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Properties

Label 24.3.h.b.5.1
Level $24$
Weight $3$
Character 24.5
Self dual yes
Analytic conductor $0.654$
Analytic rank $0$
Dimension $1$
CM discriminant -24
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,3,Mod(5,24)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(24, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 24.h (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.653952634465\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 5.1
Character \(\chi\) \(=\) 24.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -2.00000 q^{5} -6.00000 q^{6} -10.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -2.00000 q^{5} -6.00000 q^{6} -10.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -4.00000 q^{10} +10.0000 q^{11} -12.0000 q^{12} -20.0000 q^{14} +6.00000 q^{15} +16.0000 q^{16} +18.0000 q^{18} -8.00000 q^{20} +30.0000 q^{21} +20.0000 q^{22} -24.0000 q^{24} -21.0000 q^{25} -27.0000 q^{27} -40.0000 q^{28} -50.0000 q^{29} +12.0000 q^{30} +38.0000 q^{31} +32.0000 q^{32} -30.0000 q^{33} +20.0000 q^{35} +36.0000 q^{36} -16.0000 q^{40} +60.0000 q^{42} +40.0000 q^{44} -18.0000 q^{45} -48.0000 q^{48} +51.0000 q^{49} -42.0000 q^{50} +94.0000 q^{53} -54.0000 q^{54} -20.0000 q^{55} -80.0000 q^{56} -100.000 q^{58} +10.0000 q^{59} +24.0000 q^{60} +76.0000 q^{62} -90.0000 q^{63} +64.0000 q^{64} -60.0000 q^{66} +40.0000 q^{70} +72.0000 q^{72} +50.0000 q^{73} +63.0000 q^{75} -100.000 q^{77} -58.0000 q^{79} -32.0000 q^{80} +81.0000 q^{81} -134.000 q^{83} +120.000 q^{84} +150.000 q^{87} +80.0000 q^{88} -36.0000 q^{90} -114.000 q^{93} -96.0000 q^{96} -190.000 q^{97} +102.000 q^{98} +90.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.00000
\(3\) −3.00000 −1.00000
\(4\) 4.00000 1.00000
\(5\) −2.00000 −0.400000 −0.200000 − 0.979796i \(-0.564094\pi\)
−0.200000 + 0.979796i \(0.564094\pi\)
\(6\) −6.00000 −1.00000
\(7\) −10.0000 −1.42857 −0.714286 − 0.699854i \(-0.753248\pi\)
−0.714286 + 0.699854i \(0.753248\pi\)
\(8\) 8.00000 1.00000
\(9\) 9.00000 1.00000
\(10\) −4.00000 −0.400000
\(11\) 10.0000 0.909091 0.454545 − 0.890724i \(-0.349802\pi\)
0.454545 + 0.890724i \(0.349802\pi\)
\(12\) −12.0000 −1.00000
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −20.0000 −1.42857
\(15\) 6.00000 0.400000
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 18.0000 1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −8.00000 −0.400000
\(21\) 30.0000 1.42857
\(22\) 20.0000 0.909091
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −24.0000 −1.00000
\(25\) −21.0000 −0.840000
\(26\) 0 0
\(27\) −27.0000 −1.00000
\(28\) −40.0000 −1.42857
\(29\) −50.0000 −1.72414 −0.862069 − 0.506791i \(-0.830832\pi\)
−0.862069 + 0.506791i \(0.830832\pi\)
\(30\) 12.0000 0.400000
\(31\) 38.0000 1.22581 0.612903 − 0.790158i \(-0.290002\pi\)
0.612903 + 0.790158i \(0.290002\pi\)
\(32\) 32.0000 1.00000
\(33\) −30.0000 −0.909091
\(34\) 0 0
\(35\) 20.0000 0.571429
\(36\) 36.0000 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −16.0000 −0.400000
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 60.0000 1.42857
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 40.0000 0.909091
\(45\) −18.0000 −0.400000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −48.0000 −1.00000
\(49\) 51.0000 1.04082
\(50\) −42.0000 −0.840000
\(51\) 0 0
\(52\) 0 0
\(53\) 94.0000 1.77358 0.886792 − 0.462168i \(-0.152928\pi\)
0.886792 + 0.462168i \(0.152928\pi\)
\(54\) −54.0000 −1.00000
\(55\) −20.0000 −0.363636
\(56\) −80.0000 −1.42857
\(57\) 0 0
\(58\) −100.000 −1.72414
\(59\) 10.0000 0.169492 0.0847458 − 0.996403i \(-0.472992\pi\)
0.0847458 + 0.996403i \(0.472992\pi\)
\(60\) 24.0000 0.400000
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 76.0000 1.22581
\(63\) −90.0000 −1.42857
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) −60.0000 −0.909091
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 40.0000 0.571429
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 72.0000 1.00000
\(73\) 50.0000 0.684932 0.342466 − 0.939530i \(-0.388738\pi\)
0.342466 + 0.939530i \(0.388738\pi\)
\(74\) 0 0
\(75\) 63.0000 0.840000
\(76\) 0 0
\(77\) −100.000 −1.29870
\(78\) 0 0
\(79\) −58.0000 −0.734177 −0.367089 − 0.930186i \(-0.619645\pi\)
−0.367089 + 0.930186i \(0.619645\pi\)
\(80\) −32.0000 −0.400000
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) −134.000 −1.61446 −0.807229 − 0.590238i \(-0.799034\pi\)
−0.807229 + 0.590238i \(0.799034\pi\)
\(84\) 120.000 1.42857
\(85\) 0 0
\(86\) 0 0
\(87\) 150.000 1.72414
\(88\) 80.0000 0.909091
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −36.0000 −0.400000
\(91\) 0 0
\(92\) 0 0
\(93\) −114.000 −1.22581
\(94\) 0 0
\(95\) 0 0
\(96\) −96.0000 −1.00000
\(97\) −190.000 −1.95876 −0.979381 − 0.202020i \(-0.935249\pi\)
−0.979381 + 0.202020i \(0.935249\pi\)
\(98\) 102.000 1.04082
\(99\) 90.0000 0.909091
\(100\) −84.0000 −0.840000
\(101\) 190.000 1.88119 0.940594 − 0.339533i \(-0.110269\pi\)
0.940594 + 0.339533i \(0.110269\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.0970874 −0.0485437 − 0.998821i \(-0.515458\pi\)
−0.0485437 + 0.998821i \(0.515458\pi\)
\(104\) 0 0
\(105\) −60.0000 −0.571429
\(106\) 188.000 1.77358
\(107\) −86.0000 −0.803738 −0.401869 − 0.915697i \(-0.631639\pi\)
−0.401869 + 0.915697i \(0.631639\pi\)
\(108\) −108.000 −1.00000
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) −40.0000 −0.363636
\(111\) 0 0
\(112\) −160.000 −1.42857
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −200.000 −1.72414
\(117\) 0 0
\(118\) 20.0000 0.169492
\(119\) 0 0
\(120\) 48.0000 0.400000
\(121\) −21.0000 −0.173554
\(122\) 0 0
\(123\) 0 0
\(124\) 152.000 1.22581
\(125\) 92.0000 0.736000
\(126\) −180.000 −1.42857
\(127\) 230.000 1.81102 0.905512 − 0.424321i \(-0.139487\pi\)
0.905512 + 0.424321i \(0.139487\pi\)
\(128\) 128.000 1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) 250.000 1.90840 0.954198 − 0.299174i \(-0.0967112\pi\)
0.954198 + 0.299174i \(0.0967112\pi\)
\(132\) −120.000 −0.909091
\(133\) 0 0
\(134\) 0 0
\(135\) 54.0000 0.400000
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 80.0000 0.571429
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 144.000 1.00000
\(145\) 100.000 0.689655
\(146\) 100.000 0.684932
\(147\) −153.000 −1.04082
\(148\) 0 0
\(149\) −290.000 −1.94631 −0.973154 − 0.230153i \(-0.926077\pi\)
−0.973154 + 0.230153i \(0.926077\pi\)
\(150\) 126.000 0.840000
\(151\) −298.000 −1.97351 −0.986755 − 0.162218i \(-0.948135\pi\)
−0.986755 + 0.162218i \(0.948135\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −200.000 −1.29870
\(155\) −76.0000 −0.490323
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) −116.000 −0.734177
\(159\) −282.000 −1.77358
\(160\) −64.0000 −0.400000
\(161\) 0 0
\(162\) 162.000 1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 60.0000 0.363636
\(166\) −268.000 −1.61446
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 240.000 1.42857
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 46.0000 0.265896 0.132948 − 0.991123i \(-0.457556\pi\)
0.132948 + 0.991123i \(0.457556\pi\)
\(174\) 300.000 1.72414
\(175\) 210.000 1.20000
\(176\) 160.000 0.909091
\(177\) −30.0000 −0.169492
\(178\) 0 0
\(179\) −230.000 −1.28492 −0.642458 − 0.766321i \(-0.722085\pi\)
−0.642458 + 0.766321i \(0.722085\pi\)
\(180\) −72.0000 −0.400000
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) −228.000 −1.22581
\(187\) 0 0
\(188\) 0 0
\(189\) 270.000 1.42857
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −192.000 −1.00000
\(193\) 290.000 1.50259 0.751295 − 0.659966i \(-0.229429\pi\)
0.751295 + 0.659966i \(0.229429\pi\)
\(194\) −380.000 −1.95876
\(195\) 0 0
\(196\) 204.000 1.04082
\(197\) −194.000 −0.984772 −0.492386 − 0.870377i \(-0.663875\pi\)
−0.492386 + 0.870377i \(0.663875\pi\)
\(198\) 180.000 0.909091
\(199\) −202.000 −1.01508 −0.507538 − 0.861630i \(-0.669444\pi\)
−0.507538 + 0.861630i \(0.669444\pi\)
\(200\) −168.000 −0.840000
\(201\) 0 0
\(202\) 380.000 1.88119
\(203\) 500.000 2.46305
\(204\) 0 0
\(205\) 0 0
\(206\) −20.0000 −0.0970874
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) −120.000 −0.571429
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 376.000 1.77358
\(213\) 0 0
\(214\) −172.000 −0.803738
\(215\) 0 0
\(216\) −216.000 −1.00000
\(217\) −380.000 −1.75115
\(218\) 0 0
\(219\) −150.000 −0.684932
\(220\) −80.0000 −0.363636
\(221\) 0 0
\(222\) 0 0
\(223\) 230.000 1.03139 0.515695 − 0.856772i \(-0.327534\pi\)
0.515695 + 0.856772i \(0.327534\pi\)
\(224\) −320.000 −1.42857
\(225\) −189.000 −0.840000
\(226\) 0 0
\(227\) 346.000 1.52423 0.762115 − 0.647442i \(-0.224161\pi\)
0.762115 + 0.647442i \(0.224161\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 300.000 1.29870
\(232\) −400.000 −1.72414
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 40.0000 0.169492
\(237\) 174.000 0.734177
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 96.0000 0.400000
\(241\) −382.000 −1.58506 −0.792531 − 0.609831i \(-0.791237\pi\)
−0.792531 + 0.609831i \(0.791237\pi\)
\(242\) −42.0000 −0.173554
\(243\) −243.000 −1.00000
\(244\) 0 0
\(245\) −102.000 −0.416327
\(246\) 0 0
\(247\) 0 0
\(248\) 304.000 1.22581
\(249\) 402.000 1.61446
\(250\) 184.000 0.736000
\(251\) −470.000 −1.87251 −0.936255 − 0.351321i \(-0.885733\pi\)
−0.936255 + 0.351321i \(0.885733\pi\)
\(252\) −360.000 −1.42857
\(253\) 0 0
\(254\) 460.000 1.81102
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −450.000 −1.72414
\(262\) 500.000 1.90840
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −240.000 −0.909091
\(265\) −188.000 −0.709434
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 430.000 1.59851 0.799257 − 0.600990i \(-0.205227\pi\)
0.799257 + 0.600990i \(0.205227\pi\)
\(270\) 108.000 0.400000
\(271\) −58.0000 −0.214022 −0.107011 − 0.994258i \(-0.534128\pi\)
−0.107011 + 0.994258i \(0.534128\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −210.000 −0.763636
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 342.000 1.22581
\(280\) 160.000 0.571429
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 288.000 1.00000
\(289\) 289.000 1.00000
\(290\) 200.000 0.689655
\(291\) 570.000 1.95876
\(292\) 200.000 0.684932
\(293\) −386.000 −1.31741 −0.658703 − 0.752403i \(-0.728895\pi\)
−0.658703 + 0.752403i \(0.728895\pi\)
\(294\) −306.000 −1.04082
\(295\) −20.0000 −0.0677966
\(296\) 0 0
\(297\) −270.000 −0.909091
\(298\) −580.000 −1.94631
\(299\) 0 0
\(300\) 252.000 0.840000
\(301\) 0 0
\(302\) −596.000 −1.97351
\(303\) −570.000 −1.88119
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −400.000 −1.29870
\(309\) 30.0000 0.0970874
\(310\) −152.000 −0.490323
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 530.000 1.69329 0.846645 − 0.532158i \(-0.178619\pi\)
0.846645 + 0.532158i \(0.178619\pi\)
\(314\) 0 0
\(315\) 180.000 0.571429
\(316\) −232.000 −0.734177
\(317\) 334.000 1.05363 0.526814 − 0.849981i \(-0.323386\pi\)
0.526814 + 0.849981i \(0.323386\pi\)
\(318\) −564.000 −1.77358
\(319\) −500.000 −1.56740
\(320\) −128.000 −0.400000
\(321\) 258.000 0.803738
\(322\) 0 0
\(323\) 0 0
\(324\) 324.000 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 120.000 0.363636
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −536.000 −1.61446
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 480.000 1.42857
\(337\) −190.000 −0.563798 −0.281899 − 0.959444i \(-0.590964\pi\)
−0.281899 + 0.959444i \(0.590964\pi\)
\(338\) 338.000 1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 380.000 1.11437
\(342\) 0 0
\(343\) −20.0000 −0.0583090
\(344\) 0 0
\(345\) 0 0
\(346\) 92.0000 0.265896
\(347\) 106.000 0.305476 0.152738 − 0.988267i \(-0.451191\pi\)
0.152738 + 0.988267i \(0.451191\pi\)
\(348\) 600.000 1.72414
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 420.000 1.20000
\(351\) 0 0
\(352\) 320.000 0.909091
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) −60.0000 −0.169492
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −460.000 −1.28492
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −144.000 −0.400000
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 63.0000 0.173554
\(364\) 0 0
\(365\) −100.000 −0.273973
\(366\) 0 0
\(367\) 710.000 1.93460 0.967302 − 0.253626i \(-0.0816231\pi\)
0.967302 + 0.253626i \(0.0816231\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −940.000 −2.53369
\(372\) −456.000 −1.22581
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −276.000 −0.736000
\(376\) 0 0
\(377\) 0 0
\(378\) 540.000 1.42857
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −690.000 −1.81102
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −384.000 −1.00000
\(385\) 200.000 0.519481
\(386\) 580.000 1.50259
\(387\) 0 0
\(388\) −760.000 −1.95876
\(389\) 190.000 0.488432 0.244216 − 0.969721i \(-0.421469\pi\)
0.244216 + 0.969721i \(0.421469\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 408.000 1.04082
\(393\) −750.000 −1.90840
\(394\) −388.000 −0.984772
\(395\) 116.000 0.293671
\(396\) 360.000 0.909091
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −404.000 −1.01508
\(399\) 0 0
\(400\) −336.000 −0.840000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 760.000 1.88119
\(405\) −162.000 −0.400000
\(406\) 1000.00 2.46305
\(407\) 0 0
\(408\) 0 0
\(409\) −718.000 −1.75550 −0.877751 − 0.479118i \(-0.840957\pi\)
−0.877751 + 0.479118i \(0.840957\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −40.0000 −0.0970874
\(413\) −100.000 −0.242131
\(414\) 0 0
\(415\) 268.000 0.645783
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 730.000 1.74224 0.871122 − 0.491067i \(-0.163393\pi\)
0.871122 + 0.491067i \(0.163393\pi\)
\(420\) −240.000 −0.571429
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 752.000 1.77358
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −344.000 −0.803738
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −432.000 −1.00000
\(433\) −670.000 −1.54734 −0.773672 − 0.633586i \(-0.781582\pi\)
−0.773672 + 0.633586i \(0.781582\pi\)
\(434\) −760.000 −1.75115
\(435\) −300.000 −0.689655
\(436\) 0 0
\(437\) 0 0
\(438\) −300.000 −0.684932
\(439\) 278.000 0.633257 0.316629 − 0.948550i \(-0.397449\pi\)
0.316629 + 0.948550i \(0.397449\pi\)
\(440\) −160.000 −0.363636
\(441\) 459.000 1.04082
\(442\) 0 0
\(443\) −86.0000 −0.194131 −0.0970655 − 0.995278i \(-0.530946\pi\)
−0.0970655 + 0.995278i \(0.530946\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 460.000 1.03139
\(447\) 870.000 1.94631
\(448\) −640.000 −1.42857
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) −378.000 −0.840000
\(451\) 0 0
\(452\) 0 0
\(453\) 894.000 1.97351
\(454\) 692.000 1.52423
\(455\) 0 0
\(456\) 0 0
\(457\) 530.000 1.15974 0.579869 − 0.814710i \(-0.303104\pi\)
0.579869 + 0.814710i \(0.303104\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −530.000 −1.14967 −0.574837 − 0.818268i \(-0.694935\pi\)
−0.574837 + 0.818268i \(0.694935\pi\)
\(462\) 600.000 1.29870
\(463\) −250.000 −0.539957 −0.269978 − 0.962866i \(-0.587017\pi\)
−0.269978 + 0.962866i \(0.587017\pi\)
\(464\) −800.000 −1.72414
\(465\) 228.000 0.490323
\(466\) 0 0
\(467\) 634.000 1.35760 0.678801 − 0.734322i \(-0.262500\pi\)
0.678801 + 0.734322i \(0.262500\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 80.0000 0.169492
\(473\) 0 0
\(474\) 348.000 0.734177
\(475\) 0 0
\(476\) 0 0
\(477\) 846.000 1.77358
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 192.000 0.400000
\(481\) 0 0
\(482\) −764.000 −1.58506
\(483\) 0 0
\(484\) −84.0000 −0.173554
\(485\) 380.000 0.783505
\(486\) −486.000 −1.00000
\(487\) −970.000 −1.99179 −0.995893 − 0.0905356i \(-0.971142\pi\)
−0.995893 + 0.0905356i \(0.971142\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −204.000 −0.416327
\(491\) −470.000 −0.957230 −0.478615 − 0.878025i \(-0.658861\pi\)
−0.478615 + 0.878025i \(0.658861\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −180.000 −0.363636
\(496\) 608.000 1.22581
\(497\) 0 0
\(498\) 804.000 1.61446
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 368.000 0.736000
\(501\) 0 0
\(502\) −940.000 −1.87251
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −720.000 −1.42857
\(505\) −380.000 −0.752475
\(506\) 0 0
\(507\) −507.000 −1.00000
\(508\) 920.000 1.81102
\(509\) −1010.00 −1.98428 −0.992141 − 0.125121i \(-0.960068\pi\)
−0.992141 + 0.125121i \(0.960068\pi\)
\(510\) 0 0
\(511\) −500.000 −0.978474
\(512\) 512.000 1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) 20.0000 0.0388350
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −138.000 −0.265896
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −900.000 −1.72414
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 1000.00 1.90840
\(525\) −630.000 −1.20000
\(526\) 0 0
\(527\) 0 0
\(528\) −480.000 −0.909091
\(529\) 529.000 1.00000
\(530\) −376.000 −0.709434
\(531\) 90.0000 0.169492
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 172.000 0.321495
\(536\) 0 0
\(537\) 690.000 1.28492
\(538\) 860.000 1.59851
\(539\) 510.000 0.946197
\(540\) 216.000 0.400000
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) −116.000 −0.214022
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −420.000 −0.763636
\(551\) 0 0
\(552\) 0 0
\(553\) 580.000 1.04882
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −914.000 −1.64093 −0.820467 − 0.571694i \(-0.806286\pi\)
−0.820467 + 0.571694i \(0.806286\pi\)
\(558\) 684.000 1.22581
\(559\) 0 0
\(560\) 320.000 0.571429
\(561\) 0 0
\(562\) 0 0
\(563\) −326.000 −0.579041 −0.289520 − 0.957172i \(-0.593496\pi\)
−0.289520 + 0.957172i \(0.593496\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −810.000 −1.42857
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 576.000 1.00000
\(577\) 290.000 0.502600 0.251300 − 0.967909i \(-0.419142\pi\)
0.251300 + 0.967909i \(0.419142\pi\)
\(578\) 578.000 1.00000
\(579\) −870.000 −1.50259
\(580\) 400.000 0.689655
\(581\) 1340.00 2.30637
\(582\) 1140.00 1.95876
\(583\) 940.000 1.61235
\(584\) 400.000 0.684932
\(585\) 0 0
\(586\) −772.000 −1.31741
\(587\) 874.000 1.48893 0.744463 − 0.667663i \(-0.232705\pi\)
0.744463 + 0.667663i \(0.232705\pi\)
\(588\) −612.000 −1.04082
\(589\) 0 0
\(590\) −40.0000 −0.0677966
\(591\) 582.000 0.984772
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) −540.000 −0.909091
\(595\) 0 0
\(596\) −1160.00 −1.94631
\(597\) 606.000 1.01508
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 504.000 0.840000
\(601\) −1198.00 −1.99334 −0.996672 − 0.0815138i \(-0.974025\pi\)
−0.996672 + 0.0815138i \(0.974025\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1192.00 −1.97351
\(605\) 42.0000 0.0694215
\(606\) −1140.00 −1.88119
\(607\) −730.000 −1.20264 −0.601318 − 0.799010i \(-0.705357\pi\)
−0.601318 + 0.799010i \(0.705357\pi\)
\(608\) 0 0
\(609\) −1500.00 −2.46305
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −800.000 −1.29870
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 60.0000 0.0970874
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) −304.000 −0.490323
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 341.000 0.545600
\(626\) 1060.00 1.69329
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 360.000 0.571429
\(631\) 1238.00 1.96197 0.980983 − 0.194096i \(-0.0621773\pi\)
0.980983 + 0.194096i \(0.0621773\pi\)
\(632\) −464.000 −0.734177
\(633\) 0 0
\(634\) 668.000 1.05363
\(635\) −460.000 −0.724409
\(636\) −1128.00 −1.77358
\(637\) 0 0
\(638\) −1000.00 −1.56740
\(639\) 0 0
\(640\) −256.000 −0.400000
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 516.000 0.803738
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 648.000 1.00000
\(649\) 100.000 0.154083
\(650\) 0 0
\(651\) 1140.00 1.75115
\(652\) 0 0
\(653\) 1006.00 1.54058 0.770291 − 0.637693i \(-0.220111\pi\)
0.770291 + 0.637693i \(0.220111\pi\)
\(654\) 0 0
\(655\) −500.000 −0.763359
\(656\) 0 0
\(657\) 450.000 0.684932
\(658\) 0 0
\(659\) 730.000 1.10774 0.553869 − 0.832603i \(-0.313151\pi\)
0.553869 + 0.832603i \(0.313151\pi\)
\(660\) 240.000 0.363636
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −1072.00 −1.61446
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −690.000 −1.03139
\(670\) 0 0
\(671\) 0 0
\(672\) 960.000 1.42857
\(673\) −190.000 −0.282318 −0.141159 − 0.989987i \(-0.545083\pi\)
−0.141159 + 0.989987i \(0.545083\pi\)
\(674\) −380.000 −0.563798
\(675\) 567.000 0.840000
\(676\) 676.000 1.00000
\(677\) −1346.00 −1.98818 −0.994092 − 0.108545i \(-0.965381\pi\)
−0.994092 + 0.108545i \(0.965381\pi\)
\(678\) 0 0
\(679\) 1900.00 2.79823
\(680\) 0 0
\(681\) −1038.00 −1.52423
\(682\) 760.000 1.11437
\(683\) −1334.00 −1.95315 −0.976574 − 0.215182i \(-0.930965\pi\)
−0.976574 + 0.215182i \(0.930965\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −40.0000 −0.0583090
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 184.000 0.265896
\(693\) −900.000 −1.29870
\(694\) 212.000 0.305476
\(695\) 0 0
\(696\) 1200.00 1.72414
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 840.000 1.20000
\(701\) −50.0000 −0.0713267 −0.0356633 − 0.999364i \(-0.511354\pi\)
−0.0356633 + 0.999364i \(0.511354\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 640.000 0.909091
\(705\) 0 0
\(706\) 0 0
\(707\) −1900.00 −2.68741
\(708\) −120.000 −0.169492
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) −522.000 −0.734177
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −920.000 −1.28492
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −288.000 −0.400000
\(721\) 100.000 0.138696
\(722\) 722.000 1.00000
\(723\) 1146.00 1.58506
\(724\) 0 0
\(725\) 1050.00 1.44828
\(726\) 126.000 0.173554
\(727\) −1450.00 −1.99450 −0.997249 − 0.0741249i \(-0.976384\pi\)
−0.997249 + 0.0741249i \(0.976384\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) −200.000 −0.273973
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 1420.00 1.93460
\(735\) 306.000 0.416327
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1880.00 −2.53369
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) −912.000 −1.22581
\(745\) 580.000 0.778523
\(746\) 0 0
\(747\) −1206.00 −1.61446
\(748\) 0 0
\(749\) 860.000 1.14820
\(750\) −552.000 −0.736000
\(751\) −1402.00 −1.86684 −0.933422 − 0.358780i \(-0.883193\pi\)
−0.933422 + 0.358780i \(0.883193\pi\)
\(752\) 0 0
\(753\) 1410.00 1.87251
\(754\) 0 0
\(755\) 596.000 0.789404
\(756\) 1080.00 1.42857
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) −1380.00 −1.81102
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −768.000 −1.00000
\(769\) −862.000 −1.12094 −0.560468 − 0.828176i \(-0.689379\pi\)
−0.560468 + 0.828176i \(0.689379\pi\)
\(770\) 400.000 0.519481
\(771\) 0 0
\(772\) 1160.00 1.50259
\(773\) −1154.00 −1.49288 −0.746442 − 0.665450i \(-0.768240\pi\)
−0.746442 + 0.665450i \(0.768240\pi\)
\(774\) 0 0
\(775\) −798.000 −1.02968
\(776\) −1520.00 −1.95876
\(777\) 0 0
\(778\) 380.000 0.488432
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 1350.00 1.72414
\(784\) 816.000 1.04082
\(785\) 0 0
\(786\) −1500.00 −1.90840
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −776.000 −0.984772
\(789\) 0 0
\(790\) 232.000 0.293671
\(791\) 0 0
\(792\) 720.000 0.909091
\(793\) 0 0
\(794\) 0 0
\(795\) 564.000 0.709434
\(796\) −808.000 −1.01508
\(797\) 1294.00 1.62359 0.811794 − 0.583944i \(-0.198491\pi\)
0.811794 + 0.583944i \(0.198491\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −672.000 −0.840000
\(801\) 0 0
\(802\) 0 0
\(803\) 500.000 0.622665
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1290.00 −1.59851
\(808\) 1520.00 1.88119
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) −324.000 −0.400000
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 2000.00 2.46305
\(813\) 174.000 0.214022
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1436.00 −1.75550
\(819\) 0 0
\(820\) 0 0
\(821\) 670.000 0.816078 0.408039 − 0.912965i \(-0.366213\pi\)
0.408039 + 0.912965i \(0.366213\pi\)
\(822\) 0 0
\(823\) 470.000 0.571081 0.285541 − 0.958367i \(-0.407827\pi\)
0.285541 + 0.958367i \(0.407827\pi\)
\(824\) −80.0000 −0.0970874
\(825\) 630.000 0.763636
\(826\) −200.000 −0.242131
\(827\) 1546.00 1.86941 0.934704 − 0.355428i \(-0.115665\pi\)
0.934704 + 0.355428i \(0.115665\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 536.000 0.645783
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1026.00 −1.22581
\(838\) 1460.00 1.74224
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) −480.000 −0.571429
\(841\) 1659.00 1.97265
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −338.000 −0.400000
\(846\) 0 0
\(847\) 210.000 0.247934
\(848\) 1504.00 1.77358
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −688.000 −0.803738
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −864.000 −1.00000
\(865\) −92.0000 −0.106358
\(866\) −1340.00 −1.54734
\(867\) −867.000 −1.00000
\(868\) −1520.00 −1.75115
\(869\) −580.000 −0.667434
\(870\) −600.000 −0.689655
\(871\) 0 0
\(872\) 0 0
\(873\) −1710.00 −1.95876
\(874\) 0 0
\(875\) −920.000 −1.05143
\(876\) −600.000 −0.684932
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 556.000 0.633257
\(879\) 1158.00 1.31741
\(880\) −320.000 −0.363636
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 918.000 1.04082
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 60.0000 0.0677966
\(886\) −172.000 −0.194131
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −2300.00 −2.58718
\(890\) 0 0
\(891\) 810.000 0.909091
\(892\) 920.000 1.03139
\(893\) 0 0
\(894\) 1740.00 1.94631
\(895\) 460.000 0.513966
\(896\) −1280.00 −1.42857
\(897\) 0 0
\(898\) 0 0
\(899\) −1900.00 −2.11346
\(900\) −756.000 −0.840000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 1788.00 1.97351
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 1384.00 1.52423
\(909\) 1710.00 1.88119
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −1340.00 −1.46769
\(914\) 1060.00 1.15974
\(915\) 0 0
\(916\) 0 0
\(917\) −2500.00 −2.72628
\(918\) 0 0
\(919\) 662.000 0.720348 0.360174 − 0.932885i \(-0.382717\pi\)
0.360174 + 0.932885i \(0.382717\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1060.00 −1.14967
\(923\) 0 0
\(924\) 1200.00 1.29870
\(925\) 0 0
\(926\) −500.000 −0.539957
\(927\) −90.0000 −0.0970874
\(928\) −1600.00 −1.72414
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 456.000 0.490323
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 1268.00 1.35760
\(935\) 0 0
\(936\) 0 0
\(937\) 1490.00 1.59018 0.795091 − 0.606491i \(-0.207423\pi\)
0.795091 + 0.606491i \(0.207423\pi\)
\(938\) 0 0
\(939\) −1590.00 −1.69329
\(940\) 0 0
\(941\) 430.000 0.456961 0.228480 − 0.973549i \(-0.426624\pi\)
0.228480 + 0.973549i \(0.426624\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 160.000 0.169492
\(945\) −540.000 −0.571429
\(946\) 0 0
\(947\) 1306.00 1.37909 0.689546 − 0.724242i \(-0.257810\pi\)
0.689546 + 0.724242i \(0.257810\pi\)
\(948\) 696.000 0.734177
\(949\) 0 0
\(950\) 0 0
\(951\) −1002.00 −1.05363
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 1692.00 1.77358
\(955\) 0 0
\(956\) 0 0
\(957\) 1500.00 1.56740
\(958\) 0 0
\(959\) 0 0
\(960\) 384.000 0.400000
\(961\) 483.000 0.502601
\(962\) 0 0
\(963\) −774.000 −0.803738
\(964\) −1528.00 −1.58506
\(965\) −580.000 −0.601036
\(966\) 0 0
\(967\) 1910.00 1.97518 0.987590 − 0.157051i \(-0.0501987\pi\)
0.987590 + 0.157051i \(0.0501987\pi\)
\(968\) −168.000 −0.173554
\(969\) 0 0
\(970\) 760.000 0.783505
\(971\) 1930.00 1.98764 0.993821 − 0.110996i \(-0.0354042\pi\)
0.993821 + 0.110996i \(0.0354042\pi\)
\(972\) −972.000 −1.00000
\(973\) 0 0
\(974\) −1940.00 −1.99179
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −408.000 −0.416327
\(981\) 0 0
\(982\) −940.000 −0.957230
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 388.000 0.393909
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) −360.000 −0.363636
\(991\) 1382.00 1.39455 0.697275 − 0.716803i \(-0.254395\pi\)
0.697275 + 0.716803i \(0.254395\pi\)
\(992\) 1216.00 1.22581
\(993\) 0 0
\(994\) 0 0
\(995\) 404.000 0.406030
\(996\) 1608.00 1.61446
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 24.3.h.b.5.1 yes 1
3.2 odd 2 24.3.h.a.5.1 ✓ 1
4.3 odd 2 96.3.h.b.17.1 1
8.3 odd 2 96.3.h.a.17.1 1
8.5 even 2 24.3.h.a.5.1 ✓ 1
12.11 even 2 96.3.h.a.17.1 1
16.3 odd 4 768.3.e.c.257.2 2
16.5 even 4 768.3.e.d.257.2 2
16.11 odd 4 768.3.e.c.257.1 2
16.13 even 4 768.3.e.d.257.1 2
24.5 odd 2 CM 24.3.h.b.5.1 yes 1
24.11 even 2 96.3.h.b.17.1 1
48.5 odd 4 768.3.e.d.257.1 2
48.11 even 4 768.3.e.c.257.2 2
48.29 odd 4 768.3.e.d.257.2 2
48.35 even 4 768.3.e.c.257.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.3.h.a.5.1 ✓ 1 3.2 odd 2
24.3.h.a.5.1 ✓ 1 8.5 even 2
24.3.h.b.5.1 yes 1 1.1 even 1 trivial
24.3.h.b.5.1 yes 1 24.5 odd 2 CM
96.3.h.a.17.1 1 8.3 odd 2
96.3.h.a.17.1 1 12.11 even 2
96.3.h.b.17.1 1 4.3 odd 2
96.3.h.b.17.1 1 24.11 even 2
768.3.e.c.257.1 2 16.11 odd 4
768.3.e.c.257.1 2 48.35 even 4
768.3.e.c.257.2 2 16.3 odd 4
768.3.e.c.257.2 2 48.11 even 4
768.3.e.d.257.1 2 16.13 even 4
768.3.e.d.257.1 2 48.5 odd 4
768.3.e.d.257.2 2 16.5 even 4
768.3.e.d.257.2 2 48.29 odd 4