Properties

Label 55.3.d.b.54.2
Level $55$
Weight $3$
Character 55.54
Self dual yes
Analytic conductor $1.499$
Analytic rank $0$
Dimension $2$
CM discriminant -55
Inner twists $4$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [55,3,Mod(54,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("55.54");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 55.d (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: \(1.49864145398\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 54.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 55.54

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607 q^{2} +1.00000 q^{4} +5.00000 q^{5} -4.47214 q^{7} -6.70820 q^{8} +9.00000 q^{9} +11.1803 q^{10} -11.0000 q^{11} -22.3607 q^{13} -10.0000 q^{14} -19.0000 q^{16} +31.3050 q^{17} +20.1246 q^{18} +5.00000 q^{20} -24.5967 q^{22} +25.0000 q^{25} -50.0000 q^{26} -4.47214 q^{28} +18.0000 q^{31} -15.6525 q^{32} +70.0000 q^{34} -22.3607 q^{35} +9.00000 q^{36} -33.5410 q^{40} +84.9706 q^{43} -11.0000 q^{44} +45.0000 q^{45} -29.0000 q^{49} +55.9017 q^{50} -22.3607 q^{52} -55.0000 q^{55} +30.0000 q^{56} -102.000 q^{59} +40.2492 q^{62} -40.2492 q^{63} +41.0000 q^{64} -111.803 q^{65} +31.3050 q^{68} -50.0000 q^{70} -78.0000 q^{71} -60.3738 q^{72} -4.47214 q^{73} +49.1935 q^{77} -95.0000 q^{80} +81.0000 q^{81} -165.469 q^{83} +156.525 q^{85} +190.000 q^{86} +73.7902 q^{88} +2.00000 q^{89} +100.623 q^{90} +100.000 q^{91} -64.8460 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 10 q^{5} + 18 q^{9} - 22 q^{11} - 20 q^{14} - 38 q^{16} + 10 q^{20} + 50 q^{25} - 100 q^{26} + 36 q^{31} + 140 q^{34} + 18 q^{36} - 22 q^{44} + 90 q^{45} - 58 q^{49} - 110 q^{55} + 60 q^{56}+ \cdots - 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607 1.11803 0.559017 0.829156i \(-0.311179\pi\)
0.559017 + 0.829156i \(0.311179\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.00000 0.250000
\(5\) 5.00000 1.00000
\(6\) 0 0
\(7\) −4.47214 −0.638877 −0.319438 0.947607i \(-0.603494\pi\)
−0.319438 + 0.947607i \(0.603494\pi\)
\(8\) −6.70820 −0.838525
\(9\) 9.00000 1.00000
\(10\) 11.1803 1.11803
\(11\) −11.0000 −1.00000
\(12\) 0 0
\(13\) −22.3607 −1.72005 −0.860026 0.510250i \(-0.829553\pi\)
−0.860026 + 0.510250i \(0.829553\pi\)
\(14\) −10.0000 −0.714286
\(15\) 0 0
\(16\) −19.0000 −1.18750
\(17\) 31.3050 1.84147 0.920734 0.390191i \(-0.127591\pi\)
0.920734 + 0.390191i \(0.127591\pi\)
\(18\) 20.1246 1.11803
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 5.00000 0.250000
\(21\) 0 0
\(22\) −24.5967 −1.11803
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) −50.0000 −1.92308
\(27\) 0 0
\(28\) −4.47214 −0.159719
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 18.0000 0.580645 0.290323 0.956929i \(-0.406237\pi\)
0.290323 + 0.956929i \(0.406237\pi\)
\(32\) −15.6525 −0.489140
\(33\) 0 0
\(34\) 70.0000 2.05882
\(35\) −22.3607 −0.638877
\(36\) 9.00000 0.250000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −33.5410 −0.838525
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 84.9706 1.97606 0.988030 0.154262i \(-0.0492999\pi\)
0.988030 + 0.154262i \(0.0492999\pi\)
\(44\) −11.0000 −0.250000
\(45\) 45.0000 1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −29.0000 −0.591837
\(50\) 55.9017 1.11803
\(51\) 0 0
\(52\) −22.3607 −0.430013
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −55.0000 −1.00000
\(56\) 30.0000 0.535714
\(57\) 0 0
\(58\) 0 0
\(59\) −102.000 −1.72881 −0.864407 0.502793i \(-0.832306\pi\)
−0.864407 + 0.502793i \(0.832306\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 40.2492 0.649181
\(63\) −40.2492 −0.638877
\(64\) 41.0000 0.640625
\(65\) −111.803 −1.72005
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 31.3050 0.460367
\(69\) 0 0
\(70\) −50.0000 −0.714286
\(71\) −78.0000 −1.09859 −0.549296 0.835628i \(-0.685104\pi\)
−0.549296 + 0.835628i \(0.685104\pi\)
\(72\) −60.3738 −0.838525
\(73\) −4.47214 −0.0612621 −0.0306311 0.999531i \(-0.509752\pi\)
−0.0306311 + 0.999531i \(0.509752\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 49.1935 0.638877
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −95.0000 −1.18750
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) −165.469 −1.99360 −0.996801 0.0799187i \(-0.974534\pi\)
−0.996801 + 0.0799187i \(0.974534\pi\)
\(84\) 0 0
\(85\) 156.525 1.84147
\(86\) 190.000 2.20930
\(87\) 0 0
\(88\) 73.7902 0.838525
\(89\) 2.00000 0.0224719 0.0112360 0.999937i \(-0.496423\pi\)
0.0112360 + 0.999937i \(0.496423\pi\)
\(90\) 100.623 1.11803
\(91\) 100.000 1.09890
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −64.8460 −0.661694
\(99\) −99.0000 −1.00000
\(100\) 25.0000 0.250000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 150.000 1.44231
\(105\) 0 0
\(106\) 0 0
\(107\) 156.525 1.46285 0.731424 0.681923i \(-0.238856\pi\)
0.731424 + 0.681923i \(0.238856\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) −122.984 −1.11803
\(111\) 0 0
\(112\) 84.9706 0.758666
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −201.246 −1.72005
\(118\) −228.079 −1.93287
\(119\) −140.000 −1.17647
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 18.0000 0.145161
\(125\) 125.000 1.00000
\(126\) −90.0000 −0.714286
\(127\) 31.3050 0.246496 0.123248 0.992376i \(-0.460669\pi\)
0.123248 + 0.992376i \(0.460669\pi\)
\(128\) 154.289 1.20538
\(129\) 0 0
\(130\) −250.000 −1.92308
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −210.000 −1.54412
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −22.3607 −0.159719
\(141\) 0 0
\(142\) −174.413 −1.22826
\(143\) 245.967 1.72005
\(144\) −171.000 −1.18750
\(145\) 0 0
\(146\) −10.0000 −0.0684932
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 281.745 1.84147
\(154\) 110.000 0.714286
\(155\) 90.0000 0.580645
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −78.2624 −0.489140
\(161\) 0 0
\(162\) 181.122 1.11803
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −370.000 −2.22892
\(167\) −219.135 −1.31218 −0.656092 0.754681i \(-0.727792\pi\)
−0.656092 + 0.754681i \(0.727792\pi\)
\(168\) 0 0
\(169\) 331.000 1.95858
\(170\) 350.000 2.05882
\(171\) 0 0
\(172\) 84.9706 0.494015
\(173\) −237.023 −1.37008 −0.685038 0.728507i \(-0.740214\pi\)
−0.685038 + 0.728507i \(0.740214\pi\)
\(174\) 0 0
\(175\) −111.803 −0.638877
\(176\) 209.000 1.18750
\(177\) 0 0
\(178\) 4.47214 0.0251244
\(179\) −38.0000 −0.212291 −0.106145 0.994351i \(-0.533851\pi\)
−0.106145 + 0.994351i \(0.533851\pi\)
\(180\) 45.0000 0.250000
\(181\) −342.000 −1.88950 −0.944751 0.327788i \(-0.893697\pi\)
−0.944751 + 0.327788i \(0.893697\pi\)
\(182\) 223.607 1.22861
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −344.354 −1.84147
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 338.000 1.76963 0.884817 0.465939i \(-0.154283\pi\)
0.884817 + 0.465939i \(0.154283\pi\)
\(192\) 0 0
\(193\) 31.3050 0.162202 0.0811009 0.996706i \(-0.474156\pi\)
0.0811009 + 0.996706i \(0.474156\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −29.0000 −0.147959
\(197\) 84.9706 0.431323 0.215661 0.976468i \(-0.430809\pi\)
0.215661 + 0.976468i \(0.430809\pi\)
\(198\) −221.371 −1.11803
\(199\) 178.000 0.894472 0.447236 0.894416i \(-0.352408\pi\)
0.447236 + 0.894416i \(0.352408\pi\)
\(200\) −167.705 −0.838525
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 424.853 2.04256
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 350.000 1.63551
\(215\) 424.853 1.97606
\(216\) 0 0
\(217\) −80.4984 −0.370961
\(218\) 0 0
\(219\) 0 0
\(220\) −55.0000 −0.250000
\(221\) −700.000 −3.16742
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 70.0000 0.312500
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) 192.302 0.847145 0.423572 0.905862i \(-0.360776\pi\)
0.423572 + 0.905862i \(0.360776\pi\)
\(228\) 0 0
\(229\) −422.000 −1.84279 −0.921397 0.388622i \(-0.872951\pi\)
−0.921397 + 0.388622i \(0.872951\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −219.135 −0.940492 −0.470246 0.882535i \(-0.655835\pi\)
−0.470246 + 0.882535i \(0.655835\pi\)
\(234\) −450.000 −1.92308
\(235\) 0 0
\(236\) −102.000 −0.432203
\(237\) 0 0
\(238\) −313.050 −1.31533
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 270.564 1.11803
\(243\) 0 0
\(244\) 0 0
\(245\) −145.000 −0.591837
\(246\) 0 0
\(247\) 0 0
\(248\) −120.748 −0.486886
\(249\) 0 0
\(250\) 279.508 1.11803
\(251\) 282.000 1.12351 0.561753 0.827305i \(-0.310127\pi\)
0.561753 + 0.827305i \(0.310127\pi\)
\(252\) −40.2492 −0.159719
\(253\) 0 0
\(254\) 70.0000 0.275591
\(255\) 0 0
\(256\) 181.000 0.707031
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −111.803 −0.430013
\(261\) 0 0
\(262\) 0 0
\(263\) −505.351 −1.92149 −0.960744 0.277436i \(-0.910515\pi\)
−0.960744 + 0.277436i \(0.910515\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −342.000 −1.27138 −0.635688 0.771946i \(-0.719283\pi\)
−0.635688 + 0.771946i \(0.719283\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −594.794 −2.18674
\(273\) 0 0
\(274\) 0 0
\(275\) −275.000 −1.00000
\(276\) 0 0
\(277\) 371.187 1.34003 0.670013 0.742349i \(-0.266288\pi\)
0.670013 + 0.742349i \(0.266288\pi\)
\(278\) 0 0
\(279\) 162.000 0.580645
\(280\) 150.000 0.535714
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 156.525 0.553091 0.276546 0.961001i \(-0.410810\pi\)
0.276546 + 0.961001i \(0.410810\pi\)
\(284\) −78.0000 −0.274648
\(285\) 0 0
\(286\) 550.000 1.92308
\(287\) 0 0
\(288\) −140.872 −0.489140
\(289\) 691.000 2.39100
\(290\) 0 0
\(291\) 0 0
\(292\) −4.47214 −0.0153155
\(293\) 585.850 1.99949 0.999744 0.0226391i \(-0.00720686\pi\)
0.999744 + 0.0226391i \(0.00720686\pi\)
\(294\) 0 0
\(295\) −510.000 −1.72881
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −380.000 −1.26246
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 630.000 2.05882
\(307\) 478.519 1.55869 0.779346 0.626594i \(-0.215551\pi\)
0.779346 + 0.626594i \(0.215551\pi\)
\(308\) 49.1935 0.159719
\(309\) 0 0
\(310\) 201.246 0.649181
\(311\) 402.000 1.29260 0.646302 0.763082i \(-0.276315\pi\)
0.646302 + 0.763082i \(0.276315\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −201.246 −0.638877
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 205.000 0.640625
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 81.0000 0.250000
\(325\) −559.017 −1.72005
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 618.000 1.86707 0.933535 0.358487i \(-0.116707\pi\)
0.933535 + 0.358487i \(0.116707\pi\)
\(332\) −165.469 −0.498401
\(333\) 0 0
\(334\) −490.000 −1.46707
\(335\) 0 0
\(336\) 0 0
\(337\) −398.020 −1.18107 −0.590534 0.807013i \(-0.701083\pi\)
−0.590534 + 0.807013i \(0.701083\pi\)
\(338\) 740.139 2.18976
\(339\) 0 0
\(340\) 156.525 0.460367
\(341\) −198.000 −0.580645
\(342\) 0 0
\(343\) 348.827 1.01699
\(344\) −570.000 −1.65698
\(345\) 0 0
\(346\) −530.000 −1.53179
\(347\) −559.017 −1.61100 −0.805500 0.592596i \(-0.798103\pi\)
−0.805500 + 0.592596i \(0.798103\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) −250.000 −0.714286
\(351\) 0 0
\(352\) 172.177 0.489140
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −390.000 −1.09859
\(356\) 2.00000 0.00561798
\(357\) 0 0
\(358\) −84.9706 −0.237348
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −301.869 −0.838525
\(361\) 361.000 1.00000
\(362\) −764.735 −2.11253
\(363\) 0 0
\(364\) 100.000 0.274725
\(365\) −22.3607 −0.0612621
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −702.125 −1.88237 −0.941187 0.337887i \(-0.890288\pi\)
−0.941187 + 0.337887i \(0.890288\pi\)
\(374\) −770.000 −2.05882
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 538.000 1.41953 0.709763 0.704441i \(-0.248802\pi\)
0.709763 + 0.704441i \(0.248802\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 755.791 1.97851
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 245.967 0.638877
\(386\) 70.0000 0.181347
\(387\) 764.735 1.97606
\(388\) 0 0
\(389\) −102.000 −0.262211 −0.131105 0.991368i \(-0.541853\pi\)
−0.131105 + 0.991368i \(0.541853\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 194.538 0.496270
\(393\) 0 0
\(394\) 190.000 0.482234
\(395\) 0 0
\(396\) −99.0000 −0.250000
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 398.020 1.00005
\(399\) 0 0
\(400\) −475.000 −1.18750
\(401\) −782.000 −1.95012 −0.975062 0.221931i \(-0.928764\pi\)
−0.975062 + 0.221931i \(0.928764\pi\)
\(402\) 0 0
\(403\) −402.492 −0.998740
\(404\) 0 0
\(405\) 405.000 1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 456.158 1.10450
\(414\) 0 0
\(415\) −827.345 −1.99360
\(416\) 350.000 0.841346
\(417\) 0 0
\(418\) 0 0
\(419\) 442.000 1.05489 0.527446 0.849588i \(-0.323150\pi\)
0.527446 + 0.849588i \(0.323150\pi\)
\(420\) 0 0
\(421\) 138.000 0.327791 0.163895 0.986478i \(-0.447594\pi\)
0.163895 + 0.986478i \(0.447594\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 782.624 1.84147
\(426\) 0 0
\(427\) 0 0
\(428\) 156.525 0.365712
\(429\) 0 0
\(430\) 950.000 2.20930
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) −180.000 −0.414747
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 368.951 0.838525
\(441\) −261.000 −0.591837
\(442\) −1565.25 −3.54128
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 10.0000 0.0224719
\(446\) 0 0
\(447\) 0 0
\(448\) −183.358 −0.409280
\(449\) 722.000 1.60802 0.804009 0.594617i \(-0.202696\pi\)
0.804009 + 0.594617i \(0.202696\pi\)
\(450\) 503.115 1.11803
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 430.000 0.947137
\(455\) 500.000 1.09890
\(456\) 0 0
\(457\) 424.853 0.929656 0.464828 0.885401i \(-0.346116\pi\)
0.464828 + 0.885401i \(0.346116\pi\)
\(458\) −943.621 −2.06031
\(459\) 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 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −490.000 −1.05150
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −201.246 −0.430013
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 684.237 1.44965
\(473\) −934.676 −1.97606
\(474\) 0 0
\(475\) 0 0
\(476\) −140.000 −0.294118
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 121.000 0.250000
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −324.230 −0.661694
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −495.000 −1.00000
\(496\) −342.000 −0.689516
\(497\) 348.827 0.701864
\(498\) 0 0
\(499\) −982.000 −1.96794 −0.983968 0.178345i \(-0.942926\pi\)
−0.983968 + 0.178345i \(0.942926\pi\)
\(500\) 125.000 0.250000
\(501\) 0 0
\(502\) 630.571 1.25612
\(503\) 353.299 0.702383 0.351192 0.936304i \(-0.385777\pi\)
0.351192 + 0.936304i \(0.385777\pi\)
\(504\) 270.000 0.535714
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 31.3050 0.0616239
\(509\) 138.000 0.271120 0.135560 0.990769i \(-0.456717\pi\)
0.135560 + 0.990769i \(0.456717\pi\)
\(510\) 0 0
\(511\) 20.0000 0.0391389
\(512\) −212.426 −0.414895
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 750.000 1.44231
\(521\) −542.000 −1.04031 −0.520154 0.854073i \(-0.674125\pi\)
−0.520154 + 0.854073i \(0.674125\pi\)
\(522\) 0 0
\(523\) 1015.17 1.94106 0.970530 0.240978i \(-0.0774683\pi\)
0.970530 + 0.240978i \(0.0774683\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −1130.00 −2.14829
\(527\) 563.489 1.06924
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) −918.000 −1.72881
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 782.624 1.46285
\(536\) 0 0
\(537\) 0 0
\(538\) −764.735 −1.42144
\(539\) 319.000 0.591837
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −490.000 −0.900735
\(545\) 0 0
\(546\) 0 0
\(547\) −1024.12 −1.87225 −0.936124 0.351671i \(-0.885613\pi\)
−0.936124 + 0.351671i \(0.885613\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −614.919 −1.11803
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 830.000 1.49819
\(555\) 0 0
\(556\) 0 0
\(557\) −594.794 −1.06785 −0.533926 0.845531i \(-0.679284\pi\)
−0.533926 + 0.845531i \(0.679284\pi\)
\(558\) 362.243 0.649181
\(559\) −1900.00 −3.39893
\(560\) 424.853 0.758666
\(561\) 0 0
\(562\) 0 0
\(563\) −809.457 −1.43776 −0.718878 0.695136i \(-0.755344\pi\)
−0.718878 + 0.695136i \(0.755344\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 350.000 0.618375
\(567\) −362.243 −0.638877
\(568\) 523.240 0.921197
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 245.967 0.430013
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 369.000 0.640625
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 1545.12 2.67322
\(579\) 0 0
\(580\) 0 0
\(581\) 740.000 1.27367
\(582\) 0 0
\(583\) 0 0
\(584\) 30.0000 0.0513699
\(585\) −1006.23 −1.72005
\(586\) 1310.00 2.23549
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −1140.39 −1.93287
\(591\) 0 0
\(592\) 0 0
\(593\) −111.803 −0.188539 −0.0942693 0.995547i \(-0.530051\pi\)
−0.0942693 + 0.995547i \(0.530051\pi\)
\(594\) 0 0
\(595\) −700.000 −1.17647
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 802.000 1.33890 0.669449 0.742858i \(-0.266530\pi\)
0.669449 + 0.742858i \(0.266530\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) −849.706 −1.41147
\(603\) 0 0
\(604\) 0 0
\(605\) 605.000 1.00000
\(606\) 0 0
\(607\) −612.683 −1.00936 −0.504681 0.863306i \(-0.668390\pi\)
−0.504681 + 0.863306i \(0.668390\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 281.745 0.460367
\(613\) 943.621 1.53935 0.769674 0.638437i \(-0.220419\pi\)
0.769674 + 0.638437i \(0.220419\pi\)
\(614\) 1070.00 1.74267
\(615\) 0 0
\(616\) −330.000 −0.535714
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −918.000 −1.48304 −0.741519 0.670932i \(-0.765894\pi\)
−0.741519 + 0.670932i \(0.765894\pi\)
\(620\) 90.0000 0.145161
\(621\) 0 0
\(622\) 898.899 1.44518
\(623\) −8.94427 −0.0143568
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) −450.000 −0.714286
\(631\) 1042.00 1.65135 0.825674 0.564148i \(-0.190795\pi\)
0.825674 + 0.564148i \(0.190795\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 156.525 0.246496
\(636\) 0 0
\(637\) 648.460 1.01799
\(638\) 0 0
\(639\) −702.000 −1.09859
\(640\) 771.443 1.20538
\(641\) −302.000 −0.471139 −0.235569 0.971858i \(-0.575696\pi\)
−0.235569 + 0.971858i \(0.575696\pi\)
\(642\) 0 0
\(643\) 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 \(\pi\)
\(648\) −543.365 −0.838525
\(649\) 1122.00 1.72881
\(650\) −1250.00 −1.92308
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −40.2492 −0.0612621
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 442.000 0.668684 0.334342 0.942452i \(-0.391486\pi\)
0.334342 + 0.942452i \(0.391486\pi\)
\(662\) 1381.89 2.08745
\(663\) 0 0
\(664\) 1110.00 1.67169
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −219.135 −0.328046
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 961.509 1.42869 0.714346 0.699793i \(-0.246724\pi\)
0.714346 + 0.699793i \(0.246724\pi\)
\(674\) −890.000 −1.32047
\(675\) 0 0
\(676\) 331.000 0.489645
\(677\) −1346.11 −1.98835 −0.994175 0.107778i \(-0.965626\pi\)
−0.994175 + 0.107778i \(0.965626\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1050.00 −1.54412
\(681\) 0 0
\(682\) −442.741 −0.649181
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 780.000 1.13703
\(687\) 0 0
\(688\) −1614.44 −2.34657
\(689\) 0 0
\(690\) 0 0
\(691\) −598.000 −0.865412 −0.432706 0.901535i \(-0.642441\pi\)
−0.432706 + 0.901535i \(0.642441\pi\)
\(692\) −237.023 −0.342519
\(693\) 442.741 0.638877
\(694\) −1250.00 −1.80115
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −111.803 −0.159719
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −451.000 −0.640625
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1398.00 −1.97179 −0.985896 0.167361i \(-0.946475\pi\)
−0.985896 + 0.167361i \(0.946475\pi\)
\(710\) −872.067 −1.22826
\(711\) 0 0
\(712\) −13.4164 −0.0188433
\(713\) 0 0
\(714\) 0 0
\(715\) 1229.84 1.72005
\(716\) −38.0000 −0.0530726
\(717\) 0 0
\(718\) 0 0
\(719\) −718.000 −0.998609 −0.499305 0.866427i \(-0.666411\pi\)
−0.499305 + 0.866427i \(0.666411\pi\)
\(720\) −855.000 −1.18750
\(721\) 0 0
\(722\) 807.221 1.11803
\(723\) 0 0
\(724\) −342.000 −0.472376
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) −670.820 −0.921457
\(729\) 729.000 1.00000
\(730\) −50.0000 −0.0684932
\(731\) 2660.00 3.63885
\(732\) 0 0
\(733\) −594.794 −0.811452 −0.405726 0.913995i \(-0.632981\pi\)
−0.405726 + 0.913995i \(0.632981\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1149.34 −1.54689 −0.773445 0.633864i \(-0.781468\pi\)
−0.773445 + 0.633864i \(0.781468\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1570.00 −2.10456
\(747\) −1489.22 −1.99360
\(748\) −344.354 −0.460367
\(749\) −700.000 −0.934579
\(750\) 0 0
\(751\) −478.000 −0.636485 −0.318242 0.948009i \(-0.603093\pi\)
−0.318242 + 0.948009i \(0.603093\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 1203.00 1.58708
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 338.000 0.442408
\(765\) 1408.72 1.84147
\(766\) 0 0
\(767\) 2280.79 2.97365
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 550.000 0.714286
\(771\) 0 0
\(772\) 31.3050 0.0405505
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 1710.00 2.20930
\(775\) 450.000 0.580645
\(776\) 0 0
\(777\) 0 0
\(778\) −228.079 −0.293161
\(779\) 0 0
\(780\) 0 0
\(781\) 858.000 1.09859
\(782\) 0 0
\(783\) 0 0
\(784\) 551.000 0.702806
\(785\) 0 0
\(786\) 0 0
\(787\) 621.627 0.789869 0.394934 0.918709i \(-0.370767\pi\)
0.394934 + 0.918709i \(0.370767\pi\)
\(788\) 84.9706 0.107831
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 664.112 0.838525
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 178.000 0.223618
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −391.312 −0.489140
\(801\) 18.0000 0.0224719
\(802\) −1748.61 −2.18031
\(803\) 49.1935 0.0612621
\(804\) 0 0
\(805\) 0 0
\(806\) −900.000 −1.11663
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 905.608 1.11803
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 900.000 1.09890
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 1020.00 1.23487
\(827\) 1158.28 1.40058 0.700292 0.713856i \(-0.253053\pi\)
0.700292 + 0.713856i \(0.253053\pi\)
\(828\) 0 0
\(829\) −1158.00 −1.39686 −0.698432 0.715677i \(-0.746118\pi\)
−0.698432 + 0.715677i \(0.746118\pi\)
\(830\) −1850.00 −2.22892
\(831\) 0 0
\(832\) −916.788 −1.10191
\(833\) −907.844 −1.08985
\(834\) 0 0
\(835\) −1095.67 −1.31218
\(836\) 0 0
\(837\) 0 0
\(838\) 988.342 1.17941
\(839\) 1458.00 1.73778 0.868892 0.495003i \(-0.164833\pi\)
0.868892 + 0.495003i \(0.164833\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 308.577 0.366481
\(843\) 0 0
\(844\) 0 0
\(845\) 1655.00 1.95858
\(846\) 0 0
\(847\) −541.128 −0.638877
\(848\) 0 0
\(849\) 0 0
\(850\) 1750.00 2.05882
\(851\) 0 0
\(852\) 0 0
\(853\) −1346.11 −1.57809 −0.789046 0.614334i \(-0.789425\pi\)
−0.789046 + 0.614334i \(0.789425\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1050.00 −1.22664
\(857\) 1068.84 1.24719 0.623594 0.781748i \(-0.285672\pi\)
0.623594 + 0.781748i \(0.285672\pi\)
\(858\) 0 0
\(859\) −438.000 −0.509895 −0.254948 0.966955i \(-0.582058\pi\)
−0.254948 + 0.966955i \(0.582058\pi\)
\(860\) 424.853 0.494015
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −1185.12 −1.37008
\(866\) 0 0
\(867\) 0 0
\(868\) −80.4984 −0.0927401
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −559.017 −0.638877
\(876\) 0 0
\(877\) −237.023 −0.270266 −0.135133 0.990827i \(-0.543146\pi\)
−0.135133 + 0.990827i \(0.543146\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 1045.00 1.18750
\(881\) −1758.00 −1.99546 −0.997730 0.0673435i \(-0.978548\pi\)
−0.997730 + 0.0673435i \(0.978548\pi\)
\(882\) −583.614 −0.661694
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −700.000 −0.791855
\(885\) 0 0
\(886\) 0 0
\(887\) −1578.66 −1.77978 −0.889890 0.456176i \(-0.849219\pi\)
−0.889890 + 0.456176i \(0.849219\pi\)
\(888\) 0 0
\(889\) −140.000 −0.157480
\(890\) 22.3607 0.0251244
\(891\) −891.000 −1.00000
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −190.000 −0.212291
\(896\) −690.000 −0.770089
\(897\) 0 0
\(898\) 1614.44 1.79782
\(899\) 0 0
\(900\) 225.000 0.250000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1710.00 −1.88950
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 192.302 0.211786
\(909\) 0 0
\(910\) 1118.03 1.22861
\(911\) −1742.00 −1.91218 −0.956092 0.293066i \(-0.905324\pi\)
−0.956092 + 0.293066i \(0.905324\pi\)
\(912\) 0 0
\(913\) 1820.16 1.99360
\(914\) 950.000 1.03939
\(915\) 0 0
\(916\) −422.000 −0.460699
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1744.13 1.88963
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1662.00 −1.78902 −0.894510 0.447047i \(-0.852475\pi\)
−0.894510 + 0.447047i \(0.852475\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −219.135 −0.235123
\(933\) 0 0
\(934\) 0 0
\(935\) −1721.77 −1.84147
\(936\) 1350.00 1.44231
\(937\) −1793.33 −1.91390 −0.956951 0.290249i \(-0.906262\pi\)
−0.956951 + 0.290249i \(0.906262\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1938.00 2.05297
\(945\) 0 0
\(946\) −2090.00 −2.20930
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 100.000 0.105374
\(950\) 0 0
\(951\) 0 0
\(952\) 939.149 0.986501
\(953\) 782.624 0.821221 0.410611 0.911811i \(-0.365316\pi\)
0.410611 + 0.911811i \(0.365316\pi\)
\(954\) 0 0
\(955\) 1690.00 1.76963
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −637.000 −0.662851
\(962\) 0 0
\(963\) 1408.72 1.46285
\(964\) 0 0
\(965\) 156.525 0.162202
\(966\) 0 0
\(967\) 1855.94 1.91927 0.959636 0.281244i \(-0.0907471\pi\)
0.959636 + 0.281244i \(0.0907471\pi\)
\(968\) −811.693 −0.838525
\(969\) 0 0
\(970\) 0 0
\(971\) −1622.00 −1.67044 −0.835221 0.549914i \(-0.814661\pi\)
−0.835221 + 0.549914i \(0.814661\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) −22.0000 −0.0224719
\(980\) −145.000 −0.147959
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 424.853 0.431323
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) −1106.85 −1.11803
\(991\) 1938.00 1.95560 0.977800 0.209539i \(-0.0671965\pi\)
0.977800 + 0.209539i \(0.0671965\pi\)
\(992\) −281.745 −0.284017
\(993\) 0 0
\(994\) 780.000 0.784708
\(995\) 890.000 0.894472
\(996\) 0 0
\(997\) 1372.95 1.37708 0.688538 0.725200i \(-0.258253\pi\)
0.688538 + 0.725200i \(0.258253\pi\)
\(998\) −2195.82 −2.20022
\(999\) 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 55.3.d.b.54.2 yes 2
3.2 odd 2 495.3.h.b.109.1 2
4.3 odd 2 880.3.i.c.769.2 2
5.2 odd 4 275.3.c.c.76.2 2
5.3 odd 4 275.3.c.c.76.1 2
5.4 even 2 inner 55.3.d.b.54.1 2
11.10 odd 2 inner 55.3.d.b.54.1 2
15.14 odd 2 495.3.h.b.109.2 2
20.19 odd 2 880.3.i.c.769.1 2
33.32 even 2 495.3.h.b.109.2 2
44.43 even 2 880.3.i.c.769.1 2
55.32 even 4 275.3.c.c.76.1 2
55.43 even 4 275.3.c.c.76.2 2
55.54 odd 2 CM 55.3.d.b.54.2 yes 2
165.164 even 2 495.3.h.b.109.1 2
220.219 even 2 880.3.i.c.769.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.3.d.b.54.1 2 5.4 even 2 inner
55.3.d.b.54.1 2 11.10 odd 2 inner
55.3.d.b.54.2 yes 2 1.1 even 1 trivial
55.3.d.b.54.2 yes 2 55.54 odd 2 CM
275.3.c.c.76.1 2 5.3 odd 4
275.3.c.c.76.1 2 55.32 even 4
275.3.c.c.76.2 2 5.2 odd 4
275.3.c.c.76.2 2 55.43 even 4
495.3.h.b.109.1 2 3.2 odd 2
495.3.h.b.109.1 2 165.164 even 2
495.3.h.b.109.2 2 15.14 odd 2
495.3.h.b.109.2 2 33.32 even 2
880.3.i.c.769.1 2 20.19 odd 2
880.3.i.c.769.1 2 44.43 even 2
880.3.i.c.769.2 2 4.3 odd 2
880.3.i.c.769.2 2 220.219 even 2