Properties

Label 8092.2.a.n.1.1
Level $8092$
Weight $2$
Character 8092.1
Self dual yes
Analytic conductor $64.615$
Analytic rank $0$
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] = [8092,2,Mod(1,8092)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8092, 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("8092.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8092 = 2^{2} \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8092.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: \(64.6149453156\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 476)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 8092.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.302776 q^{3} +3.30278 q^{5} -1.00000 q^{7} -2.90833 q^{9} +O(q^{10})\) \(q-0.302776 q^{3} +3.30278 q^{5} -1.00000 q^{7} -2.90833 q^{9} -4.60555 q^{11} -6.60555 q^{13} -1.00000 q^{15} -6.00000 q^{19} +0.302776 q^{21} +2.60555 q^{23} +5.90833 q^{25} +1.78890 q^{27} +8.60555 q^{29} +6.69722 q^{31} +1.39445 q^{33} -3.30278 q^{35} -7.21110 q^{37} +2.00000 q^{39} -9.51388 q^{41} -4.30278 q^{43} -9.60555 q^{45} +10.0000 q^{47} +1.00000 q^{49} +0.697224 q^{53} -15.2111 q^{55} +1.81665 q^{57} +5.21110 q^{59} +4.30278 q^{61} +2.90833 q^{63} -21.8167 q^{65} +2.69722 q^{67} -0.788897 q^{69} +2.00000 q^{71} +13.5139 q^{73} -1.78890 q^{75} +4.60555 q^{77} +6.00000 q^{79} +8.18335 q^{81} +9.21110 q^{83} -2.60555 q^{87} -2.00000 q^{89} +6.60555 q^{91} -2.02776 q^{93} -19.8167 q^{95} +2.09167 q^{97} +13.3944 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 3 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + 3 q^{5} - 2 q^{7} + 5 q^{9} - 2 q^{11} - 6 q^{13} - 2 q^{15} - 12 q^{19} - 3 q^{21} - 2 q^{23} + q^{25} + 18 q^{27} + 10 q^{29} + 17 q^{31} + 10 q^{33} - 3 q^{35} + 4 q^{39} - q^{41} - 5 q^{43} - 12 q^{45} + 20 q^{47} + 2 q^{49} + 5 q^{53} - 16 q^{55} - 18 q^{57} - 4 q^{59} + 5 q^{61} - 5 q^{63} - 22 q^{65} + 9 q^{67} - 16 q^{69} + 4 q^{71} + 9 q^{73} - 18 q^{75} + 2 q^{77} + 12 q^{79} + 38 q^{81} + 4 q^{83} + 2 q^{87} - 4 q^{89} + 6 q^{91} + 32 q^{93} - 18 q^{95} + 15 q^{97} + 34 q^{99}+O(q^{100}) \) 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
\(3\) −0.302776 −0.174808 −0.0874038 0.996173i \(-0.527857\pi\)
−0.0874038 + 0.996173i \(0.527857\pi\)
\(4\) 0 0
\(5\) 3.30278 1.47705 0.738523 0.674228i \(-0.235524\pi\)
0.738523 + 0.674228i \(0.235524\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.90833 −0.969442
\(10\) 0 0
\(11\) −4.60555 −1.38863 −0.694313 0.719673i \(-0.744292\pi\)
−0.694313 + 0.719673i \(0.744292\pi\)
\(12\) 0 0
\(13\) −6.60555 −1.83205 −0.916025 0.401121i \(-0.868621\pi\)
−0.916025 + 0.401121i \(0.868621\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0.302776 0.0660711
\(22\) 0 0
\(23\) 2.60555 0.543295 0.271647 0.962397i \(-0.412432\pi\)
0.271647 + 0.962397i \(0.412432\pi\)
\(24\) 0 0
\(25\) 5.90833 1.18167
\(26\) 0 0
\(27\) 1.78890 0.344273
\(28\) 0 0
\(29\) 8.60555 1.59801 0.799005 0.601324i \(-0.205360\pi\)
0.799005 + 0.601324i \(0.205360\pi\)
\(30\) 0 0
\(31\) 6.69722 1.20286 0.601429 0.798927i \(-0.294598\pi\)
0.601429 + 0.798927i \(0.294598\pi\)
\(32\) 0 0
\(33\) 1.39445 0.242742
\(34\) 0 0
\(35\) −3.30278 −0.558271
\(36\) 0 0
\(37\) −7.21110 −1.18550 −0.592749 0.805387i \(-0.701957\pi\)
−0.592749 + 0.805387i \(0.701957\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −9.51388 −1.48582 −0.742909 0.669392i \(-0.766555\pi\)
−0.742909 + 0.669392i \(0.766555\pi\)
\(42\) 0 0
\(43\) −4.30278 −0.656167 −0.328084 0.944649i \(-0.606403\pi\)
−0.328084 + 0.944649i \(0.606403\pi\)
\(44\) 0 0
\(45\) −9.60555 −1.43191
\(46\) 0 0
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.697224 0.0957711 0.0478856 0.998853i \(-0.484752\pi\)
0.0478856 + 0.998853i \(0.484752\pi\)
\(54\) 0 0
\(55\) −15.2111 −2.05106
\(56\) 0 0
\(57\) 1.81665 0.240622
\(58\) 0 0
\(59\) 5.21110 0.678428 0.339214 0.940709i \(-0.389839\pi\)
0.339214 + 0.940709i \(0.389839\pi\)
\(60\) 0 0
\(61\) 4.30278 0.550914 0.275457 0.961313i \(-0.411171\pi\)
0.275457 + 0.961313i \(0.411171\pi\)
\(62\) 0 0
\(63\) 2.90833 0.366415
\(64\) 0 0
\(65\) −21.8167 −2.70602
\(66\) 0 0
\(67\) 2.69722 0.329518 0.164759 0.986334i \(-0.447315\pi\)
0.164759 + 0.986334i \(0.447315\pi\)
\(68\) 0 0
\(69\) −0.788897 −0.0949721
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 13.5139 1.58168 0.790840 0.612023i \(-0.209644\pi\)
0.790840 + 0.612023i \(0.209644\pi\)
\(74\) 0 0
\(75\) −1.78890 −0.206564
\(76\) 0 0
\(77\) 4.60555 0.524851
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 8.18335 0.909261
\(82\) 0 0
\(83\) 9.21110 1.01105 0.505525 0.862812i \(-0.331299\pi\)
0.505525 + 0.862812i \(0.331299\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.60555 −0.279344
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 6.60555 0.692450
\(92\) 0 0
\(93\) −2.02776 −0.210269
\(94\) 0 0
\(95\) −19.8167 −2.03315
\(96\) 0 0
\(97\) 2.09167 0.212377 0.106189 0.994346i \(-0.466135\pi\)
0.106189 + 0.994346i \(0.466135\pi\)
\(98\) 0 0
\(99\) 13.3944 1.34619
\(100\) 0 0
\(101\) −12.4222 −1.23606 −0.618028 0.786156i \(-0.712068\pi\)
−0.618028 + 0.786156i \(0.712068\pi\)
\(102\) 0 0
\(103\) 13.2111 1.30173 0.650864 0.759194i \(-0.274407\pi\)
0.650864 + 0.759194i \(0.274407\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 9.39445 0.908196 0.454098 0.890952i \(-0.349962\pi\)
0.454098 + 0.890952i \(0.349962\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 2.18335 0.207234
\(112\) 0 0
\(113\) −19.2111 −1.80723 −0.903614 0.428347i \(-0.859096\pi\)
−0.903614 + 0.428347i \(0.859096\pi\)
\(114\) 0 0
\(115\) 8.60555 0.802472
\(116\) 0 0
\(117\) 19.2111 1.77607
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.2111 0.928282
\(122\) 0 0
\(123\) 2.88057 0.259732
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −5.11943 −0.454276 −0.227138 0.973863i \(-0.572937\pi\)
−0.227138 + 0.973863i \(0.572937\pi\)
\(128\) 0 0
\(129\) 1.30278 0.114703
\(130\) 0 0
\(131\) 10.4222 0.910592 0.455296 0.890340i \(-0.349533\pi\)
0.455296 + 0.890340i \(0.349533\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) 0 0
\(135\) 5.90833 0.508508
\(136\) 0 0
\(137\) −3.09167 −0.264139 −0.132070 0.991240i \(-0.542162\pi\)
−0.132070 + 0.991240i \(0.542162\pi\)
\(138\) 0 0
\(139\) 1.48612 0.126051 0.0630256 0.998012i \(-0.479925\pi\)
0.0630256 + 0.998012i \(0.479925\pi\)
\(140\) 0 0
\(141\) −3.02776 −0.254983
\(142\) 0 0
\(143\) 30.4222 2.54403
\(144\) 0 0
\(145\) 28.4222 2.36034
\(146\) 0 0
\(147\) −0.302776 −0.0249725
\(148\) 0 0
\(149\) 4.90833 0.402106 0.201053 0.979580i \(-0.435564\pi\)
0.201053 + 0.979580i \(0.435564\pi\)
\(150\) 0 0
\(151\) 14.1194 1.14902 0.574511 0.818497i \(-0.305192\pi\)
0.574511 + 0.818497i \(0.305192\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 22.1194 1.77668
\(156\) 0 0
\(157\) −0.183346 −0.0146326 −0.00731631 0.999973i \(-0.502329\pi\)
−0.00731631 + 0.999973i \(0.502329\pi\)
\(158\) 0 0
\(159\) −0.211103 −0.0167415
\(160\) 0 0
\(161\) −2.60555 −0.205346
\(162\) 0 0
\(163\) 21.8167 1.70881 0.854406 0.519606i \(-0.173921\pi\)
0.854406 + 0.519606i \(0.173921\pi\)
\(164\) 0 0
\(165\) 4.60555 0.358542
\(166\) 0 0
\(167\) 2.11943 0.164006 0.0820032 0.996632i \(-0.473868\pi\)
0.0820032 + 0.996632i \(0.473868\pi\)
\(168\) 0 0
\(169\) 30.6333 2.35641
\(170\) 0 0
\(171\) 17.4500 1.33443
\(172\) 0 0
\(173\) −4.90833 −0.373173 −0.186587 0.982439i \(-0.559742\pi\)
−0.186587 + 0.982439i \(0.559742\pi\)
\(174\) 0 0
\(175\) −5.90833 −0.446628
\(176\) 0 0
\(177\) −1.57779 −0.118594
\(178\) 0 0
\(179\) 11.0917 0.829031 0.414515 0.910042i \(-0.363951\pi\)
0.414515 + 0.910042i \(0.363951\pi\)
\(180\) 0 0
\(181\) 21.6333 1.60799 0.803996 0.594635i \(-0.202704\pi\)
0.803996 + 0.594635i \(0.202704\pi\)
\(182\) 0 0
\(183\) −1.30278 −0.0963039
\(184\) 0 0
\(185\) −23.8167 −1.75104
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.78890 −0.130123
\(190\) 0 0
\(191\) −23.7250 −1.71668 −0.858340 0.513082i \(-0.828504\pi\)
−0.858340 + 0.513082i \(0.828504\pi\)
\(192\) 0 0
\(193\) 5.39445 0.388301 0.194150 0.980972i \(-0.437805\pi\)
0.194150 + 0.980972i \(0.437805\pi\)
\(194\) 0 0
\(195\) 6.60555 0.473033
\(196\) 0 0
\(197\) 23.0278 1.64066 0.820330 0.571891i \(-0.193790\pi\)
0.820330 + 0.571891i \(0.193790\pi\)
\(198\) 0 0
\(199\) −13.5139 −0.957973 −0.478987 0.877822i \(-0.658996\pi\)
−0.478987 + 0.877822i \(0.658996\pi\)
\(200\) 0 0
\(201\) −0.816654 −0.0576023
\(202\) 0 0
\(203\) −8.60555 −0.603991
\(204\) 0 0
\(205\) −31.4222 −2.19462
\(206\) 0 0
\(207\) −7.57779 −0.526693
\(208\) 0 0
\(209\) 27.6333 1.91144
\(210\) 0 0
\(211\) 27.0278 1.86067 0.930334 0.366714i \(-0.119517\pi\)
0.930334 + 0.366714i \(0.119517\pi\)
\(212\) 0 0
\(213\) −0.605551 −0.0414917
\(214\) 0 0
\(215\) −14.2111 −0.969189
\(216\) 0 0
\(217\) −6.69722 −0.454637
\(218\) 0 0
\(219\) −4.09167 −0.276490
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −21.6333 −1.44867 −0.724337 0.689446i \(-0.757854\pi\)
−0.724337 + 0.689446i \(0.757854\pi\)
\(224\) 0 0
\(225\) −17.1833 −1.14556
\(226\) 0 0
\(227\) −0.0916731 −0.00608456 −0.00304228 0.999995i \(-0.500968\pi\)
−0.00304228 + 0.999995i \(0.500968\pi\)
\(228\) 0 0
\(229\) −5.81665 −0.384375 −0.192188 0.981358i \(-0.561558\pi\)
−0.192188 + 0.981358i \(0.561558\pi\)
\(230\) 0 0
\(231\) −1.39445 −0.0917480
\(232\) 0 0
\(233\) −13.2111 −0.865488 −0.432744 0.901517i \(-0.642455\pi\)
−0.432744 + 0.901517i \(0.642455\pi\)
\(234\) 0 0
\(235\) 33.0278 2.15449
\(236\) 0 0
\(237\) −1.81665 −0.118004
\(238\) 0 0
\(239\) −12.3028 −0.795800 −0.397900 0.917429i \(-0.630261\pi\)
−0.397900 + 0.917429i \(0.630261\pi\)
\(240\) 0 0
\(241\) −13.9083 −0.895914 −0.447957 0.894055i \(-0.647848\pi\)
−0.447957 + 0.894055i \(0.647848\pi\)
\(242\) 0 0
\(243\) −7.84441 −0.503219
\(244\) 0 0
\(245\) 3.30278 0.211007
\(246\) 0 0
\(247\) 39.6333 2.52181
\(248\) 0 0
\(249\) −2.78890 −0.176739
\(250\) 0 0
\(251\) 8.78890 0.554750 0.277375 0.960762i \(-0.410536\pi\)
0.277375 + 0.960762i \(0.410536\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −13.8167 −0.861859 −0.430930 0.902386i \(-0.641814\pi\)
−0.430930 + 0.902386i \(0.641814\pi\)
\(258\) 0 0
\(259\) 7.21110 0.448076
\(260\) 0 0
\(261\) −25.0278 −1.54918
\(262\) 0 0
\(263\) 22.4222 1.38261 0.691306 0.722562i \(-0.257036\pi\)
0.691306 + 0.722562i \(0.257036\pi\)
\(264\) 0 0
\(265\) 2.30278 0.141458
\(266\) 0 0
\(267\) 0.605551 0.0370591
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −31.8167 −1.93272 −0.966362 0.257186i \(-0.917205\pi\)
−0.966362 + 0.257186i \(0.917205\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) −27.2111 −1.64089
\(276\) 0 0
\(277\) −25.0278 −1.50377 −0.751886 0.659293i \(-0.770856\pi\)
−0.751886 + 0.659293i \(0.770856\pi\)
\(278\) 0 0
\(279\) −19.4777 −1.16610
\(280\) 0 0
\(281\) −16.6972 −0.996073 −0.498036 0.867156i \(-0.665945\pi\)
−0.498036 + 0.867156i \(0.665945\pi\)
\(282\) 0 0
\(283\) −6.69722 −0.398109 −0.199054 0.979988i \(-0.563787\pi\)
−0.199054 + 0.979988i \(0.563787\pi\)
\(284\) 0 0
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) 9.51388 0.561586
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) −0.633308 −0.0371252
\(292\) 0 0
\(293\) 1.81665 0.106130 0.0530650 0.998591i \(-0.483101\pi\)
0.0530650 + 0.998591i \(0.483101\pi\)
\(294\) 0 0
\(295\) 17.2111 1.00207
\(296\) 0 0
\(297\) −8.23886 −0.478067
\(298\) 0 0
\(299\) −17.2111 −0.995344
\(300\) 0 0
\(301\) 4.30278 0.248008
\(302\) 0 0
\(303\) 3.76114 0.216072
\(304\) 0 0
\(305\) 14.2111 0.813725
\(306\) 0 0
\(307\) −26.0000 −1.48390 −0.741949 0.670456i \(-0.766098\pi\)
−0.741949 + 0.670456i \(0.766098\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 15.7250 0.891682 0.445841 0.895112i \(-0.352905\pi\)
0.445841 + 0.895112i \(0.352905\pi\)
\(312\) 0 0
\(313\) 5.72498 0.323595 0.161798 0.986824i \(-0.448271\pi\)
0.161798 + 0.986824i \(0.448271\pi\)
\(314\) 0 0
\(315\) 9.60555 0.541212
\(316\) 0 0
\(317\) 12.6056 0.707998 0.353999 0.935246i \(-0.384822\pi\)
0.353999 + 0.935246i \(0.384822\pi\)
\(318\) 0 0
\(319\) −39.6333 −2.21904
\(320\) 0 0
\(321\) −2.84441 −0.158759
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −39.0278 −2.16487
\(326\) 0 0
\(327\) −0.605551 −0.0334871
\(328\) 0 0
\(329\) −10.0000 −0.551318
\(330\) 0 0
\(331\) −0.275019 −0.0151164 −0.00755821 0.999971i \(-0.502406\pi\)
−0.00755821 + 0.999971i \(0.502406\pi\)
\(332\) 0 0
\(333\) 20.9722 1.14927
\(334\) 0 0
\(335\) 8.90833 0.486714
\(336\) 0 0
\(337\) −10.7889 −0.587709 −0.293854 0.955850i \(-0.594938\pi\)
−0.293854 + 0.955850i \(0.594938\pi\)
\(338\) 0 0
\(339\) 5.81665 0.315917
\(340\) 0 0
\(341\) −30.8444 −1.67032
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −2.60555 −0.140278
\(346\) 0 0
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) 0 0
\(349\) 19.2111 1.02835 0.514173 0.857686i \(-0.328099\pi\)
0.514173 + 0.857686i \(0.328099\pi\)
\(350\) 0 0
\(351\) −11.8167 −0.630726
\(352\) 0 0
\(353\) 4.60555 0.245129 0.122564 0.992461i \(-0.460888\pi\)
0.122564 + 0.992461i \(0.460888\pi\)
\(354\) 0 0
\(355\) 6.60555 0.350586
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −17.4861 −0.922882 −0.461441 0.887171i \(-0.652667\pi\)
−0.461441 + 0.887171i \(0.652667\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −3.09167 −0.162271
\(364\) 0 0
\(365\) 44.6333 2.33621
\(366\) 0 0
\(367\) −26.7250 −1.39503 −0.697516 0.716569i \(-0.745712\pi\)
−0.697516 + 0.716569i \(0.745712\pi\)
\(368\) 0 0
\(369\) 27.6695 1.44041
\(370\) 0 0
\(371\) −0.697224 −0.0361981
\(372\) 0 0
\(373\) 12.3028 0.637014 0.318507 0.947921i \(-0.396819\pi\)
0.318507 + 0.947921i \(0.396819\pi\)
\(374\) 0 0
\(375\) −0.908327 −0.0469058
\(376\) 0 0
\(377\) −56.8444 −2.92764
\(378\) 0 0
\(379\) 28.2389 1.45053 0.725266 0.688468i \(-0.241717\pi\)
0.725266 + 0.688468i \(0.241717\pi\)
\(380\) 0 0
\(381\) 1.55004 0.0794109
\(382\) 0 0
\(383\) −3.02776 −0.154711 −0.0773556 0.997004i \(-0.524648\pi\)
−0.0773556 + 0.997004i \(0.524648\pi\)
\(384\) 0 0
\(385\) 15.2111 0.775230
\(386\) 0 0
\(387\) 12.5139 0.636116
\(388\) 0 0
\(389\) −14.0917 −0.714476 −0.357238 0.934013i \(-0.616282\pi\)
−0.357238 + 0.934013i \(0.616282\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −3.15559 −0.159178
\(394\) 0 0
\(395\) 19.8167 0.997084
\(396\) 0 0
\(397\) 6.09167 0.305732 0.152866 0.988247i \(-0.451150\pi\)
0.152866 + 0.988247i \(0.451150\pi\)
\(398\) 0 0
\(399\) −1.81665 −0.0909464
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) −44.2389 −2.20369
\(404\) 0 0
\(405\) 27.0278 1.34302
\(406\) 0 0
\(407\) 33.2111 1.64621
\(408\) 0 0
\(409\) −3.02776 −0.149713 −0.0748565 0.997194i \(-0.523850\pi\)
−0.0748565 + 0.997194i \(0.523850\pi\)
\(410\) 0 0
\(411\) 0.936083 0.0461736
\(412\) 0 0
\(413\) −5.21110 −0.256422
\(414\) 0 0
\(415\) 30.4222 1.49337
\(416\) 0 0
\(417\) −0.449961 −0.0220347
\(418\) 0 0
\(419\) 1.30278 0.0636448 0.0318224 0.999494i \(-0.489869\pi\)
0.0318224 + 0.999494i \(0.489869\pi\)
\(420\) 0 0
\(421\) −0.908327 −0.0442691 −0.0221346 0.999755i \(-0.507046\pi\)
−0.0221346 + 0.999755i \(0.507046\pi\)
\(422\) 0 0
\(423\) −29.0833 −1.41408
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.30278 −0.208226
\(428\) 0 0
\(429\) −9.21110 −0.444716
\(430\) 0 0
\(431\) −18.8444 −0.907703 −0.453852 0.891077i \(-0.649950\pi\)
−0.453852 + 0.891077i \(0.649950\pi\)
\(432\) 0 0
\(433\) 11.6333 0.559061 0.279531 0.960137i \(-0.409821\pi\)
0.279531 + 0.960137i \(0.409821\pi\)
\(434\) 0 0
\(435\) −8.60555 −0.412605
\(436\) 0 0
\(437\) −15.6333 −0.747843
\(438\) 0 0
\(439\) 0.302776 0.0144507 0.00722535 0.999974i \(-0.497700\pi\)
0.00722535 + 0.999974i \(0.497700\pi\)
\(440\) 0 0
\(441\) −2.90833 −0.138492
\(442\) 0 0
\(443\) −10.4222 −0.495174 −0.247587 0.968866i \(-0.579638\pi\)
−0.247587 + 0.968866i \(0.579638\pi\)
\(444\) 0 0
\(445\) −6.60555 −0.313133
\(446\) 0 0
\(447\) −1.48612 −0.0702911
\(448\) 0 0
\(449\) 32.2389 1.52145 0.760723 0.649077i \(-0.224845\pi\)
0.760723 + 0.649077i \(0.224845\pi\)
\(450\) 0 0
\(451\) 43.8167 2.06325
\(452\) 0 0
\(453\) −4.27502 −0.200858
\(454\) 0 0
\(455\) 21.8167 1.02278
\(456\) 0 0
\(457\) 6.09167 0.284956 0.142478 0.989798i \(-0.454493\pi\)
0.142478 + 0.989798i \(0.454493\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.21110 0.429004 0.214502 0.976724i \(-0.431187\pi\)
0.214502 + 0.976724i \(0.431187\pi\)
\(462\) 0 0
\(463\) 36.5416 1.69823 0.849117 0.528205i \(-0.177135\pi\)
0.849117 + 0.528205i \(0.177135\pi\)
\(464\) 0 0
\(465\) −6.69722 −0.310576
\(466\) 0 0
\(467\) 17.2111 0.796435 0.398217 0.917291i \(-0.369629\pi\)
0.398217 + 0.917291i \(0.369629\pi\)
\(468\) 0 0
\(469\) −2.69722 −0.124546
\(470\) 0 0
\(471\) 0.0555128 0.00255789
\(472\) 0 0
\(473\) 19.8167 0.911171
\(474\) 0 0
\(475\) −35.4500 −1.62656
\(476\) 0 0
\(477\) −2.02776 −0.0928446
\(478\) 0 0
\(479\) 24.3028 1.11042 0.555211 0.831709i \(-0.312637\pi\)
0.555211 + 0.831709i \(0.312637\pi\)
\(480\) 0 0
\(481\) 47.6333 2.17189
\(482\) 0 0
\(483\) 0.788897 0.0358961
\(484\) 0 0
\(485\) 6.90833 0.313691
\(486\) 0 0
\(487\) −13.6333 −0.617784 −0.308892 0.951097i \(-0.599958\pi\)
−0.308892 + 0.951097i \(0.599958\pi\)
\(488\) 0 0
\(489\) −6.60555 −0.298713
\(490\) 0 0
\(491\) 30.6972 1.38535 0.692673 0.721252i \(-0.256433\pi\)
0.692673 + 0.721252i \(0.256433\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 44.2389 1.98839
\(496\) 0 0
\(497\) −2.00000 −0.0897123
\(498\) 0 0
\(499\) 6.00000 0.268597 0.134298 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) 0 0
\(501\) −0.641712 −0.0286696
\(502\) 0 0
\(503\) −10.8806 −0.485141 −0.242570 0.970134i \(-0.577991\pi\)
−0.242570 + 0.970134i \(0.577991\pi\)
\(504\) 0 0
\(505\) −41.0278 −1.82571
\(506\) 0 0
\(507\) −9.27502 −0.411918
\(508\) 0 0
\(509\) 3.39445 0.150456 0.0752281 0.997166i \(-0.476031\pi\)
0.0752281 + 0.997166i \(0.476031\pi\)
\(510\) 0 0
\(511\) −13.5139 −0.597819
\(512\) 0 0
\(513\) −10.7334 −0.473891
\(514\) 0 0
\(515\) 43.6333 1.92271
\(516\) 0 0
\(517\) −46.0555 −2.02552
\(518\) 0 0
\(519\) 1.48612 0.0652335
\(520\) 0 0
\(521\) −7.33053 −0.321156 −0.160578 0.987023i \(-0.551336\pi\)
−0.160578 + 0.987023i \(0.551336\pi\)
\(522\) 0 0
\(523\) 17.0278 0.744572 0.372286 0.928118i \(-0.378574\pi\)
0.372286 + 0.928118i \(0.378574\pi\)
\(524\) 0 0
\(525\) 1.78890 0.0780739
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −16.2111 −0.704831
\(530\) 0 0
\(531\) −15.1556 −0.657697
\(532\) 0 0
\(533\) 62.8444 2.72209
\(534\) 0 0
\(535\) 31.0278 1.34145
\(536\) 0 0
\(537\) −3.35829 −0.144921
\(538\) 0 0
\(539\) −4.60555 −0.198375
\(540\) 0 0
\(541\) −3.81665 −0.164091 −0.0820454 0.996629i \(-0.526145\pi\)
−0.0820454 + 0.996629i \(0.526145\pi\)
\(542\) 0 0
\(543\) −6.55004 −0.281089
\(544\) 0 0
\(545\) 6.60555 0.282951
\(546\) 0 0
\(547\) 39.4500 1.68676 0.843379 0.537319i \(-0.180563\pi\)
0.843379 + 0.537319i \(0.180563\pi\)
\(548\) 0 0
\(549\) −12.5139 −0.534079
\(550\) 0 0
\(551\) −51.6333 −2.19965
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) 0 0
\(555\) 7.21110 0.306094
\(556\) 0 0
\(557\) 11.2111 0.475030 0.237515 0.971384i \(-0.423667\pi\)
0.237515 + 0.971384i \(0.423667\pi\)
\(558\) 0 0
\(559\) 28.4222 1.20213
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 32.2389 1.35871 0.679353 0.733812i \(-0.262261\pi\)
0.679353 + 0.733812i \(0.262261\pi\)
\(564\) 0 0
\(565\) −63.4500 −2.66936
\(566\) 0 0
\(567\) −8.18335 −0.343668
\(568\) 0 0
\(569\) −41.5139 −1.74035 −0.870176 0.492741i \(-0.835995\pi\)
−0.870176 + 0.492741i \(0.835995\pi\)
\(570\) 0 0
\(571\) 33.6333 1.40751 0.703755 0.710443i \(-0.251505\pi\)
0.703755 + 0.710443i \(0.251505\pi\)
\(572\) 0 0
\(573\) 7.18335 0.300089
\(574\) 0 0
\(575\) 15.3944 0.641993
\(576\) 0 0
\(577\) 39.0278 1.62475 0.812373 0.583138i \(-0.198175\pi\)
0.812373 + 0.583138i \(0.198175\pi\)
\(578\) 0 0
\(579\) −1.63331 −0.0678779
\(580\) 0 0
\(581\) −9.21110 −0.382141
\(582\) 0 0
\(583\) −3.21110 −0.132990
\(584\) 0 0
\(585\) 63.4500 2.62333
\(586\) 0 0
\(587\) −17.2111 −0.710378 −0.355189 0.934794i \(-0.615584\pi\)
−0.355189 + 0.934794i \(0.615584\pi\)
\(588\) 0 0
\(589\) −40.1833 −1.65573
\(590\) 0 0
\(591\) −6.97224 −0.286800
\(592\) 0 0
\(593\) 9.63331 0.395593 0.197796 0.980243i \(-0.436622\pi\)
0.197796 + 0.980243i \(0.436622\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.09167 0.167461
\(598\) 0 0
\(599\) −32.5416 −1.32962 −0.664808 0.747015i \(-0.731486\pi\)
−0.664808 + 0.747015i \(0.731486\pi\)
\(600\) 0 0
\(601\) −19.2111 −0.783637 −0.391819 0.920042i \(-0.628154\pi\)
−0.391819 + 0.920042i \(0.628154\pi\)
\(602\) 0 0
\(603\) −7.84441 −0.319449
\(604\) 0 0
\(605\) 33.7250 1.37112
\(606\) 0 0
\(607\) 16.0917 0.653141 0.326570 0.945173i \(-0.394107\pi\)
0.326570 + 0.945173i \(0.394107\pi\)
\(608\) 0 0
\(609\) 2.60555 0.105582
\(610\) 0 0
\(611\) −66.0555 −2.67232
\(612\) 0 0
\(613\) −30.9361 −1.24950 −0.624748 0.780826i \(-0.714798\pi\)
−0.624748 + 0.780826i \(0.714798\pi\)
\(614\) 0 0
\(615\) 9.51388 0.383637
\(616\) 0 0
\(617\) −4.60555 −0.185413 −0.0927063 0.995694i \(-0.529552\pi\)
−0.0927063 + 0.995694i \(0.529552\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) 4.66106 0.187042
\(622\) 0 0
\(623\) 2.00000 0.0801283
\(624\) 0 0
\(625\) −19.6333 −0.785332
\(626\) 0 0
\(627\) −8.36669 −0.334134
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 13.1194 0.522276 0.261138 0.965301i \(-0.415902\pi\)
0.261138 + 0.965301i \(0.415902\pi\)
\(632\) 0 0
\(633\) −8.18335 −0.325259
\(634\) 0 0
\(635\) −16.9083 −0.670986
\(636\) 0 0
\(637\) −6.60555 −0.261721
\(638\) 0 0
\(639\) −5.81665 −0.230103
\(640\) 0 0
\(641\) 7.57779 0.299305 0.149652 0.988739i \(-0.452185\pi\)
0.149652 + 0.988739i \(0.452185\pi\)
\(642\) 0 0
\(643\) −27.3305 −1.07781 −0.538905 0.842366i \(-0.681162\pi\)
−0.538905 + 0.842366i \(0.681162\pi\)
\(644\) 0 0
\(645\) 4.30278 0.169422
\(646\) 0 0
\(647\) −27.3944 −1.07699 −0.538493 0.842630i \(-0.681006\pi\)
−0.538493 + 0.842630i \(0.681006\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 2.02776 0.0794740
\(652\) 0 0
\(653\) −2.00000 −0.0782660 −0.0391330 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) 0 0
\(655\) 34.4222 1.34499
\(656\) 0 0
\(657\) −39.3028 −1.53335
\(658\) 0 0
\(659\) 7.54163 0.293780 0.146890 0.989153i \(-0.453074\pi\)
0.146890 + 0.989153i \(0.453074\pi\)
\(660\) 0 0
\(661\) 15.6333 0.608065 0.304033 0.952662i \(-0.401667\pi\)
0.304033 + 0.952662i \(0.401667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 19.8167 0.768457
\(666\) 0 0
\(667\) 22.4222 0.868191
\(668\) 0 0
\(669\) 6.55004 0.253239
\(670\) 0 0
\(671\) −19.8167 −0.765013
\(672\) 0 0
\(673\) 20.8444 0.803493 0.401746 0.915751i \(-0.368403\pi\)
0.401746 + 0.915751i \(0.368403\pi\)
\(674\) 0 0
\(675\) 10.5694 0.406816
\(676\) 0 0
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 0 0
\(679\) −2.09167 −0.0802710
\(680\) 0 0
\(681\) 0.0277564 0.00106363
\(682\) 0 0
\(683\) 11.0278 0.421965 0.210983 0.977490i \(-0.432334\pi\)
0.210983 + 0.977490i \(0.432334\pi\)
\(684\) 0 0
\(685\) −10.2111 −0.390146
\(686\) 0 0
\(687\) 1.76114 0.0671917
\(688\) 0 0
\(689\) −4.60555 −0.175458
\(690\) 0 0
\(691\) −4.30278 −0.163685 −0.0818426 0.996645i \(-0.526080\pi\)
−0.0818426 + 0.996645i \(0.526080\pi\)
\(692\) 0 0
\(693\) −13.3944 −0.508813
\(694\) 0 0
\(695\) 4.90833 0.186183
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 4.00000 0.151294
\(700\) 0 0
\(701\) −11.2111 −0.423437 −0.211719 0.977331i \(-0.567906\pi\)
−0.211719 + 0.977331i \(0.567906\pi\)
\(702\) 0 0
\(703\) 43.2666 1.63183
\(704\) 0 0
\(705\) −10.0000 −0.376622
\(706\) 0 0
\(707\) 12.4222 0.467185
\(708\) 0 0
\(709\) 24.1833 0.908225 0.454112 0.890944i \(-0.349956\pi\)
0.454112 + 0.890944i \(0.349956\pi\)
\(710\) 0 0
\(711\) −17.4500 −0.654425
\(712\) 0 0
\(713\) 17.4500 0.653506
\(714\) 0 0
\(715\) 100.478 3.75765
\(716\) 0 0
\(717\) 3.72498 0.139112
\(718\) 0 0
\(719\) 2.48612 0.0927167 0.0463583 0.998925i \(-0.485238\pi\)
0.0463583 + 0.998925i \(0.485238\pi\)
\(720\) 0 0
\(721\) −13.2111 −0.492007
\(722\) 0 0
\(723\) 4.21110 0.156613
\(724\) 0 0
\(725\) 50.8444 1.88831
\(726\) 0 0
\(727\) 49.2666 1.82720 0.913599 0.406617i \(-0.133292\pi\)
0.913599 + 0.406617i \(0.133292\pi\)
\(728\) 0 0
\(729\) −22.1749 −0.821294
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −15.6333 −0.577429 −0.288715 0.957415i \(-0.593228\pi\)
−0.288715 + 0.957415i \(0.593228\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) −12.4222 −0.457578
\(738\) 0 0
\(739\) −24.9083 −0.916268 −0.458134 0.888883i \(-0.651482\pi\)
−0.458134 + 0.888883i \(0.651482\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) 33.3944 1.22512 0.612562 0.790423i \(-0.290139\pi\)
0.612562 + 0.790423i \(0.290139\pi\)
\(744\) 0 0
\(745\) 16.2111 0.593929
\(746\) 0 0
\(747\) −26.7889 −0.980155
\(748\) 0 0
\(749\) −9.39445 −0.343266
\(750\) 0 0
\(751\) 51.2111 1.86872 0.934360 0.356331i \(-0.115972\pi\)
0.934360 + 0.356331i \(0.115972\pi\)
\(752\) 0 0
\(753\) −2.66106 −0.0969746
\(754\) 0 0
\(755\) 46.6333 1.69716
\(756\) 0 0
\(757\) 23.0917 0.839281 0.419641 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(758\) 0 0
\(759\) 3.63331 0.131881
\(760\) 0 0
\(761\) 46.6056 1.68945 0.844725 0.535201i \(-0.179764\pi\)
0.844725 + 0.535201i \(0.179764\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −34.4222 −1.24291
\(768\) 0 0
\(769\) 11.8167 0.426119 0.213060 0.977039i \(-0.431657\pi\)
0.213060 + 0.977039i \(0.431657\pi\)
\(770\) 0 0
\(771\) 4.18335 0.150660
\(772\) 0 0
\(773\) −6.84441 −0.246176 −0.123088 0.992396i \(-0.539280\pi\)
−0.123088 + 0.992396i \(0.539280\pi\)
\(774\) 0 0
\(775\) 39.5694 1.42137
\(776\) 0 0
\(777\) −2.18335 −0.0783271
\(778\) 0 0
\(779\) 57.0833 2.04522
\(780\) 0 0
\(781\) −9.21110 −0.329599
\(782\) 0 0
\(783\) 15.3944 0.550153
\(784\) 0 0
\(785\) −0.605551 −0.0216131
\(786\) 0 0
\(787\) 14.4222 0.514096 0.257048 0.966399i \(-0.417250\pi\)
0.257048 + 0.966399i \(0.417250\pi\)
\(788\) 0 0
\(789\) −6.78890 −0.241691
\(790\) 0 0
\(791\) 19.2111 0.683068
\(792\) 0 0
\(793\) −28.4222 −1.00930
\(794\) 0 0
\(795\) −0.697224 −0.0247280
\(796\) 0 0
\(797\) 14.7889 0.523850 0.261925 0.965088i \(-0.415643\pi\)
0.261925 + 0.965088i \(0.415643\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 5.81665 0.205521
\(802\) 0 0
\(803\) −62.2389 −2.19636
\(804\) 0 0
\(805\) −8.60555 −0.303306
\(806\) 0 0
\(807\) −1.81665 −0.0639492
\(808\) 0 0
\(809\) −45.4500 −1.59794 −0.798968 0.601374i \(-0.794620\pi\)
−0.798968 + 0.601374i \(0.794620\pi\)
\(810\) 0 0
\(811\) 38.3583 1.34694 0.673471 0.739214i \(-0.264803\pi\)
0.673471 + 0.739214i \(0.264803\pi\)
\(812\) 0 0
\(813\) 9.63331 0.337855
\(814\) 0 0
\(815\) 72.0555 2.52399
\(816\) 0 0
\(817\) 25.8167 0.903210
\(818\) 0 0
\(819\) −19.2111 −0.671290
\(820\) 0 0
\(821\) −40.4222 −1.41074 −0.705372 0.708837i \(-0.749220\pi\)
−0.705372 + 0.708837i \(0.749220\pi\)
\(822\) 0 0
\(823\) 5.02776 0.175257 0.0876283 0.996153i \(-0.472071\pi\)
0.0876283 + 0.996153i \(0.472071\pi\)
\(824\) 0 0
\(825\) 8.23886 0.286840
\(826\) 0 0
\(827\) −18.8444 −0.655284 −0.327642 0.944802i \(-0.606254\pi\)
−0.327642 + 0.944802i \(0.606254\pi\)
\(828\) 0 0
\(829\) −29.3944 −1.02091 −0.510456 0.859904i \(-0.670523\pi\)
−0.510456 + 0.859904i \(0.670523\pi\)
\(830\) 0 0
\(831\) 7.57779 0.262871
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 7.00000 0.242245
\(836\) 0 0
\(837\) 11.9806 0.414112
\(838\) 0 0
\(839\) −21.2111 −0.732289 −0.366144 0.930558i \(-0.619322\pi\)
−0.366144 + 0.930558i \(0.619322\pi\)
\(840\) 0 0
\(841\) 45.0555 1.55364
\(842\) 0 0
\(843\) 5.05551 0.174121
\(844\) 0 0
\(845\) 101.175 3.48052
\(846\) 0 0
\(847\) −10.2111 −0.350858
\(848\) 0 0
\(849\) 2.02776 0.0695924
\(850\) 0 0
\(851\) −18.7889 −0.644075
\(852\) 0 0
\(853\) 19.2111 0.657776 0.328888 0.944369i \(-0.393326\pi\)
0.328888 + 0.944369i \(0.393326\pi\)
\(854\) 0 0
\(855\) 57.6333 1.97102
\(856\) 0 0
\(857\) −12.5139 −0.427466 −0.213733 0.976892i \(-0.568562\pi\)
−0.213733 + 0.976892i \(0.568562\pi\)
\(858\) 0 0
\(859\) −45.6333 −1.55699 −0.778494 0.627652i \(-0.784016\pi\)
−0.778494 + 0.627652i \(0.784016\pi\)
\(860\) 0 0
\(861\) −2.88057 −0.0981696
\(862\) 0 0
\(863\) 8.93608 0.304188 0.152094 0.988366i \(-0.451398\pi\)
0.152094 + 0.988366i \(0.451398\pi\)
\(864\) 0 0
\(865\) −16.2111 −0.551194
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −27.6333 −0.937396
\(870\) 0 0
\(871\) −17.8167 −0.603694
\(872\) 0 0
\(873\) −6.08327 −0.205887
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) 6.18335 0.208797 0.104398 0.994536i \(-0.466708\pi\)
0.104398 + 0.994536i \(0.466708\pi\)
\(878\) 0 0
\(879\) −0.550039 −0.0185523
\(880\) 0 0
\(881\) 7.11943 0.239860 0.119930 0.992782i \(-0.461733\pi\)
0.119930 + 0.992782i \(0.461733\pi\)
\(882\) 0 0
\(883\) 55.7250 1.87529 0.937647 0.347588i \(-0.112999\pi\)
0.937647 + 0.347588i \(0.112999\pi\)
\(884\) 0 0
\(885\) −5.21110 −0.175169
\(886\) 0 0
\(887\) 12.2750 0.412155 0.206077 0.978536i \(-0.433930\pi\)
0.206077 + 0.978536i \(0.433930\pi\)
\(888\) 0 0
\(889\) 5.11943 0.171700
\(890\) 0 0
\(891\) −37.6888 −1.26262
\(892\) 0 0
\(893\) −60.0000 −2.00782
\(894\) 0 0
\(895\) 36.6333 1.22452
\(896\) 0 0
\(897\) 5.21110 0.173994
\(898\) 0 0
\(899\) 57.6333 1.92218
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −1.30278 −0.0433537
\(904\) 0 0
\(905\) 71.4500 2.37508
\(906\) 0 0
\(907\) 3.57779 0.118799 0.0593994 0.998234i \(-0.481081\pi\)
0.0593994 + 0.998234i \(0.481081\pi\)
\(908\) 0 0
\(909\) 36.1278 1.19828
\(910\) 0 0
\(911\) 10.4222 0.345303 0.172652 0.984983i \(-0.444767\pi\)
0.172652 + 0.984983i \(0.444767\pi\)
\(912\) 0 0
\(913\) −42.4222 −1.40397
\(914\) 0 0
\(915\) −4.30278 −0.142245
\(916\) 0 0
\(917\) −10.4222 −0.344172
\(918\) 0 0
\(919\) −10.4861 −0.345905 −0.172953 0.984930i \(-0.555331\pi\)
−0.172953 + 0.984930i \(0.555331\pi\)
\(920\) 0 0
\(921\) 7.87217 0.259397
\(922\) 0 0
\(923\) −13.2111 −0.434849
\(924\) 0 0
\(925\) −42.6056 −1.40086
\(926\) 0 0
\(927\) −38.4222 −1.26195
\(928\) 0 0
\(929\) 24.6972 0.810290 0.405145 0.914253i \(-0.367221\pi\)
0.405145 + 0.914253i \(0.367221\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) −4.76114 −0.155873
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −25.2111 −0.823611 −0.411805 0.911272i \(-0.635102\pi\)
−0.411805 + 0.911272i \(0.635102\pi\)
\(938\) 0 0
\(939\) −1.73338 −0.0565669
\(940\) 0 0
\(941\) 0.669468 0.0218240 0.0109120 0.999940i \(-0.496527\pi\)
0.0109120 + 0.999940i \(0.496527\pi\)
\(942\) 0 0
\(943\) −24.7889 −0.807238
\(944\) 0 0
\(945\) −5.90833 −0.192198
\(946\) 0 0
\(947\) −20.2389 −0.657675 −0.328837 0.944387i \(-0.606657\pi\)
−0.328837 + 0.944387i \(0.606657\pi\)
\(948\) 0 0
\(949\) −89.2666 −2.89772
\(950\) 0 0
\(951\) −3.81665 −0.123763
\(952\) 0 0
\(953\) 26.7527 0.866606 0.433303 0.901248i \(-0.357348\pi\)
0.433303 + 0.901248i \(0.357348\pi\)
\(954\) 0 0
\(955\) −78.3583 −2.53561
\(956\) 0 0
\(957\) 12.0000 0.387905
\(958\) 0 0
\(959\) 3.09167 0.0998353
\(960\) 0 0
\(961\) 13.8528 0.446865
\(962\) 0 0
\(963\) −27.3221 −0.880443
\(964\) 0 0
\(965\) 17.8167 0.573538
\(966\) 0 0
\(967\) −20.3028 −0.652893 −0.326447 0.945216i \(-0.605851\pi\)
−0.326447 + 0.945216i \(0.605851\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −20.2389 −0.649496 −0.324748 0.945801i \(-0.605279\pi\)
−0.324748 + 0.945801i \(0.605279\pi\)
\(972\) 0 0
\(973\) −1.48612 −0.0476429
\(974\) 0 0
\(975\) 11.8167 0.378436
\(976\) 0 0
\(977\) 12.5139 0.400354 0.200177 0.979760i \(-0.435848\pi\)
0.200177 + 0.979760i \(0.435848\pi\)
\(978\) 0 0
\(979\) 9.21110 0.294388
\(980\) 0 0
\(981\) −5.81665 −0.185711
\(982\) 0 0
\(983\) −14.5139 −0.462921 −0.231460 0.972844i \(-0.574350\pi\)
−0.231460 + 0.972844i \(0.574350\pi\)
\(984\) 0 0
\(985\) 76.0555 2.42333
\(986\) 0 0
\(987\) 3.02776 0.0963745
\(988\) 0 0
\(989\) −11.2111 −0.356492
\(990\) 0 0
\(991\) −6.00000 −0.190596 −0.0952981 0.995449i \(-0.530380\pi\)
−0.0952981 + 0.995449i \(0.530380\pi\)
\(992\) 0 0
\(993\) 0.0832691 0.00264247
\(994\) 0 0
\(995\) −44.6333 −1.41497
\(996\) 0 0
\(997\) −32.9083 −1.04222 −0.521109 0.853490i \(-0.674481\pi\)
−0.521109 + 0.853490i \(0.674481\pi\)
\(998\) 0 0
\(999\) −12.8999 −0.408136
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 8092.2.a.n.1.1 2
17.16 even 2 476.2.a.a.1.2 2
51.50 odd 2 4284.2.a.p.1.2 2
68.67 odd 2 1904.2.a.l.1.1 2
119.118 odd 2 3332.2.a.n.1.1 2
136.67 odd 2 7616.2.a.m.1.2 2
136.101 even 2 7616.2.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.a.1.2 2 17.16 even 2
1904.2.a.l.1.1 2 68.67 odd 2
3332.2.a.n.1.1 2 119.118 odd 2
4284.2.a.p.1.2 2 51.50 odd 2
7616.2.a.m.1.2 2 136.67 odd 2
7616.2.a.z.1.1 2 136.101 even 2
8092.2.a.n.1.1 2 1.1 even 1 trivial