Properties

Label 8100.2.a.be.1.6
Level $8100$
Weight $2$
Character 8100.1
Self dual yes
Analytic conductor $64.679$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8100,2,Mod(1,8100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8100.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.a (trivial)

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: \(64.6788256372\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.28356903014400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 37x^{6} + 399x^{4} - 1195x^{2} + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1620)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(4.46736\) of defining polynomial
Character \(\chi\) \(=\) 8100.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.28148 q^{7} +O(q^{10})\) \(q+1.28148 q^{7} +4.14474 q^{11} -6.52202 q^{13} +5.98507 q^{17} +7.17891 q^{19} -7.53098 q^{23} +5.19615 q^{29} +5.17891 q^{31} -5.24054 q^{37} +0.680643 q^{41} +1.28148 q^{43} +5.31139 q^{47} -5.35782 q^{49} +2.21958 q^{53} -7.60885 q^{59} -2.17891 q^{61} +15.6070 q^{67} +5.50603 q^{71} +7.80350 q^{73} +5.31139 q^{77} +6.00000 q^{79} -9.75056 q^{83} -10.0215 q^{89} -8.35782 q^{91} -14.3255 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{19} - 4 q^{31} + 48 q^{49} + 28 q^{61} + 48 q^{79} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 1.28148 0.484353 0.242176 0.970232i \(-0.422139\pi\)
0.242176 + 0.970232i \(0.422139\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.14474 1.24969 0.624844 0.780750i \(-0.285163\pi\)
0.624844 + 0.780750i \(0.285163\pi\)
\(12\) 0 0
\(13\) −6.52202 −1.80888 −0.904441 0.426598i \(-0.859712\pi\)
−0.904441 + 0.426598i \(0.859712\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.98507 1.45159 0.725797 0.687909i \(-0.241471\pi\)
0.725797 + 0.687909i \(0.241471\pi\)
\(18\) 0 0
\(19\) 7.17891 1.64695 0.823477 0.567349i \(-0.192031\pi\)
0.823477 + 0.567349i \(0.192031\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.53098 −1.57032 −0.785159 0.619295i \(-0.787419\pi\)
−0.785159 + 0.619295i \(0.787419\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.19615 0.964901 0.482451 0.875923i \(-0.339747\pi\)
0.482451 + 0.875923i \(0.339747\pi\)
\(30\) 0 0
\(31\) 5.17891 0.930159 0.465080 0.885269i \(-0.346026\pi\)
0.465080 + 0.885269i \(0.346026\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.24054 −0.861540 −0.430770 0.902462i \(-0.641758\pi\)
−0.430770 + 0.902462i \(0.641758\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.680643 0.106299 0.0531493 0.998587i \(-0.483074\pi\)
0.0531493 + 0.998587i \(0.483074\pi\)
\(42\) 0 0
\(43\) 1.28148 0.195423 0.0977117 0.995215i \(-0.468848\pi\)
0.0977117 + 0.995215i \(0.468848\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.31139 0.774747 0.387373 0.921923i \(-0.373382\pi\)
0.387373 + 0.921923i \(0.373382\pi\)
\(48\) 0 0
\(49\) −5.35782 −0.765402
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.21958 0.304883 0.152442 0.988312i \(-0.451286\pi\)
0.152442 + 0.988312i \(0.451286\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.60885 −0.990587 −0.495294 0.868726i \(-0.664940\pi\)
−0.495294 + 0.868726i \(0.664940\pi\)
\(60\) 0 0
\(61\) −2.17891 −0.278981 −0.139490 0.990223i \(-0.544546\pi\)
−0.139490 + 0.990223i \(0.544546\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 15.6070 1.90670 0.953349 0.301871i \(-0.0976113\pi\)
0.953349 + 0.301871i \(0.0976113\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.50603 0.653446 0.326723 0.945120i \(-0.394056\pi\)
0.326723 + 0.945120i \(0.394056\pi\)
\(72\) 0 0
\(73\) 7.80350 0.913330 0.456665 0.889639i \(-0.349044\pi\)
0.456665 + 0.889639i \(0.349044\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.31139 0.605290
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.75056 −1.07026 −0.535132 0.844769i \(-0.679738\pi\)
−0.535132 + 0.844769i \(0.679738\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0215 −1.06228 −0.531141 0.847284i \(-0.678236\pi\)
−0.531141 + 0.847284i \(0.678236\pi\)
\(90\) 0 0
\(91\) −8.35782 −0.876137
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.3255 −1.45454 −0.727268 0.686354i \(-0.759210\pi\)
−0.727268 + 0.686354i \(0.759210\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.680643 −0.0677265 −0.0338633 0.999426i \(-0.510781\pi\)
−0.0338633 + 0.999426i \(0.510781\pi\)
\(102\) 0 0
\(103\) 2.56295 0.252535 0.126268 0.991996i \(-0.459700\pi\)
0.126268 + 0.991996i \(0.459700\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.07688 0.853881 0.426941 0.904280i \(-0.359591\pi\)
0.426941 + 0.904280i \(0.359591\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 7.66973 0.703083
\(120\) 0 0
\(121\) 6.17891 0.561719
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.28148 0.113713 0.0568563 0.998382i \(-0.481892\pi\)
0.0568563 + 0.998382i \(0.481892\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.50603 0.481064 0.240532 0.970641i \(-0.422678\pi\)
0.240532 + 0.970641i \(0.422678\pi\)
\(132\) 0 0
\(133\) 9.19961 0.797707
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.2965 −0.965122 −0.482561 0.875862i \(-0.660293\pi\)
−0.482561 + 0.875862i \(0.660293\pi\)
\(138\) 0 0
\(139\) 17.1789 1.45710 0.728548 0.684995i \(-0.240196\pi\)
0.728548 + 0.684995i \(0.240196\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −27.0321 −2.26054
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −19.7332 −1.61661 −0.808303 0.588766i \(-0.799614\pi\)
−0.808303 + 0.588766i \(0.799614\pi\)
\(150\) 0 0
\(151\) 5.17891 0.421454 0.210727 0.977545i \(-0.432417\pi\)
0.210727 + 0.977545i \(0.432417\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.24054 0.418241 0.209120 0.977890i \(-0.432940\pi\)
0.209120 + 0.977890i \(0.432940\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.65078 −0.760588
\(162\) 0 0
\(163\) 11.7626 0.921315 0.460657 0.887578i \(-0.347614\pi\)
0.460657 + 0.887578i \(0.347614\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.21958 0.171757 0.0858783 0.996306i \(-0.472630\pi\)
0.0858783 + 0.996306i \(0.472630\pi\)
\(168\) 0 0
\(169\) 29.5367 2.27206
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.673677 0.0512187 0.0256094 0.999672i \(-0.491847\pi\)
0.0256094 + 0.999672i \(0.491847\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −14.5370 −1.08655 −0.543275 0.839555i \(-0.682816\pi\)
−0.543275 + 0.839555i \(0.682816\pi\)
\(180\) 0 0
\(181\) 15.1789 1.12824 0.564120 0.825693i \(-0.309216\pi\)
0.564120 + 0.825693i \(0.309216\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 24.8066 1.81404
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 26.2906 1.90232 0.951162 0.308692i \(-0.0998913\pi\)
0.951162 + 0.308692i \(0.0998913\pi\)
\(192\) 0 0
\(193\) −12.9294 −0.930679 −0.465339 0.885132i \(-0.654068\pi\)
−0.465339 + 0.885132i \(0.654068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.20466 −0.584558 −0.292279 0.956333i \(-0.594414\pi\)
−0.292279 + 0.956333i \(0.594414\pi\)
\(198\) 0 0
\(199\) −2.35782 −0.167141 −0.0835706 0.996502i \(-0.526632\pi\)
−0.0835706 + 0.996502i \(0.526632\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.65875 0.467353
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 29.7547 2.05818
\(210\) 0 0
\(211\) 4.82109 0.331898 0.165949 0.986134i \(-0.446931\pi\)
0.165949 + 0.986134i \(0.446931\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.63665 0.450525
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −39.0348 −2.62576
\(222\) 0 0
\(223\) −7.91813 −0.530237 −0.265119 0.964216i \(-0.585411\pi\)
−0.265119 + 0.964216i \(0.585411\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.7207 1.44165 0.720827 0.693115i \(-0.243762\pi\)
0.720827 + 0.693115i \(0.243762\pi\)
\(228\) 0 0
\(229\) 18.1789 1.20130 0.600648 0.799514i \(-0.294909\pi\)
0.600648 + 0.799514i \(0.294909\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.0470 1.37884 0.689418 0.724363i \(-0.257866\pi\)
0.689418 + 0.724363i \(0.257866\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −22.1459 −1.43250 −0.716249 0.697844i \(-0.754143\pi\)
−0.716249 + 0.697844i \(0.754143\pi\)
\(240\) 0 0
\(241\) 19.0000 1.22390 0.611949 0.790897i \(-0.290386\pi\)
0.611949 + 0.790897i \(0.290386\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −46.8210 −2.97915
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −17.3205 −1.09326 −0.546630 0.837374i \(-0.684090\pi\)
−0.546630 + 0.837374i \(0.684090\pi\)
\(252\) 0 0
\(253\) −31.2140 −1.96241
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.85730 −0.427747 −0.213873 0.976861i \(-0.568608\pi\)
−0.213873 + 0.976861i \(0.568608\pi\)
\(258\) 0 0
\(259\) −6.71563 −0.417289
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.0620 0.928760 0.464380 0.885636i \(-0.346277\pi\)
0.464380 + 0.885636i \(0.346277\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.28001 0.565812 0.282906 0.959148i \(-0.408702\pi\)
0.282906 + 0.959148i \(0.408702\pi\)
\(270\) 0 0
\(271\) 0.357817 0.0217358 0.0108679 0.999941i \(-0.496541\pi\)
0.0108679 + 0.999941i \(0.496541\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.7626 −0.706744 −0.353372 0.935483i \(-0.614965\pi\)
−0.353372 + 0.935483i \(0.614965\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.23808 −0.431788 −0.215894 0.976417i \(-0.569267\pi\)
−0.215894 + 0.976417i \(0.569267\pi\)
\(282\) 0 0
\(283\) 28.6510 1.70313 0.851563 0.524252i \(-0.175655\pi\)
0.851563 + 0.524252i \(0.175655\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.872228 0.0514860
\(288\) 0 0
\(289\) 18.8211 1.10712
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −17.9552 −1.04895 −0.524477 0.851424i \(-0.675739\pi\)
−0.524477 + 0.851424i \(0.675739\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 49.1172 2.84052
\(300\) 0 0
\(301\) 1.64218 0.0946539
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −16.8885 −0.963876 −0.481938 0.876205i \(-0.660067\pi\)
−0.481938 + 0.876205i \(0.660067\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.97013 0.508650 0.254325 0.967119i \(-0.418147\pi\)
0.254325 + 0.967119i \(0.418147\pi\)
\(312\) 0 0
\(313\) −9.08497 −0.513513 −0.256757 0.966476i \(-0.582654\pi\)
−0.256757 + 0.966476i \(0.582654\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.1687 −0.683462 −0.341731 0.939798i \(-0.611013\pi\)
−0.341731 + 0.939798i \(0.611013\pi\)
\(318\) 0 0
\(319\) 21.5367 1.20583
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 42.9663 2.39071
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.80643 0.375251
\(330\) 0 0
\(331\) 1.53673 0.0844660 0.0422330 0.999108i \(-0.486553\pi\)
0.0422330 + 0.999108i \(0.486553\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.56295 −0.139613 −0.0698065 0.997561i \(-0.522238\pi\)
−0.0698065 + 0.997561i \(0.522238\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 21.4653 1.16241
\(342\) 0 0
\(343\) −15.8363 −0.855078
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.21958 0.119153 0.0595767 0.998224i \(-0.481025\pi\)
0.0595767 + 0.998224i \(0.481025\pi\)
\(348\) 0 0
\(349\) 8.82109 0.472182 0.236091 0.971731i \(-0.424134\pi\)
0.236091 + 0.971731i \(0.424134\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27.9043 1.48520 0.742599 0.669737i \(-0.233593\pi\)
0.742599 + 0.669737i \(0.233593\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −19.3624 −1.02191 −0.510955 0.859607i \(-0.670708\pi\)
−0.510955 + 0.859607i \(0.670708\pi\)
\(360\) 0 0
\(361\) 32.5367 1.71246
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.56295 −0.133785 −0.0668926 0.997760i \(-0.521308\pi\)
−0.0668926 + 0.997760i \(0.521308\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.84434 0.147671
\(372\) 0 0
\(373\) 16.8885 0.874452 0.437226 0.899352i \(-0.355961\pi\)
0.437226 + 0.899352i \(0.355961\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −33.8894 −1.74539
\(378\) 0 0
\(379\) 22.3578 1.14844 0.574222 0.818700i \(-0.305305\pi\)
0.574222 + 0.818700i \(0.305305\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.9701 0.611646 0.305823 0.952088i \(-0.401068\pi\)
0.305823 + 0.952088i \(0.401068\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.4951 0.633528 0.316764 0.948504i \(-0.397404\pi\)
0.316764 + 0.948504i \(0.397404\pi\)
\(390\) 0 0
\(391\) −45.0735 −2.27946
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 24.6920 1.23925 0.619627 0.784896i \(-0.287284\pi\)
0.619627 + 0.784896i \(0.287284\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0635 0.602421 0.301210 0.953558i \(-0.402609\pi\)
0.301210 + 0.953558i \(0.402609\pi\)
\(402\) 0 0
\(403\) −33.7769 −1.68255
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −21.7207 −1.07666
\(408\) 0 0
\(409\) 18.1789 0.898889 0.449445 0.893308i \(-0.351622\pi\)
0.449445 + 0.893308i \(0.351622\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −9.75056 −0.479794
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.65078 0.471471 0.235736 0.971817i \(-0.424250\pi\)
0.235736 + 0.971817i \(0.424250\pi\)
\(420\) 0 0
\(421\) −15.7156 −0.765933 −0.382967 0.923762i \(-0.625098\pi\)
−0.382967 + 0.923762i \(0.625098\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.79222 −0.135125
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −31.1160 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(432\) 0 0
\(433\) −16.7738 −0.806099 −0.403050 0.915178i \(-0.632050\pi\)
−0.403050 + 0.915178i \(0.632050\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −54.0642 −2.58624
\(438\) 0 0
\(439\) −15.5367 −0.741527 −0.370764 0.928727i \(-0.620904\pi\)
−0.370764 + 0.928727i \(0.620904\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −20.3734 −0.967967 −0.483984 0.875077i \(-0.660811\pi\)
−0.483984 + 0.875077i \(0.660811\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 33.8386 1.59694 0.798471 0.602033i \(-0.205642\pi\)
0.798471 + 0.602033i \(0.205642\pi\)
\(450\) 0 0
\(451\) 2.82109 0.132840
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.114633 0.00536233 0.00268116 0.999996i \(-0.499147\pi\)
0.00268116 + 0.999996i \(0.499147\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.97013 −0.417781 −0.208890 0.977939i \(-0.566985\pi\)
−0.208890 + 0.977939i \(0.566985\pi\)
\(462\) 0 0
\(463\) −7.68886 −0.357332 −0.178666 0.983910i \(-0.557178\pi\)
−0.178666 + 0.983910i \(0.557178\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 29.2517 1.35361 0.676803 0.736164i \(-0.263365\pi\)
0.676803 + 0.736164i \(0.263365\pi\)
\(468\) 0 0
\(469\) 20.0000 0.923514
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.31139 0.244218
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 31.8576 1.45561 0.727804 0.685785i \(-0.240541\pi\)
0.727804 + 0.685785i \(0.240541\pi\)
\(480\) 0 0
\(481\) 34.1789 1.55842
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 33.7769 1.53058 0.765290 0.643686i \(-0.222596\pi\)
0.765290 + 0.643686i \(0.222596\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −23.4463 −1.05812 −0.529058 0.848586i \(-0.677455\pi\)
−0.529058 + 0.848586i \(0.677455\pi\)
\(492\) 0 0
\(493\) 31.0993 1.40064
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.05585 0.316498
\(498\) 0 0
\(499\) −7.17891 −0.321372 −0.160686 0.987006i \(-0.551371\pi\)
−0.160686 + 0.987006i \(0.551371\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −40.7467 −1.81681 −0.908403 0.418096i \(-0.862698\pi\)
−0.908403 + 0.418096i \(0.862698\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −31.7967 −1.40936 −0.704681 0.709524i \(-0.748910\pi\)
−0.704681 + 0.709524i \(0.748910\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 22.0144 0.968191
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 22.1459 0.970229 0.485115 0.874451i \(-0.338778\pi\)
0.485115 + 0.874451i \(0.338778\pi\)
\(522\) 0 0
\(523\) 41.6951 1.82320 0.911599 0.411081i \(-0.134849\pi\)
0.911599 + 0.411081i \(0.134849\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 30.9961 1.35021
\(528\) 0 0
\(529\) 33.7156 1.46590
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.43917 −0.192282
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −22.2068 −0.956514
\(540\) 0 0
\(541\) −19.3578 −0.832258 −0.416129 0.909306i \(-0.636613\pi\)
−0.416129 + 0.909306i \(0.636613\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5.12591 0.219168 0.109584 0.993978i \(-0.465048\pi\)
0.109584 + 0.993978i \(0.465048\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 37.3027 1.58915
\(552\) 0 0
\(553\) 7.68886 0.326964
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.41813 −0.102460 −0.0512298 0.998687i \(-0.516314\pi\)
−0.0512298 + 0.998687i \(0.516314\pi\)
\(558\) 0 0
\(559\) −8.35782 −0.353498
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.31139 0.223849 0.111924 0.993717i \(-0.464299\pi\)
0.111924 + 0.993717i \(0.464299\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21.7751 0.912861 0.456430 0.889759i \(-0.349128\pi\)
0.456430 + 0.889759i \(0.349128\pi\)
\(570\) 0 0
\(571\) −6.82109 −0.285454 −0.142727 0.989762i \(-0.545587\pi\)
−0.142727 + 0.989762i \(0.545587\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5.24054 −0.218167 −0.109083 0.994033i \(-0.534792\pi\)
−0.109083 + 0.994033i \(0.534792\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.4951 −0.518385
\(582\) 0 0
\(583\) 9.19961 0.381009
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −47.4055 −1.95663 −0.978316 0.207117i \(-0.933592\pi\)
−0.978316 + 0.207117i \(0.933592\pi\)
\(588\) 0 0
\(589\) 37.1789 1.53193
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −11.2965 −0.463890 −0.231945 0.972729i \(-0.574509\pi\)
−0.231945 + 0.972729i \(0.574509\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −36.6829 −1.49882 −0.749412 0.662104i \(-0.769664\pi\)
−0.749412 + 0.662104i \(0.769664\pi\)
\(600\) 0 0
\(601\) 19.3578 0.789622 0.394811 0.918762i \(-0.370810\pi\)
0.394811 + 0.918762i \(0.370810\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1.28148 −0.0520135 −0.0260068 0.999662i \(-0.508279\pi\)
−0.0260068 + 0.999662i \(0.508279\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −34.6410 −1.40143
\(612\) 0 0
\(613\) 38.9028 1.57127 0.785636 0.618690i \(-0.212336\pi\)
0.785636 + 0.618690i \(0.212336\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.7976 1.23986 0.619932 0.784655i \(-0.287160\pi\)
0.619932 + 0.784655i \(0.287160\pi\)
\(618\) 0 0
\(619\) 34.3578 1.38096 0.690479 0.723353i \(-0.257400\pi\)
0.690479 + 0.723353i \(0.257400\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.8424 −0.514519
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −31.3650 −1.25061
\(630\) 0 0
\(631\) 11.5367 0.459270 0.229635 0.973277i \(-0.426247\pi\)
0.229635 + 0.973277i \(0.426247\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 34.9438 1.38452
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7.91872 0.312771 0.156385 0.987696i \(-0.450016\pi\)
0.156385 + 0.987696i \(0.450016\pi\)
\(642\) 0 0
\(643\) −9.19961 −0.362797 −0.181399 0.983410i \(-0.558062\pi\)
−0.181399 + 0.983410i \(0.558062\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −13.7146 −0.539177 −0.269588 0.962976i \(-0.586888\pi\)
−0.269588 + 0.962976i \(0.586888\pi\)
\(648\) 0 0
\(649\) −31.5367 −1.23792
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.0941 1.64727 0.823634 0.567122i \(-0.191943\pi\)
0.823634 + 0.567122i \(0.191943\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −24.1269 −0.939852 −0.469926 0.882706i \(-0.655719\pi\)
−0.469926 + 0.882706i \(0.655719\pi\)
\(660\) 0 0
\(661\) 44.4313 1.72818 0.864088 0.503341i \(-0.167896\pi\)
0.864088 + 0.503341i \(0.167896\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −39.1321 −1.51520
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9.03102 −0.348639
\(672\) 0 0
\(673\) 32.6101 1.25703 0.628513 0.777799i \(-0.283664\pi\)
0.628513 + 0.777799i \(0.283664\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 50.9724 1.95903 0.979514 0.201376i \(-0.0645412\pi\)
0.979514 + 0.201376i \(0.0645412\pi\)
\(678\) 0 0
\(679\) −18.3578 −0.704508
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 28.3795 1.08591 0.542955 0.839762i \(-0.317305\pi\)
0.542955 + 0.839762i \(0.317305\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −14.4762 −0.551498
\(690\) 0 0
\(691\) 14.7156 0.559809 0.279905 0.960028i \(-0.409697\pi\)
0.279905 + 0.960028i \(0.409697\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.07370 0.154302
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14.2272 0.537353 0.268676 0.963230i \(-0.413414\pi\)
0.268676 + 0.963230i \(0.413414\pi\)
\(702\) 0 0
\(703\) −37.6214 −1.41892
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.872228 −0.0328035
\(708\) 0 0
\(709\) −18.5367 −0.696161 −0.348081 0.937465i \(-0.613166\pi\)
−0.348081 + 0.937465i \(0.613166\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −39.0022 −1.46065
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −31.7358 −1.18355 −0.591773 0.806105i \(-0.701572\pi\)
−0.591773 + 0.806105i \(0.701572\pi\)
\(720\) 0 0
\(721\) 3.28437 0.122316
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −31.4432 −1.16617 −0.583083 0.812413i \(-0.698154\pi\)
−0.583083 + 0.812413i \(0.698154\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.66973 0.283675
\(732\) 0 0
\(733\) −5.12591 −0.189330 −0.0946649 0.995509i \(-0.530178\pi\)
−0.0946649 + 0.995509i \(0.530178\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 64.6870 2.38278
\(738\) 0 0
\(739\) 33.8945 1.24683 0.623415 0.781891i \(-0.285745\pi\)
0.623415 + 0.781891i \(0.285745\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −38.1300 −1.39885 −0.699427 0.714704i \(-0.746562\pi\)
−0.699427 + 0.714704i \(0.746562\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.3578 0.596905 0.298453 0.954424i \(-0.403530\pi\)
0.298453 + 0.954424i \(0.403530\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −28.6510 −1.04134 −0.520670 0.853758i \(-0.674318\pi\)
−0.520670 + 0.853758i \(0.674318\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.73205 0.0627868 0.0313934 0.999507i \(-0.490006\pi\)
0.0313934 + 0.999507i \(0.490006\pi\)
\(762\) 0 0
\(763\) 8.97034 0.324748
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 49.6250 1.79186
\(768\) 0 0
\(769\) −19.7156 −0.710964 −0.355482 0.934683i \(-0.615683\pi\)
−0.355482 + 0.934683i \(0.615683\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.63772 0.166807 0.0834036 0.996516i \(-0.473421\pi\)
0.0834036 + 0.996516i \(0.473421\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.88627 0.175069
\(780\) 0 0
\(781\) 22.8211 0.816603
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 45.7688 1.63148 0.815740 0.578419i \(-0.196330\pi\)
0.815740 + 0.578419i \(0.196330\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11.6318 0.413580
\(792\) 0 0
\(793\) 14.2109 0.504643
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 26.3584 0.933663 0.466832 0.884346i \(-0.345395\pi\)
0.466832 + 0.884346i \(0.345395\pi\)
\(798\) 0 0
\(799\) 31.7891 1.12462
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 32.3435 1.14138
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.19615 0.182687 0.0913435 0.995819i \(-0.470884\pi\)
0.0913435 + 0.995819i \(0.470884\pi\)
\(810\) 0 0
\(811\) −49.8945 −1.75203 −0.876017 0.482280i \(-0.839809\pi\)
−0.876017 + 0.482280i \(0.839809\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 9.19961 0.321853
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.248992 0.00868988 0.00434494 0.999991i \(-0.498617\pi\)
0.00434494 + 0.999991i \(0.498617\pi\)
\(822\) 0 0
\(823\) −26.0881 −0.909373 −0.454687 0.890652i \(-0.650249\pi\)
−0.454687 + 0.890652i \(0.650249\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.9701 −0.416243 −0.208121 0.978103i \(-0.566735\pi\)
−0.208121 + 0.978103i \(0.566735\pi\)
\(828\) 0 0
\(829\) 7.17891 0.249334 0.124667 0.992199i \(-0.460214\pi\)
0.124667 + 0.992199i \(0.460214\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −32.0669 −1.11105
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −22.0850 −0.762459 −0.381230 0.924480i \(-0.624499\pi\)
−0.381230 + 0.924480i \(0.624499\pi\)
\(840\) 0 0
\(841\) −2.00000 −0.0689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 7.91813 0.272070
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 39.4664 1.35289
\(852\) 0 0
\(853\) −40.4136 −1.38373 −0.691867 0.722025i \(-0.743212\pi\)
−0.691867 + 0.722025i \(0.743212\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.02103 −0.0690371 −0.0345186 0.999404i \(-0.510990\pi\)
−0.0345186 + 0.999404i \(0.510990\pi\)
\(858\) 0 0
\(859\) 4.46327 0.152285 0.0761425 0.997097i \(-0.475740\pi\)
0.0761425 + 0.997097i \(0.475740\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.21958 −0.0755555 −0.0377777 0.999286i \(-0.512028\pi\)
−0.0377777 + 0.999286i \(0.512028\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 24.8685 0.843605
\(870\) 0 0
\(871\) −101.789 −3.44899
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −9.31424 −0.314520 −0.157260 0.987557i \(-0.550266\pi\)
−0.157260 + 0.987557i \(0.550266\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 43.6111 1.46930 0.734648 0.678448i \(-0.237347\pi\)
0.734648 + 0.678448i \(0.237347\pi\)
\(882\) 0 0
\(883\) 1.51074 0.0508406 0.0254203 0.999677i \(-0.491908\pi\)
0.0254203 + 0.999677i \(0.491908\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −6.65875 −0.223579 −0.111789 0.993732i \(-0.535658\pi\)
−0.111789 + 0.993732i \(0.535658\pi\)
\(888\) 0 0
\(889\) 1.64218 0.0550771
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 38.1300 1.27597
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 26.9104 0.897512
\(900\) 0 0
\(901\) 13.2844 0.442566
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −15.3777 −0.510609 −0.255304 0.966861i \(-0.582176\pi\)
−0.255304 + 0.966861i \(0.582176\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −23.4463 −0.776810 −0.388405 0.921489i \(-0.626974\pi\)
−0.388405 + 0.921489i \(0.626974\pi\)
\(912\) 0 0
\(913\) −40.4136 −1.33749
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.05585 0.233005
\(918\) 0 0
\(919\) −31.1789 −1.02850 −0.514249 0.857641i \(-0.671929\pi\)
−0.514249 + 0.857641i \(0.671929\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −35.9104 −1.18201
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −26.6005 −0.872735 −0.436367 0.899769i \(-0.643735\pi\)
−0.436367 + 0.899769i \(0.643735\pi\)
\(930\) 0 0
\(931\) −38.4633 −1.26058
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −3.95906 −0.129337 −0.0646685 0.997907i \(-0.520599\pi\)
−0.0646685 + 0.997907i \(0.520599\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 31.3650 1.02247 0.511235 0.859441i \(-0.329188\pi\)
0.511235 + 0.859441i \(0.329188\pi\)
\(942\) 0 0
\(943\) −5.12591 −0.166923
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −27.0321 −0.878425 −0.439213 0.898383i \(-0.644743\pi\)
−0.439213 + 0.898383i \(0.644743\pi\)
\(948\) 0 0
\(949\) −50.8945 −1.65211
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −26.3584 −0.853833 −0.426917 0.904291i \(-0.640400\pi\)
−0.426917 + 0.904291i \(0.640400\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.4762 −0.467460
\(960\) 0 0
\(961\) −4.17891 −0.134803
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 32.4955 1.04498 0.522492 0.852644i \(-0.325003\pi\)
0.522492 + 0.852644i \(0.325003\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −37.3027 −1.19710 −0.598550 0.801085i \(-0.704256\pi\)
−0.598550 + 0.801085i \(0.704256\pi\)
\(972\) 0 0
\(973\) 22.0144 0.705748
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −18.1538 −0.580790 −0.290395 0.956907i \(-0.593787\pi\)
−0.290395 + 0.956907i \(0.593787\pi\)
\(978\) 0 0
\(979\) −41.5367 −1.32752
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −48.7528 −1.55497 −0.777487 0.628900i \(-0.783506\pi\)
−0.777487 + 0.628900i \(0.783506\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9.65078 −0.306877
\(990\) 0 0
\(991\) 1.53673 0.0488157 0.0244078 0.999702i \(-0.492230\pi\)
0.0244078 + 0.999702i \(0.492230\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −44.1434 −1.39804 −0.699018 0.715105i \(-0.746379\pi\)
−0.699018 + 0.715105i \(0.746379\pi\)
\(998\) 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 8100.2.a.be.1.6 8
3.2 odd 2 inner 8100.2.a.be.1.5 8
5.2 odd 4 1620.2.d.e.649.5 yes 8
5.3 odd 4 1620.2.d.e.649.6 yes 8
5.4 even 2 inner 8100.2.a.be.1.4 8
15.2 even 4 1620.2.d.e.649.4 yes 8
15.8 even 4 1620.2.d.e.649.3 8
15.14 odd 2 inner 8100.2.a.be.1.3 8
45.2 even 12 1620.2.r.h.109.3 16
45.7 odd 12 1620.2.r.h.109.6 16
45.13 odd 12 1620.2.r.h.1189.6 16
45.22 odd 12 1620.2.r.h.1189.1 16
45.23 even 12 1620.2.r.h.1189.3 16
45.32 even 12 1620.2.r.h.1189.8 16
45.38 even 12 1620.2.r.h.109.8 16
45.43 odd 12 1620.2.r.h.109.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.d.e.649.3 8 15.8 even 4
1620.2.d.e.649.4 yes 8 15.2 even 4
1620.2.d.e.649.5 yes 8 5.2 odd 4
1620.2.d.e.649.6 yes 8 5.3 odd 4
1620.2.r.h.109.1 16 45.43 odd 12
1620.2.r.h.109.3 16 45.2 even 12
1620.2.r.h.109.6 16 45.7 odd 12
1620.2.r.h.109.8 16 45.38 even 12
1620.2.r.h.1189.1 16 45.22 odd 12
1620.2.r.h.1189.3 16 45.23 even 12
1620.2.r.h.1189.6 16 45.13 odd 12
1620.2.r.h.1189.8 16 45.32 even 12
8100.2.a.be.1.3 8 15.14 odd 2 inner
8100.2.a.be.1.4 8 5.4 even 2 inner
8100.2.a.be.1.5 8 3.2 odd 2 inner
8100.2.a.be.1.6 8 1.1 even 1 trivial