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Properties

Label 5400.2.a.br.1.1
Level $5400$
Weight $2$
Character 5400.1
Self dual yes
Analytic conductor $43.119$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [5400,2,Mod(1,5400)] 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("5400.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,4,0,0,0,-2,0,-4,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(43.1192170915\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5400.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{7} -2.00000 q^{11} -4.00000 q^{13} +1.00000 q^{17} -5.00000 q^{19} +5.00000 q^{23} -8.00000 q^{29} +7.00000 q^{31} +6.00000 q^{37} -6.00000 q^{41} +2.00000 q^{43} +8.00000 q^{47} +9.00000 q^{49} +9.00000 q^{53} -4.00000 q^{59} +13.0000 q^{61} +10.0000 q^{67} +6.00000 q^{71} +6.00000 q^{73} -8.00000 q^{77} +9.00000 q^{79} -17.0000 q^{83} +6.00000 q^{89} -16.0000 q^{91} +8.00000 q^{97} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 − 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 − 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.00000 1.04257 0.521286 − 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 − 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 − 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) 9.00000 1.01258 0.506290 − 0.862364i \(-0.331017\pi\)
0.506290 + 0.862364i \(0.331017\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −17.0000 −1.86599 −0.932996 − 0.359886i \(-0.882816\pi\)
−0.932996 + 0.359886i \(0.882816\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −16.0000 −1.67726
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 − 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 17.0000 1.62830 0.814152 − 0.580651i \(-0.197202\pi\)
0.814152 + 0.580651i \(0.197202\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 − 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) −20.0000 −1.73422
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.0000 0.939793 0.469897 − 0.882721i \(-0.344291\pi\)
0.469897 + 0.882721i \(0.344291\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.0000 1.14692 0.573462 − 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 − 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 20.0000 1.57622
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 − 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.0000 −1.16073 −0.580367 − 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.00000 0.532200 0.266100 − 0.963945i \(-0.414265\pi\)
0.266100 + 0.963945i \(0.414265\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 − 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 − 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 26.0000 1.88129 0.940647 − 0.339387i \(-0.110219\pi\)
0.940647 + 0.339387i \(0.110219\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 − 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −25.0000 −1.78118 −0.890588 − 0.454811i \(-0.849707\pi\)
−0.890588 + 0.454811i \(0.849707\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −32.0000 −2.24596
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 − 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 28.0000 1.90076
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 − 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.00000 −0.597351 −0.298675 − 0.954355i \(-0.596545\pi\)
−0.298675 + 0.954355i \(0.596545\pi\)
\(228\) 0 0
\(229\) 19.0000 1.25556 0.627778 − 0.778393i \(-0.283965\pi\)
0.627778 + 0.778393i \(0.283965\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 − 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 − 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 20.0000 1.27257
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −10.0000 −0.628695
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.00000 0.187135 0.0935674 − 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 26.0000 1.58525 0.792624 − 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) 15.0000 0.911185 0.455593 − 0.890188i \(-0.349427\pi\)
0.455593 + 0.890188i \(0.349427\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 − 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 − 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.00000 −0.0584206 −0.0292103 − 0.999573i \(-0.509299\pi\)
−0.0292103 + 0.999573i \(0.509299\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −20.0000 −1.15663
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −10.0000 −0.570730 −0.285365 − 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 − 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) 28.0000 1.58265 0.791327 − 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 33.0000 1.85346 0.926732 − 0.375722i \(-0.122605\pi\)
0.926732 + 0.375722i \(0.122605\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.00000 −0.278207
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.0000 1.08947 0.544735 − 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −14.0000 −0.758143
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.0000 −1.07366 −0.536828 − 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) 9.00000 0.481759 0.240879 − 0.970555i \(-0.422564\pi\)
0.240879 + 0.970555i \(0.422564\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 34.0000 1.80964 0.904819 − 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −30.0000 −1.58334 −0.791670 − 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 − 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 36.0000 1.86903
\(372\) 0 0
\(373\) 20.0000 1.03556 0.517780 − 0.855514i \(-0.326758\pi\)
0.517780 + 0.855514i \(0.326758\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 32.0000 1.64808
\(378\) 0 0
\(379\) 13.0000 0.667765 0.333883 − 0.942615i \(-0.391641\pi\)
0.333883 + 0.942615i \(0.391641\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.0000 1.07305 0.536525 − 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 32.0000 1.60603 0.803017 − 0.595956i \(-0.203227\pi\)
0.803017 + 0.595956i \(0.203227\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −28.0000 −1.39478
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) −23.0000 −1.13728 −0.568638 − 0.822588i \(-0.692530\pi\)
−0.568638 + 0.822588i \(0.692530\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 − 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) 15.0000 0.731055 0.365528 − 0.930800i \(-0.380889\pi\)
0.365528 + 0.930800i \(0.380889\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 52.0000 2.51646
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000 0.963366 0.481683 − 0.876346i \(-0.340026\pi\)
0.481683 + 0.876346i \(0.340026\pi\)
\(432\) 0 0
\(433\) −20.0000 −0.961139 −0.480569 − 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −25.0000 −1.19591
\(438\) 0 0
\(439\) −5.00000 −0.238637 −0.119318 − 0.992856i \(-0.538071\pi\)
−0.119318 + 0.992856i \(0.538071\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.00000 0.142534 0.0712672 − 0.997457i \(-0.477296\pi\)
0.0712672 + 0.997457i \(0.477296\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 − 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −6.00000 −0.278844 −0.139422 − 0.990233i \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.0000 1.43451 0.717254 − 0.696811i \(-0.245399\pi\)
0.717254 + 0.696811i \(0.245399\pi\)
\(468\) 0 0
\(469\) 40.0000 1.84703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −26.0000 −1.18797 −0.593985 − 0.804476i \(-0.702446\pi\)
−0.593985 + 0.804476i \(0.702446\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −34.0000 −1.54069 −0.770344 − 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 − 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) −1.00000 −0.0447661 −0.0223831 − 0.999749i \(-0.507125\pi\)
−0.0223831 + 0.999749i \(0.507125\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −27.0000 −1.20387 −0.601935 − 0.798545i \(-0.705603\pi\)
−0.601935 + 0.798545i \(0.705603\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.0000 1.24108 0.620539 − 0.784176i \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 − 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) 8.00000 0.349816 0.174908 − 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.00000 0.304925
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 − 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 10.0000 0.427569 0.213785 − 0.976881i \(-0.431421\pi\)
0.213785 + 0.976881i \(0.431421\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 40.0000 1.70406
\(552\) 0 0
\(553\) 36.0000 1.53088
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22.0000 0.932170 0.466085 − 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −28.0000 −1.17382 −0.586911 − 0.809652i \(-0.699656\pi\)
−0.586911 + 0.809652i \(0.699656\pi\)
\(570\) 0 0
\(571\) −5.00000 −0.209243 −0.104622 − 0.994512i \(-0.533363\pi\)
−0.104622 + 0.994512i \(0.533363\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 − 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −68.0000 −2.82112
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.0000 1.36206 0.681028 − 0.732257i \(-0.261533\pi\)
0.681028 + 0.732257i \(0.261533\pi\)
\(588\) 0 0
\(589\) −35.0000 −1.44215
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.0000 0.615976 0.307988 − 0.951390i \(-0.400344\pi\)
0.307988 + 0.951390i \(0.400344\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 − 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −47.0000 −1.91717 −0.958585 − 0.284807i \(-0.908071\pi\)
−0.958585 + 0.284807i \(0.908071\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −30.0000 −1.21766 −0.608831 − 0.793300i \(-0.708361\pi\)
−0.608831 + 0.793300i \(0.708361\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −32.0000 −1.29458
\(612\) 0 0
\(613\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27.0000 −1.08698 −0.543490 − 0.839416i \(-0.682897\pi\)
−0.543490 + 0.839416i \(0.682897\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 1.00000 0.0398094 0.0199047 − 0.999802i \(-0.493664\pi\)
0.0199047 + 0.999802i \(0.493664\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −36.0000 −1.42637
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 48.0000 1.89589 0.947943 − 0.318440i \(-0.103159\pi\)
0.947943 + 0.318440i \(0.103159\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 − 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 33.0000 1.29736 0.648682 − 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25.0000 0.978326 0.489163 − 0.872192i \(-0.337302\pi\)
0.489163 + 0.872192i \(0.337302\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 − 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −40.0000 −1.54881
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −26.0000 −1.00372
\(672\) 0 0
\(673\) 36.0000 1.38770 0.693849 − 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −34.0000 −1.30673 −0.653363 − 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) 0 0
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −9.00000 −0.344375 −0.172188 − 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 − 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 − 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) −30.0000 −1.13147
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −48.0000 −1.80523
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 − 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 35.0000 1.31076
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 34.0000 1.26799 0.633993 − 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 4.00000 0.148352 0.0741759 − 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.00000 0.0739727
\(732\) 0 0
\(733\) −24.0000 −0.886460 −0.443230 − 0.896408i \(-0.646168\pi\)
−0.443230 + 0.896408i \(0.646168\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −20.0000 −0.736709
\(738\) 0 0
\(739\) 9.00000 0.331070 0.165535 − 0.986204i \(-0.447065\pi\)
0.165535 + 0.986204i \(0.447065\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 − 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) −39.0000 −1.42313 −0.711565 − 0.702620i \(-0.752013\pi\)
−0.711565 + 0.702620i \(0.752013\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 − 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24.0000 0.869999 0.435000 − 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) 0 0
\(763\) 68.0000 2.46177
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16.0000 0.577727
\(768\) 0 0
\(769\) −35.0000 −1.26213 −0.631066 − 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 33.0000 1.18693 0.593464 − 0.804861i \(-0.297760\pi\)
0.593464 + 0.804861i \(0.297760\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 30.0000 1.07486
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −38.0000 −1.35455 −0.677277 − 0.735728i \(-0.736840\pi\)
−0.677277 + 0.735728i \(0.736840\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) −52.0000 −1.84657
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −21.0000 −0.743858 −0.371929 − 0.928261i \(-0.621304\pi\)
−0.371929 + 0.928261i \(0.621304\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −10.0000 −0.349856
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.00000 −0.0698005 −0.0349002 − 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 0 0
\(823\) 50.0000 1.74289 0.871445 − 0.490493i \(-0.163183\pi\)
0.871445 + 0.490493i \(0.163183\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.00000 −0.104320 −0.0521601 − 0.998639i \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 − 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.00000 0.311832
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −28.0000 −0.966667 −0.483334 − 0.875436i \(-0.660574\pi\)
−0.483334 + 0.875436i \(0.660574\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −28.0000 −0.962091
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 30.0000 1.02839
\(852\) 0 0
\(853\) −32.0000 −1.09566 −0.547830 − 0.836590i \(-0.684546\pi\)
−0.547830 + 0.836590i \(0.684546\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 45.0000 1.53717 0.768585 − 0.639747i \(-0.220961\pi\)
0.768585 + 0.639747i \(0.220961\pi\)
\(858\) 0 0
\(859\) −31.0000 −1.05771 −0.528853 − 0.848713i \(-0.677378\pi\)
−0.528853 + 0.848713i \(0.677378\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −39.0000 −1.32758 −0.663788 − 0.747921i \(-0.731052\pi\)
−0.663788 + 0.747921i \(0.731052\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −18.0000 −0.610608
\(870\) 0 0
\(871\) −40.0000 −1.35535
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 − 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 24.0000 0.808581 0.404290 − 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 17.0000 0.570804 0.285402 − 0.958408i \(-0.407873\pi\)
0.285402 + 0.958408i \(0.407873\pi\)
\(888\) 0 0
\(889\) 72.0000 2.41480
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −40.0000 −1.33855
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −56.0000 −1.86770
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 − 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) 34.0000 1.12524
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −24.0000 −0.792550
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 − 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −38.0000 −1.24674 −0.623370 − 0.781927i \(-0.714237\pi\)
−0.623370 + 0.781927i \(0.714237\pi\)
\(930\) 0 0
\(931\) −45.0000 −1.47482
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −28.0000 −0.914720 −0.457360 − 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −38.0000 −1.23876 −0.619382 − 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) 0 0
\(943\) −30.0000 −0.976934
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.0000 0.357452 0.178726 − 0.983899i \(-0.442802\pi\)
0.178726 + 0.983899i \(0.442802\pi\)
\(948\) 0 0
\(949\) −24.0000 −0.779073
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 44.0000 1.42083
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 42.0000 1.35063 0.675314 − 0.737530i \(-0.264008\pi\)
0.675314 + 0.737530i \(0.264008\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 26.0000 0.834380 0.417190 − 0.908819i \(-0.363015\pi\)
0.417190 + 0.908819i \(0.363015\pi\)
\(972\) 0 0
\(973\) 48.0000 1.53881
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −38.0000 −1.21573 −0.607864 − 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −27.0000 −0.861166 −0.430583 − 0.902551i \(-0.641692\pi\)
−0.430583 + 0.902551i \(0.641692\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 10.0000 0.317982
\(990\) 0 0
\(991\) 5.00000 0.158830 0.0794151 − 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 − 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 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 5400.2.a.br.1.1 1
3.2 odd 2 5400.2.a.bu.1.1 1
5.2 odd 4 5400.2.f.l.649.2 2
5.3 odd 4 5400.2.f.l.649.1 2
5.4 even 2 1080.2.a.g.1.1 yes 1
15.2 even 4 5400.2.f.s.649.2 2
15.8 even 4 5400.2.f.s.649.1 2
15.14 odd 2 1080.2.a.a.1.1 ✓ 1
20.19 odd 2 2160.2.a.w.1.1 1
40.19 odd 2 8640.2.a.bd.1.1 1
40.29 even 2 8640.2.a.a.1.1 1
45.4 even 6 3240.2.q.l.2161.1 2
45.14 odd 6 3240.2.q.w.2161.1 2
45.29 odd 6 3240.2.q.w.1081.1 2
45.34 even 6 3240.2.q.l.1081.1 2
60.59 even 2 2160.2.a.l.1.1 1
120.29 odd 2 8640.2.a.bf.1.1 1
120.59 even 2 8640.2.a.cg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.2.a.a.1.1 ✓ 1 15.14 odd 2
1080.2.a.g.1.1 yes 1 5.4 even 2
2160.2.a.l.1.1 1 60.59 even 2
2160.2.a.w.1.1 1 20.19 odd 2
3240.2.q.l.1081.1 2 45.34 even 6
3240.2.q.l.2161.1 2 45.4 even 6
3240.2.q.w.1081.1 2 45.29 odd 6
3240.2.q.w.2161.1 2 45.14 odd 6
5400.2.a.br.1.1 1 1.1 even 1 trivial
5400.2.a.bu.1.1 1 3.2 odd 2
5400.2.f.l.649.1 2 5.3 odd 4
5400.2.f.l.649.2 2 5.2 odd 4
5400.2.f.s.649.1 2 15.8 even 4
5400.2.f.s.649.2 2 15.2 even 4
8640.2.a.a.1.1 1 40.29 even 2
8640.2.a.bd.1.1 1 40.19 odd 2
8640.2.a.bf.1.1 1 120.29 odd 2
8640.2.a.cg.1.1 1 120.59 even 2