Properties

Label 4032.2.a.bo.1.2
Level $4032$
Weight $2$
Character 4032.1
Self dual yes
Analytic conductor $32.196$
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] = [4032,2,Mod(1,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, 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("4032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.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: \(32.1956820950\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2016)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 4032.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.23607 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+1.23607 q^{5} -1.00000 q^{7} -1.23607 q^{11} +4.47214 q^{13} +5.23607 q^{17} -6.47214 q^{19} -7.70820 q^{23} -3.47214 q^{25} -10.4721 q^{29} -6.47214 q^{31} -1.23607 q^{35} -4.47214 q^{37} +2.76393 q^{41} +4.94427 q^{43} +10.4721 q^{47} +1.00000 q^{49} -8.00000 q^{53} -1.52786 q^{55} -2.47214 q^{59} -4.47214 q^{61} +5.52786 q^{65} -8.00000 q^{67} +2.76393 q^{71} +2.94427 q^{73} +1.23607 q^{77} +4.94427 q^{79} +4.94427 q^{83} +6.47214 q^{85} +15.7082 q^{89} -4.47214 q^{91} -8.00000 q^{95} +10.9443 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{7} + 2 q^{11} + 6 q^{17} - 4 q^{19} - 2 q^{23} + 2 q^{25} - 12 q^{29} - 4 q^{31} + 2 q^{35} + 10 q^{41} - 8 q^{43} + 12 q^{47} + 2 q^{49} - 16 q^{53} - 12 q^{55} + 4 q^{59} + 20 q^{65} - 16 q^{67} + 10 q^{71} - 12 q^{73} - 2 q^{77} - 8 q^{79} - 8 q^{83} + 4 q^{85} + 18 q^{89} - 16 q^{95} + 4 q^{97}+O(q^{100}) \) 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
\(5\) 1.23607 0.552786 0.276393 0.961045i \(-0.410861\pi\)
0.276393 + 0.961045i \(0.410861\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.23607 −0.372689 −0.186344 0.982485i \(-0.559664\pi\)
−0.186344 + 0.982485i \(0.559664\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.23607 1.26993 0.634967 0.772540i \(-0.281014\pi\)
0.634967 + 0.772540i \(0.281014\pi\)
\(18\) 0 0
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.70820 −1.60727 −0.803636 0.595121i \(-0.797104\pi\)
−0.803636 + 0.595121i \(0.797104\pi\)
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.4721 −1.94463 −0.972313 0.233681i \(-0.924923\pi\)
−0.972313 + 0.233681i \(0.924923\pi\)
\(30\) 0 0
\(31\) −6.47214 −1.16243 −0.581215 0.813750i \(-0.697422\pi\)
−0.581215 + 0.813750i \(0.697422\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.23607 −0.208934
\(36\) 0 0
\(37\) −4.47214 −0.735215 −0.367607 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.76393 0.431654 0.215827 0.976432i \(-0.430755\pi\)
0.215827 + 0.976432i \(0.430755\pi\)
\(42\) 0 0
\(43\) 4.94427 0.753994 0.376997 0.926214i \(-0.376957\pi\)
0.376997 + 0.926214i \(0.376957\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.4721 1.52752 0.763759 0.645501i \(-0.223352\pi\)
0.763759 + 0.645501i \(0.223352\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) −1.52786 −0.206017
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.47214 −0.321845 −0.160922 0.986967i \(-0.551447\pi\)
−0.160922 + 0.986967i \(0.551447\pi\)
\(60\) 0 0
\(61\) −4.47214 −0.572598 −0.286299 0.958140i \(-0.592425\pi\)
−0.286299 + 0.958140i \(0.592425\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.52786 0.685647
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.76393 0.328018 0.164009 0.986459i \(-0.447557\pi\)
0.164009 + 0.986459i \(0.447557\pi\)
\(72\) 0 0
\(73\) 2.94427 0.344601 0.172300 0.985044i \(-0.444880\pi\)
0.172300 + 0.985044i \(0.444880\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.23607 0.140863
\(78\) 0 0
\(79\) 4.94427 0.556274 0.278137 0.960541i \(-0.410283\pi\)
0.278137 + 0.960541i \(0.410283\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.94427 0.542704 0.271352 0.962480i \(-0.412529\pi\)
0.271352 + 0.962480i \(0.412529\pi\)
\(84\) 0 0
\(85\) 6.47214 0.702002
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.7082 1.66507 0.832533 0.553975i \(-0.186890\pi\)
0.832533 + 0.553975i \(0.186890\pi\)
\(90\) 0 0
\(91\) −4.47214 −0.468807
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 10.9443 1.11122 0.555611 0.831442i \(-0.312484\pi\)
0.555611 + 0.831442i \(0.312484\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.29180 −0.427050 −0.213525 0.976938i \(-0.568494\pi\)
−0.213525 + 0.976938i \(0.568494\pi\)
\(102\) 0 0
\(103\) −9.52786 −0.938808 −0.469404 0.882983i \(-0.655531\pi\)
−0.469404 + 0.882983i \(0.655531\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.76393 0.653894 0.326947 0.945043i \(-0.393980\pi\)
0.326947 + 0.945043i \(0.393980\pi\)
\(108\) 0 0
\(109\) −6.94427 −0.665141 −0.332570 0.943078i \(-0.607916\pi\)
−0.332570 + 0.943078i \(0.607916\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.94427 −0.465118 −0.232559 0.972582i \(-0.574710\pi\)
−0.232559 + 0.972582i \(0.574710\pi\)
\(114\) 0 0
\(115\) −9.52786 −0.888478
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.23607 −0.479990
\(120\) 0 0
\(121\) −9.47214 −0.861103
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.4721 −0.936656
\(126\) 0 0
\(127\) −17.8885 −1.58735 −0.793676 0.608341i \(-0.791835\pi\)
−0.793676 + 0.608341i \(0.791835\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) 6.47214 0.561205
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.4721 0.894695 0.447347 0.894360i \(-0.352369\pi\)
0.447347 + 0.894360i \(0.352369\pi\)
\(138\) 0 0
\(139\) −12.9443 −1.09792 −0.548959 0.835849i \(-0.684976\pi\)
−0.548959 + 0.835849i \(0.684976\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.52786 −0.462263
\(144\) 0 0
\(145\) −12.9443 −1.07496
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.8885 −1.46549 −0.732743 0.680505i \(-0.761760\pi\)
−0.732743 + 0.680505i \(0.761760\pi\)
\(150\) 0 0
\(151\) −20.9443 −1.70442 −0.852210 0.523199i \(-0.824738\pi\)
−0.852210 + 0.523199i \(0.824738\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −7.52786 −0.600789 −0.300394 0.953815i \(-0.597118\pi\)
−0.300394 + 0.953815i \(0.597118\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.70820 0.607492
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.4164 1.19296 0.596479 0.802629i \(-0.296566\pi\)
0.596479 + 0.802629i \(0.296566\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.29180 −0.326299 −0.163150 0.986601i \(-0.552165\pi\)
−0.163150 + 0.986601i \(0.552165\pi\)
\(174\) 0 0
\(175\) 3.47214 0.262469
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 22.1803 1.65784 0.828918 0.559370i \(-0.188957\pi\)
0.828918 + 0.559370i \(0.188957\pi\)
\(180\) 0 0
\(181\) −0.472136 −0.0350936 −0.0175468 0.999846i \(-0.505586\pi\)
−0.0175468 + 0.999846i \(0.505586\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.52786 −0.406417
\(186\) 0 0
\(187\) −6.47214 −0.473289
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.65248 0.336641 0.168321 0.985732i \(-0.446166\pi\)
0.168321 + 0.985732i \(0.446166\pi\)
\(192\) 0 0
\(193\) 3.52786 0.253941 0.126971 0.991906i \(-0.459475\pi\)
0.126971 + 0.991906i \(0.459475\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.9443 0.922241 0.461121 0.887337i \(-0.347448\pi\)
0.461121 + 0.887337i \(0.347448\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 10.4721 0.735000
\(204\) 0 0
\(205\) 3.41641 0.238612
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −24.0000 −1.65223 −0.826114 0.563503i \(-0.809453\pi\)
−0.826114 + 0.563503i \(0.809453\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.11146 0.416798
\(216\) 0 0
\(217\) 6.47214 0.439357
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 23.4164 1.57516
\(222\) 0 0
\(223\) 4.94427 0.331093 0.165546 0.986202i \(-0.447061\pi\)
0.165546 + 0.986202i \(0.447061\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 28.3607 1.88236 0.941182 0.337899i \(-0.109716\pi\)
0.941182 + 0.337899i \(0.109716\pi\)
\(228\) 0 0
\(229\) −3.52786 −0.233128 −0.116564 0.993183i \(-0.537188\pi\)
−0.116564 + 0.993183i \(0.537188\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −20.3607 −1.33387 −0.666936 0.745115i \(-0.732395\pi\)
−0.666936 + 0.745115i \(0.732395\pi\)
\(234\) 0 0
\(235\) 12.9443 0.844391
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.2361 0.856170 0.428085 0.903738i \(-0.359188\pi\)
0.428085 + 0.903738i \(0.359188\pi\)
\(240\) 0 0
\(241\) −27.8885 −1.79646 −0.898230 0.439527i \(-0.855146\pi\)
−0.898230 + 0.439527i \(0.855146\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.23607 0.0789695
\(246\) 0 0
\(247\) −28.9443 −1.84168
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −23.4164 −1.47803 −0.739015 0.673689i \(-0.764709\pi\)
−0.739015 + 0.673689i \(0.764709\pi\)
\(252\) 0 0
\(253\) 9.52786 0.599012
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.2918 −1.01625 −0.508127 0.861282i \(-0.669662\pi\)
−0.508127 + 0.861282i \(0.669662\pi\)
\(258\) 0 0
\(259\) 4.47214 0.277885
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5.81966 −0.358856 −0.179428 0.983771i \(-0.557425\pi\)
−0.179428 + 0.983771i \(0.557425\pi\)
\(264\) 0 0
\(265\) −9.88854 −0.607448
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.23607 0.0753644 0.0376822 0.999290i \(-0.488003\pi\)
0.0376822 + 0.999290i \(0.488003\pi\)
\(270\) 0 0
\(271\) 19.4164 1.17946 0.589731 0.807599i \(-0.299234\pi\)
0.589731 + 0.807599i \(0.299234\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.29180 0.258805
\(276\) 0 0
\(277\) 20.4721 1.23005 0.615026 0.788507i \(-0.289146\pi\)
0.615026 + 0.788507i \(0.289146\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 15.4164 0.919666 0.459833 0.888005i \(-0.347909\pi\)
0.459833 + 0.888005i \(0.347909\pi\)
\(282\) 0 0
\(283\) 6.47214 0.384729 0.192364 0.981324i \(-0.438384\pi\)
0.192364 + 0.981324i \(0.438384\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.76393 −0.163150
\(288\) 0 0
\(289\) 10.4164 0.612730
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.81966 0.573671 0.286835 0.957980i \(-0.407397\pi\)
0.286835 + 0.957980i \(0.407397\pi\)
\(294\) 0 0
\(295\) −3.05573 −0.177911
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −34.4721 −1.99358
\(300\) 0 0
\(301\) −4.94427 −0.284983
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.52786 −0.316525
\(306\) 0 0
\(307\) −9.52786 −0.543784 −0.271892 0.962328i \(-0.587649\pi\)
−0.271892 + 0.962328i \(0.587649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −15.4164 −0.874184 −0.437092 0.899417i \(-0.643992\pi\)
−0.437092 + 0.899417i \(0.643992\pi\)
\(312\) 0 0
\(313\) 19.8885 1.12417 0.562083 0.827081i \(-0.310000\pi\)
0.562083 + 0.827081i \(0.310000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.05573 0.171627 0.0858134 0.996311i \(-0.472651\pi\)
0.0858134 + 0.996311i \(0.472651\pi\)
\(318\) 0 0
\(319\) 12.9443 0.724740
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −33.8885 −1.88561
\(324\) 0 0
\(325\) −15.5279 −0.861331
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −10.4721 −0.577348
\(330\) 0 0
\(331\) −17.8885 −0.983243 −0.491622 0.870809i \(-0.663596\pi\)
−0.491622 + 0.870809i \(0.663596\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −9.88854 −0.540269
\(336\) 0 0
\(337\) 23.8885 1.30129 0.650646 0.759381i \(-0.274498\pi\)
0.650646 + 0.759381i \(0.274498\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 25.2361 1.35474 0.677372 0.735641i \(-0.263119\pi\)
0.677372 + 0.735641i \(0.263119\pi\)
\(348\) 0 0
\(349\) 11.5279 0.617072 0.308536 0.951213i \(-0.400161\pi\)
0.308536 + 0.951213i \(0.400161\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.1803 0.541845 0.270922 0.962601i \(-0.412671\pi\)
0.270922 + 0.962601i \(0.412671\pi\)
\(354\) 0 0
\(355\) 3.41641 0.181324
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −21.2361 −1.12080 −0.560398 0.828223i \(-0.689352\pi\)
−0.560398 + 0.828223i \(0.689352\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.63932 0.190491
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) −23.8885 −1.23690 −0.618451 0.785823i \(-0.712239\pi\)
−0.618451 + 0.785823i \(0.712239\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −46.8328 −2.41201
\(378\) 0 0
\(379\) 24.0000 1.23280 0.616399 0.787434i \(-0.288591\pi\)
0.616399 + 0.787434i \(0.288591\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) 1.52786 0.0778672
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.4721 −0.530958 −0.265479 0.964117i \(-0.585530\pi\)
−0.265479 + 0.964117i \(0.585530\pi\)
\(390\) 0 0
\(391\) −40.3607 −2.04113
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.11146 0.307501
\(396\) 0 0
\(397\) 8.47214 0.425204 0.212602 0.977139i \(-0.431806\pi\)
0.212602 + 0.977139i \(0.431806\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −26.4721 −1.32196 −0.660978 0.750406i \(-0.729858\pi\)
−0.660978 + 0.750406i \(0.729858\pi\)
\(402\) 0 0
\(403\) −28.9443 −1.44182
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.52786 0.274006
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.47214 0.121646
\(414\) 0 0
\(415\) 6.11146 0.300000
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.47214 −0.120772 −0.0603859 0.998175i \(-0.519233\pi\)
−0.0603859 + 0.998175i \(0.519233\pi\)
\(420\) 0 0
\(421\) −5.41641 −0.263980 −0.131990 0.991251i \(-0.542137\pi\)
−0.131990 + 0.991251i \(0.542137\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −18.1803 −0.881876
\(426\) 0 0
\(427\) 4.47214 0.216422
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −31.1246 −1.49922 −0.749610 0.661880i \(-0.769759\pi\)
−0.749610 + 0.661880i \(0.769759\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 49.8885 2.38649
\(438\) 0 0
\(439\) −30.8328 −1.47157 −0.735785 0.677215i \(-0.763187\pi\)
−0.735785 + 0.677215i \(0.763187\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 30.1803 1.43391 0.716956 0.697119i \(-0.245535\pi\)
0.716956 + 0.697119i \(0.245535\pi\)
\(444\) 0 0
\(445\) 19.4164 0.920426
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.94427 −0.233335 −0.116667 0.993171i \(-0.537221\pi\)
−0.116667 + 0.993171i \(0.537221\pi\)
\(450\) 0 0
\(451\) −3.41641 −0.160872
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.52786 −0.259150
\(456\) 0 0
\(457\) −35.8885 −1.67880 −0.839398 0.543518i \(-0.817092\pi\)
−0.839398 + 0.543518i \(0.817092\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.70820 0.172708 0.0863541 0.996265i \(-0.472478\pi\)
0.0863541 + 0.996265i \(0.472478\pi\)
\(462\) 0 0
\(463\) 17.8885 0.831351 0.415676 0.909513i \(-0.363545\pi\)
0.415676 + 0.909513i \(0.363545\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −34.4721 −1.59518 −0.797590 0.603200i \(-0.793892\pi\)
−0.797590 + 0.603200i \(0.793892\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.11146 −0.281005
\(474\) 0 0
\(475\) 22.4721 1.03109
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −0.583592 −0.0266650 −0.0133325 0.999911i \(-0.504244\pi\)
−0.0133325 + 0.999911i \(0.504244\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.5279 0.614269
\(486\) 0 0
\(487\) 30.8328 1.39717 0.698584 0.715528i \(-0.253814\pi\)
0.698584 + 0.715528i \(0.253814\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 16.0689 0.725179 0.362589 0.931949i \(-0.381893\pi\)
0.362589 + 0.931949i \(0.381893\pi\)
\(492\) 0 0
\(493\) −54.8328 −2.46955
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.76393 −0.123979
\(498\) 0 0
\(499\) −11.0557 −0.494922 −0.247461 0.968898i \(-0.579596\pi\)
−0.247461 + 0.968898i \(0.579596\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 11.0557 0.492951 0.246475 0.969149i \(-0.420728\pi\)
0.246475 + 0.969149i \(0.420728\pi\)
\(504\) 0 0
\(505\) −5.30495 −0.236067
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 35.1246 1.55687 0.778436 0.627725i \(-0.216014\pi\)
0.778436 + 0.627725i \(0.216014\pi\)
\(510\) 0 0
\(511\) −2.94427 −0.130247
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −11.7771 −0.518960
\(516\) 0 0
\(517\) −12.9443 −0.569288
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.1246 0.662621 0.331311 0.943522i \(-0.392509\pi\)
0.331311 + 0.943522i \(0.392509\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −33.8885 −1.47621
\(528\) 0 0
\(529\) 36.4164 1.58332
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.3607 0.535400
\(534\) 0 0
\(535\) 8.36068 0.361464
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.23607 −0.0532412
\(540\) 0 0
\(541\) 6.36068 0.273467 0.136733 0.990608i \(-0.456340\pi\)
0.136733 + 0.990608i \(0.456340\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.58359 −0.367681
\(546\) 0 0
\(547\) −14.8328 −0.634205 −0.317103 0.948391i \(-0.602710\pi\)
−0.317103 + 0.948391i \(0.602710\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 67.7771 2.88740
\(552\) 0 0
\(553\) −4.94427 −0.210252
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.05573 0.129475 0.0647377 0.997902i \(-0.479379\pi\)
0.0647377 + 0.997902i \(0.479379\pi\)
\(558\) 0 0
\(559\) 22.1115 0.935215
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −33.3050 −1.40364 −0.701818 0.712356i \(-0.747628\pi\)
−0.701818 + 0.712356i \(0.747628\pi\)
\(564\) 0 0
\(565\) −6.11146 −0.257111
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.4721 1.10977 0.554885 0.831927i \(-0.312762\pi\)
0.554885 + 0.831927i \(0.312762\pi\)
\(570\) 0 0
\(571\) 11.0557 0.462668 0.231334 0.972874i \(-0.425691\pi\)
0.231334 + 0.972874i \(0.425691\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 26.7639 1.11613
\(576\) 0 0
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −4.94427 −0.205123
\(582\) 0 0
\(583\) 9.88854 0.409542
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.41641 0.306108 0.153054 0.988218i \(-0.451089\pi\)
0.153054 + 0.988218i \(0.451089\pi\)
\(588\) 0 0
\(589\) 41.8885 1.72599
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.1246 0.949614 0.474807 0.880090i \(-0.342518\pi\)
0.474807 + 0.880090i \(0.342518\pi\)
\(594\) 0 0
\(595\) −6.47214 −0.265332
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −29.2361 −1.19455 −0.597277 0.802035i \(-0.703751\pi\)
−0.597277 + 0.802035i \(0.703751\pi\)
\(600\) 0 0
\(601\) 35.8885 1.46392 0.731962 0.681345i \(-0.238605\pi\)
0.731962 + 0.681345i \(0.238605\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11.7082 −0.476006
\(606\) 0 0
\(607\) 11.0557 0.448738 0.224369 0.974504i \(-0.427968\pi\)
0.224369 + 0.974504i \(0.427968\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 46.8328 1.89465
\(612\) 0 0
\(613\) 40.8328 1.64922 0.824611 0.565700i \(-0.191394\pi\)
0.824611 + 0.565700i \(0.191394\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −37.5279 −1.51081 −0.755407 0.655255i \(-0.772561\pi\)
−0.755407 + 0.655255i \(0.772561\pi\)
\(618\) 0 0
\(619\) 28.9443 1.16337 0.581684 0.813415i \(-0.302394\pi\)
0.581684 + 0.813415i \(0.302394\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −15.7082 −0.629336
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −23.4164 −0.933673
\(630\) 0 0
\(631\) 17.8885 0.712132 0.356066 0.934461i \(-0.384118\pi\)
0.356066 + 0.934461i \(0.384118\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −22.1115 −0.877466
\(636\) 0 0
\(637\) 4.47214 0.177192
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.2492 −1.19477 −0.597386 0.801954i \(-0.703794\pi\)
−0.597386 + 0.801954i \(0.703794\pi\)
\(642\) 0 0
\(643\) 3.41641 0.134730 0.0673650 0.997728i \(-0.478541\pi\)
0.0673650 + 0.997728i \(0.478541\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.5279 −0.846348 −0.423174 0.906049i \(-0.639084\pi\)
−0.423174 + 0.906049i \(0.639084\pi\)
\(648\) 0 0
\(649\) 3.05573 0.119948
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 15.4164 0.603291 0.301645 0.953420i \(-0.402464\pi\)
0.301645 + 0.953420i \(0.402464\pi\)
\(654\) 0 0
\(655\) −19.7771 −0.772755
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.652476 0.0254169 0.0127084 0.999919i \(-0.495955\pi\)
0.0127084 + 0.999919i \(0.495955\pi\)
\(660\) 0 0
\(661\) −15.5279 −0.603964 −0.301982 0.953314i \(-0.597648\pi\)
−0.301982 + 0.953314i \(0.597648\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 80.7214 3.12554
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.52786 0.213401
\(672\) 0 0
\(673\) −19.3050 −0.744151 −0.372076 0.928202i \(-0.621354\pi\)
−0.372076 + 0.928202i \(0.621354\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −45.0132 −1.73000 −0.864998 0.501775i \(-0.832680\pi\)
−0.864998 + 0.501775i \(0.832680\pi\)
\(678\) 0 0
\(679\) −10.9443 −0.420003
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 14.1803 0.542596 0.271298 0.962495i \(-0.412547\pi\)
0.271298 + 0.962495i \(0.412547\pi\)
\(684\) 0 0
\(685\) 12.9443 0.494575
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −35.7771 −1.36300
\(690\) 0 0
\(691\) −6.11146 −0.232491 −0.116245 0.993221i \(-0.537086\pi\)
−0.116245 + 0.993221i \(0.537086\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) 14.4721 0.548171
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −15.4164 −0.582270 −0.291135 0.956682i \(-0.594033\pi\)
−0.291135 + 0.956682i \(0.594033\pi\)
\(702\) 0 0
\(703\) 28.9443 1.09165
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.29180 0.161410
\(708\) 0 0
\(709\) −14.9443 −0.561244 −0.280622 0.959818i \(-0.590541\pi\)
−0.280622 + 0.959818i \(0.590541\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 49.8885 1.86834
\(714\) 0 0
\(715\) −6.83282 −0.255533
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 36.9443 1.37779 0.688894 0.724862i \(-0.258096\pi\)
0.688894 + 0.724862i \(0.258096\pi\)
\(720\) 0 0
\(721\) 9.52786 0.354836
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 36.3607 1.35040
\(726\) 0 0
\(727\) 32.3607 1.20019 0.600096 0.799928i \(-0.295129\pi\)
0.600096 + 0.799928i \(0.295129\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 25.8885 0.957522
\(732\) 0 0
\(733\) −15.3050 −0.565301 −0.282651 0.959223i \(-0.591214\pi\)
−0.282651 + 0.959223i \(0.591214\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.88854 0.364249
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 35.4853 1.30183 0.650915 0.759151i \(-0.274386\pi\)
0.650915 + 0.759151i \(0.274386\pi\)
\(744\) 0 0
\(745\) −22.1115 −0.810101
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6.76393 −0.247149
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −25.8885 −0.942181
\(756\) 0 0
\(757\) 1.05573 0.0383711 0.0191855 0.999816i \(-0.493893\pi\)
0.0191855 + 0.999816i \(0.493893\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −10.1803 −0.369037 −0.184519 0.982829i \(-0.559073\pi\)
−0.184519 + 0.982829i \(0.559073\pi\)
\(762\) 0 0
\(763\) 6.94427 0.251400
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −11.0557 −0.399199
\(768\) 0 0
\(769\) −7.88854 −0.284468 −0.142234 0.989833i \(-0.545429\pi\)
−0.142234 + 0.989833i \(0.545429\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −25.2361 −0.907678 −0.453839 0.891084i \(-0.649946\pi\)
−0.453839 + 0.891084i \(0.649946\pi\)
\(774\) 0 0
\(775\) 22.4721 0.807223
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17.8885 −0.640924
\(780\) 0 0
\(781\) −3.41641 −0.122249
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.30495 −0.332108
\(786\) 0 0
\(787\) −54.8328 −1.95458 −0.977289 0.211909i \(-0.932032\pi\)
−0.977289 + 0.211909i \(0.932032\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.94427 0.175798
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32.6525 1.15661 0.578305 0.815821i \(-0.303714\pi\)
0.578305 + 0.815821i \(0.303714\pi\)
\(798\) 0 0
\(799\) 54.8328 1.93985
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.63932 −0.128429
\(804\) 0 0
\(805\) 9.52786 0.335813
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 32.0000 1.12506 0.562530 0.826777i \(-0.309828\pi\)
0.562530 + 0.826777i \(0.309828\pi\)
\(810\) 0 0
\(811\) 41.8885 1.47091 0.735453 0.677576i \(-0.236969\pi\)
0.735453 + 0.677576i \(0.236969\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −9.88854 −0.346381
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −19.0557 −0.665049 −0.332525 0.943095i \(-0.607900\pi\)
−0.332525 + 0.943095i \(0.607900\pi\)
\(822\) 0 0
\(823\) −36.9443 −1.28780 −0.643898 0.765111i \(-0.722684\pi\)
−0.643898 + 0.765111i \(0.722684\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −38.1803 −1.32766 −0.663830 0.747883i \(-0.731070\pi\)
−0.663830 + 0.747883i \(0.731070\pi\)
\(828\) 0 0
\(829\) −40.4721 −1.40566 −0.702828 0.711360i \(-0.748080\pi\)
−0.702828 + 0.711360i \(0.748080\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.23607 0.181419
\(834\) 0 0
\(835\) 19.0557 0.659451
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −5.52786 −0.190843 −0.0954215 0.995437i \(-0.530420\pi\)
−0.0954215 + 0.995437i \(0.530420\pi\)
\(840\) 0 0
\(841\) 80.6656 2.78157
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8.65248 0.297654
\(846\) 0 0
\(847\) 9.47214 0.325466
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 34.4721 1.18169
\(852\) 0 0
\(853\) 45.4164 1.55503 0.777514 0.628866i \(-0.216480\pi\)
0.777514 + 0.628866i \(0.216480\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10.1803 −0.347754 −0.173877 0.984767i \(-0.555629\pi\)
−0.173877 + 0.984767i \(0.555629\pi\)
\(858\) 0 0
\(859\) 25.5279 0.870999 0.435500 0.900189i \(-0.356572\pi\)
0.435500 + 0.900189i \(0.356572\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −21.8197 −0.742750 −0.371375 0.928483i \(-0.621114\pi\)
−0.371375 + 0.928483i \(0.621114\pi\)
\(864\) 0 0
\(865\) −5.30495 −0.180374
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.11146 −0.207317
\(870\) 0 0
\(871\) −35.7771 −1.21226
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 10.4721 0.354023
\(876\) 0 0
\(877\) −44.8328 −1.51390 −0.756948 0.653475i \(-0.773311\pi\)
−0.756948 + 0.653475i \(0.773311\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 39.1246 1.31814 0.659071 0.752081i \(-0.270950\pi\)
0.659071 + 0.752081i \(0.270950\pi\)
\(882\) 0 0
\(883\) 4.94427 0.166388 0.0831940 0.996533i \(-0.473488\pi\)
0.0831940 + 0.996533i \(0.473488\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 21.5279 0.722835 0.361417 0.932404i \(-0.382293\pi\)
0.361417 + 0.932404i \(0.382293\pi\)
\(888\) 0 0
\(889\) 17.8885 0.599963
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −67.7771 −2.26807
\(894\) 0 0
\(895\) 27.4164 0.916429
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 67.7771 2.26049
\(900\) 0 0
\(901\) −41.8885 −1.39551
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −0.583592 −0.0193993
\(906\) 0 0
\(907\) −4.94427 −0.164172 −0.0820859 0.996625i \(-0.526158\pi\)
−0.0820859 + 0.996625i \(0.526158\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −15.1246 −0.501101 −0.250550 0.968104i \(-0.580612\pi\)
−0.250550 + 0.968104i \(0.580612\pi\)
\(912\) 0 0
\(913\) −6.11146 −0.202260
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) 27.0557 0.892486 0.446243 0.894912i \(-0.352762\pi\)
0.446243 + 0.894912i \(0.352762\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 12.3607 0.406857
\(924\) 0 0
\(925\) 15.5279 0.510553
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 50.7639 1.66551 0.832755 0.553641i \(-0.186762\pi\)
0.832755 + 0.553641i \(0.186762\pi\)
\(930\) 0 0
\(931\) −6.47214 −0.212116
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −2.94427 −0.0961852 −0.0480926 0.998843i \(-0.515314\pi\)
−0.0480926 + 0.998843i \(0.515314\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −52.4296 −1.70915 −0.854577 0.519324i \(-0.826184\pi\)
−0.854577 + 0.519324i \(0.826184\pi\)
\(942\) 0 0
\(943\) −21.3050 −0.693785
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11.1246 −0.361501 −0.180751 0.983529i \(-0.557853\pi\)
−0.180751 + 0.983529i \(0.557853\pi\)
\(948\) 0 0
\(949\) 13.1672 0.427425
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 51.7771 1.67722 0.838612 0.544729i \(-0.183367\pi\)
0.838612 + 0.544729i \(0.183367\pi\)
\(954\) 0 0
\(955\) 5.75078 0.186091
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.4721 −0.338163
\(960\) 0 0
\(961\) 10.8885 0.351243
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.36068 0.140375
\(966\) 0 0
\(967\) −59.7771 −1.92230 −0.961151 0.276024i \(-0.910983\pi\)
−0.961151 + 0.276024i \(0.910983\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −25.8885 −0.830803 −0.415401 0.909638i \(-0.636359\pi\)
−0.415401 + 0.909638i \(0.636359\pi\)
\(972\) 0 0
\(973\) 12.9443 0.414974
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.2492 0.967758 0.483879 0.875135i \(-0.339227\pi\)
0.483879 + 0.875135i \(0.339227\pi\)
\(978\) 0 0
\(979\) −19.4164 −0.620551
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.94427 −0.157698 −0.0788489 0.996887i \(-0.525124\pi\)
−0.0788489 + 0.996887i \(0.525124\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −38.1115 −1.21187
\(990\) 0 0
\(991\) 62.8328 1.99595 0.997975 0.0636060i \(-0.0202601\pi\)
0.997975 + 0.0636060i \(0.0202601\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −9.88854 −0.313488
\(996\) 0 0
\(997\) −5.63932 −0.178599 −0.0892995 0.996005i \(-0.528463\pi\)
−0.0892995 + 0.996005i \(0.528463\pi\)
\(998\) 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 4032.2.a.bo.1.2 2
3.2 odd 2 4032.2.a.bu.1.1 2
4.3 odd 2 4032.2.a.bp.1.2 2
8.3 odd 2 2016.2.a.v.1.1 yes 2
8.5 even 2 2016.2.a.u.1.1 yes 2
12.11 even 2 4032.2.a.bx.1.1 2
24.5 odd 2 2016.2.a.p.1.2 2
24.11 even 2 2016.2.a.q.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2016.2.a.p.1.2 2 24.5 odd 2
2016.2.a.q.1.2 yes 2 24.11 even 2
2016.2.a.u.1.1 yes 2 8.5 even 2
2016.2.a.v.1.1 yes 2 8.3 odd 2
4032.2.a.bo.1.2 2 1.1 even 1 trivial
4032.2.a.bp.1.2 2 4.3 odd 2
4032.2.a.bu.1.1 2 3.2 odd 2
4032.2.a.bx.1.1 2 12.11 even 2