Properties

Label 7920.2.a.cm.1.1
Level $7920$
Weight $2$
Character 7920.1
Self dual yes
Analytic conductor $63.242$
Analytic rank $0$
Dimension $4$
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] = [7920,2,Mod(1,7920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7920.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7920.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: \(63.2415184009\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.48704.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 495)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.852061\) of defining polynomial
Character \(\chi\) \(=\) 7920.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} -4.90749 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -4.90749 q^{7} -1.00000 q^{11} -4.61162 q^{13} -3.76776 q^{17} -4.84386 q^{19} -0.860262 q^{23} +1.00000 q^{25} +10.3794 q^{29} -7.81499 q^{31} +4.90749 q^{35} -4.67525 q^{37} -2.97113 q^{41} +0.907494 q^{43} -13.2232 q^{47} +17.0835 q^{49} -5.13974 q^{53} +1.00000 q^{55} -12.5480 q^{59} -11.2232 q^{61} +4.61162 q^{65} -4.26851 q^{67} -5.13974 q^{71} -4.61162 q^{73} +4.90749 q^{77} +0.843861 q^{79} -5.75135 q^{83} +3.76776 q^{85} -1.40825 q^{89} +22.6315 q^{91} +4.84386 q^{95} +8.54798 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 4 q^{7} - 4 q^{11} + 8 q^{13} - 4 q^{17} - 4 q^{19} - 8 q^{23} + 4 q^{25} + 4 q^{29} + 4 q^{35} + 8 q^{37} + 4 q^{41} - 12 q^{43} + 20 q^{49} - 16 q^{53} + 4 q^{55} - 24 q^{59} + 8 q^{61} - 8 q^{65} - 16 q^{71} + 8 q^{73} + 4 q^{77} - 12 q^{79} + 8 q^{83} + 4 q^{85} + 16 q^{89} + 16 q^{91} + 4 q^{95} + 8 q^{97}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.90749 −1.85486 −0.927429 0.373999i \(-0.877986\pi\)
−0.927429 + 0.373999i \(0.877986\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.61162 −1.27903 −0.639516 0.768778i \(-0.720865\pi\)
−0.639516 + 0.768778i \(0.720865\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.76776 −0.913815 −0.456907 0.889514i \(-0.651043\pi\)
−0.456907 + 0.889514i \(0.651043\pi\)
\(18\) 0 0
\(19\) −4.84386 −1.11126 −0.555629 0.831430i \(-0.687522\pi\)
−0.555629 + 0.831430i \(0.687522\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.860262 −0.179377 −0.0896885 0.995970i \(-0.528587\pi\)
−0.0896885 + 0.995970i \(0.528587\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.3794 1.92740 0.963700 0.266986i \(-0.0860277\pi\)
0.963700 + 0.266986i \(0.0860277\pi\)
\(30\) 0 0
\(31\) −7.81499 −1.40361 −0.701807 0.712368i \(-0.747623\pi\)
−0.701807 + 0.712368i \(0.747623\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.90749 0.829518
\(36\) 0 0
\(37\) −4.67525 −0.768606 −0.384303 0.923207i \(-0.625558\pi\)
−0.384303 + 0.923207i \(0.625558\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.97113 −0.464012 −0.232006 0.972714i \(-0.574529\pi\)
−0.232006 + 0.972714i \(0.574529\pi\)
\(42\) 0 0
\(43\) 0.907494 0.138391 0.0691957 0.997603i \(-0.477957\pi\)
0.0691957 + 0.997603i \(0.477957\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −13.2232 −1.92881 −0.964403 0.264436i \(-0.914814\pi\)
−0.964403 + 0.264436i \(0.914814\pi\)
\(48\) 0 0
\(49\) 17.0835 2.44050
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.13974 −0.705997 −0.352999 0.935624i \(-0.614838\pi\)
−0.352999 + 0.935624i \(0.614838\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.5480 −1.63361 −0.816804 0.576915i \(-0.804256\pi\)
−0.816804 + 0.576915i \(0.804256\pi\)
\(60\) 0 0
\(61\) −11.2232 −1.43699 −0.718494 0.695533i \(-0.755168\pi\)
−0.718494 + 0.695533i \(0.755168\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.61162 0.572001
\(66\) 0 0
\(67\) −4.26851 −0.521481 −0.260741 0.965409i \(-0.583967\pi\)
−0.260741 + 0.965409i \(0.583967\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.13974 −0.609975 −0.304987 0.952356i \(-0.598652\pi\)
−0.304987 + 0.952356i \(0.598652\pi\)
\(72\) 0 0
\(73\) −4.61162 −0.539749 −0.269874 0.962896i \(-0.586982\pi\)
−0.269874 + 0.962896i \(0.586982\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.90749 0.559261
\(78\) 0 0
\(79\) 0.843861 0.0949417 0.0474709 0.998873i \(-0.484884\pi\)
0.0474709 + 0.998873i \(0.484884\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.75135 −0.631293 −0.315647 0.948877i \(-0.602221\pi\)
−0.315647 + 0.948877i \(0.602221\pi\)
\(84\) 0 0
\(85\) 3.76776 0.408670
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.40825 −0.149274 −0.0746368 0.997211i \(-0.523780\pi\)
−0.0746368 + 0.997211i \(0.523780\pi\)
\(90\) 0 0
\(91\) 22.6315 2.37242
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.84386 0.496970
\(96\) 0 0
\(97\) 8.54798 0.867916 0.433958 0.900933i \(-0.357117\pi\)
0.433958 + 0.900933i \(0.357117\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.56438 0.454173 0.227087 0.973875i \(-0.427080\pi\)
0.227087 + 0.973875i \(0.427080\pi\)
\(102\) 0 0
\(103\) 4.73300 0.466356 0.233178 0.972434i \(-0.425088\pi\)
0.233178 + 0.972434i \(0.425088\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.34461 0.710030 0.355015 0.934861i \(-0.384476\pi\)
0.355015 + 0.934861i \(0.384476\pi\)
\(108\) 0 0
\(109\) 9.40825 0.901146 0.450573 0.892739i \(-0.351220\pi\)
0.450573 + 0.892739i \(0.351220\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −10.9547 −1.03053 −0.515267 0.857030i \(-0.672307\pi\)
−0.515267 + 0.857030i \(0.672307\pi\)
\(114\) 0 0
\(115\) 0.860262 0.0802198
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 18.4902 1.69500
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.62802 −0.588141 −0.294071 0.955784i \(-0.595010\pi\)
−0.294071 + 0.955784i \(0.595010\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.2232 1.15532 0.577660 0.816278i \(-0.303966\pi\)
0.577660 + 0.816278i \(0.303966\pi\)
\(132\) 0 0
\(133\) 23.7712 2.06123
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 12.9711 1.10020 0.550098 0.835100i \(-0.314590\pi\)
0.550098 + 0.835100i \(0.314590\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.61162 0.385643
\(144\) 0 0
\(145\) −10.3794 −0.861960
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.0273705 0.00224228 0.00112114 0.999999i \(-0.499643\pi\)
0.00112114 + 0.999999i \(0.499643\pi\)
\(150\) 0 0
\(151\) −4.84386 −0.394188 −0.197094 0.980385i \(-0.563150\pi\)
−0.197094 + 0.980385i \(0.563150\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.81499 0.627715
\(156\) 0 0
\(157\) 2.12727 0.169774 0.0848872 0.996391i \(-0.472947\pi\)
0.0848872 + 0.996391i \(0.472947\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.22173 0.332719
\(162\) 0 0
\(163\) −10.0835 −0.789800 −0.394900 0.918724i \(-0.629221\pi\)
−0.394900 + 0.918724i \(0.629221\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.7514 1.37364 0.686821 0.726827i \(-0.259006\pi\)
0.686821 + 0.726827i \(0.259006\pi\)
\(168\) 0 0
\(169\) 8.26700 0.635923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −15.7678 −1.19880 −0.599400 0.800450i \(-0.704594\pi\)
−0.599400 + 0.800450i \(0.704594\pi\)
\(174\) 0 0
\(175\) −4.90749 −0.370972
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.40825 0.553718 0.276859 0.960911i \(-0.410706\pi\)
0.276859 + 0.960911i \(0.410706\pi\)
\(180\) 0 0
\(181\) 7.26700 0.540152 0.270076 0.962839i \(-0.412951\pi\)
0.270076 + 0.962839i \(0.412951\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.67525 0.343731
\(186\) 0 0
\(187\) 3.76776 0.275526
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.87123 −0.207755 −0.103877 0.994590i \(-0.533125\pi\)
−0.103877 + 0.994590i \(0.533125\pi\)
\(192\) 0 0
\(193\) 18.8911 1.35981 0.679905 0.733300i \(-0.262021\pi\)
0.679905 + 0.733300i \(0.262021\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.0472 −1.00082 −0.500412 0.865787i \(-0.666818\pi\)
−0.500412 + 0.865787i \(0.666818\pi\)
\(198\) 0 0
\(199\) −3.40825 −0.241604 −0.120802 0.992677i \(-0.538547\pi\)
−0.120802 + 0.992677i \(0.538547\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −50.9367 −3.57506
\(204\) 0 0
\(205\) 2.97113 0.207512
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.84386 0.335057
\(210\) 0 0
\(211\) 24.4738 1.68485 0.842424 0.538815i \(-0.181128\pi\)
0.842424 + 0.538815i \(0.181128\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.907494 −0.0618906
\(216\) 0 0
\(217\) 38.3520 2.60350
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 17.3754 1.16880
\(222\) 0 0
\(223\) 11.5355 0.772475 0.386237 0.922399i \(-0.373775\pi\)
0.386237 + 0.922399i \(0.373775\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.15960 −0.0769653 −0.0384827 0.999259i \(-0.512252\pi\)
−0.0384827 + 0.999259i \(0.512252\pi\)
\(228\) 0 0
\(229\) −24.4465 −1.61547 −0.807734 0.589547i \(-0.799306\pi\)
−0.807734 + 0.589547i \(0.799306\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.45548 −0.619449 −0.309724 0.950826i \(-0.600237\pi\)
−0.309724 + 0.950826i \(0.600237\pi\)
\(234\) 0 0
\(235\) 13.2232 0.862589
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −14.9438 −0.966631 −0.483316 0.875446i \(-0.660568\pi\)
−0.483316 + 0.875446i \(0.660568\pi\)
\(240\) 0 0
\(241\) 7.81499 0.503408 0.251704 0.967804i \(-0.419009\pi\)
0.251704 + 0.967804i \(0.419009\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −17.0835 −1.09142
\(246\) 0 0
\(247\) 22.3380 1.42133
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.41921 0.215819 0.107909 0.994161i \(-0.465584\pi\)
0.107909 + 0.994161i \(0.465584\pi\)
\(252\) 0 0
\(253\) 0.860262 0.0540842
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.53551 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(258\) 0 0
\(259\) 22.9438 1.42566
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −0.190899 −0.0117713 −0.00588567 0.999983i \(-0.501873\pi\)
−0.00588567 + 0.999983i \(0.501873\pi\)
\(264\) 0 0
\(265\) 5.13974 0.315732
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.4465 −0.880817 −0.440408 0.897798i \(-0.645166\pi\)
−0.440408 + 0.897798i \(0.645166\pi\)
\(270\) 0 0
\(271\) 9.97263 0.605794 0.302897 0.953023i \(-0.402046\pi\)
0.302897 + 0.953023i \(0.402046\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 12.5788 0.755788 0.377894 0.925849i \(-0.376648\pi\)
0.377894 + 0.925849i \(0.376648\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.6917 0.995740 0.497870 0.867252i \(-0.334116\pi\)
0.497870 + 0.867252i \(0.334116\pi\)
\(282\) 0 0
\(283\) 21.5937 1.28361 0.641806 0.766867i \(-0.278185\pi\)
0.641806 + 0.766867i \(0.278185\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.5808 0.860677
\(288\) 0 0
\(289\) −2.80402 −0.164942
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −33.0855 −1.93287 −0.966436 0.256906i \(-0.917297\pi\)
−0.966436 + 0.256906i \(0.917297\pi\)
\(294\) 0 0
\(295\) 12.5480 0.730572
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.96720 0.229429
\(300\) 0 0
\(301\) −4.45352 −0.256697
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 11.2232 0.642640
\(306\) 0 0
\(307\) −7.72398 −0.440831 −0.220416 0.975406i \(-0.570741\pi\)
−0.220416 + 0.975406i \(0.570741\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.1780 −0.690549 −0.345274 0.938502i \(-0.612214\pi\)
−0.345274 + 0.938502i \(0.612214\pi\)
\(312\) 0 0
\(313\) −2.95473 −0.167011 −0.0835055 0.996507i \(-0.526612\pi\)
−0.0835055 + 0.996507i \(0.526612\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −27.4917 −1.54409 −0.772045 0.635568i \(-0.780766\pi\)
−0.772045 + 0.635568i \(0.780766\pi\)
\(318\) 0 0
\(319\) −10.3794 −0.581133
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 18.2505 1.01548
\(324\) 0 0
\(325\) −4.61162 −0.255806
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 64.8929 3.57766
\(330\) 0 0
\(331\) −16.5737 −0.910975 −0.455487 0.890242i \(-0.650535\pi\)
−0.455487 + 0.890242i \(0.650535\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.26851 0.233213
\(336\) 0 0
\(337\) 4.64442 0.252998 0.126499 0.991967i \(-0.459626\pi\)
0.126499 + 0.991967i \(0.459626\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7.81499 0.423205
\(342\) 0 0
\(343\) −49.4847 −2.67192
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 23.6936 1.27194 0.635970 0.771714i \(-0.280600\pi\)
0.635970 + 0.771714i \(0.280600\pi\)
\(348\) 0 0
\(349\) −21.7572 −1.16464 −0.582319 0.812960i \(-0.697855\pi\)
−0.582319 + 0.812960i \(0.697855\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.49024 −0.345440 −0.172720 0.984971i \(-0.555256\pi\)
−0.172720 + 0.984971i \(0.555256\pi\)
\(354\) 0 0
\(355\) 5.13974 0.272789
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 19.1655 1.01152 0.505758 0.862676i \(-0.331213\pi\)
0.505758 + 0.862676i \(0.331213\pi\)
\(360\) 0 0
\(361\) 4.46299 0.234894
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.61162 0.241383
\(366\) 0 0
\(367\) 10.1850 0.531653 0.265827 0.964021i \(-0.414355\pi\)
0.265827 + 0.964021i \(0.414355\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 25.2232 1.30952
\(372\) 0 0
\(373\) 19.0184 0.984733 0.492367 0.870388i \(-0.336132\pi\)
0.492367 + 0.870388i \(0.336132\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −47.8657 −2.46521
\(378\) 0 0
\(379\) −11.0710 −0.568680 −0.284340 0.958723i \(-0.591774\pi\)
−0.284340 + 0.958723i \(0.591774\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.7330 −0.650626 −0.325313 0.945606i \(-0.605470\pi\)
−0.325313 + 0.945606i \(0.605470\pi\)
\(384\) 0 0
\(385\) −4.90749 −0.250109
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 32.7587 1.66093 0.830467 0.557068i \(-0.188074\pi\)
0.830467 + 0.557068i \(0.188074\pi\)
\(390\) 0 0
\(391\) 3.24126 0.163917
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.843861 −0.0424592
\(396\) 0 0
\(397\) −15.0820 −0.756943 −0.378472 0.925613i \(-0.623550\pi\)
−0.378472 + 0.925613i \(0.623550\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.7259 −0.935129 −0.467564 0.883959i \(-0.654868\pi\)
−0.467564 + 0.883959i \(0.654868\pi\)
\(402\) 0 0
\(403\) 36.0397 1.79527
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.67525 0.231744
\(408\) 0 0
\(409\) 24.4793 1.21042 0.605211 0.796065i \(-0.293089\pi\)
0.605211 + 0.796065i \(0.293089\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 61.5791 3.03011
\(414\) 0 0
\(415\) 5.75135 0.282323
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.12877 0.445970 0.222985 0.974822i \(-0.428420\pi\)
0.222985 + 0.974822i \(0.428420\pi\)
\(420\) 0 0
\(421\) −13.5465 −0.660215 −0.330108 0.943943i \(-0.607085\pi\)
−0.330108 + 0.943943i \(0.607085\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.76776 −0.182763
\(426\) 0 0
\(427\) 55.0779 2.66541
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −26.4465 −1.27388 −0.636941 0.770913i \(-0.719800\pi\)
−0.636941 + 0.770913i \(0.719800\pi\)
\(432\) 0 0
\(433\) −16.1780 −0.777463 −0.388732 0.921351i \(-0.627087\pi\)
−0.388732 + 0.921351i \(0.627087\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.16699 0.199334
\(438\) 0 0
\(439\) 7.02887 0.335470 0.167735 0.985832i \(-0.446355\pi\)
0.167735 + 0.985832i \(0.446355\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −24.3630 −1.15752 −0.578760 0.815498i \(-0.696463\pi\)
−0.578760 + 0.815498i \(0.696463\pi\)
\(444\) 0 0
\(445\) 1.40825 0.0667572
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 38.3887 1.81168 0.905838 0.423625i \(-0.139242\pi\)
0.905838 + 0.423625i \(0.139242\pi\)
\(450\) 0 0
\(451\) 2.97113 0.139905
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −22.6315 −1.06098
\(456\) 0 0
\(457\) 21.9621 1.02734 0.513672 0.857987i \(-0.328285\pi\)
0.513672 + 0.857987i \(0.328285\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −27.5698 −1.28405 −0.642027 0.766682i \(-0.721906\pi\)
−0.642027 + 0.766682i \(0.721906\pi\)
\(462\) 0 0
\(463\) 24.8860 1.15655 0.578275 0.815842i \(-0.303726\pi\)
0.578275 + 0.815842i \(0.303726\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −29.0273 −1.34322 −0.671610 0.740904i \(-0.734397\pi\)
−0.671610 + 0.740904i \(0.734397\pi\)
\(468\) 0 0
\(469\) 20.9477 0.967274
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.907494 −0.0417266
\(474\) 0 0
\(475\) −4.84386 −0.222252
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 29.4450 1.34537 0.672687 0.739927i \(-0.265140\pi\)
0.672687 + 0.739927i \(0.265140\pi\)
\(480\) 0 0
\(481\) 21.5605 0.983072
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.54798 −0.388144
\(486\) 0 0
\(487\) −27.5285 −1.24743 −0.623717 0.781650i \(-0.714378\pi\)
−0.623717 + 0.781650i \(0.714378\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 22.2795 1.00546 0.502729 0.864444i \(-0.332329\pi\)
0.502729 + 0.864444i \(0.332329\pi\)
\(492\) 0 0
\(493\) −39.1069 −1.76129
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 25.2232 1.13142
\(498\) 0 0
\(499\) 35.0382 1.56853 0.784263 0.620428i \(-0.213041\pi\)
0.784263 + 0.620428i \(0.213041\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −33.9731 −1.51478 −0.757392 0.652960i \(-0.773527\pi\)
−0.757392 + 0.652960i \(0.773527\pi\)
\(504\) 0 0
\(505\) −4.56438 −0.203112
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −13.3505 −0.591750 −0.295875 0.955227i \(-0.595611\pi\)
−0.295875 + 0.955227i \(0.595611\pi\)
\(510\) 0 0
\(511\) 22.6315 1.00116
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.73300 −0.208561
\(516\) 0 0
\(517\) 13.2232 0.581557
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.46599 0.327091 0.163546 0.986536i \(-0.447707\pi\)
0.163546 + 0.986536i \(0.447707\pi\)
\(522\) 0 0
\(523\) 15.9785 0.698692 0.349346 0.936994i \(-0.386404\pi\)
0.349346 + 0.936994i \(0.386404\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 29.4450 1.28264
\(528\) 0 0
\(529\) −22.2599 −0.967824
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 13.7017 0.593486
\(534\) 0 0
\(535\) −7.34461 −0.317535
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −17.0835 −0.735838
\(540\) 0 0
\(541\) −5.46299 −0.234872 −0.117436 0.993080i \(-0.537468\pi\)
−0.117436 + 0.993080i \(0.537468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.40825 −0.403005
\(546\) 0 0
\(547\) −38.8895 −1.66279 −0.831397 0.555679i \(-0.812458\pi\)
−0.831397 + 0.555679i \(0.812458\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −50.2762 −2.14184
\(552\) 0 0
\(553\) −4.14124 −0.176103
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.58425 0.278983 0.139492 0.990223i \(-0.455453\pi\)
0.139492 + 0.990223i \(0.455453\pi\)
\(558\) 0 0
\(559\) −4.18501 −0.177007
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −25.0323 −1.05499 −0.527494 0.849559i \(-0.676868\pi\)
−0.527494 + 0.849559i \(0.676868\pi\)
\(564\) 0 0
\(565\) 10.9547 0.460869
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −34.5066 −1.44659 −0.723297 0.690537i \(-0.757374\pi\)
−0.723297 + 0.690537i \(0.757374\pi\)
\(570\) 0 0
\(571\) 18.6588 0.780848 0.390424 0.920635i \(-0.372328\pi\)
0.390424 + 0.920635i \(0.372328\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.860262 −0.0358754
\(576\) 0 0
\(577\) −37.6697 −1.56821 −0.784105 0.620628i \(-0.786878\pi\)
−0.784105 + 0.620628i \(0.786878\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 28.2247 1.17096
\(582\) 0 0
\(583\) 5.13974 0.212866
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −11.0242 −0.455019 −0.227510 0.973776i \(-0.573058\pi\)
−0.227510 + 0.973776i \(0.573058\pi\)
\(588\) 0 0
\(589\) 37.8547 1.55978
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −39.4703 −1.62085 −0.810425 0.585843i \(-0.800764\pi\)
−0.810425 + 0.585843i \(0.800764\pi\)
\(594\) 0 0
\(595\) −18.4902 −0.758026
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.44646 0.0999598 0.0499799 0.998750i \(-0.484084\pi\)
0.0499799 + 0.998750i \(0.484084\pi\)
\(600\) 0 0
\(601\) −37.6697 −1.53658 −0.768290 0.640103i \(-0.778892\pi\)
−0.768290 + 0.640103i \(0.778892\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −6.62802 −0.269023 −0.134511 0.990912i \(-0.542947\pi\)
−0.134511 + 0.990912i \(0.542947\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 60.9805 2.46701
\(612\) 0 0
\(613\) 15.6498 0.632091 0.316045 0.948744i \(-0.397645\pi\)
0.316045 + 0.948744i \(0.397645\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 27.7463 1.11702 0.558511 0.829497i \(-0.311373\pi\)
0.558511 + 0.829497i \(0.311373\pi\)
\(618\) 0 0
\(619\) −16.5737 −0.666154 −0.333077 0.942900i \(-0.608087\pi\)
−0.333077 + 0.942900i \(0.608087\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.91095 0.276882
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 17.6152 0.702364
\(630\) 0 0
\(631\) 37.8547 1.50697 0.753486 0.657464i \(-0.228371\pi\)
0.753486 + 0.657464i \(0.228371\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.62802 0.263025
\(636\) 0 0
\(637\) −78.7825 −3.12148
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.9423 −0.471691 −0.235845 0.971791i \(-0.575786\pi\)
−0.235845 + 0.971791i \(0.575786\pi\)
\(642\) 0 0
\(643\) −10.3380 −0.407692 −0.203846 0.979003i \(-0.565344\pi\)
−0.203846 + 0.979003i \(0.565344\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11.0125 0.432945 0.216472 0.976289i \(-0.430545\pi\)
0.216472 + 0.976289i \(0.430545\pi\)
\(648\) 0 0
\(649\) 12.5480 0.492551
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13.2810 −0.519725 −0.259862 0.965646i \(-0.583677\pi\)
−0.259862 + 0.965646i \(0.583677\pi\)
\(654\) 0 0
\(655\) −13.2232 −0.516674
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −10.2247 −0.398299 −0.199150 0.979969i \(-0.563818\pi\)
−0.199150 + 0.979969i \(0.563818\pi\)
\(660\) 0 0
\(661\) 17.0710 0.663986 0.331993 0.943282i \(-0.392279\pi\)
0.331993 + 0.943282i \(0.392279\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −23.7712 −0.921808
\(666\) 0 0
\(667\) −8.92898 −0.345731
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 11.2232 0.433268
\(672\) 0 0
\(673\) 27.8926 1.07518 0.537590 0.843206i \(-0.319335\pi\)
0.537590 + 0.843206i \(0.319335\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.4922 0.749146 0.374573 0.927197i \(-0.377789\pi\)
0.374573 + 0.927197i \(0.377789\pi\)
\(678\) 0 0
\(679\) −41.9492 −1.60986
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −26.9438 −1.03097 −0.515487 0.856897i \(-0.672389\pi\)
−0.515487 + 0.856897i \(0.672389\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 23.7025 0.902993
\(690\) 0 0
\(691\) −41.2849 −1.57055 −0.785276 0.619146i \(-0.787479\pi\)
−0.785276 + 0.619146i \(0.787479\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.9711 −0.492023
\(696\) 0 0
\(697\) 11.1945 0.424021
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −41.1631 −1.55471 −0.777354 0.629064i \(-0.783438\pi\)
−0.777354 + 0.629064i \(0.783438\pi\)
\(702\) 0 0
\(703\) 22.6463 0.854120
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −22.3997 −0.842427
\(708\) 0 0
\(709\) −19.8040 −0.743756 −0.371878 0.928282i \(-0.621286\pi\)
−0.371878 + 0.928282i \(0.621286\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.72294 0.251776
\(714\) 0 0
\(715\) −4.61162 −0.172465
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −42.6682 −1.59126 −0.795628 0.605786i \(-0.792859\pi\)
−0.795628 + 0.605786i \(0.792859\pi\)
\(720\) 0 0
\(721\) −23.2271 −0.865024
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 10.3794 0.385480
\(726\) 0 0
\(727\) 39.0054 1.44663 0.723315 0.690518i \(-0.242617\pi\)
0.723315 + 0.690518i \(0.242617\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.41921 −0.126464
\(732\) 0 0
\(733\) −12.2744 −0.453365 −0.226683 0.973969i \(-0.572788\pi\)
−0.226683 + 0.973969i \(0.572788\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.26851 0.157232
\(738\) 0 0
\(739\) −34.0343 −1.25197 −0.625986 0.779834i \(-0.715303\pi\)
−0.625986 + 0.779834i \(0.715303\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.2854 0.597452 0.298726 0.954339i \(-0.403438\pi\)
0.298726 + 0.954339i \(0.403438\pi\)
\(744\) 0 0
\(745\) −0.0273705 −0.00100278
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −36.0436 −1.31701
\(750\) 0 0
\(751\) −14.0577 −0.512974 −0.256487 0.966548i \(-0.582565\pi\)
−0.256487 + 0.966548i \(0.582565\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.84386 0.176286
\(756\) 0 0
\(757\) 38.2286 1.38944 0.694722 0.719278i \(-0.255527\pi\)
0.694722 + 0.719278i \(0.255527\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.52616 0.345323 0.172662 0.984981i \(-0.444763\pi\)
0.172662 + 0.984981i \(0.444763\pi\)
\(762\) 0 0
\(763\) −46.1709 −1.67150
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 57.8665 2.08944
\(768\) 0 0
\(769\) 2.24276 0.0808760 0.0404380 0.999182i \(-0.487125\pi\)
0.0404380 + 0.999182i \(0.487125\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17.7025 0.636715 0.318357 0.947971i \(-0.396869\pi\)
0.318357 + 0.947971i \(0.396869\pi\)
\(774\) 0 0
\(775\) −7.81499 −0.280723
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 14.3917 0.515637
\(780\) 0 0
\(781\) 5.13974 0.183914
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.12727 −0.0759254
\(786\) 0 0
\(787\) 17.4445 0.621830 0.310915 0.950438i \(-0.399365\pi\)
0.310915 + 0.950438i \(0.399365\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 53.7602 1.91149
\(792\) 0 0
\(793\) 51.7572 1.83795
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.2093 −0.397052 −0.198526 0.980096i \(-0.563615\pi\)
−0.198526 + 0.980096i \(0.563615\pi\)
\(798\) 0 0
\(799\) 49.8219 1.76257
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.61162 0.162740
\(804\) 0 0
\(805\) −4.22173 −0.148796
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −43.1381 −1.51666 −0.758328 0.651874i \(-0.773983\pi\)
−0.758328 + 0.651874i \(0.773983\pi\)
\(810\) 0 0
\(811\) −11.0289 −0.387276 −0.193638 0.981073i \(-0.562029\pi\)
−0.193638 + 0.981073i \(0.562029\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 10.0835 0.353209
\(816\) 0 0
\(817\) −4.39577 −0.153789
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −14.9164 −0.520585 −0.260293 0.965530i \(-0.583819\pi\)
−0.260293 + 0.965530i \(0.583819\pi\)
\(822\) 0 0
\(823\) 41.4589 1.44517 0.722584 0.691283i \(-0.242954\pi\)
0.722584 + 0.691283i \(0.242954\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 19.4171 0.675200 0.337600 0.941290i \(-0.390385\pi\)
0.337600 + 0.941290i \(0.390385\pi\)
\(828\) 0 0
\(829\) −17.8290 −0.619225 −0.309613 0.950863i \(-0.600199\pi\)
−0.309613 + 0.950863i \(0.600199\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −64.3664 −2.23016
\(834\) 0 0
\(835\) −17.7514 −0.614311
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12.2935 0.424417 0.212209 0.977224i \(-0.431934\pi\)
0.212209 + 0.977224i \(0.431934\pi\)
\(840\) 0 0
\(841\) 78.7314 2.71487
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8.26700 −0.284394
\(846\) 0 0
\(847\) −4.90749 −0.168623
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.02194 0.137870
\(852\) 0 0
\(853\) −18.0776 −0.618966 −0.309483 0.950905i \(-0.600156\pi\)
−0.309483 + 0.950905i \(0.600156\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 55.3649 1.89123 0.945615 0.325288i \(-0.105461\pi\)
0.945615 + 0.325288i \(0.105461\pi\)
\(858\) 0 0
\(859\) 0.943756 0.0322005 0.0161003 0.999870i \(-0.494875\pi\)
0.0161003 + 0.999870i \(0.494875\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −23.6370 −0.804614 −0.402307 0.915505i \(-0.631792\pi\)
−0.402307 + 0.915505i \(0.631792\pi\)
\(864\) 0 0
\(865\) 15.7678 0.536120
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −0.843861 −0.0286260
\(870\) 0 0
\(871\) 19.6847 0.666991
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.90749 0.165904
\(876\) 0 0
\(877\) −16.6841 −0.563383 −0.281692 0.959505i \(-0.590896\pi\)
−0.281692 + 0.959505i \(0.590896\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 16.2217 0.546524 0.273262 0.961940i \(-0.411897\pi\)
0.273262 + 0.961940i \(0.411897\pi\)
\(882\) 0 0
\(883\) −35.0710 −1.18023 −0.590117 0.807318i \(-0.700918\pi\)
−0.590117 + 0.807318i \(0.700918\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.1581 0.542536 0.271268 0.962504i \(-0.412557\pi\)
0.271268 + 0.962504i \(0.412557\pi\)
\(888\) 0 0
\(889\) 32.5270 1.09092
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 64.0515 2.14340
\(894\) 0 0
\(895\) −7.40825 −0.247630
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −81.1147 −2.70533
\(900\) 0 0
\(901\) 19.3653 0.645151
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.26700 −0.241563
\(906\) 0 0
\(907\) −41.9820 −1.39399 −0.696994 0.717077i \(-0.745480\pi\)
−0.696994 + 0.717077i \(0.745480\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −57.2302 −1.89612 −0.948060 0.318092i \(-0.896958\pi\)
−0.948060 + 0.318092i \(0.896958\pi\)
\(912\) 0 0
\(913\) 5.75135 0.190342
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −64.8929 −2.14295
\(918\) 0 0
\(919\) 0.346569 0.0114323 0.00571613 0.999984i \(-0.498180\pi\)
0.00571613 + 0.999984i \(0.498180\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 23.7025 0.780177
\(924\) 0 0
\(925\) −4.67525 −0.153721
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.66278 0.251408 0.125704 0.992068i \(-0.459881\pi\)
0.125704 + 0.992068i \(0.459881\pi\)
\(930\) 0 0
\(931\) −82.7501 −2.71202
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.76776 −0.123219
\(936\) 0 0
\(937\) 34.0894 1.11365 0.556826 0.830629i \(-0.312019\pi\)
0.556826 + 0.830629i \(0.312019\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 28.1944 0.919110 0.459555 0.888149i \(-0.348009\pi\)
0.459555 + 0.888149i \(0.348009\pi\)
\(942\) 0 0
\(943\) 2.55595 0.0832331
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 29.7602 0.967078 0.483539 0.875323i \(-0.339351\pi\)
0.483539 + 0.875323i \(0.339351\pi\)
\(948\) 0 0
\(949\) 21.2670 0.690356
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 20.8309 0.674780 0.337390 0.941365i \(-0.390456\pi\)
0.337390 + 0.941365i \(0.390456\pi\)
\(954\) 0 0
\(955\) 2.87123 0.0929109
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −29.4450 −0.950827
\(960\) 0 0
\(961\) 30.0740 0.970130
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −18.8911 −0.608126
\(966\) 0 0
\(967\) 30.7225 0.987968 0.493984 0.869471i \(-0.335540\pi\)
0.493984 + 0.869471i \(0.335540\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 21.9095 0.703108 0.351554 0.936168i \(-0.385653\pi\)
0.351554 + 0.936168i \(0.385653\pi\)
\(972\) 0 0
\(973\) −63.6557 −2.04071
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 14.1889 0.453944 0.226972 0.973901i \(-0.427117\pi\)
0.226972 + 0.973901i \(0.427117\pi\)
\(978\) 0 0
\(979\) 1.40825 0.0450077
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −5.52454 −0.176206 −0.0881028 0.996111i \(-0.528080\pi\)
−0.0881028 + 0.996111i \(0.528080\pi\)
\(984\) 0 0
\(985\) 14.0472 0.447582
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.780682 −0.0248243
\(990\) 0 0
\(991\) −7.93048 −0.251920 −0.125960 0.992035i \(-0.540201\pi\)
−0.125960 + 0.992035i \(0.540201\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.40825 0.108049
\(996\) 0 0
\(997\) −52.0596 −1.64874 −0.824372 0.566049i \(-0.808471\pi\)
−0.824372 + 0.566049i \(0.808471\pi\)
\(998\) 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 7920.2.a.cm.1.1 4
3.2 odd 2 7920.2.a.cn.1.1 4
4.3 odd 2 495.2.a.f.1.2 4
12.11 even 2 495.2.a.g.1.3 yes 4
20.3 even 4 2475.2.c.t.199.6 8
20.7 even 4 2475.2.c.t.199.3 8
20.19 odd 2 2475.2.a.bj.1.3 4
44.43 even 2 5445.2.a.bs.1.3 4
60.23 odd 4 2475.2.c.s.199.3 8
60.47 odd 4 2475.2.c.s.199.6 8
60.59 even 2 2475.2.a.bf.1.2 4
132.131 odd 2 5445.2.a.bh.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.a.f.1.2 4 4.3 odd 2
495.2.a.g.1.3 yes 4 12.11 even 2
2475.2.a.bf.1.2 4 60.59 even 2
2475.2.a.bj.1.3 4 20.19 odd 2
2475.2.c.s.199.3 8 60.23 odd 4
2475.2.c.s.199.6 8 60.47 odd 4
2475.2.c.t.199.3 8 20.7 even 4
2475.2.c.t.199.6 8 20.3 even 4
5445.2.a.bh.1.2 4 132.131 odd 2
5445.2.a.bs.1.3 4 44.43 even 2
7920.2.a.cm.1.1 4 1.1 even 1 trivial
7920.2.a.cn.1.1 4 3.2 odd 2