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Properties

Label 4200.2.a.ba.1.1
Level $4200$
Weight $2$
Character 4200.1
Self dual yes
Analytic conductor $33.537$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4200,2,Mod(1,4200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4200.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: \(33.5371688489\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} -2.00000 q^{13} -4.00000 q^{17} +6.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{27} +1.00000 q^{29} +2.00000 q^{31} +5.00000 q^{33} -5.00000 q^{37} -2.00000 q^{39} +7.00000 q^{43} +2.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} +6.00000 q^{53} +6.00000 q^{57} -6.00000 q^{59} -1.00000 q^{63} -3.00000 q^{67} +1.00000 q^{69} +9.00000 q^{71} +12.0000 q^{73} -5.00000 q^{77} +3.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +1.00000 q^{87} +6.00000 q^{89} +2.00000 q^{91} +2.00000 q^{93} +10.0000 q^{97} +5.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514 0.104257 − 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 − 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 7.00000 1.06749 0.533745 − 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 − 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 12.0000 1.40449 0.702247 − 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 3.00000 0.337526 0.168763 − 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 − 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 − 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) −5.00000 −0.474579
\(112\) 0 0
\(113\) 21.0000 1.97551 0.987757 − 0.156001i \(-0.0498603\pi\)
0.987757 + 0.156001i \(0.0498603\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 − 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) 7.00000 0.616316
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 0 0
\(143\) −10.0000 −0.836242
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 11.0000 0.901155 0.450578 − 0.892737i \(-0.351218\pi\)
0.450578 + 0.892737i \(0.351218\pi\)
\(150\) 0 0
\(151\) 17.0000 1.38344 0.691720 − 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 − 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.0000 −1.08335 −0.541676 − 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 0 0
\(173\) 22.0000 1.67263 0.836315 − 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 − 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 − 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −20.0000 −1.46254
\(188\) 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
\(193\) 13.0000 0.935760 0.467880 − 0.883792i \(-0.345018\pi\)
0.467880 + 0.883792i \(0.345018\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.00000 −0.213741 −0.106871 − 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −3.00000 −0.211604
\(202\) 0 0
\(203\) −1.00000 −0.0701862
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 30.0000 2.07514
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 9.00000 0.616670
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 − 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.0000 1.46019 0.730096 − 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) 3.00000 0.196537 0.0982683 − 0.995160i \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 3.00000 0.194871
\(238\) 0 0
\(239\) 28.0000 1.81117 0.905585 − 0.424165i \(-0.139432\pi\)
0.905585 + 0.424165i \(0.139432\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 − 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 5.00000 0.314347
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 5.00000 0.310685
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 23.0000 1.41824 0.709120 − 0.705087i \(-0.249092\pi\)
0.709120 + 0.705087i \(0.249092\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 − 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 − 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 5.00000 0.298275 0.149137 − 0.988816i \(-0.452350\pi\)
0.149137 + 0.988816i \(0.452350\pi\)
\(282\) 0 0
\(283\) −22.0000 −1.30776 −0.653882 − 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) −2.00000 −0.116841 −0.0584206 − 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −7.00000 −0.403473
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −24.0000 −1.36975 −0.684876 − 0.728659i \(-0.740144\pi\)
−0.684876 + 0.728659i \(0.740144\pi\)
\(308\) 0 0
\(309\) −10.0000 −0.568880
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 − 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 33.0000 1.85346 0.926732 − 0.375722i \(-0.122605\pi\)
0.926732 + 0.375722i \(0.122605\pi\)
\(318\) 0 0
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −11.0000 −0.608301
\(328\) 0 0
\(329\) −2.00000 −0.110264
\(330\) 0 0
\(331\) −5.00000 −0.274825 −0.137412 − 0.990514i \(-0.543879\pi\)
−0.137412 + 0.990514i \(0.543879\pi\)
\(332\) 0 0
\(333\) −5.00000 −0.273998
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 − 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 0 0
\(339\) 21.0000 1.14056
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 27.0000 1.44944 0.724718 − 0.689046i \(-0.241970\pi\)
0.724718 + 0.689046i \(0.241970\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) −27.0000 −1.42501 −0.712503 − 0.701669i \(-0.752438\pi\)
−0.712503 + 0.701669i \(0.752438\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 − 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) −3.00000 −0.155334 −0.0776671 − 0.996979i \(-0.524747\pi\)
−0.0776671 + 0.996979i \(0.524747\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 − 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) −7.00000 −0.358621
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 − 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.00000 0.355830
\(388\) 0 0
\(389\) 9.00000 0.456318 0.228159 − 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) 5.00000 0.249688 0.124844 − 0.992176i \(-0.460157\pi\)
0.124844 + 0.992176i \(0.460157\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −25.0000 −1.23920
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 − 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 − 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) −29.0000 −1.41337 −0.706687 − 0.707527i \(-0.749811\pi\)
−0.706687 + 0.707527i \(0.749811\pi\)
\(422\) 0 0
\(423\) 2.00000 0.0972433
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −10.0000 −0.482805
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 − 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 − 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.00000 0.287019
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 − 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 11.0000 0.520282
\(448\) 0 0
\(449\) 19.0000 0.896665 0.448333 − 0.893867i \(-0.352018\pi\)
0.448333 + 0.893867i \(0.352018\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 17.0000 0.798730
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15.0000 −0.701670 −0.350835 − 0.936437i \(-0.614102\pi\)
−0.350835 + 0.936437i \(0.614102\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 − 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 − 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) 3.00000 0.138527
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) 35.0000 1.60930
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 − 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −13.0000 −0.589086 −0.294543 − 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −31.0000 −1.39901 −0.699505 − 0.714628i \(-0.746596\pi\)
−0.699505 + 0.714628i \(0.746596\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.00000 −0.403705
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −14.0000 −0.625474
\(502\) 0 0
\(503\) −14.0000 −0.624229 −0.312115 − 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) 0 0
\(513\) 6.00000 0.264906
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 10.0000 0.439799
\(518\) 0 0
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 32.0000 1.39926 0.699631 − 0.714504i \(-0.253348\pi\)
0.699631 + 0.714504i \(0.253348\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 20.0000 0.863064
\(538\) 0 0
\(539\) 5.00000 0.215365
\(540\) 0 0
\(541\) 23.0000 0.988847 0.494424 − 0.869221i \(-0.335379\pi\)
0.494424 + 0.869221i \(0.335379\pi\)
\(542\) 0 0
\(543\) 16.0000 0.686626
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 13.0000 0.555840 0.277920 − 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) −3.00000 −0.127573
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3.00000 −0.127114 −0.0635570 − 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) 0 0
\(559\) −14.0000 −0.592137
\(560\) 0 0
\(561\) −20.0000 −0.844401
\(562\) 0 0
\(563\) −38.0000 −1.60151 −0.800755 − 0.598993i \(-0.795568\pi\)
−0.800755 + 0.598993i \(0.795568\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −27.0000 −1.13190 −0.565949 − 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) 0 0
\(571\) −1.00000 −0.0418487 −0.0209243 − 0.999781i \(-0.506661\pi\)
−0.0209243 + 0.999781i \(0.506661\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) 13.0000 0.540262
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 30.0000 1.24247
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.0000 0.660391 0.330195 − 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −3.00000 −0.123404
\(592\) 0 0
\(593\) 16.0000 0.657041 0.328521 − 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) −29.0000 −1.18491 −0.592454 − 0.805604i \(-0.701841\pi\)
−0.592454 + 0.805604i \(0.701841\pi\)
\(600\) 0 0
\(601\) 48.0000 1.95796 0.978980 − 0.203954i \(-0.0653794\pi\)
0.978980 + 0.203954i \(0.0653794\pi\)
\(602\) 0 0
\(603\) −3.00000 −0.122169
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −2.00000 −0.0811775 −0.0405887 − 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 0 0
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) −4.00000 −0.161823
\(612\) 0 0
\(613\) 35.0000 1.41364 0.706818 − 0.707395i \(-0.250130\pi\)
0.706818 + 0.707395i \(0.250130\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −17.0000 −0.684394 −0.342197 − 0.939628i \(-0.611171\pi\)
−0.342197 + 0.939628i \(0.611171\pi\)
\(618\) 0 0
\(619\) 22.0000 0.884255 0.442127 − 0.896952i \(-0.354224\pi\)
0.442127 + 0.896952i \(0.354224\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 30.0000 1.19808
\(628\) 0 0
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) 23.0000 0.915616 0.457808 − 0.889051i \(-0.348635\pi\)
0.457808 + 0.889051i \(0.348635\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 9.00000 0.356034
\(640\) 0 0
\(641\) 41.0000 1.61940 0.809701 − 0.586842i \(-0.199629\pi\)
0.809701 + 0.586842i \(0.199629\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 − 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) −30.0000 −1.17760
\(650\) 0 0
\(651\) −2.00000 −0.0783862
\(652\) 0 0
\(653\) 26.0000 1.01746 0.508729 − 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 − 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 − 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 8.00000 0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 − 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 22.0000 0.843042
\(682\) 0 0
\(683\) −5.00000 −0.191320 −0.0956598 − 0.995414i \(-0.530496\pi\)
−0.0956598 + 0.995414i \(0.530496\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −24.0000 −0.913003 −0.456502 − 0.889723i \(-0.650898\pi\)
−0.456502 + 0.889723i \(0.650898\pi\)
\(692\) 0 0
\(693\) −5.00000 −0.189934
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 3.00000 0.113470
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 − 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −30.0000 −1.13147
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 − 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) 3.00000 0.112509
\(712\) 0 0
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 28.0000 1.04568
\(718\) 0 0
\(719\) −36.0000 −1.34257 −0.671287 − 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 10.0000 0.372419
\(722\) 0 0
\(723\) −2.00000 −0.0743808
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −28.0000 −1.03562
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 − 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −15.0000 −0.552532
\(738\) 0 0
\(739\) −7.00000 −0.257499 −0.128750 − 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) −28.0000 −1.02722 −0.513610 − 0.858024i \(-0.671692\pi\)
−0.513610 + 0.858024i \(0.671692\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 − 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 15.0000 0.545184 0.272592 − 0.962130i \(-0.412119\pi\)
0.272592 + 0.962130i \(0.412119\pi\)
\(758\) 0 0
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) 16.0000 0.580000 0.290000 − 0.957027i \(-0.406345\pi\)
0.290000 + 0.957027i \(0.406345\pi\)
\(762\) 0 0
\(763\) 11.0000 0.398227
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) −44.0000 −1.58668 −0.793340 − 0.608778i \(-0.791660\pi\)
−0.793340 + 0.608778i \(0.791660\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) −38.0000 −1.36677 −0.683383 − 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 5.00000 0.179374
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 45.0000 1.61023
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 23.0000 0.818822
\(790\) 0 0
\(791\) −21.0000 −0.746674
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.00000 0.283375 0.141687 − 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 60.0000 2.11735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) −49.0000 −1.72275 −0.861374 − 0.507971i \(-0.830396\pi\)
−0.861374 + 0.507971i \(0.830396\pi\)
\(810\) 0 0
\(811\) −54.0000 −1.89620 −0.948098 − 0.317978i \(-0.896996\pi\)
−0.948098 + 0.317978i \(0.896996\pi\)
\(812\) 0 0
\(813\) 22.0000 0.771574
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 42.0000 1.46939
\(818\) 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
\(823\) −3.00000 −0.104573 −0.0522867 − 0.998632i \(-0.516651\pi\)
−0.0522867 + 0.998632i \(0.516651\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.0000 1.07798 0.538988 − 0.842314i \(-0.318807\pi\)
0.538988 + 0.842314i \(0.318807\pi\)
\(828\) 0 0
\(829\) −56.0000 −1.94496 −0.972480 − 0.232986i \(-0.925151\pi\)
−0.972480 + 0.232986i \(0.925151\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) −4.00000 −0.138592
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 0 0
\(839\) −52.0000 −1.79524 −0.897620 − 0.440771i \(-0.854705\pi\)
−0.897620 + 0.440771i \(0.854705\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 5.00000 0.172209
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 0 0
\(849\) −22.0000 −0.755038
\(850\) 0 0
\(851\) −5.00000 −0.171398
\(852\) 0 0
\(853\) −56.0000 −1.91740 −0.958702 − 0.284413i \(-0.908201\pi\)
−0.958702 + 0.284413i \(0.908201\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −32.0000 −1.09310 −0.546550 − 0.837427i \(-0.684059\pi\)
−0.546550 + 0.837427i \(0.684059\pi\)
\(858\) 0 0
\(859\) −54.0000 −1.84246 −0.921228 − 0.389023i \(-0.872813\pi\)
−0.921228 + 0.389023i \(0.872813\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −41.0000 −1.39566 −0.697828 − 0.716265i \(-0.745850\pi\)
−0.697828 + 0.716265i \(0.745850\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) 6.00000 0.203302
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 − 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −2.00000 −0.0674583
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 − 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) −51.0000 −1.71629 −0.858143 − 0.513410i \(-0.828382\pi\)
−0.858143 + 0.513410i \(0.828382\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 7.00000 0.234772
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 0 0
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 0 0
\(899\) 2.00000 0.0667037
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) −7.00000 −0.232945
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 − 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 27.0000 0.894550 0.447275 − 0.894397i \(-0.352395\pi\)
0.447275 + 0.894397i \(0.352395\pi\)
\(912\) 0 0
\(913\) −30.0000 −0.992855
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) 3.00000 0.0989609 0.0494804 − 0.998775i \(-0.484243\pi\)
0.0494804 + 0.998775i \(0.484243\pi\)
\(920\) 0 0
\(921\) −24.0000 −0.790827
\(922\) 0 0
\(923\) −18.0000 −0.592477
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −10.0000 −0.328443
\(928\) 0 0
\(929\) 40.0000 1.31236 0.656179 − 0.754606i \(-0.272172\pi\)
0.656179 + 0.754606i \(0.272172\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −42.0000 −1.36916 −0.684580 − 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 − 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) −24.0000 −0.779073
\(950\) 0 0
\(951\) 33.0000 1.07010
\(952\) 0 0
\(953\) 57.0000 1.84641 0.923206 − 0.384307i \(-0.125559\pi\)
0.923206 + 0.384307i \(0.125559\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 5.00000 0.161627
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −22.0000 −0.706014 −0.353007 − 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) 0 0
\(973\) −16.0000 −0.512936
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23.0000 0.735835 0.367918 − 0.929858i \(-0.380071\pi\)
0.367918 + 0.929858i \(0.380071\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 0 0
\(981\) −11.0000 −0.351203
\(982\) 0 0
\(983\) 30.0000 0.956851 0.478426 − 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.00000 −0.0636607
\(988\) 0 0
\(989\) 7.00000 0.222587
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 − 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 0 0
\(993\) −5.00000 −0.158670
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −2.00000 −0.0633406 −0.0316703 − 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 0 0
\(999\) −5.00000 −0.158193
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 4200.2.a.ba.1.1 yes 1
4.3 odd 2 8400.2.a.n.1.1 1
5.2 odd 4 4200.2.t.t.1849.1 2
5.3 odd 4 4200.2.t.t.1849.2 2
5.4 even 2 4200.2.a.o.1.1 ✓ 1
20.19 odd 2 8400.2.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4200.2.a.o.1.1 ✓ 1 5.4 even 2
4200.2.a.ba.1.1 yes 1 1.1 even 1 trivial
4200.2.t.t.1849.1 2 5.2 odd 4
4200.2.t.t.1849.2 2 5.3 odd 4
8400.2.a.n.1.1 1 4.3 odd 2
8400.2.a.bk.1.1 1 20.19 odd 2