477 0.862840i \(-0.668683\pi\)
0.505477 0.862840i \(-0.331317\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) 9.00000i 0.589610i 0.955557 + 0.294805i \(0.0952546\pi\)
−0.955557 + 0.294805i \(0.904745\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.00000i 0.0649570i
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 12.0000i 0.763542i
\(248\) 0 0
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) − 1.00000i − 0.0628695i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 22.0000i − 1.37232i −0.727450 0.686161i \(-0.759294\pi\)
0.727450 0.686161i \(-0.240706\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) − 9.00000i − 0.554964i −0.960731 0.277482i \(-0.910500\pi\)
0.960731 0.277482i \(-0.0894999\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.00000i 0.122398i
\(268\) 0 0
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) 0 0
\(273\) − 2.00000i − 0.121046i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 17.0000 1.01413 0.507067 0.861906i \(-0.330729\pi\)
0.507067 + 0.861906i \(0.330729\pi\)
\(282\) 0 0
\(283\) 10.0000i 0.594438i 0.954809 + 0.297219i \(0.0960592\pi\)
−0.954809 + 0.297219i \(0.903941\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 8.00000i − 0.472225i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) − 18.0000i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1.00000i − 0.0580259i
\(298\) 0 0
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) 12.0000i 0.689382i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 20.0000i − 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 0 0
\(309\) 2.00000 0.113776
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 15.0000i − 0.842484i −0.906948 0.421242i \(-0.861594\pi\)
0.906948 0.421242i \(-0.138406\pi\)
\(318\) 0 0
\(319\) 1.00000 0.0559893
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 9.00000i 0.497701i
\(328\) 0 0
\(329\) −2.00000 −0.110264
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) 0 0
\(333\) − 7.00000i − 0.383598i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.0000i 0.980522i 0.871576 + 0.490261i \(0.163099\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 23.0000i − 1.23470i −0.786687 0.617352i \(-0.788205\pi\)
0.786687 0.617352i \(-0.211795\pi\)
\(348\) 0 0
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.0000 1.53056 0.765281 0.643697i \(-0.222600\pi\)
0.765281 + 0.643697i \(0.222600\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 10.0000i 0.524864i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) 0 0
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) 14.0000 0.726844
\(372\) 0 0
\(373\) 15.0000i 0.776671i 0.921518 + 0.388335i \(0.126950\pi\)
−0.921518 + 0.388335i \(0.873050\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.00000i 0.103005i
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 11.0000 0.563547
\(382\) 0 0
\(383\) 14.0000i 0.715367i 0.933843 + 0.357683i \(0.116433\pi\)
−0.933843 + 0.357683i \(0.883567\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.00000i 0.0508329i
\(388\) 0 0
\(389\) −25.0000 −1.26755 −0.633775 0.773517i \(-0.718496\pi\)
−0.633775 + 0.773517i \(0.718496\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 8.00000i 0.403547i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 26.0000i − 1.30490i −0.757831 0.652451i \(-0.773741\pi\)
0.757831 0.652451i \(-0.226259\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) 1.00000 0.0499376 0.0249688 0.999688i \(-0.492051\pi\)
0.0249688 + 0.999688i \(0.492051\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 7.00000i − 0.346977i
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) − 10.0000i − 0.492068i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.00000i − 0.195881i
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) 31.0000 1.51085 0.755424 0.655237i \(-0.227431\pi\)
0.755424 + 0.655237i \(0.227431\pi\)
\(422\) 0 0
\(423\) − 2.00000i − 0.0972433i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 4.00000i 0.192228i 0.995370 + 0.0961139i \(0.0306413\pi\)
−0.995370 + 0.0961139i \(0.969359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 6.00000i − 0.287019i
\(438\) 0 0
\(439\) −38.0000 −1.81364 −0.906821 0.421517i \(-0.861498\pi\)
−0.906821 + 0.421517i \(0.861498\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 40.0000i 1.90046i 0.311553 + 0.950229i \(0.399151\pi\)
−0.311553 + 0.950229i \(0.600849\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 21.0000i − 0.993266i
\(448\) 0 0
\(449\) 25.0000 1.17982 0.589911 0.807468i \(-0.299163\pi\)
0.589911 + 0.807468i \(0.299163\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) − 3.00000i − 0.140952i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 33.0000i 1.54367i 0.635820 + 0.771837i \(0.280662\pi\)
−0.635820 + 0.771837i \(0.719338\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 30.0000i − 1.38823i −0.719862 0.694117i \(-0.755795\pi\)
0.719862 0.694117i \(-0.244205\pi\)
\(468\) 0 0
\(469\) −3.00000 −0.138527
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 1.00000i 0.0459800i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 14.0000i 0.641016i
\(478\) 0 0
\(479\) 2.00000 0.0913823 0.0456912 0.998956i \(-0.485451\pi\)
0.0456912 + 0.998956i \(0.485451\pi\)
\(480\) 0 0
\(481\) 14.0000 0.638345
\(482\) 0 0
\(483\) 1.00000i 0.0455016i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 23.0000i − 1.04223i −0.853487 0.521115i \(-0.825516\pi\)
0.853487 0.521115i \(-0.174484\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −13.0000 −0.586682 −0.293341 0.956008i \(-0.594767\pi\)
−0.293341 + 0.956008i \(0.594767\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 9.00000i − 0.403705i
\(498\) 0 0
\(499\) 44.0000 1.96971 0.984855 0.173379i \(-0.0554684\pi\)
0.984855 + 0.173379i \(0.0554684\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 0 0
\(503\) − 6.00000i − 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) 0 0
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) − 6.00000i − 0.264906i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.00000i − 0.0879599i
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 0 0
\(533\) 16.0000i 0.693037i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 4.00000i − 0.172613i
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) 0 0
\(543\) − 12.0000i − 0.514969i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 37.0000i − 1.58201i −0.611812 0.791003i \(-0.709559\pi\)
0.611812 0.791003i \(-0.290441\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) − 1.00000i − 0.0425243i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 3.00000i − 0.127114i −0.997978 0.0635570i \(-0.979756\pi\)
0.997978 0.0635570i \(-0.0202445\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.0000i 0.421450i 0.977545 + 0.210725i \(0.0675824\pi\)
−0.977545 + 0.210725i \(0.932418\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 39.0000 1.63497 0.817483 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) 0 0
\(571\) 25.0000 1.04622 0.523109 0.852266i \(-0.324772\pi\)
0.523109 + 0.852266i \(0.324772\pi\)
\(572\) 0 0
\(573\) 24.0000i 1.00261i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 22.0000i − 0.915872i −0.888985 0.457936i \(-0.848589\pi\)
0.888985 0.457936i \(-0.151411\pi\)
\(578\) 0 0
\(579\) −5.00000 −0.207793
\(580\) 0 0
\(581\) −2.00000 −0.0829740
\(582\) 0 0
\(583\) 14.0000i 0.579821i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −11.0000 −0.452480
\(592\) 0 0
\(593\) − 8.00000i − 0.328521i −0.986417 0.164260i \(-0.947476\pi\)
0.986417 0.164260i \(-0.0525237\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 24.0000i 0.982255i
\(598\) 0 0
\(599\) −5.00000 −0.204294 −0.102147 0.994769i \(-0.532571\pi\)
−0.102147 + 0.994769i \(0.532571\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) − 3.00000i − 0.122169i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) 0 0
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) − 23.0000i − 0.928961i −0.885583 0.464481i \(-0.846241\pi\)
0.885583 0.464481i \(-0.153759\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 13.0000i − 0.523360i −0.965155 0.261680i \(-0.915723\pi\)
0.965155 0.261680i \(-0.0842766\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) − 2.00000i − 0.0801283i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 6.00000i − 0.239617i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −35.0000 −1.39333 −0.696664 0.717398i \(-0.745333\pi\)
−0.696664 + 0.717398i \(0.745333\pi\)
\(632\) 0 0
\(633\) 4.00000i 0.158986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) 9.00000 0.356034
\(640\) 0 0
\(641\) −3.00000 −0.118493 −0.0592464 0.998243i \(-0.518870\pi\)
−0.0592464 + 0.998243i \(0.518870\pi\)
\(642\) 0 0
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.0000i 0.550397i 0.961387 + 0.275198i \(0.0887435\pi\)
−0.961387 + 0.275198i \(0.911256\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) −2.00000 −0.0783862
\(652\) 0 0
\(653\) − 2.00000i − 0.0782660i −0.999234 0.0391330i \(-0.987540\pi\)
0.999234 0.0391330i \(-0.0124596\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 1.00000i − 0.0387202i
\(668\) 0 0
\(669\) 12.0000 0.463947
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 38.0000i − 1.46046i −0.683202 0.730229i \(-0.739413\pi\)
0.683202 0.730229i \(-0.260587\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) −26.0000 −0.996322
\(682\) 0 0
\(683\) 7.00000i 0.267848i 0.990992 + 0.133924i \(0.0427577\pi\)
−0.990992 + 0.133924i \(0.957242\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) 0 0
\(689\) −28.0000 −1.06672
\(690\) 0 0
\(691\) −24.0000 −0.913003 −0.456502 0.889723i \(-0.650898\pi\)
−0.456502 + 0.889723i \(0.650898\pi\)
\(692\) 0 0
\(693\) 1.00000i 0.0379869i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 9.00000 0.340411
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) − 42.0000i − 1.58406i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 12.0000i − 0.451306i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 1.00000 0.0375029
\(712\) 0 0
\(713\) − 2.00000i − 0.0749006i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 12.0000i − 0.448148i
\(718\) 0 0
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) −2.00000 −0.0744839
\(722\) 0 0
\(723\) − 10.0000i − 0.371904i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) − 4.00000i − 0.147743i −0.997268 0.0738717i \(-0.976464\pi\)
0.997268 0.0738717i \(-0.0235355\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 3.00000i − 0.110506i
\(738\) 0 0
\(739\) −47.0000 −1.72892 −0.864461 0.502699i \(-0.832340\pi\)
−0.864461 + 0.502699i \(0.832340\pi\)
\(740\) 0 0
\(741\) 12.0000 0.440831
\(742\) 0 0
\(743\) 4.00000i 0.146746i 0.997305 + 0.0733729i \(0.0233763\pi\)
−0.997305 + 0.0733729i \(0.976624\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 2.00000i − 0.0731762i
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 0 0
\(753\) 20.0000i 0.728841i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 43.0000i 1.56286i 0.623992 + 0.781431i \(0.285510\pi\)
−0.623992 + 0.781431i \(0.714490\pi\)
\(758\) 0 0
\(759\) −1.00000 −0.0362977
\(760\) 0 0
\(761\) 24.0000 0.869999 0.435000 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) 0 0
\(763\) − 9.00000i − 0.325822i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.0000i 0.722158i
\(768\) 0 0
\(769\) 44.0000 1.58668 0.793340 0.608778i \(-0.208340\pi\)
0.793340 + 0.608778i \(0.208340\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 0 0
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 7.00000i 0.251124i
\(778\) 0 0
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) 9.00000 0.322045
\(782\) 0 0
\(783\) − 1.00000i − 0.0357371i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 4.00000i − 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) 0 0
\(789\) −9.00000 −0.320408
\(790\) 0 0
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 36.0000i 1.27519i 0.770374 + 0.637593i \(0.220070\pi\)
−0.770374 + 0.637593i \(0.779930\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.0000i 0.492823i
\(808\) 0 0
\(809\) −51.0000 −1.79306 −0.896532 0.442978i \(-0.853922\pi\)
−0.896532 + 0.442978i \(0.853922\pi\)
\(810\) 0 0
\(811\) 30.0000 1.05344 0.526721 0.850038i \(-0.323421\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(812\) 0 0
\(813\) − 14.0000i − 0.491001i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6.00000i 0.209913i
\(818\) 0 0
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 9.00000i 0.313720i 0.987621 + 0.156860i \(0.0501372\pi\)
−0.987621 + 0.156860i \(0.949863\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 11.0000i − 0.382507i −0.981541 0.191254i \(-0.938745\pi\)
0.981541 0.191254i \(-0.0612553\pi\)
\(828\) 0 0
\(829\) −28.0000 −0.972480 −0.486240 0.873825i \(-0.661632\pi\)
−0.486240 + 0.873825i \(0.661632\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 2.00000i − 0.0691301i
\(838\) 0 0
\(839\) −32.0000 −1.10476 −0.552381 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) − 17.0000i − 0.585511i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 10.0000i − 0.343604i
\(848\) 0 0
\(849\) 10.0000 0.343199
\(850\) 0 0
\(851\) −7.00000 −0.239957
\(852\) 0 0
\(853\) 16.0000i 0.547830i 0.961754 + 0.273915i \(0.0883186\pi\)
−0.961754 + 0.273915i \(0.911681\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.0000i 1.09310i 0.837427 + 0.546550i \(0.184059\pi\)
−0.837427 + 0.546550i \(0.815941\pi\)
\(858\) 0 0
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) 31.0000i 1.05525i 0.849477 + 0.527626i \(0.176918\pi\)
−0.849477 + 0.527626i \(0.823082\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 17.0000i − 0.577350i
\(868\) 0 0
\(869\) 1.00000 0.0339227
\(870\) 0 0
\(871\) 6.00000 0.203302
\(872\) 0 0
\(873\) − 10.0000i − 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34.0000i 1.14810i 0.818821 + 0.574049i \(0.194628\pi\)
−0.818821 + 0.574049i \(0.805372\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 0 0
\(883\) − 51.0000i − 1.71629i −0.513410 0.858143i \(-0.671618\pi\)
0.513410 0.858143i \(-0.328382\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.0000i 0.537227i 0.963248 + 0.268614i \(0.0865655\pi\)
−0.963248 + 0.268614i \(0.913434\pi\)
\(888\) 0 0
\(889\) −11.0000 −0.368928
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) − 12.0000i − 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 2.00000i − 0.0667781i
\(898\) 0 0
\(899\) 2.00000 0.0667037
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) − 1.00000i − 0.0332779i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 44.0000i − 1.46100i −0.682915 0.730498i \(-0.739288\pi\)
0.682915 0.730498i \(-0.260712\pi\)
\(908\) 0 0
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −11.0000 −0.364446 −0.182223 0.983257i \(-0.558329\pi\)
−0.182223 + 0.983257i \(0.558329\pi\)
\(912\) 0 0
\(913\) − 2.00000i − 0.0661903i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 8.00000i − 0.264183i
\(918\) 0 0
\(919\) −9.00000 −0.296883 −0.148441 0.988921i \(-0.547426\pi\)
−0.148441 + 0.988921i \(0.547426\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) 18.0000i 0.592477i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 2.00000i − 0.0656886i
\(928\) 0 0
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 12.0000i 0.392862i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 48.0000i − 1.56809i −0.620703 0.784046i \(-0.713153\pi\)
0.620703 0.784046i \(-0.286847\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 0 0
\(943\) − 8.00000i − 0.260516i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.0000i 0.649913i 0.945729 + 0.324956i \(0.105350\pi\)
−0.945729 + 0.324956i \(0.894650\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −15.0000 −0.486408
\(952\) 0 0
\(953\) 43.0000i 1.39291i 0.717602 + 0.696453i \(0.245240\pi\)
−0.717602 + 0.696453i \(0.754760\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 1.00000i − 0.0323254i
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) − 4.00000i − 0.128898i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) 4.00000i 0.128234i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 51.0000i 1.63163i 0.578310 + 0.815817i \(0.303713\pi\)
−0.578310 + 0.815817i \(0.696287\pi\)
\(978\) 0 0
\(979\) 2.00000 0.0639203
\(980\) 0 0
\(981\) 9.00000 0.287348
\(982\) 0 0
\(983\) − 2.00000i − 0.0637901i −0.999491 0.0318950i \(-0.989846\pi\)
0.999491 0.0318950i \(-0.0101542\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2.00000i 0.0636607i
\(988\) 0 0
\(989\) 1.00000 0.0317982
\(990\) 0 0
\(991\) −35.0000 −1.11181 −0.555906 0.831245i \(-0.687628\pi\)
−0.555906 + 0.831245i \(0.687628\pi\)
\(992\) 0 0
\(993\) − 5.00000i − 0.158670i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10.0000i 0.316703i 0.987383 + 0.158352i \(0.0506179\pi\)
−0.987383 + 0.158352i \(0.949382\pi\)
\(998\) 0 0
\(999\) −7.00000 −0.221470
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 2100.2.k.d.1849.1 2
3.2 odd 2 6300.2.k.k.6049.2 2
5.2 odd 4 2100.2.a.c.1.1 1
5.3 odd 4 2100.2.a.p.1.1 yes 1
5.4 even 2 inner 2100.2.k.d.1849.2 2
15.2 even 4 6300.2.a.k.1.1 1
15.8 even 4 6300.2.a.z.1.1 1
15.14 odd 2 6300.2.k.k.6049.1 2
20.3 even 4 8400.2.a.h.1.1 1
20.7 even 4 8400.2.a.cp.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.a.c.1.1 1 5.2 odd 4
2100.2.a.p.1.1 yes 1 5.3 odd 4
2100.2.k.d.1849.1 2 1.1 even 1 trivial
2100.2.k.d.1849.2 2 5.4 even 2 inner
6300.2.a.k.1.1 1 15.2 even 4
6300.2.a.z.1.1 1 15.8 even 4
6300.2.k.k.6049.1 2 15.14 odd 2
6300.2.k.k.6049.2 2 3.2 odd 2
8400.2.a.h.1.1 1 20.3 even 4
8400.2.a.cp.1.1 1 20.7 even 4