Properties

Label 7200.2.d.t.2449.1
Level $7200$
Weight $2$
Character 7200.2449
Analytic conductor $57.492$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7200,2,Mod(2449,7200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7200.2449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.d (of order \(2\), degree \(1\), not 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: no
Analytic conductor: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.214798336.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2449.1
Root \(-0.565036 - 1.29643i\) of defining polynomial
Character \(\chi\) \(=\) 7200.2449
Dual form 7200.2.d.t.2449.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.72294i q^{7} +O(q^{10})\) \(q-4.72294i q^{7} +3.93012i q^{11} +3.46733 q^{13} -3.51575i q^{17} +5.44133i q^{19} +7.11585i q^{23} +3.66998i q^{29} -5.23414 q^{31} -0.414376 q^{37} -3.00454 q^{41} +5.34450 q^{43} -0.925579i q^{47} -15.3061 q^{49} -0.233196 q^{53} +14.3805i q^{59} +0.118290i q^{61} -13.4504 q^{67} +2.19027 q^{71} +0.563219i q^{73} +18.5617 q^{77} -10.2746 q^{79} -11.3490 q^{83} +8.88265 q^{89} -16.3760i q^{91} +7.27462i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{31} + 8 q^{43} - 8 q^{53} - 24 q^{67} - 40 q^{71} + 24 q^{77} - 16 q^{79} - 32 q^{83}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.72294i − 1.78510i −0.450947 0.892551i \(-0.648914\pi\)
0.450947 0.892551i \(-0.351086\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.93012i 1.18498i 0.805579 + 0.592488i \(0.201854\pi\)
−0.805579 + 0.592488i \(0.798146\pi\)
\(12\) 0 0
\(13\) 3.46733 0.961665 0.480833 0.876812i \(-0.340334\pi\)
0.480833 + 0.876812i \(0.340334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.51575i − 0.852694i −0.904560 0.426347i \(-0.859800\pi\)
0.904560 0.426347i \(-0.140200\pi\)
\(18\) 0 0
\(19\) 5.44133i 1.24833i 0.781294 + 0.624163i \(0.214560\pi\)
−0.781294 + 0.624163i \(0.785440\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.11585i 1.48376i 0.670534 + 0.741878i \(0.266065\pi\)
−0.670534 + 0.741878i \(0.733935\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.66998i 0.681498i 0.940154 + 0.340749i \(0.110681\pi\)
−0.940154 + 0.340749i \(0.889319\pi\)
\(30\) 0 0
\(31\) −5.23414 −0.940079 −0.470039 0.882645i \(-0.655760\pi\)
−0.470039 + 0.882645i \(0.655760\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.414376 −0.0681231 −0.0340615 0.999420i \(-0.510844\pi\)
−0.0340615 + 0.999420i \(0.510844\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00454 −0.469231 −0.234616 0.972088i \(-0.575383\pi\)
−0.234616 + 0.972088i \(0.575383\pi\)
\(42\) 0 0
\(43\) 5.34450 0.815029 0.407514 0.913199i \(-0.366396\pi\)
0.407514 + 0.913199i \(0.366396\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 0.925579i − 0.135010i −0.997719 0.0675048i \(-0.978496\pi\)
0.997719 0.0675048i \(-0.0215038\pi\)
\(48\) 0 0
\(49\) −15.3061 −2.18659
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.233196 −0.0320320 −0.0160160 0.999872i \(-0.505098\pi\)
−0.0160160 + 0.999872i \(0.505098\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.3805i 1.87219i 0.351752 + 0.936093i \(0.385586\pi\)
−0.351752 + 0.936093i \(0.614414\pi\)
\(60\) 0 0
\(61\) 0.118290i 0.0151454i 0.999971 + 0.00757271i \(0.00241049\pi\)
−0.999971 + 0.00757271i \(0.997590\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.4504 −1.64323 −0.821615 0.570043i \(-0.806927\pi\)
−0.821615 + 0.570043i \(0.806927\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.19027 0.259937 0.129969 0.991518i \(-0.458512\pi\)
0.129969 + 0.991518i \(0.458512\pi\)
\(72\) 0 0
\(73\) 0.563219i 0.0659197i 0.999457 + 0.0329599i \(0.0104934\pi\)
−0.999457 + 0.0329599i \(0.989507\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 18.5617 2.11530
\(78\) 0 0
\(79\) −10.2746 −1.15599 −0.577993 0.816042i \(-0.696164\pi\)
−0.577993 + 0.816042i \(0.696164\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.3490 −1.24572 −0.622860 0.782334i \(-0.714029\pi\)
−0.622860 + 0.782334i \(0.714029\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.88265 0.941559 0.470780 0.882251i \(-0.343973\pi\)
0.470780 + 0.882251i \(0.343973\pi\)
\(90\) 0 0
\(91\) − 16.3760i − 1.71667i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.27462i 0.738626i 0.929305 + 0.369313i \(0.120407\pi\)
−0.929305 + 0.369313i \(0.879593\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.23320i 0.421219i 0.977570 + 0.210609i \(0.0675448\pi\)
−0.977570 + 0.210609i \(0.932455\pi\)
\(102\) 0 0
\(103\) 0.0429270i 0.00422972i 0.999998 + 0.00211486i \(0.000673181\pi\)
−0.999998 + 0.00211486i \(0.999327\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.4728 −1.49581 −0.747907 0.663804i \(-0.768941\pi\)
−0.747907 + 0.663804i \(0.768941\pi\)
\(108\) 0 0
\(109\) − 12.9561i − 1.24097i −0.784217 0.620486i \(-0.786935\pi\)
0.784217 0.620486i \(-0.213065\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.86025i 0.363141i 0.983378 + 0.181571i \(0.0581181\pi\)
−0.983378 + 0.181571i \(0.941882\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −16.6046 −1.52215
\(120\) 0 0
\(121\) −4.44587 −0.404170
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 18.3805i 1.63101i 0.578751 + 0.815505i \(0.303540\pi\)
−0.578751 + 0.815505i \(0.696460\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.41892i 0.298713i 0.988783 + 0.149356i \(0.0477201\pi\)
−0.988783 + 0.149356i \(0.952280\pi\)
\(132\) 0 0
\(133\) 25.6990 2.22839
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.3714i 1.39871i 0.714776 + 0.699354i \(0.246529\pi\)
−0.714776 + 0.699354i \(0.753471\pi\)
\(138\) 0 0
\(139\) 1.95707i 0.165997i 0.996550 + 0.0829984i \(0.0264496\pi\)
−0.996550 + 0.0829984i \(0.973550\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.6271i 1.13955i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 12.0968i − 0.991011i −0.868605 0.495505i \(-0.834983\pi\)
0.868605 0.495505i \(-0.165017\pi\)
\(150\) 0 0
\(151\) 4.87178 0.396460 0.198230 0.980156i \(-0.436481\pi\)
0.198230 + 0.980156i \(0.436481\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 21.6561 1.72835 0.864173 0.503195i \(-0.167842\pi\)
0.864173 + 0.503195i \(0.167842\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 33.6077 2.64866
\(162\) 0 0
\(163\) 16.2362 1.27172 0.635860 0.771804i \(-0.280645\pi\)
0.635860 + 0.771804i \(0.280645\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 6.69238i − 0.517872i −0.965894 0.258936i \(-0.916628\pi\)
0.965894 0.258936i \(-0.0833719\pi\)
\(168\) 0 0
\(169\) −0.977595 −0.0751996
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −22.4220 −1.70471 −0.852355 0.522963i \(-0.824827\pi\)
−0.852355 + 0.522963i \(0.824827\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 0.148842i − 0.0111250i −0.999985 0.00556249i \(-0.998229\pi\)
0.999985 0.00556249i \(-0.00177060\pi\)
\(180\) 0 0
\(181\) − 10.3929i − 0.772499i −0.922394 0.386250i \(-0.873770\pi\)
0.922394 0.386250i \(-0.126230\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 13.8173 1.01042
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.23320 0.451018 0.225509 0.974241i \(-0.427595\pi\)
0.225509 + 0.974241i \(0.427595\pi\)
\(192\) 0 0
\(193\) 0.391971i 0.0282147i 0.999900 + 0.0141074i \(0.00449066\pi\)
−0.999900 + 0.0141074i \(0.995509\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.96616 0.425071 0.212536 0.977153i \(-0.431828\pi\)
0.212536 + 0.977153i \(0.431828\pi\)
\(198\) 0 0
\(199\) −17.9322 −1.27118 −0.635591 0.772026i \(-0.719243\pi\)
−0.635591 + 0.772026i \(0.719243\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 17.3331 1.21654
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −21.3851 −1.47924
\(210\) 0 0
\(211\) 6.51575i 0.448563i 0.974524 + 0.224281i \(0.0720034\pi\)
−0.974524 + 0.224281i \(0.927997\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 24.7205i 1.67814i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 12.1903i − 0.820006i
\(222\) 0 0
\(223\) 3.14640i 0.210699i 0.994435 + 0.105349i \(0.0335961\pi\)
−0.994435 + 0.105349i \(0.966404\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.92103 0.260248 0.130124 0.991498i \(-0.458462\pi\)
0.130124 + 0.991498i \(0.458462\pi\)
\(228\) 0 0
\(229\) 25.6899i 1.69764i 0.528683 + 0.848820i \(0.322686\pi\)
−0.528683 + 0.848820i \(0.677314\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 25.7565i 1.68737i 0.536841 + 0.843683i \(0.319617\pi\)
−0.536841 + 0.843683i \(0.680383\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.8727 1.02672 0.513360 0.858173i \(-0.328400\pi\)
0.513360 + 0.858173i \(0.328400\pi\)
\(240\) 0 0
\(241\) 28.1664 1.81436 0.907178 0.420748i \(-0.138232\pi\)
0.907178 + 0.420748i \(0.138232\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 18.8669i 1.20047i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.66004i 0.294139i 0.989126 + 0.147070i \(0.0469842\pi\)
−0.989126 + 0.147070i \(0.953016\pi\)
\(252\) 0 0
\(253\) −27.9662 −1.75822
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.33996i 0.208341i 0.994559 + 0.104170i \(0.0332187\pi\)
−0.994559 + 0.104170i \(0.966781\pi\)
\(258\) 0 0
\(259\) 1.95707i 0.121607i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 27.5932i − 1.70147i −0.525595 0.850735i \(-0.676157\pi\)
0.525595 0.850735i \(-0.323843\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 21.8727i 1.33360i 0.745235 + 0.666802i \(0.232337\pi\)
−0.745235 + 0.666802i \(0.767663\pi\)
\(270\) 0 0
\(271\) 8.78583 0.533701 0.266850 0.963738i \(-0.414017\pi\)
0.266850 + 0.963738i \(0.414017\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.3838 0.623904 0.311952 0.950098i \(-0.399017\pi\)
0.311952 + 0.950098i \(0.399017\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −17.0584 −1.01762 −0.508811 0.860878i \(-0.669915\pi\)
−0.508811 + 0.860878i \(0.669915\pi\)
\(282\) 0 0
\(283\) 3.54724 0.210862 0.105431 0.994427i \(-0.466378\pi\)
0.105431 + 0.994427i \(0.466378\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.1903i 0.837625i
\(288\) 0 0
\(289\) 4.63952 0.272913
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.72538 −0.334480 −0.167240 0.985916i \(-0.553485\pi\)
−0.167240 + 0.985916i \(0.553485\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.6730i 1.42688i
\(300\) 0 0
\(301\) − 25.2417i − 1.45491i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.3760 0.706335 0.353168 0.935560i \(-0.385105\pi\)
0.353168 + 0.935560i \(0.385105\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −18.2746 −1.03626 −0.518129 0.855302i \(-0.673371\pi\)
−0.518129 + 0.855302i \(0.673371\pi\)
\(312\) 0 0
\(313\) 12.7114i 0.718491i 0.933243 + 0.359246i \(0.116966\pi\)
−0.933243 + 0.359246i \(0.883034\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.8602 0.890800 0.445400 0.895332i \(-0.353061\pi\)
0.445400 + 0.895332i \(0.353061\pi\)
\(318\) 0 0
\(319\) −14.4235 −0.807559
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 19.1303 1.06444
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.37145 −0.241006
\(330\) 0 0
\(331\) 23.0315i 1.26593i 0.774182 + 0.632963i \(0.218161\pi\)
−0.774182 + 0.632963i \(0.781839\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 0.860247i − 0.0468606i −0.999725 0.0234303i \(-0.992541\pi\)
0.999725 0.0234303i \(-0.00745878\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 20.5708i − 1.11397i
\(342\) 0 0
\(343\) 39.2293i 2.11818i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.2856 1.73318 0.866591 0.499019i \(-0.166306\pi\)
0.866591 + 0.499019i \(0.166306\pi\)
\(348\) 0 0
\(349\) 0.742899i 0.0397665i 0.999802 + 0.0198832i \(0.00632945\pi\)
−0.999802 + 0.0198832i \(0.993671\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 32.6392i − 1.73721i −0.495506 0.868604i \(-0.665018\pi\)
0.495506 0.868604i \(-0.334982\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.71056 0.143058 0.0715290 0.997439i \(-0.477212\pi\)
0.0715290 + 0.997439i \(0.477212\pi\)
\(360\) 0 0
\(361\) −10.6080 −0.558317
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 0.680008i − 0.0354961i −0.999842 0.0177481i \(-0.994350\pi\)
0.999842 0.0177481i \(-0.00564968\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.10137i 0.0571803i
\(372\) 0 0
\(373\) −3.47642 −0.180002 −0.0900012 0.995942i \(-0.528687\pi\)
−0.0900012 + 0.995942i \(0.528687\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.7250i 0.655373i
\(378\) 0 0
\(379\) 23.0650i 1.18477i 0.805655 + 0.592385i \(0.201813\pi\)
−0.805655 + 0.592385i \(0.798187\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.81544i 0.501545i 0.968046 + 0.250773i \(0.0806847\pi\)
−0.968046 + 0.250773i \(0.919315\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 9.86175i − 0.500010i −0.968244 0.250005i \(-0.919568\pi\)
0.968244 0.250005i \(-0.0804323\pi\)
\(390\) 0 0
\(391\) 25.0175 1.26519
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −12.7783 −0.641326 −0.320663 0.947193i \(-0.603906\pi\)
−0.320663 + 0.947193i \(0.603906\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.17325 −0.158465 −0.0792323 0.996856i \(-0.525247\pi\)
−0.0792323 + 0.996856i \(0.525247\pi\)
\(402\) 0 0
\(403\) −18.1485 −0.904041
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1.62855i − 0.0807243i
\(408\) 0 0
\(409\) −12.1125 −0.598923 −0.299461 0.954108i \(-0.596807\pi\)
−0.299461 + 0.954108i \(0.596807\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 67.9184 3.34204
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 4.24767i − 0.207512i −0.994603 0.103756i \(-0.966914\pi\)
0.994603 0.103756i \(-0.0330862\pi\)
\(420\) 0 0
\(421\) − 3.77928i − 0.184191i −0.995750 0.0920953i \(-0.970644\pi\)
0.995750 0.0920953i \(-0.0293564\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.558674 0.0270361
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 25.6271 1.23441 0.617206 0.786802i \(-0.288265\pi\)
0.617206 + 0.786802i \(0.288265\pi\)
\(432\) 0 0
\(433\) 2.03149i 0.0976274i 0.998808 + 0.0488137i \(0.0155441\pi\)
−0.998808 + 0.0488137i \(0.984456\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −38.7196 −1.85221
\(438\) 0 0
\(439\) −22.4864 −1.07322 −0.536608 0.843832i \(-0.680294\pi\)
−0.536608 + 0.843832i \(0.680294\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.39385 −0.161247 −0.0806234 0.996745i \(-0.525691\pi\)
−0.0806234 + 0.996745i \(0.525691\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.17780 −0.102777 −0.0513883 0.998679i \(-0.516365\pi\)
−0.0513883 + 0.998679i \(0.516365\pi\)
\(450\) 0 0
\(451\) − 11.8082i − 0.556028i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 3.57653i − 0.167303i −0.996495 0.0836516i \(-0.973342\pi\)
0.996495 0.0836516i \(-0.0266583\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 13.1158i − 0.610866i −0.952214 0.305433i \(-0.901199\pi\)
0.952214 0.305433i \(-0.0988012\pi\)
\(462\) 0 0
\(463\) − 3.21417i − 0.149375i −0.997207 0.0746877i \(-0.976204\pi\)
0.997207 0.0746877i \(-0.0237960\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.3016 −1.40219 −0.701095 0.713068i \(-0.747305\pi\)
−0.701095 + 0.713068i \(0.747305\pi\)
\(468\) 0 0
\(469\) 63.5254i 2.93333i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 21.0045i 0.965790i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −35.0896 −1.60328 −0.801642 0.597804i \(-0.796040\pi\)
−0.801642 + 0.597804i \(0.796040\pi\)
\(480\) 0 0
\(481\) −1.43678 −0.0655116
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 36.9117i − 1.67263i −0.548250 0.836315i \(-0.684706\pi\)
0.548250 0.836315i \(-0.315294\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 20.9867i − 0.947116i −0.880763 0.473558i \(-0.842969\pi\)
0.880763 0.473558i \(-0.157031\pi\)
\(492\) 0 0
\(493\) 12.9027 0.581109
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 10.3445i − 0.464014i
\(498\) 0 0
\(499\) 15.9906i 0.715836i 0.933753 + 0.357918i \(0.116513\pi\)
−0.933753 + 0.357918i \(0.883487\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 37.6023i 1.67660i 0.545206 + 0.838302i \(0.316451\pi\)
−0.545206 + 0.838302i \(0.683549\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 12.8579i − 0.569917i −0.958540 0.284958i \(-0.908020\pi\)
0.958540 0.284958i \(-0.0919797\pi\)
\(510\) 0 0
\(511\) 2.66004 0.117673
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.63764 0.159983
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −37.0015 −1.62107 −0.810534 0.585692i \(-0.800823\pi\)
−0.810534 + 0.585692i \(0.800823\pi\)
\(522\) 0 0
\(523\) 17.4952 0.765013 0.382506 0.923953i \(-0.375061\pi\)
0.382506 + 0.923953i \(0.375061\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.4019i 0.801600i
\(528\) 0 0
\(529\) −27.6353 −1.20153
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −10.4178 −0.451243
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 60.1549i − 2.59106i
\(540\) 0 0
\(541\) 38.1225i 1.63901i 0.573069 + 0.819507i \(0.305753\pi\)
−0.573069 + 0.819507i \(0.694247\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 35.7406 1.52816 0.764078 0.645124i \(-0.223194\pi\)
0.764078 + 0.645124i \(0.223194\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −19.9695 −0.850731
\(552\) 0 0
\(553\) 48.5264i 2.06355i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.65516 −0.112503 −0.0562514 0.998417i \(-0.517915\pi\)
−0.0562514 + 0.998417i \(0.517915\pi\)
\(558\) 0 0
\(559\) 18.5312 0.783785
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.3107 0.855992 0.427996 0.903781i \(-0.359220\pi\)
0.427996 + 0.903781i \(0.359220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −28.4274 −1.19174 −0.595868 0.803082i \(-0.703192\pi\)
−0.595868 + 0.803082i \(0.703192\pi\)
\(570\) 0 0
\(571\) 16.1485i 0.675794i 0.941183 + 0.337897i \(0.109716\pi\)
−0.941183 + 0.337897i \(0.890284\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 30.3600i − 1.26390i −0.775008 0.631952i \(-0.782254\pi\)
0.775008 0.631952i \(-0.217746\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 53.6008i 2.22374i
\(582\) 0 0
\(583\) − 0.916490i − 0.0379571i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.74070 −0.113121 −0.0565604 0.998399i \(-0.518013\pi\)
−0.0565604 + 0.998399i \(0.518013\pi\)
\(588\) 0 0
\(589\) − 28.4807i − 1.17352i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 22.2586i − 0.914053i −0.889453 0.457027i \(-0.848914\pi\)
0.889453 0.457027i \(-0.151086\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −48.4526 −1.97972 −0.989860 0.142046i \(-0.954632\pi\)
−0.989860 + 0.142046i \(0.954632\pi\)
\(600\) 0 0
\(601\) 5.26553 0.214786 0.107393 0.994217i \(-0.465750\pi\)
0.107393 + 0.994217i \(0.465750\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.38288i 0.299662i 0.988712 + 0.149831i \(0.0478730\pi\)
−0.988712 + 0.149831i \(0.952127\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 3.20929i − 0.129834i
\(612\) 0 0
\(613\) −2.64607 −0.106874 −0.0534369 0.998571i \(-0.517018\pi\)
−0.0534369 + 0.998571i \(0.517018\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.0136i 0.845977i 0.906135 + 0.422989i \(0.139019\pi\)
−0.906135 + 0.422989i \(0.860981\pi\)
\(618\) 0 0
\(619\) 24.0874i 0.968154i 0.875025 + 0.484077i \(0.160845\pi\)
−0.875025 + 0.484077i \(0.839155\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 41.9522i − 1.68078i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.45684i 0.0580881i
\(630\) 0 0
\(631\) −25.2094 −1.00357 −0.501785 0.864992i \(-0.667323\pi\)
−0.501785 + 0.864992i \(0.667323\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −53.0714 −2.10277
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.4755 0.729738 0.364869 0.931059i \(-0.381114\pi\)
0.364869 + 0.931059i \(0.381114\pi\)
\(642\) 0 0
\(643\) 0.636984 0.0251202 0.0125601 0.999921i \(-0.496002\pi\)
0.0125601 + 0.999921i \(0.496002\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 32.0182i − 1.25876i −0.777096 0.629382i \(-0.783308\pi\)
0.777096 0.629382i \(-0.216692\pi\)
\(648\) 0 0
\(649\) −56.5173 −2.21850
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25.9769 1.01656 0.508278 0.861193i \(-0.330282\pi\)
0.508278 + 0.861193i \(0.330282\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 21.3422i 0.831372i 0.909508 + 0.415686i \(0.136459\pi\)
−0.909508 + 0.415686i \(0.863541\pi\)
\(660\) 0 0
\(661\) 14.8397i 0.577198i 0.957450 + 0.288599i \(0.0931895\pi\)
−0.957450 + 0.288599i \(0.906810\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −26.1150 −1.01118
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.464893 −0.0179470
\(672\) 0 0
\(673\) 18.1167i 0.698347i 0.937058 + 0.349174i \(0.113538\pi\)
−0.937058 + 0.349174i \(0.886462\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −30.5617 −1.17458 −0.587291 0.809376i \(-0.699806\pi\)
−0.587291 + 0.809376i \(0.699806\pi\)
\(678\) 0 0
\(679\) 34.3576 1.31852
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22.2027 −0.849564 −0.424782 0.905296i \(-0.639649\pi\)
−0.424782 + 0.905296i \(0.639649\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.808569 −0.0308040
\(690\) 0 0
\(691\) 12.6890i 0.482712i 0.970437 + 0.241356i \(0.0775922\pi\)
−0.970437 + 0.241356i \(0.922408\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 10.5632i 0.400110i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 34.9241i − 1.31906i −0.751676 0.659532i \(-0.770754\pi\)
0.751676 0.659532i \(-0.229246\pi\)
\(702\) 0 0
\(703\) − 2.25476i − 0.0850398i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 19.9931 0.751918
\(708\) 0 0
\(709\) − 5.65610i − 0.212419i −0.994344 0.106210i \(-0.966129\pi\)
0.994344 0.106210i \(-0.0338715\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 37.2453i − 1.39485i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 20.0844 0.749020 0.374510 0.927223i \(-0.377811\pi\)
0.374510 + 0.927223i \(0.377811\pi\)
\(720\) 0 0
\(721\) 0.202741 0.00755048
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 13.0424i 0.483715i 0.970312 + 0.241857i \(0.0777566\pi\)
−0.970312 + 0.241857i \(0.922243\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 18.7899i − 0.694970i
\(732\) 0 0
\(733\) −34.8917 −1.28876 −0.644378 0.764707i \(-0.722884\pi\)
−0.644378 + 0.764707i \(0.722884\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 52.8618i − 1.94719i
\(738\) 0 0
\(739\) − 41.5040i − 1.52675i −0.645957 0.763374i \(-0.723541\pi\)
0.645957 0.763374i \(-0.276459\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.63764i 0.280198i 0.990138 + 0.140099i \(0.0447421\pi\)
−0.990138 + 0.140099i \(0.955258\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 73.0771i 2.67018i
\(750\) 0 0
\(751\) 19.4029 0.708023 0.354012 0.935241i \(-0.384817\pi\)
0.354012 + 0.935241i \(0.384817\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 43.7959 1.59179 0.795894 0.605436i \(-0.207001\pi\)
0.795894 + 0.605436i \(0.207001\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −9.95519 −0.360875 −0.180438 0.983586i \(-0.557751\pi\)
−0.180438 + 0.983586i \(0.557751\pi\)
\(762\) 0 0
\(763\) −61.1910 −2.21526
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 49.8621i 1.80042i
\(768\) 0 0
\(769\) 17.9008 0.645520 0.322760 0.946481i \(-0.395389\pi\)
0.322760 + 0.946481i \(0.395389\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.8182 −0.748777 −0.374389 0.927272i \(-0.622147\pi\)
−0.374389 + 0.927272i \(0.622147\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 16.3487i − 0.585753i
\(780\) 0 0
\(781\) 8.60803i 0.308019i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 27.2047 0.969744 0.484872 0.874585i \(-0.338866\pi\)
0.484872 + 0.874585i \(0.338866\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 18.2317 0.648244
\(792\) 0 0
\(793\) 0.410149i 0.0145648i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.4344 −0.582138 −0.291069 0.956702i \(-0.594011\pi\)
−0.291069 + 0.956702i \(0.594011\pi\)
\(798\) 0 0
\(799\) −3.25410 −0.115122
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.21352 −0.0781134
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 28.7698 1.01149 0.505747 0.862682i \(-0.331217\pi\)
0.505747 + 0.862682i \(0.331217\pi\)
\(810\) 0 0
\(811\) − 14.3512i − 0.503940i −0.967735 0.251970i \(-0.918922\pi\)
0.967735 0.251970i \(-0.0810785\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 29.0812i 1.01742i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 4.97271i − 0.173549i −0.996228 0.0867744i \(-0.972344\pi\)
0.996228 0.0867744i \(-0.0276559\pi\)
\(822\) 0 0
\(823\) 20.3625i 0.709791i 0.934906 + 0.354895i \(0.115483\pi\)
−0.934906 + 0.354895i \(0.884517\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0039 1.25198 0.625989 0.779832i \(-0.284696\pi\)
0.625989 + 0.779832i \(0.284696\pi\)
\(828\) 0 0
\(829\) 7.68666i 0.266969i 0.991051 + 0.133484i \(0.0426166\pi\)
−0.991051 + 0.133484i \(0.957383\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 53.8124i 1.86449i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 25.3725 0.875956 0.437978 0.898986i \(-0.355695\pi\)
0.437978 + 0.898986i \(0.355695\pi\)
\(840\) 0 0
\(841\) 15.5313 0.535561
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 20.9976i 0.721485i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 2.94864i − 0.101078i
\(852\) 0 0
\(853\) 20.1365 0.689460 0.344730 0.938702i \(-0.387971\pi\)
0.344730 + 0.938702i \(0.387971\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 42.0380i − 1.43599i −0.696047 0.717996i \(-0.745060\pi\)
0.696047 0.717996i \(-0.254940\pi\)
\(858\) 0 0
\(859\) − 12.4095i − 0.423406i −0.977334 0.211703i \(-0.932099\pi\)
0.977334 0.211703i \(-0.0679010\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 21.8154i − 0.742606i −0.928512 0.371303i \(-0.878911\pi\)
0.928512 0.371303i \(-0.121089\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 40.3805i − 1.36982i
\(870\) 0 0
\(871\) −46.6371 −1.58024
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −5.83690 −0.197098 −0.0985491 0.995132i \(-0.531420\pi\)
−0.0985491 + 0.995132i \(0.531420\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 11.9613 0.402986 0.201493 0.979490i \(-0.435421\pi\)
0.201493 + 0.979490i \(0.435421\pi\)
\(882\) 0 0
\(883\) 3.74810 0.126134 0.0630668 0.998009i \(-0.479912\pi\)
0.0630668 + 0.998009i \(0.479912\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 42.8101i 1.43742i 0.695309 + 0.718711i \(0.255268\pi\)
−0.695309 + 0.718711i \(0.744732\pi\)
\(888\) 0 0
\(889\) 86.8101 2.91152
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.03638 0.168536
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 19.2092i − 0.640662i
\(900\) 0 0
\(901\) 0.819859i 0.0273135i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 6.97759 0.231687 0.115844 0.993267i \(-0.463043\pi\)
0.115844 + 0.993267i \(0.463043\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.2837 0.804555 0.402278 0.915518i \(-0.368219\pi\)
0.402278 + 0.915518i \(0.368219\pi\)
\(912\) 0 0
\(913\) − 44.6031i − 1.47615i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.1473 0.533232
\(918\) 0 0
\(919\) −1.25720 −0.0414712 −0.0207356 0.999785i \(-0.506601\pi\)
−0.0207356 + 0.999785i \(0.506601\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 7.59440 0.249973
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 10.4933 0.344275 0.172138 0.985073i \(-0.444933\pi\)
0.172138 + 0.985073i \(0.444933\pi\)
\(930\) 0 0
\(931\) − 83.2856i − 2.72957i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 12.9680i 0.423648i 0.977308 + 0.211824i \(0.0679403\pi\)
−0.977308 + 0.211824i \(0.932060\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 44.9947i 1.46678i 0.679806 + 0.733392i \(0.262064\pi\)
−0.679806 + 0.733392i \(0.737936\pi\)
\(942\) 0 0
\(943\) − 21.3799i − 0.696225i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.6282 0.995283 0.497642 0.867383i \(-0.334199\pi\)
0.497642 + 0.867383i \(0.334199\pi\)
\(948\) 0 0
\(949\) 1.95287i 0.0633927i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 17.5717i 0.569202i 0.958646 + 0.284601i \(0.0918610\pi\)
−0.958646 + 0.284601i \(0.908139\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 77.3213 2.49683
\(960\) 0 0
\(961\) −3.60380 −0.116252
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 4.50932i − 0.145010i −0.997368 0.0725050i \(-0.976901\pi\)
0.997368 0.0725050i \(-0.0230993\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 35.8512i − 1.15052i −0.817971 0.575259i \(-0.804901\pi\)
0.817971 0.575259i \(-0.195099\pi\)
\(972\) 0 0
\(973\) 9.24313 0.296321
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 43.9704i 1.40674i 0.710826 + 0.703368i \(0.248322\pi\)
−0.710826 + 0.703368i \(0.751678\pi\)
\(978\) 0 0
\(979\) 34.9099i 1.11573i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.64187i 0.148053i 0.997256 + 0.0740263i \(0.0235849\pi\)
−0.997256 + 0.0740263i \(0.976415\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 38.0307i 1.20930i
\(990\) 0 0
\(991\) 52.9117 1.68080 0.840398 0.541970i \(-0.182321\pi\)
0.840398 + 0.541970i \(0.182321\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 39.4228 1.24853 0.624266 0.781211i \(-0.285398\pi\)
0.624266 + 0.781211i \(0.285398\pi\)
\(998\) 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 7200.2.d.t.2449.1 8
3.2 odd 2 2400.2.d.h.49.1 8
4.3 odd 2 1800.2.d.t.1549.2 8
5.2 odd 4 7200.2.k.s.3601.8 8
5.3 odd 4 7200.2.k.r.3601.2 8
5.4 even 2 7200.2.d.s.2449.8 8
8.3 odd 2 1800.2.d.s.1549.8 8
8.5 even 2 7200.2.d.s.2449.1 8
12.11 even 2 600.2.d.g.349.7 8
15.2 even 4 2400.2.k.e.1201.4 8
15.8 even 4 2400.2.k.d.1201.5 8
15.14 odd 2 2400.2.d.g.49.8 8
20.3 even 4 1800.2.k.t.901.6 8
20.7 even 4 1800.2.k.q.901.3 8
20.19 odd 2 1800.2.d.s.1549.7 8
24.5 odd 2 2400.2.d.g.49.1 8
24.11 even 2 600.2.d.h.349.1 8
40.3 even 4 1800.2.k.t.901.5 8
40.13 odd 4 7200.2.k.r.3601.1 8
40.19 odd 2 1800.2.d.t.1549.1 8
40.27 even 4 1800.2.k.q.901.4 8
40.29 even 2 inner 7200.2.d.t.2449.8 8
40.37 odd 4 7200.2.k.s.3601.7 8
60.23 odd 4 600.2.k.d.301.3 8
60.47 odd 4 600.2.k.e.301.6 yes 8
60.59 even 2 600.2.d.h.349.2 8
120.29 odd 2 2400.2.d.h.49.8 8
120.53 even 4 2400.2.k.d.1201.1 8
120.59 even 2 600.2.d.g.349.8 8
120.77 even 4 2400.2.k.e.1201.8 8
120.83 odd 4 600.2.k.d.301.4 yes 8
120.107 odd 4 600.2.k.e.301.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.d.g.349.7 8 12.11 even 2
600.2.d.g.349.8 8 120.59 even 2
600.2.d.h.349.1 8 24.11 even 2
600.2.d.h.349.2 8 60.59 even 2
600.2.k.d.301.3 8 60.23 odd 4
600.2.k.d.301.4 yes 8 120.83 odd 4
600.2.k.e.301.5 yes 8 120.107 odd 4
600.2.k.e.301.6 yes 8 60.47 odd 4
1800.2.d.s.1549.7 8 20.19 odd 2
1800.2.d.s.1549.8 8 8.3 odd 2
1800.2.d.t.1549.1 8 40.19 odd 2
1800.2.d.t.1549.2 8 4.3 odd 2
1800.2.k.q.901.3 8 20.7 even 4
1800.2.k.q.901.4 8 40.27 even 4
1800.2.k.t.901.5 8 40.3 even 4
1800.2.k.t.901.6 8 20.3 even 4
2400.2.d.g.49.1 8 24.5 odd 2
2400.2.d.g.49.8 8 15.14 odd 2
2400.2.d.h.49.1 8 3.2 odd 2
2400.2.d.h.49.8 8 120.29 odd 2
2400.2.k.d.1201.1 8 120.53 even 4
2400.2.k.d.1201.5 8 15.8 even 4
2400.2.k.e.1201.4 8 15.2 even 4
2400.2.k.e.1201.8 8 120.77 even 4
7200.2.d.s.2449.1 8 8.5 even 2
7200.2.d.s.2449.8 8 5.4 even 2
7200.2.d.t.2449.1 8 1.1 even 1 trivial
7200.2.d.t.2449.8 8 40.29 even 2 inner
7200.2.k.r.3601.1 8 40.13 odd 4
7200.2.k.r.3601.2 8 5.3 odd 4
7200.2.k.s.3601.7 8 40.37 odd 4
7200.2.k.s.3601.8 8 5.2 odd 4