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Properties

Label 228.3.b.d.227.1
Level $228$
Weight $3$
Character 228.227
Self dual yes
Analytic conductor $6.213$
Analytic rank $0$
Dimension $1$
CM discriminant -228
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [228,3,Mod(227,228)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("228.227"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(228, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 228.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.21255002741\)
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 227.1
Character \(\chi\) \(=\) 228.227

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +8.00000 q^{8} +9.00000 q^{9} -16.0000 q^{11} +12.0000 q^{12} +16.0000 q^{16} +18.0000 q^{18} -19.0000 q^{19} -32.0000 q^{22} +8.00000 q^{23} +24.0000 q^{24} +25.0000 q^{25} +27.0000 q^{27} -56.0000 q^{29} -14.0000 q^{31} +32.0000 q^{32} -48.0000 q^{33} +36.0000 q^{36} -38.0000 q^{38} -32.0000 q^{41} -64.0000 q^{44} +16.0000 q^{46} +56.0000 q^{47} +48.0000 q^{48} +49.0000 q^{49} +50.0000 q^{50} -8.00000 q^{53} +54.0000 q^{54} -57.0000 q^{57} -112.000 q^{58} -106.000 q^{61} -28.0000 q^{62} +64.0000 q^{64} -96.0000 q^{66} +58.0000 q^{67} +24.0000 q^{69} +72.0000 q^{72} -82.0000 q^{73} +75.0000 q^{75} -76.0000 q^{76} -146.000 q^{79} +81.0000 q^{81} -64.0000 q^{82} +128.000 q^{83} -168.000 q^{87} -128.000 q^{88} +64.0000 q^{89} +32.0000 q^{92} -42.0000 q^{93} +112.000 q^{94} +96.0000 q^{96} +98.0000 q^{98} -144.000 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/228\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\) \(115\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 6.00000 1.00000
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 8.00000 1.00000
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) −16.0000 −1.45455 −0.727273 − 0.686349i \(-0.759212\pi\)
−0.727273 + 0.686349i \(0.759212\pi\)
\(12\) 12.0000 1.00000
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 18.0000 1.00000
\(19\) −19.0000 −1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) −32.0000 −1.45455
\(23\) 8.00000 0.347826 0.173913 − 0.984761i \(-0.444359\pi\)
0.173913 + 0.984761i \(0.444359\pi\)
\(24\) 24.0000 1.00000
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 27.0000 1.00000
\(28\) 0 0
\(29\) −56.0000 −1.93103 −0.965517 − 0.260339i \(-0.916166\pi\)
−0.965517 + 0.260339i \(0.916166\pi\)
\(30\) 0 0
\(31\) −14.0000 −0.451613 −0.225806 − 0.974172i \(-0.572502\pi\)
−0.225806 + 0.974172i \(0.572502\pi\)
\(32\) 32.0000 1.00000
\(33\) −48.0000 −1.45455
\(34\) 0 0
\(35\) 0 0
\(36\) 36.0000 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −38.0000 −1.00000
\(39\) 0 0
\(40\) 0 0
\(41\) −32.0000 −0.780488 −0.390244 − 0.920712i \(-0.627609\pi\)
−0.390244 + 0.920712i \(0.627609\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −64.0000 −1.45455
\(45\) 0 0
\(46\) 16.0000 0.347826
\(47\) 56.0000 1.19149 0.595745 − 0.803174i \(-0.296857\pi\)
0.595745 + 0.803174i \(0.296857\pi\)
\(48\) 48.0000 1.00000
\(49\) 49.0000 1.00000
\(50\) 50.0000 1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) −8.00000 −0.150943 −0.0754717 − 0.997148i \(-0.524046\pi\)
−0.0754717 + 0.997148i \(0.524046\pi\)
\(54\) 54.0000 1.00000
\(55\) 0 0
\(56\) 0 0
\(57\) −57.0000 −1.00000
\(58\) −112.000 −1.93103
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −106.000 −1.73770 −0.868852 − 0.495071i \(-0.835142\pi\)
−0.868852 + 0.495071i \(0.835142\pi\)
\(62\) −28.0000 −0.451613
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) −96.0000 −1.45455
\(67\) 58.0000 0.865672 0.432836 − 0.901473i \(-0.357513\pi\)
0.432836 + 0.901473i \(0.357513\pi\)
\(68\) 0 0
\(69\) 24.0000 0.347826
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 72.0000 1.00000
\(73\) −82.0000 −1.12329 −0.561644 − 0.827379i \(-0.689831\pi\)
−0.561644 + 0.827379i \(0.689831\pi\)
\(74\) 0 0
\(75\) 75.0000 1.00000
\(76\) −76.0000 −1.00000
\(77\) 0 0
\(78\) 0 0
\(79\) −146.000 −1.84810 −0.924051 − 0.382270i \(-0.875142\pi\)
−0.924051 + 0.382270i \(0.875142\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) −64.0000 −0.780488
\(83\) 128.000 1.54217 0.771084 − 0.636733i \(-0.219715\pi\)
0.771084 + 0.636733i \(0.219715\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −168.000 −1.93103
\(88\) −128.000 −1.45455
\(89\) 64.0000 0.719101 0.359551 − 0.933126i \(-0.382930\pi\)
0.359551 + 0.933126i \(0.382930\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 32.0000 0.347826
\(93\) −42.0000 −0.451613
\(94\) 112.000 1.19149
\(95\) 0 0
\(96\) 96.0000 1.00000
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 98.0000 1.00000
\(99\) −144.000 −1.45455
\(100\) 100.000 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −98.0000 −0.951456 −0.475728 − 0.879592i \(-0.657815\pi\)
−0.475728 + 0.879592i \(0.657815\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −16.0000 −0.150943
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 108.000 1.00000
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 112.000 0.991150 0.495575 − 0.868565i \(-0.334957\pi\)
0.495575 + 0.868565i \(0.334957\pi\)
\(114\) −114.000 −1.00000
\(115\) 0 0
\(116\) −224.000 −1.93103
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 135.000 1.11570
\(122\) −212.000 −1.73770
\(123\) −96.0000 −0.780488
\(124\) −56.0000 −0.451613
\(125\) 0 0
\(126\) 0 0
\(127\) 178.000 1.40157 0.700787 − 0.713370i \(-0.252832\pi\)
0.700787 + 0.713370i \(0.252832\pi\)
\(128\) 128.000 1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) 224.000 1.70992 0.854962 − 0.518691i \(-0.173580\pi\)
0.854962 + 0.518691i \(0.173580\pi\)
\(132\) −192.000 −1.45455
\(133\) 0 0
\(134\) 116.000 0.865672
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 48.0000 0.347826
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 168.000 1.19149
\(142\) 0 0
\(143\) 0 0
\(144\) 144.000 1.00000
\(145\) 0 0
\(146\) −164.000 −1.12329
\(147\) 147.000 1.00000
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 150.000 1.00000
\(151\) −2.00000 −0.0132450 −0.00662252 − 0.999978i \(-0.502108\pi\)
−0.00662252 + 0.999978i \(0.502108\pi\)
\(152\) −152.000 −1.00000
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 86.0000 0.547771 0.273885 − 0.961762i \(-0.411691\pi\)
0.273885 + 0.961762i \(0.411691\pi\)
\(158\) −292.000 −1.84810
\(159\) −24.0000 −0.150943
\(160\) 0 0
\(161\) 0 0
\(162\) 162.000 1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −128.000 −0.780488
\(165\) 0 0
\(166\) 256.000 1.54217
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) −171.000 −1.00000
\(172\) 0 0
\(173\) 232.000 1.34104 0.670520 − 0.741891i \(-0.266071\pi\)
0.670520 + 0.741891i \(0.266071\pi\)
\(174\) −336.000 −1.93103
\(175\) 0 0
\(176\) −256.000 −1.45455
\(177\) 0 0
\(178\) 128.000 0.719101
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) −318.000 −1.73770
\(184\) 64.0000 0.347826
\(185\) 0 0
\(186\) −84.0000 −0.451613
\(187\) 0 0
\(188\) 224.000 1.19149
\(189\) 0 0
\(190\) 0 0
\(191\) 344.000 1.80105 0.900524 − 0.434807i \(-0.143183\pi\)
0.900524 + 0.434807i \(0.143183\pi\)
\(192\) 192.000 1.00000
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −288.000 −1.45455
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 200.000 1.00000
\(201\) 174.000 0.865672
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −196.000 −0.951456
\(207\) 72.0000 0.347826
\(208\) 0 0
\(209\) 304.000 1.45455
\(210\) 0 0
\(211\) 346.000 1.63981 0.819905 − 0.572499i \(-0.194026\pi\)
0.819905 + 0.572499i \(0.194026\pi\)
\(212\) −32.0000 −0.150943
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 216.000 1.00000
\(217\) 0 0
\(218\) 0 0
\(219\) −246.000 −1.12329
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 142.000 0.636771 0.318386 − 0.947961i \(-0.396859\pi\)
0.318386 + 0.947961i \(0.396859\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) 224.000 0.991150
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) −228.000 −1.00000
\(229\) −454.000 −1.98253 −0.991266 − 0.131875i \(-0.957900\pi\)
−0.991266 + 0.131875i \(0.957900\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −448.000 −1.93103
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −438.000 −1.84810
\(238\) 0 0
\(239\) −472.000 −1.97490 −0.987448 − 0.157946i \(-0.949513\pi\)
−0.987448 + 0.157946i \(0.949513\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 270.000 1.11570
\(243\) 243.000 1.00000
\(244\) −424.000 −1.73770
\(245\) 0 0
\(246\) −192.000 −0.780488
\(247\) 0 0
\(248\) −112.000 −0.451613
\(249\) 384.000 1.54217
\(250\) 0 0
\(251\) −448.000 −1.78486 −0.892430 − 0.451185i \(-0.851001\pi\)
−0.892430 + 0.451185i \(0.851001\pi\)
\(252\) 0 0
\(253\) −128.000 −0.505929
\(254\) 356.000 1.40157
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) −512.000 −1.99222 −0.996109 − 0.0881304i \(-0.971911\pi\)
−0.996109 + 0.0881304i \(0.971911\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −504.000 −1.93103
\(262\) 448.000 1.70992
\(263\) 488.000 1.85551 0.927757 − 0.373186i \(-0.121735\pi\)
0.927757 + 0.373186i \(0.121735\pi\)
\(264\) −384.000 −1.45455
\(265\) 0 0
\(266\) 0 0
\(267\) 192.000 0.719101
\(268\) 232.000 0.865672
\(269\) −488.000 −1.81413 −0.907063 − 0.420994i \(-0.861681\pi\)
−0.907063 + 0.420994i \(0.861681\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −400.000 −1.45455
\(276\) 96.0000 0.347826
\(277\) −358.000 −1.29242 −0.646209 − 0.763160i \(-0.723647\pi\)
−0.646209 + 0.763160i \(0.723647\pi\)
\(278\) 0 0
\(279\) −126.000 −0.451613
\(280\) 0 0
\(281\) −464.000 −1.65125 −0.825623 − 0.564223i \(-0.809176\pi\)
−0.825623 + 0.564223i \(0.809176\pi\)
\(282\) 336.000 1.19149
\(283\) 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
\(290\) 0 0
\(291\) 0 0
\(292\) −328.000 −1.12329
\(293\) 472.000 1.61092 0.805461 − 0.592649i \(-0.201918\pi\)
0.805461 + 0.592649i \(0.201918\pi\)
\(294\) 294.000 1.00000
\(295\) 0 0
\(296\) 0 0
\(297\) −432.000 −1.45455
\(298\) 0 0
\(299\) 0 0
\(300\) 300.000 1.00000
\(301\) 0 0
\(302\) −4.00000 −0.0132450
\(303\) 0 0
\(304\) −304.000 −1.00000
\(305\) 0 0
\(306\) 0 0
\(307\) −602.000 −1.96091 −0.980456 − 0.196738i \(-0.936965\pi\)
−0.980456 + 0.196738i \(0.936965\pi\)
\(308\) 0 0
\(309\) −294.000 −0.951456
\(310\) 0 0
\(311\) −328.000 −1.05466 −0.527331 − 0.849660i \(-0.676807\pi\)
−0.527331 + 0.849660i \(0.676807\pi\)
\(312\) 0 0
\(313\) 398.000 1.27157 0.635783 − 0.771868i \(-0.280677\pi\)
0.635783 + 0.771868i \(0.280677\pi\)
\(314\) 172.000 0.547771
\(315\) 0 0
\(316\) −584.000 −1.84810
\(317\) −392.000 −1.23659 −0.618297 − 0.785945i \(-0.712177\pi\)
−0.618297 + 0.785945i \(0.712177\pi\)
\(318\) −48.0000 −0.150943
\(319\) 896.000 2.80878
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 324.000 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −256.000 −0.780488
\(329\) 0 0
\(330\) 0 0
\(331\) −554.000 −1.67372 −0.836858 − 0.547420i \(-0.815610\pi\)
−0.836858 + 0.547420i \(0.815610\pi\)
\(332\) 512.000 1.54217
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 338.000 1.00000
\(339\) 336.000 0.991150
\(340\) 0 0
\(341\) 224.000 0.656891
\(342\) −342.000 −1.00000
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 464.000 1.34104
\(347\) 656.000 1.89049 0.945245 − 0.326362i \(-0.105823\pi\)
0.945245 + 0.326362i \(0.105823\pi\)
\(348\) −672.000 −1.93103
\(349\) −214.000 −0.613181 −0.306590 − 0.951842i \(-0.599188\pi\)
−0.306590 + 0.951842i \(0.599188\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −512.000 −1.45455
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 256.000 0.719101
\(357\) 0 0
\(358\) 0 0
\(359\) −232.000 −0.646240 −0.323120 − 0.946358i \(-0.604732\pi\)
−0.323120 + 0.946358i \(0.604732\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 405.000 1.11570
\(364\) 0 0
\(365\) 0 0
\(366\) −636.000 −1.73770
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 128.000 0.347826
\(369\) −288.000 −0.780488
\(370\) 0 0
\(371\) 0 0
\(372\) −168.000 −0.451613
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 448.000 1.19149
\(377\) 0 0
\(378\) 0 0
\(379\) −458.000 −1.20844 −0.604222 − 0.796816i \(-0.706516\pi\)
−0.604222 + 0.796816i \(0.706516\pi\)
\(380\) 0 0
\(381\) 534.000 1.40157
\(382\) 688.000 1.80105
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 384.000 1.00000
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 392.000 1.00000
\(393\) 672.000 1.70992
\(394\) 0 0
\(395\) 0 0
\(396\) −576.000 −1.45455
\(397\) −118.000 −0.297229 −0.148615 − 0.988895i \(-0.547481\pi\)
−0.148615 + 0.988895i \(0.547481\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 400.000 1.00000
\(401\) −224.000 −0.558603 −0.279302 − 0.960203i \(-0.590103\pi\)
−0.279302 + 0.960203i \(0.590103\pi\)
\(402\) 348.000 0.865672
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −392.000 −0.951456
\(413\) 0 0
\(414\) 144.000 0.347826
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 608.000 1.45455
\(419\) −112.000 −0.267303 −0.133652 − 0.991028i \(-0.542670\pi\)
−0.133652 + 0.991028i \(0.542670\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 692.000 1.63981
\(423\) 504.000 1.19149
\(424\) −64.0000 −0.150943
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 432.000 1.00000
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −152.000 −0.347826
\(438\) −492.000 −1.12329
\(439\) 574.000 1.30752 0.653759 − 0.756703i \(-0.273191\pi\)
0.653759 + 0.756703i \(0.273191\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) 848.000 1.91422 0.957111 − 0.289723i \(-0.0935631\pi\)
0.957111 + 0.289723i \(0.0935631\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 284.000 0.636771
\(447\) 0 0
\(448\) 0 0
\(449\) 784.000 1.74610 0.873051 − 0.487629i \(-0.162138\pi\)
0.873051 + 0.487629i \(0.162138\pi\)
\(450\) 450.000 1.00000
\(451\) 512.000 1.13525
\(452\) 448.000 0.991150
\(453\) −6.00000 −0.0132450
\(454\) 0 0
\(455\) 0 0
\(456\) −456.000 −1.00000
\(457\) 686.000 1.50109 0.750547 − 0.660817i \(-0.229790\pi\)
0.750547 + 0.660817i \(0.229790\pi\)
\(458\) −908.000 −1.98253
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −896.000 −1.93103
\(465\) 0 0
\(466\) 0 0
\(467\) −928.000 −1.98715 −0.993576 − 0.113167i \(-0.963901\pi\)
−0.993576 + 0.113167i \(0.963901\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 258.000 0.547771
\(472\) 0 0
\(473\) 0 0
\(474\) −876.000 −1.84810
\(475\) −475.000 −1.00000
\(476\) 0 0
\(477\) −72.0000 −0.150943
\(478\) −944.000 −1.97490
\(479\) −904.000 −1.88727 −0.943633 − 0.330995i \(-0.892616\pi\)
−0.943633 + 0.330995i \(0.892616\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 540.000 1.11570
\(485\) 0 0
\(486\) 486.000 1.00000
\(487\) −926.000 −1.90144 −0.950719 − 0.310055i \(-0.899653\pi\)
−0.950719 + 0.310055i \(0.899653\pi\)
\(488\) −848.000 −1.73770
\(489\) 0 0
\(490\) 0 0
\(491\) 32.0000 0.0651731 0.0325866 − 0.999469i \(-0.489626\pi\)
0.0325866 + 0.999469i \(0.489626\pi\)
\(492\) −384.000 −0.780488
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −224.000 −0.451613
\(497\) 0 0
\(498\) 768.000 1.54217
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −896.000 −1.78486
\(503\) −856.000 −1.70179 −0.850895 − 0.525336i \(-0.823939\pi\)
−0.850895 + 0.525336i \(0.823939\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −256.000 −0.505929
\(507\) 507.000 1.00000
\(508\) 712.000 1.40157
\(509\) 904.000 1.77603 0.888016 − 0.459813i \(-0.152084\pi\)
0.888016 + 0.459813i \(0.152084\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 1.00000
\(513\) −513.000 −1.00000
\(514\) −1024.00 −1.99222
\(515\) 0 0
\(516\) 0 0
\(517\) −896.000 −1.73308
\(518\) 0 0
\(519\) 696.000 1.34104
\(520\) 0 0
\(521\) 16.0000 0.0307102 0.0153551 − 0.999882i \(-0.495112\pi\)
0.0153551 + 0.999882i \(0.495112\pi\)
\(522\) −1008.00 −1.93103
\(523\) −854.000 −1.63289 −0.816444 − 0.577425i \(-0.804058\pi\)
−0.816444 + 0.577425i \(0.804058\pi\)
\(524\) 896.000 1.70992
\(525\) 0 0
\(526\) 976.000 1.85551
\(527\) 0 0
\(528\) −768.000 −1.45455
\(529\) −465.000 −0.879017
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 384.000 0.719101
\(535\) 0 0
\(536\) 464.000 0.865672
\(537\) 0 0
\(538\) −976.000 −1.81413
\(539\) −784.000 −1.45455
\(540\) 0 0
\(541\) 854.000 1.57856 0.789279 − 0.614035i \(-0.210454\pi\)
0.789279 + 0.614035i \(0.210454\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −122.000 −0.223035 −0.111517 − 0.993762i \(-0.535571\pi\)
−0.111517 + 0.993762i \(0.535571\pi\)
\(548\) 0 0
\(549\) −954.000 −1.73770
\(550\) −800.000 −1.45455
\(551\) 1064.00 1.93103
\(552\) 192.000 0.347826
\(553\) 0 0
\(554\) −716.000 −1.29242
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) −252.000 −0.451613
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −928.000 −1.65125
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 672.000 1.19149
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 112.000 0.196837 0.0984183 − 0.995145i \(-0.468622\pi\)
0.0984183 + 0.995145i \(0.468622\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 1032.00 1.80105
\(574\) 0 0
\(575\) 200.000 0.347826
\(576\) 576.000 1.00000
\(577\) −898.000 −1.55633 −0.778163 − 0.628062i \(-0.783848\pi\)
−0.778163 + 0.628062i \(0.783848\pi\)
\(578\) 578.000 1.00000
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 128.000 0.219554
\(584\) −656.000 −1.12329
\(585\) 0 0
\(586\) 944.000 1.61092
\(587\) −688.000 −1.17206 −0.586031 − 0.810289i \(-0.699310\pi\)
−0.586031 + 0.810289i \(0.699310\pi\)
\(588\) 588.000 1.00000
\(589\) 266.000 0.451613
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) −864.000 −1.45455
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 600.000 1.00000
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 522.000 0.865672
\(604\) −8.00000 −0.0132450
\(605\) 0 0
\(606\) 0 0
\(607\) 1138.00 1.87479 0.937397 − 0.348263i \(-0.113228\pi\)
0.937397 + 0.348263i \(0.113228\pi\)
\(608\) −608.000 −1.00000
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −826.000 −1.34747 −0.673736 − 0.738972i \(-0.735311\pi\)
−0.673736 + 0.738972i \(0.735311\pi\)
\(614\) −1204.00 −1.96091
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) −588.000 −0.951456
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 216.000 0.347826
\(622\) −656.000 −1.05466
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 796.000 1.27157
\(627\) 912.000 1.45455
\(628\) 344.000 0.547771
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) −1168.00 −1.84810
\(633\) 1038.00 1.63981
\(634\) −784.000 −1.23659
\(635\) 0 0
\(636\) −96.0000 −0.150943
\(637\) 0 0
\(638\) 1792.00 2.80878
\(639\) 0 0
\(640\) 0 0
\(641\) 1168.00 1.82215 0.911076 − 0.412237i \(-0.135253\pi\)
0.911076 + 0.412237i \(0.135253\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −568.000 −0.877898 −0.438949 − 0.898512i \(-0.644649\pi\)
−0.438949 + 0.898512i \(0.644649\pi\)
\(648\) 648.000 1.00000
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −512.000 −0.780488
\(657\) −738.000 −1.12329
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −1108.00 −1.67372
\(663\) 0 0
\(664\) 1024.00 1.54217
\(665\) 0 0
\(666\) 0 0
\(667\) −448.000 −0.671664
\(668\) 0 0
\(669\) 426.000 0.636771
\(670\) 0 0
\(671\) 1696.00 2.52757
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 675.000 1.00000
\(676\) 676.000 1.00000
\(677\) 328.000 0.484490 0.242245 − 0.970215i \(-0.422116\pi\)
0.242245 + 0.970215i \(0.422116\pi\)
\(678\) 672.000 0.991150
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 448.000 0.656891
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −684.000 −1.00000
\(685\) 0 0
\(686\) 0 0
\(687\) −1362.00 −1.98253
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 928.000 1.34104
\(693\) 0 0
\(694\) 1312.00 1.89049
\(695\) 0 0
\(696\) −1344.00 −1.93103
\(697\) 0 0
\(698\) −428.000 −0.613181
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1024.00 −1.45455
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −634.000 −0.894217 −0.447109 − 0.894480i \(-0.647546\pi\)
−0.447109 + 0.894480i \(0.647546\pi\)
\(710\) 0 0
\(711\) −1314.00 −1.84810
\(712\) 512.000 0.719101
\(713\) −112.000 −0.157083
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1416.00 −1.97490
\(718\) −464.000 −0.646240
\(719\) −424.000 −0.589708 −0.294854 − 0.955542i \(-0.595271\pi\)
−0.294854 + 0.955542i \(0.595271\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 722.000 1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) −1400.00 −1.93103
\(726\) 810.000 1.11570
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −1272.00 −1.73770
\(733\) 1238.00 1.68895 0.844475 − 0.535595i \(-0.179913\pi\)
0.844475 + 0.535595i \(0.179913\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 256.000 0.347826
\(737\) −928.000 −1.25916
\(738\) −576.000 −0.780488
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) −336.000 −0.451613
\(745\) 0 0
\(746\) 0 0
\(747\) 1152.00 1.54217
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1198.00 1.59521 0.797603 − 0.603183i \(-0.206101\pi\)
0.797603 + 0.603183i \(0.206101\pi\)
\(752\) 896.000 1.19149
\(753\) −1344.00 −1.78486
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 602.000 0.795244 0.397622 − 0.917549i \(-0.369835\pi\)
0.397622 + 0.917549i \(0.369835\pi\)
\(758\) −916.000 −1.20844
\(759\) −384.000 −0.505929
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 1068.00 1.40157
\(763\) 0 0
\(764\) 1376.00 1.80105
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 768.000 1.00000
\(769\) −514.000 −0.668401 −0.334200 − 0.942502i \(-0.608466\pi\)
−0.334200 + 0.942502i \(0.608466\pi\)
\(770\) 0 0
\(771\) −1536.00 −1.99222
\(772\) 0 0
\(773\) −1304.00 −1.68693 −0.843467 − 0.537181i \(-0.819489\pi\)
−0.843467 + 0.537181i \(0.819489\pi\)
\(774\) 0 0
\(775\) −350.000 −0.451613
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 608.000 0.780488
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1512.00 −1.93103
\(784\) 784.000 1.00000
\(785\) 0 0
\(786\) 1344.00 1.70992
\(787\) 1498.00 1.90343 0.951715 − 0.306982i \(-0.0993191\pi\)
0.951715 + 0.306982i \(0.0993191\pi\)
\(788\) 0 0
\(789\) 1464.00 1.85551
\(790\) 0 0
\(791\) 0 0
\(792\) −1152.00 −1.45455
\(793\) 0 0
\(794\) −236.000 −0.297229
\(795\) 0 0
\(796\) 0 0
\(797\) −1256.00 −1.57591 −0.787955 − 0.615733i \(-0.788860\pi\)
−0.787955 + 0.615733i \(0.788860\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 800.000 1.00000
\(801\) 576.000 0.719101
\(802\) −448.000 −0.558603
\(803\) 1312.00 1.63387
\(804\) 696.000 0.865672
\(805\) 0 0
\(806\) 0 0
\(807\) −1464.00 −1.81413
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 406.000 0.500617 0.250308 − 0.968166i \(-0.419468\pi\)
0.250308 + 0.968166i \(0.419468\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) −784.000 −0.951456
\(825\) −1200.00 −1.45455
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 288.000 0.347826
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) −1074.00 −1.29242
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 1216.00 1.45455
\(837\) −378.000 −0.451613
\(838\) −224.000 −0.267303
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2295.00 2.72889
\(842\) 0 0
\(843\) −1392.00 −1.65125
\(844\) 1384.00 1.63981
\(845\) 0 0
\(846\) 1008.00 1.19149
\(847\) 0 0
\(848\) −128.000 −0.150943
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 794.000 0.930832 0.465416 − 0.885092i \(-0.345905\pi\)
0.465416 + 0.885092i \(0.345905\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1136.00 −1.32555 −0.662777 − 0.748817i \(-0.730622\pi\)
−0.662777 + 0.748817i \(0.730622\pi\)
\(858\) 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
\(865\) 0 0
\(866\) 0 0
\(867\) 867.000 1.00000
\(868\) 0 0
\(869\) 2336.00 2.68815
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) −304.000 −0.347826
\(875\) 0 0
\(876\) −984.000 −1.12329
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 1148.00 1.30752
\(879\) 1416.00 1.61092
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 882.000 1.00000
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1696.00 1.91422
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1296.00 −1.45455
\(892\) 568.000 0.636771
\(893\) −1064.00 −1.19149
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1568.00 1.74610
\(899\) 784.000 0.872080
\(900\) 900.000 1.00000
\(901\) 0 0
\(902\) 1024.00 1.13525
\(903\) 0 0
\(904\) 896.000 0.991150
\(905\) 0 0
\(906\) −12.0000 −0.0132450
\(907\) −86.0000 −0.0948181 −0.0474090 − 0.998876i \(-0.515096\pi\)
−0.0474090 + 0.998876i \(0.515096\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −912.000 −1.00000
\(913\) −2048.00 −2.24315
\(914\) 1372.00 1.50109
\(915\) 0 0
\(916\) −1816.00 −1.98253
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −1806.00 −1.96091
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −882.000 −0.951456
\(928\) −1792.00 −1.93103
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −931.000 −1.00000
\(932\) 0 0
\(933\) −984.000 −1.05466
\(934\) −1856.00 −1.98715
\(935\) 0 0
\(936\) 0 0
\(937\) −1774.00 −1.89328 −0.946638 − 0.322298i \(-0.895545\pi\)
−0.946638 + 0.322298i \(0.895545\pi\)
\(938\) 0 0
\(939\) 1194.00 1.27157
\(940\) 0 0
\(941\) 856.000 0.909671 0.454835 − 0.890576i \(-0.349698\pi\)
0.454835 + 0.890576i \(0.349698\pi\)
\(942\) 516.000 0.547771
\(943\) −256.000 −0.271474
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1856.00 1.95987 0.979937 − 0.199309i \(-0.0638698\pi\)
0.979937 + 0.199309i \(0.0638698\pi\)
\(948\) −1752.00 −1.84810
\(949\) 0 0
\(950\) −950.000 −1.00000
\(951\) −1176.00 −1.23659
\(952\) 0 0
\(953\) 1792.00 1.88038 0.940189 − 0.340654i \(-0.110648\pi\)
0.940189 + 0.340654i \(0.110648\pi\)
\(954\) −144.000 −0.150943
\(955\) 0 0
\(956\) −1888.00 −1.97490
\(957\) 2688.00 2.80878
\(958\) −1808.00 −1.88727
\(959\) 0 0
\(960\) 0 0
\(961\) −765.000 −0.796046
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 1080.00 1.11570
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 972.000 1.00000
\(973\) 0 0
\(974\) −1852.00 −1.90144
\(975\) 0 0
\(976\) −1696.00 −1.73770
\(977\) −896.000 −0.917093 −0.458547 − 0.888670i \(-0.651630\pi\)
−0.458547 + 0.888670i \(0.651630\pi\)
\(978\) 0 0
\(979\) −1024.00 −1.04597
\(980\) 0 0
\(981\) 0 0
\(982\) 64.0000 0.0651731
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −768.000 −0.780488
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1906.00 1.92331 0.961655 − 0.274262i \(-0.0884337\pi\)
0.961655 + 0.274262i \(0.0884337\pi\)
\(992\) −448.000 −0.451613
\(993\) −1662.00 −1.67372
\(994\) 0 0
\(995\) 0 0
\(996\) 1536.00 1.54217
\(997\) −58.0000 −0.0581745 −0.0290873 − 0.999577i \(-0.509260\pi\)
−0.0290873 + 0.999577i \(0.509260\pi\)
\(998\) 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 228.3.b.d.227.1 yes 1
3.2 odd 2 228.3.b.b.227.1 yes 1
4.3 odd 2 228.3.b.c.227.1 yes 1
12.11 even 2 228.3.b.a.227.1 ✓ 1
19.18 odd 2 228.3.b.a.227.1 ✓ 1
57.56 even 2 228.3.b.c.227.1 yes 1
76.75 even 2 228.3.b.b.227.1 yes 1
228.227 odd 2 CM 228.3.b.d.227.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
228.3.b.a.227.1 ✓ 1 12.11 even 2
228.3.b.a.227.1 ✓ 1 19.18 odd 2
228.3.b.b.227.1 yes 1 3.2 odd 2
228.3.b.b.227.1 yes 1 76.75 even 2
228.3.b.c.227.1 yes 1 4.3 odd 2
228.3.b.c.227.1 yes 1 57.56 even 2
228.3.b.d.227.1 yes 1 1.1 even 1 trivial
228.3.b.d.227.1 yes 1 228.227 odd 2 CM