Properties

Label 3168.2.a.bd.1.2
Level $3168$
Weight $2$
Character 3168.1
Self dual yes
Analytic conductor $25.297$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3168,2,Mod(1,3168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3168.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3168.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: \(25.2966073603\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 352)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 3168.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.561553 q^{5} +1.00000 q^{11} +2.00000 q^{13} -7.12311 q^{17} +1.12311 q^{19} -7.68466 q^{23} -4.68466 q^{25} -7.12311 q^{29} +5.43845 q^{31} -5.68466 q^{37} +8.24621 q^{41} +1.12311 q^{43} -4.00000 q^{47} -7.00000 q^{49} -8.24621 q^{53} +0.561553 q^{55} -0.315342 q^{59} +9.36932 q^{61} +1.12311 q^{65} -7.68466 q^{67} +15.6847 q^{71} -6.00000 q^{73} +13.1231 q^{79} -11.3693 q^{83} -4.00000 q^{85} -0.561553 q^{89} +0.630683 q^{95} +5.68466 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} + 2 q^{11} + 4 q^{13} - 6 q^{17} - 6 q^{19} - 3 q^{23} + 3 q^{25} - 6 q^{29} + 15 q^{31} + q^{37} - 6 q^{43} - 8 q^{47} - 14 q^{49} - 3 q^{55} - 13 q^{59} - 6 q^{61} - 6 q^{65} - 3 q^{67}+ \cdots - q^{97}+O(q^{100}) \) 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
\(5\) 0.561553 0.251134 0.125567 0.992085i \(-0.459925\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.12311 −1.72761 −0.863803 0.503829i \(-0.831924\pi\)
−0.863803 + 0.503829i \(0.831924\pi\)
\(18\) 0 0
\(19\) 1.12311 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.68466 −1.60236 −0.801181 0.598422i \(-0.795795\pi\)
−0.801181 + 0.598422i \(0.795795\pi\)
\(24\) 0 0
\(25\) −4.68466 −0.936932
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.12311 −1.32273 −0.661364 0.750065i \(-0.730022\pi\)
−0.661364 + 0.750065i \(0.730022\pi\)
\(30\) 0 0
\(31\) 5.43845 0.976774 0.488387 0.872627i \(-0.337585\pi\)
0.488387 + 0.872627i \(0.337585\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.68466 −0.934552 −0.467276 0.884111i \(-0.654765\pi\)
−0.467276 + 0.884111i \(0.654765\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.24621 1.28784 0.643921 0.765092i \(-0.277307\pi\)
0.643921 + 0.765092i \(0.277307\pi\)
\(42\) 0 0
\(43\) 1.12311 0.171272 0.0856360 0.996326i \(-0.472708\pi\)
0.0856360 + 0.996326i \(0.472708\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.24621 −1.13270 −0.566352 0.824163i \(-0.691646\pi\)
−0.566352 + 0.824163i \(0.691646\pi\)
\(54\) 0 0
\(55\) 0.561553 0.0757198
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.315342 −0.0410540 −0.0205270 0.999789i \(-0.506534\pi\)
−0.0205270 + 0.999789i \(0.506534\pi\)
\(60\) 0 0
\(61\) 9.36932 1.19962 0.599809 0.800143i \(-0.295243\pi\)
0.599809 + 0.800143i \(0.295243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.12311 0.139304
\(66\) 0 0
\(67\) −7.68466 −0.938830 −0.469415 0.882978i \(-0.655535\pi\)
−0.469415 + 0.882978i \(0.655535\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 15.6847 1.86143 0.930713 0.365750i \(-0.119187\pi\)
0.930713 + 0.365750i \(0.119187\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 13.1231 1.47646 0.738232 0.674546i \(-0.235661\pi\)
0.738232 + 0.674546i \(0.235661\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.3693 −1.24794 −0.623972 0.781446i \(-0.714482\pi\)
−0.623972 + 0.781446i \(0.714482\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.561553 −0.0595245 −0.0297622 0.999557i \(-0.509475\pi\)
−0.0297622 + 0.999557i \(0.509475\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.630683 0.0647067
\(96\) 0 0
\(97\) 5.68466 0.577190 0.288595 0.957451i \(-0.406812\pi\)
0.288595 + 0.957451i \(0.406812\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.87689 −0.485269 −0.242635 0.970118i \(-0.578012\pi\)
−0.242635 + 0.970118i \(0.578012\pi\)
\(102\) 0 0
\(103\) 14.2462 1.40372 0.701860 0.712314i \(-0.252353\pi\)
0.701860 + 0.712314i \(0.252353\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.4384 −1.07604 −0.538019 0.842933i \(-0.680827\pi\)
−0.538019 + 0.842933i \(0.680827\pi\)
\(114\) 0 0
\(115\) −4.31534 −0.402408
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.43845 −0.486430
\(126\) 0 0
\(127\) −15.3693 −1.36381 −0.681903 0.731442i \(-0.738847\pi\)
−0.681903 + 0.731442i \(0.738847\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.6847 −1.16916 −0.584580 0.811336i \(-0.698741\pi\)
−0.584580 + 0.811336i \(0.698741\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.87689 −0.399531 −0.199765 0.979844i \(-0.564018\pi\)
−0.199765 + 0.979844i \(0.564018\pi\)
\(150\) 0 0
\(151\) −15.3693 −1.25074 −0.625369 0.780329i \(-0.715051\pi\)
−0.625369 + 0.780329i \(0.715051\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.05398 0.245301
\(156\) 0 0
\(157\) 1.68466 0.134450 0.0672252 0.997738i \(-0.478585\pi\)
0.0672252 + 0.997738i \(0.478585\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.36932 0.570255 0.285127 0.958490i \(-0.407964\pi\)
0.285127 + 0.958490i \(0.407964\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.68466 0.574378 0.287189 0.957874i \(-0.407279\pi\)
0.287189 + 0.957874i \(0.407279\pi\)
\(180\) 0 0
\(181\) 25.0540 1.86225 0.931124 0.364704i \(-0.118830\pi\)
0.931124 + 0.364704i \(0.118830\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.19224 −0.234698
\(186\) 0 0
\(187\) −7.12311 −0.520893
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.68466 0.556042 0.278021 0.960575i \(-0.410321\pi\)
0.278021 + 0.960575i \(0.410321\pi\)
\(192\) 0 0
\(193\) −13.3693 −0.962344 −0.481172 0.876626i \(-0.659789\pi\)
−0.481172 + 0.876626i \(0.659789\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.24621 0.587518 0.293759 0.955879i \(-0.405094\pi\)
0.293759 + 0.955879i \(0.405094\pi\)
\(198\) 0 0
\(199\) 14.2462 1.00989 0.504944 0.863152i \(-0.331513\pi\)
0.504944 + 0.863152i \(0.331513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.63068 0.323421
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.12311 0.0776868
\(210\) 0 0
\(211\) −16.4924 −1.13539 −0.567693 0.823241i \(-0.692164\pi\)
−0.567693 + 0.823241i \(0.692164\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.630683 0.0430122
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −14.2462 −0.958304
\(222\) 0 0
\(223\) 7.68466 0.514603 0.257301 0.966331i \(-0.417167\pi\)
0.257301 + 0.966331i \(0.417167\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) −21.6847 −1.43296 −0.716481 0.697606i \(-0.754248\pi\)
−0.716481 + 0.697606i \(0.754248\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.12311 −0.466650 −0.233325 0.972399i \(-0.574961\pi\)
−0.233325 + 0.972399i \(0.574961\pi\)
\(234\) 0 0
\(235\) −2.24621 −0.146527
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.630683 −0.0407955 −0.0203977 0.999792i \(-0.506493\pi\)
−0.0203977 + 0.999792i \(0.506493\pi\)
\(240\) 0 0
\(241\) 2.63068 0.169457 0.0847286 0.996404i \(-0.472998\pi\)
0.0847286 + 0.996404i \(0.472998\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.93087 −0.251134
\(246\) 0 0
\(247\) 2.24621 0.142923
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −23.0540 −1.45515 −0.727577 0.686026i \(-0.759354\pi\)
−0.727577 + 0.686026i \(0.759354\pi\)
\(252\) 0 0
\(253\) −7.68466 −0.483130
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.24621 0.514385 0.257192 0.966360i \(-0.417203\pi\)
0.257192 + 0.966360i \(0.417203\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) −4.63068 −0.284461
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.4924 −0.639734 −0.319867 0.947462i \(-0.603638\pi\)
−0.319867 + 0.947462i \(0.603638\pi\)
\(270\) 0 0
\(271\) 13.1231 0.797172 0.398586 0.917131i \(-0.369501\pi\)
0.398586 + 0.917131i \(0.369501\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.68466 −0.282496
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −9.75379 −0.579803 −0.289901 0.957057i \(-0.593622\pi\)
−0.289901 + 0.957057i \(0.593622\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 33.7386 1.98463
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.4924 0.612974 0.306487 0.951875i \(-0.400846\pi\)
0.306487 + 0.951875i \(0.400846\pi\)
\(294\) 0 0
\(295\) −0.177081 −0.0103101
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −15.3693 −0.888831
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.26137 0.301265
\(306\) 0 0
\(307\) −27.3693 −1.56205 −0.781025 0.624500i \(-0.785303\pi\)
−0.781025 + 0.624500i \(0.785303\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 18.7386 1.06257 0.531285 0.847193i \(-0.321709\pi\)
0.531285 + 0.847193i \(0.321709\pi\)
\(312\) 0 0
\(313\) 6.31534 0.356964 0.178482 0.983943i \(-0.442881\pi\)
0.178482 + 0.983943i \(0.442881\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.0540 1.63183 0.815917 0.578169i \(-0.196233\pi\)
0.815917 + 0.578169i \(0.196233\pi\)
\(318\) 0 0
\(319\) −7.12311 −0.398817
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) −9.36932 −0.519716
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −16.3153 −0.896772 −0.448386 0.893840i \(-0.648001\pi\)
−0.448386 + 0.893840i \(0.648001\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.31534 −0.235772
\(336\) 0 0
\(337\) −6.63068 −0.361196 −0.180598 0.983557i \(-0.557803\pi\)
−0.180598 + 0.983557i \(0.557803\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.43845 0.294508
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 27.3693 1.46926 0.734631 0.678467i \(-0.237355\pi\)
0.734631 + 0.678467i \(0.237355\pi\)
\(348\) 0 0
\(349\) −13.3693 −0.715643 −0.357822 0.933790i \(-0.616480\pi\)
−0.357822 + 0.933790i \(0.616480\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −24.5616 −1.30728 −0.653640 0.756806i \(-0.726759\pi\)
−0.653640 + 0.756806i \(0.726759\pi\)
\(354\) 0 0
\(355\) 8.80776 0.467468
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.3693 0.811162 0.405581 0.914059i \(-0.367069\pi\)
0.405581 + 0.914059i \(0.367069\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.36932 −0.176358
\(366\) 0 0
\(367\) 0.946025 0.0493821 0.0246910 0.999695i \(-0.492140\pi\)
0.0246910 + 0.999695i \(0.492140\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −21.3693 −1.10646 −0.553231 0.833028i \(-0.686605\pi\)
−0.553231 + 0.833028i \(0.686605\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −14.2462 −0.733717
\(378\) 0 0
\(379\) −18.5616 −0.953443 −0.476721 0.879054i \(-0.658175\pi\)
−0.476721 + 0.879054i \(0.658175\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.31534 0.424894 0.212447 0.977173i \(-0.431857\pi\)
0.212447 + 0.977173i \(0.431857\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 26.8078 1.35921 0.679604 0.733579i \(-0.262152\pi\)
0.679604 + 0.733579i \(0.262152\pi\)
\(390\) 0 0
\(391\) 54.7386 2.76825
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.36932 0.370791
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.4924 −1.12322 −0.561609 0.827403i \(-0.689817\pi\)
−0.561609 + 0.827403i \(0.689817\pi\)
\(402\) 0 0
\(403\) 10.8769 0.541817
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.68466 −0.281778
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.38447 −0.313401
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −18.7386 −0.915442 −0.457721 0.889096i \(-0.651334\pi\)
−0.457721 + 0.889096i \(0.651334\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 33.3693 1.61865
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −38.7386 −1.86597 −0.932987 0.359910i \(-0.882808\pi\)
−0.932987 + 0.359910i \(0.882808\pi\)
\(432\) 0 0
\(433\) 5.68466 0.273187 0.136594 0.990627i \(-0.456385\pi\)
0.136594 + 0.990627i \(0.456385\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.63068 −0.412862
\(438\) 0 0
\(439\) 26.2462 1.25266 0.626332 0.779557i \(-0.284556\pi\)
0.626332 + 0.779557i \(0.284556\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8.31534 −0.395074 −0.197537 0.980295i \(-0.563294\pi\)
−0.197537 + 0.980295i \(0.563294\pi\)
\(444\) 0 0
\(445\) −0.315342 −0.0149486
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.80776 −0.132507 −0.0662533 0.997803i \(-0.521105\pi\)
−0.0662533 + 0.997803i \(0.521105\pi\)
\(450\) 0 0
\(451\) 8.24621 0.388299
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 41.3693 1.93518 0.967588 0.252536i \(-0.0812646\pi\)
0.967588 + 0.252536i \(0.0812646\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 19.1231 0.890652 0.445326 0.895369i \(-0.353088\pi\)
0.445326 + 0.895369i \(0.353088\pi\)
\(462\) 0 0
\(463\) −33.9309 −1.57690 −0.788451 0.615098i \(-0.789116\pi\)
−0.788451 + 0.615098i \(0.789116\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.31534 −0.384788 −0.192394 0.981318i \(-0.561625\pi\)
−0.192394 + 0.981318i \(0.561625\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.12311 0.0516405
\(474\) 0 0
\(475\) −5.26137 −0.241408
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0.630683 0.0288166 0.0144083 0.999896i \(-0.495414\pi\)
0.0144083 + 0.999896i \(0.495414\pi\)
\(480\) 0 0
\(481\) −11.3693 −0.518396
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.19224 0.144952
\(486\) 0 0
\(487\) 31.6847 1.43577 0.717884 0.696162i \(-0.245111\pi\)
0.717884 + 0.696162i \(0.245111\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −26.7386 −1.20670 −0.603349 0.797477i \(-0.706167\pi\)
−0.603349 + 0.797477i \(0.706167\pi\)
\(492\) 0 0
\(493\) 50.7386 2.28515
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 40.4924 1.81269 0.906345 0.422539i \(-0.138861\pi\)
0.906345 + 0.422539i \(0.138861\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 15.3693 0.685284 0.342642 0.939466i \(-0.388678\pi\)
0.342642 + 0.939466i \(0.388678\pi\)
\(504\) 0 0
\(505\) −2.73863 −0.121868
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.6847 0.606562 0.303281 0.952901i \(-0.401918\pi\)
0.303281 + 0.952901i \(0.401918\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.0540 0.747148 0.373574 0.927600i \(-0.378132\pi\)
0.373574 + 0.927600i \(0.378132\pi\)
\(522\) 0 0
\(523\) −34.1080 −1.49144 −0.745718 0.666261i \(-0.767894\pi\)
−0.745718 + 0.666261i \(0.767894\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −38.7386 −1.68748
\(528\) 0 0
\(529\) 36.0540 1.56756
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 16.4924 0.714366
\(534\) 0 0
\(535\) −6.73863 −0.291337
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7.00000 −0.301511
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −7.86174 −0.336760
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −33.3693 −1.41390 −0.706952 0.707262i \(-0.749930\pi\)
−0.706952 + 0.707262i \(0.749930\pi\)
\(558\) 0 0
\(559\) 2.24621 0.0950046
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.7386 0.789739 0.394870 0.918737i \(-0.370790\pi\)
0.394870 + 0.918737i \(0.370790\pi\)
\(564\) 0 0
\(565\) −6.42329 −0.270230
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21.3693 0.895848 0.447924 0.894072i \(-0.352163\pi\)
0.447924 + 0.894072i \(0.352163\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 36.0000 1.50130
\(576\) 0 0
\(577\) 5.68466 0.236655 0.118328 0.992975i \(-0.462247\pi\)
0.118328 + 0.992975i \(0.462247\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −8.24621 −0.341523
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 6.10795 0.251674
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 25.8617 1.06201 0.531007 0.847367i \(-0.321814\pi\)
0.531007 + 0.847367i \(0.321814\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −34.7386 −1.41938 −0.709691 0.704513i \(-0.751165\pi\)
−0.709691 + 0.704513i \(0.751165\pi\)
\(600\) 0 0
\(601\) 32.1080 1.30971 0.654855 0.755754i \(-0.272730\pi\)
0.654855 + 0.755754i \(0.272730\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.561553 0.0228304
\(606\) 0 0
\(607\) −37.1231 −1.50678 −0.753390 0.657574i \(-0.771583\pi\)
−0.753390 + 0.657574i \(0.771583\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 17.3693 0.701540 0.350770 0.936462i \(-0.385920\pi\)
0.350770 + 0.936462i \(0.385920\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.2462 1.29818 0.649092 0.760710i \(-0.275149\pi\)
0.649092 + 0.760710i \(0.275149\pi\)
\(618\) 0 0
\(619\) 7.68466 0.308873 0.154436 0.988003i \(-0.450644\pi\)
0.154436 + 0.988003i \(0.450644\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 40.4924 1.61454
\(630\) 0 0
\(631\) 38.4233 1.52961 0.764804 0.644264i \(-0.222836\pi\)
0.764804 + 0.644264i \(0.222836\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.63068 −0.342498
\(636\) 0 0
\(637\) −14.0000 −0.554700
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 27.9309 1.10320 0.551602 0.834108i \(-0.314017\pi\)
0.551602 + 0.834108i \(0.314017\pi\)
\(642\) 0 0
\(643\) 42.5616 1.67846 0.839232 0.543774i \(-0.183005\pi\)
0.839232 + 0.543774i \(0.183005\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 31.6847 1.24565 0.622826 0.782360i \(-0.285984\pi\)
0.622826 + 0.782360i \(0.285984\pi\)
\(648\) 0 0
\(649\) −0.315342 −0.0123782
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13.6847 0.535522 0.267761 0.963485i \(-0.413716\pi\)
0.267761 + 0.963485i \(0.413716\pi\)
\(654\) 0 0
\(655\) 2.24621 0.0877667
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.6307 −0.492022 −0.246011 0.969267i \(-0.579120\pi\)
−0.246011 + 0.969267i \(0.579120\pi\)
\(660\) 0 0
\(661\) −21.0540 −0.818905 −0.409452 0.912331i \(-0.634280\pi\)
−0.409452 + 0.912331i \(0.634280\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 54.7386 2.11949
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9.36932 0.361698
\(672\) 0 0
\(673\) 24.7386 0.953604 0.476802 0.879011i \(-0.341796\pi\)
0.476802 + 0.879011i \(0.341796\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −37.8617 −1.45514 −0.727572 0.686031i \(-0.759351\pi\)
−0.727572 + 0.686031i \(0.759351\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −7.68466 −0.293616
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −16.4924 −0.628311
\(690\) 0 0
\(691\) 25.3002 0.962464 0.481232 0.876593i \(-0.340189\pi\)
0.481232 + 0.876593i \(0.340189\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.73863 −0.255611
\(696\) 0 0
\(697\) −58.7386 −2.22488
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 47.6155 1.79841 0.899207 0.437524i \(-0.144144\pi\)
0.899207 + 0.437524i \(0.144144\pi\)
\(702\) 0 0
\(703\) −6.38447 −0.240795
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 10.3153 0.387401 0.193700 0.981061i \(-0.437951\pi\)
0.193700 + 0.981061i \(0.437951\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −41.7926 −1.56515
\(714\) 0 0
\(715\) 1.12311 0.0420018
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.3153 0.608460 0.304230 0.952599i \(-0.401601\pi\)
0.304230 + 0.952599i \(0.401601\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 33.3693 1.23931
\(726\) 0 0
\(727\) −16.3153 −0.605103 −0.302551 0.953133i \(-0.597838\pi\)
−0.302551 + 0.953133i \(0.597838\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −36.1080 −1.33368 −0.666839 0.745202i \(-0.732353\pi\)
−0.666839 + 0.745202i \(0.732353\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.68466 −0.283068
\(738\) 0 0
\(739\) −29.6155 −1.08942 −0.544712 0.838623i \(-0.683361\pi\)
−0.544712 + 0.838623i \(0.683361\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −30.7386 −1.12769 −0.563846 0.825880i \(-0.690679\pi\)
−0.563846 + 0.825880i \(0.690679\pi\)
\(744\) 0 0
\(745\) −2.73863 −0.100336
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −12.1771 −0.444348 −0.222174 0.975007i \(-0.571315\pi\)
−0.222174 + 0.975007i \(0.571315\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.63068 −0.314103
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.75379 0.136075 0.0680374 0.997683i \(-0.478326\pi\)
0.0680374 + 0.997683i \(0.478326\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −0.630683 −0.0227726
\(768\) 0 0
\(769\) −38.0000 −1.37032 −0.685158 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −8.24621 −0.296596 −0.148298 0.988943i \(-0.547379\pi\)
−0.148298 + 0.988943i \(0.547379\pi\)
\(774\) 0 0
\(775\) −25.4773 −0.915170
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.26137 0.331823
\(780\) 0 0
\(781\) 15.6847 0.561241
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.946025 0.0337651
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 18.7386 0.665428
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.5616 0.870015 0.435007 0.900427i \(-0.356746\pi\)
0.435007 + 0.900427i \(0.356746\pi\)
\(798\) 0 0
\(799\) 28.4924 1.00799
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 8.24621 0.289921 0.144961 0.989437i \(-0.453694\pi\)
0.144961 + 0.989437i \(0.453694\pi\)
\(810\) 0 0
\(811\) −51.3693 −1.80382 −0.901910 0.431923i \(-0.857835\pi\)
−0.901910 + 0.431923i \(0.857835\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.73863 0.236044
\(816\) 0 0
\(817\) 1.26137 0.0441296
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.738634 −0.0257785 −0.0128892 0.999917i \(-0.504103\pi\)
−0.0128892 + 0.999917i \(0.504103\pi\)
\(822\) 0 0
\(823\) −18.5616 −0.647015 −0.323508 0.946226i \(-0.604862\pi\)
−0.323508 + 0.946226i \(0.604862\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.36932 0.117163 0.0585813 0.998283i \(-0.481342\pi\)
0.0585813 + 0.998283i \(0.481342\pi\)
\(828\) 0 0
\(829\) 17.6847 0.614214 0.307107 0.951675i \(-0.400639\pi\)
0.307107 + 0.951675i \(0.400639\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 49.8617 1.72761
\(834\) 0 0
\(835\) 4.13826 0.143210
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −30.4233 −1.05033 −0.525164 0.851001i \(-0.675996\pi\)
−0.525164 + 0.851001i \(0.675996\pi\)
\(840\) 0 0
\(841\) 21.7386 0.749608
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5.05398 −0.173862
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 43.6847 1.49749
\(852\) 0 0
\(853\) 40.7386 1.39486 0.697432 0.716651i \(-0.254326\pi\)
0.697432 + 0.716651i \(0.254326\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −4.87689 −0.166592 −0.0832958 0.996525i \(-0.526545\pi\)
−0.0832958 + 0.996525i \(0.526545\pi\)
\(858\) 0 0
\(859\) −57.9309 −1.97658 −0.988288 0.152601i \(-0.951235\pi\)
−0.988288 + 0.152601i \(0.951235\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.7386 0.637871 0.318935 0.947777i \(-0.396675\pi\)
0.318935 + 0.947777i \(0.396675\pi\)
\(864\) 0 0
\(865\) −10.1080 −0.343681
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 13.1231 0.445171
\(870\) 0 0
\(871\) −15.3693 −0.520769
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −7.26137 −0.245199 −0.122599 0.992456i \(-0.539123\pi\)
−0.122599 + 0.992456i \(0.539123\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 19.3002 0.650240 0.325120 0.945673i \(-0.394595\pi\)
0.325120 + 0.945673i \(0.394595\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 22.1080 0.742312 0.371156 0.928570i \(-0.378961\pi\)
0.371156 + 0.928570i \(0.378961\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −4.49242 −0.150333
\(894\) 0 0
\(895\) 4.31534 0.144246
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −38.7386 −1.29201
\(900\) 0 0
\(901\) 58.7386 1.95687
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 14.0691 0.467674
\(906\) 0 0
\(907\) 16.4924 0.547622 0.273811 0.961784i \(-0.411716\pi\)
0.273811 + 0.961784i \(0.411716\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 26.7386 0.885890 0.442945 0.896549i \(-0.353934\pi\)
0.442945 + 0.896549i \(0.353934\pi\)
\(912\) 0 0
\(913\) −11.3693 −0.376269
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −17.6155 −0.581083 −0.290541 0.956862i \(-0.593835\pi\)
−0.290541 + 0.956862i \(0.593835\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 31.3693 1.03253
\(924\) 0 0
\(925\) 26.6307 0.875611
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22.4924 −0.737952 −0.368976 0.929439i \(-0.620292\pi\)
−0.368976 + 0.929439i \(0.620292\pi\)
\(930\) 0 0
\(931\) −7.86174 −0.257658
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) −20.7386 −0.677502 −0.338751 0.940876i \(-0.610004\pi\)
−0.338751 + 0.940876i \(0.610004\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8.24621 0.268819 0.134409 0.990926i \(-0.457086\pi\)
0.134409 + 0.990926i \(0.457086\pi\)
\(942\) 0 0
\(943\) −63.3693 −2.06359
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 31.6847 1.02961 0.514807 0.857306i \(-0.327864\pi\)
0.514807 + 0.857306i \(0.327864\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 47.6155 1.54242 0.771209 0.636582i \(-0.219652\pi\)
0.771209 + 0.636582i \(0.219652\pi\)
\(954\) 0 0
\(955\) 4.31534 0.139641
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1.42329 −0.0459127
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −7.50758 −0.241677
\(966\) 0 0
\(967\) 30.7386 0.988488 0.494244 0.869323i \(-0.335445\pi\)
0.494244 + 0.869323i \(0.335445\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 16.3153 0.523584 0.261792 0.965124i \(-0.415687\pi\)
0.261792 + 0.965124i \(0.415687\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 50.0388 1.60088 0.800442 0.599410i \(-0.204598\pi\)
0.800442 + 0.599410i \(0.204598\pi\)
\(978\) 0 0
\(979\) −0.561553 −0.0179473
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 16.3153 0.520379 0.260189 0.965558i \(-0.416215\pi\)
0.260189 + 0.965558i \(0.416215\pi\)
\(984\) 0 0
\(985\) 4.63068 0.147546
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.63068 −0.274440
\(990\) 0 0
\(991\) −5.26137 −0.167133 −0.0835664 0.996502i \(-0.526631\pi\)
−0.0835664 + 0.996502i \(0.526631\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) 25.3693 0.803454 0.401727 0.915759i \(-0.368410\pi\)
0.401727 + 0.915759i \(0.368410\pi\)
\(998\) 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 3168.2.a.bd.1.2 2
3.2 odd 2 352.2.a.g.1.1 2
4.3 odd 2 3168.2.a.bc.1.2 2
8.3 odd 2 6336.2.a.cw.1.1 2
8.5 even 2 6336.2.a.cv.1.1 2
12.11 even 2 352.2.a.h.1.2 yes 2
15.14 odd 2 8800.2.a.be.1.2 2
24.5 odd 2 704.2.a.o.1.2 2
24.11 even 2 704.2.a.n.1.1 2
33.32 even 2 3872.2.a.p.1.1 2
48.5 odd 4 2816.2.c.t.1409.4 4
48.11 even 4 2816.2.c.s.1409.1 4
48.29 odd 4 2816.2.c.t.1409.1 4
48.35 even 4 2816.2.c.s.1409.4 4
60.59 even 2 8800.2.a.bd.1.1 2
132.131 odd 2 3872.2.a.ba.1.2 2
264.131 odd 2 7744.2.a.bw.1.1 2
264.197 even 2 7744.2.a.cm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
352.2.a.g.1.1 2 3.2 odd 2
352.2.a.h.1.2 yes 2 12.11 even 2
704.2.a.n.1.1 2 24.11 even 2
704.2.a.o.1.2 2 24.5 odd 2
2816.2.c.s.1409.1 4 48.11 even 4
2816.2.c.s.1409.4 4 48.35 even 4
2816.2.c.t.1409.1 4 48.29 odd 4
2816.2.c.t.1409.4 4 48.5 odd 4
3168.2.a.bc.1.2 2 4.3 odd 2
3168.2.a.bd.1.2 2 1.1 even 1 trivial
3872.2.a.p.1.1 2 33.32 even 2
3872.2.a.ba.1.2 2 132.131 odd 2
6336.2.a.cv.1.1 2 8.5 even 2
6336.2.a.cw.1.1 2 8.3 odd 2
7744.2.a.bw.1.1 2 264.131 odd 2
7744.2.a.cm.1.2 2 264.197 even 2
8800.2.a.bd.1.1 2 60.59 even 2
8800.2.a.be.1.2 2 15.14 odd 2