Properties

Label 1140.2.a.f.1.3
Level $1140$
Weight $2$
Character 1140.1
Self dual yes
Analytic conductor $9.103$
Analytic rank $0$
Dimension $3$
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] = [1140,2,Mod(1,1140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1140, 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("1140.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1140.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: \(9.10294583043\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.140435\) of defining polynomial
Character \(\chi\) \(=\) 1140.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +4.98028 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +4.98028 q^{7} +1.00000 q^{9} -5.26115 q^{11} +6.98028 q^{13} -1.00000 q^{15} +0.280871 q^{17} +1.00000 q^{19} -4.98028 q^{21} +0.280871 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.26115 q^{29} +2.28087 q^{31} +5.26115 q^{33} +4.98028 q^{35} -2.98028 q^{37} -6.98028 q^{39} +0.738851 q^{41} -5.54202 q^{43} +1.00000 q^{45} -4.28087 q^{47} +17.8032 q^{49} -0.280871 q^{51} +5.71913 q^{53} -5.26115 q^{55} -1.00000 q^{57} +10.5223 q^{59} -2.24143 q^{61} +4.98028 q^{63} +6.98028 q^{65} -0.280871 q^{69} +14.5223 q^{71} +5.43826 q^{73} -1.00000 q^{75} -26.2020 q^{77} -16.4829 q^{79} +1.00000 q^{81} +10.8032 q^{83} +0.280871 q^{85} +3.26115 q^{87} +16.6600 q^{89} +34.7637 q^{91} -2.28087 q^{93} +1.00000 q^{95} +2.98028 q^{97} -5.26115 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 2 q^{11} + 6 q^{13} - 3 q^{15} + 2 q^{17} + 3 q^{19} + 2 q^{23} + 3 q^{25} - 3 q^{27} + 4 q^{29} + 8 q^{31} + 2 q^{33} + 6 q^{37} - 6 q^{39} + 16 q^{41} - 4 q^{43} + 3 q^{45} - 14 q^{47} + 27 q^{49} - 2 q^{51} + 16 q^{53} - 2 q^{55} - 3 q^{57} + 4 q^{59} + 22 q^{61} + 6 q^{65} - 2 q^{69} + 16 q^{71} + 14 q^{73} - 3 q^{75} - 20 q^{77} + 8 q^{79} + 3 q^{81} + 6 q^{83} + 2 q^{85} - 4 q^{87} + 4 q^{89} + 48 q^{91} - 8 q^{93} + 3 q^{95} - 6 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.98028 1.88237 0.941184 0.337894i \(-0.109715\pi\)
0.941184 + 0.337894i \(0.109715\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.26115 −1.58630 −0.793148 0.609029i \(-0.791559\pi\)
−0.793148 + 0.609029i \(0.791559\pi\)
\(12\) 0 0
\(13\) 6.98028 1.93598 0.967990 0.250987i \(-0.0807552\pi\)
0.967990 + 0.250987i \(0.0807552\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0.280871 0.0681212 0.0340606 0.999420i \(-0.489156\pi\)
0.0340606 + 0.999420i \(0.489156\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −4.98028 −1.08679
\(22\) 0 0
\(23\) 0.280871 0.0585656 0.0292828 0.999571i \(-0.490678\pi\)
0.0292828 + 0.999571i \(0.490678\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.26115 −0.605580 −0.302790 0.953057i \(-0.597918\pi\)
−0.302790 + 0.953057i \(0.597918\pi\)
\(30\) 0 0
\(31\) 2.28087 0.409656 0.204828 0.978798i \(-0.434336\pi\)
0.204828 + 0.978798i \(0.434336\pi\)
\(32\) 0 0
\(33\) 5.26115 0.915848
\(34\) 0 0
\(35\) 4.98028 0.841821
\(36\) 0 0
\(37\) −2.98028 −0.489955 −0.244977 0.969529i \(-0.578781\pi\)
−0.244977 + 0.969529i \(0.578781\pi\)
\(38\) 0 0
\(39\) −6.98028 −1.11774
\(40\) 0 0
\(41\) 0.738851 0.115389 0.0576946 0.998334i \(-0.481625\pi\)
0.0576946 + 0.998334i \(0.481625\pi\)
\(42\) 0 0
\(43\) −5.54202 −0.845150 −0.422575 0.906328i \(-0.638874\pi\)
−0.422575 + 0.906328i \(0.638874\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −4.28087 −0.624429 −0.312215 0.950012i \(-0.601071\pi\)
−0.312215 + 0.950012i \(0.601071\pi\)
\(48\) 0 0
\(49\) 17.8032 2.54331
\(50\) 0 0
\(51\) −0.280871 −0.0393298
\(52\) 0 0
\(53\) 5.71913 0.785583 0.392791 0.919628i \(-0.371509\pi\)
0.392791 + 0.919628i \(0.371509\pi\)
\(54\) 0 0
\(55\) −5.26115 −0.709413
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 10.5223 1.36989 0.684943 0.728596i \(-0.259827\pi\)
0.684943 + 0.728596i \(0.259827\pi\)
\(60\) 0 0
\(61\) −2.24143 −0.286985 −0.143493 0.989651i \(-0.545833\pi\)
−0.143493 + 0.989651i \(0.545833\pi\)
\(62\) 0 0
\(63\) 4.98028 0.627456
\(64\) 0 0
\(65\) 6.98028 0.865797
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −0.280871 −0.0338129
\(70\) 0 0
\(71\) 14.5223 1.72348 0.861740 0.507350i \(-0.169375\pi\)
0.861740 + 0.507350i \(0.169375\pi\)
\(72\) 0 0
\(73\) 5.43826 0.636500 0.318250 0.948007i \(-0.396905\pi\)
0.318250 + 0.948007i \(0.396905\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −26.2020 −2.98599
\(78\) 0 0
\(79\) −16.4829 −1.85447 −0.927233 0.374485i \(-0.877819\pi\)
−0.927233 + 0.374485i \(0.877819\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.8032 1.18580 0.592901 0.805275i \(-0.297983\pi\)
0.592901 + 0.805275i \(0.297983\pi\)
\(84\) 0 0
\(85\) 0.280871 0.0304647
\(86\) 0 0
\(87\) 3.26115 0.349632
\(88\) 0 0
\(89\) 16.6600 1.76595 0.882976 0.469418i \(-0.155536\pi\)
0.882976 + 0.469418i \(0.155536\pi\)
\(90\) 0 0
\(91\) 34.7637 3.64423
\(92\) 0 0
\(93\) −2.28087 −0.236515
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 2.98028 0.302601 0.151301 0.988488i \(-0.451654\pi\)
0.151301 + 0.988488i \(0.451654\pi\)
\(98\) 0 0
\(99\) −5.26115 −0.528765
\(100\) 0 0
\(101\) 13.0840 1.30191 0.650955 0.759116i \(-0.274369\pi\)
0.650955 + 0.759116i \(0.274369\pi\)
\(102\) 0 0
\(103\) 4.56174 0.449482 0.224741 0.974419i \(-0.427846\pi\)
0.224741 + 0.974419i \(0.427846\pi\)
\(104\) 0 0
\(105\) −4.98028 −0.486025
\(106\) 0 0
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 0 0
\(109\) −4.52230 −0.433158 −0.216579 0.976265i \(-0.569490\pi\)
−0.216579 + 0.976265i \(0.569490\pi\)
\(110\) 0 0
\(111\) 2.98028 0.282875
\(112\) 0 0
\(113\) −0.241427 −0.0227115 −0.0113557 0.999936i \(-0.503615\pi\)
−0.0113557 + 0.999936i \(0.503615\pi\)
\(114\) 0 0
\(115\) 0.280871 0.0261913
\(116\) 0 0
\(117\) 6.98028 0.645327
\(118\) 0 0
\(119\) 1.39881 0.128229
\(120\) 0 0
\(121\) 16.6797 1.51634
\(122\) 0 0
\(123\) −0.738851 −0.0666200
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 5.54202 0.487948
\(130\) 0 0
\(131\) −15.7834 −1.37901 −0.689503 0.724283i \(-0.742171\pi\)
−0.689503 + 0.724283i \(0.742171\pi\)
\(132\) 0 0
\(133\) 4.98028 0.431845
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −10.5223 −0.892490 −0.446245 0.894911i \(-0.647239\pi\)
−0.446245 + 0.894911i \(0.647239\pi\)
\(140\) 0 0
\(141\) 4.28087 0.360514
\(142\) 0 0
\(143\) −36.7243 −3.07104
\(144\) 0 0
\(145\) −3.26115 −0.270824
\(146\) 0 0
\(147\) −17.8032 −1.46838
\(148\) 0 0
\(149\) −5.43826 −0.445519 −0.222760 0.974873i \(-0.571507\pi\)
−0.222760 + 0.974873i \(0.571507\pi\)
\(150\) 0 0
\(151\) −14.2020 −1.15574 −0.577870 0.816128i \(-0.696116\pi\)
−0.577870 + 0.816128i \(0.696116\pi\)
\(152\) 0 0
\(153\) 0.280871 0.0227071
\(154\) 0 0
\(155\) 2.28087 0.183204
\(156\) 0 0
\(157\) −15.3988 −1.22896 −0.614480 0.788933i \(-0.710634\pi\)
−0.614480 + 0.788933i \(0.710634\pi\)
\(158\) 0 0
\(159\) −5.71913 −0.453556
\(160\) 0 0
\(161\) 1.39881 0.110242
\(162\) 0 0
\(163\) −18.9408 −1.48356 −0.741780 0.670643i \(-0.766018\pi\)
−0.741780 + 0.670643i \(0.766018\pi\)
\(164\) 0 0
\(165\) 5.26115 0.409580
\(166\) 0 0
\(167\) 11.6797 0.903801 0.451901 0.892068i \(-0.350746\pi\)
0.451901 + 0.892068i \(0.350746\pi\)
\(168\) 0 0
\(169\) 35.7243 2.74802
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −16.2414 −1.23481 −0.617406 0.786644i \(-0.711817\pi\)
−0.617406 + 0.786644i \(0.711817\pi\)
\(174\) 0 0
\(175\) 4.98028 0.376474
\(176\) 0 0
\(177\) −10.5223 −0.790904
\(178\) 0 0
\(179\) −21.9606 −1.64141 −0.820705 0.571353i \(-0.806419\pi\)
−0.820705 + 0.571353i \(0.806419\pi\)
\(180\) 0 0
\(181\) 19.9606 1.48366 0.741828 0.670590i \(-0.233959\pi\)
0.741828 + 0.670590i \(0.233959\pi\)
\(182\) 0 0
\(183\) 2.24143 0.165691
\(184\) 0 0
\(185\) −2.98028 −0.219114
\(186\) 0 0
\(187\) −1.47770 −0.108060
\(188\) 0 0
\(189\) −4.98028 −0.362262
\(190\) 0 0
\(191\) −19.2217 −1.39083 −0.695417 0.718607i \(-0.744780\pi\)
−0.695417 + 0.718607i \(0.744780\pi\)
\(192\) 0 0
\(193\) −6.41854 −0.462016 −0.231008 0.972952i \(-0.574202\pi\)
−0.231008 + 0.972952i \(0.574202\pi\)
\(194\) 0 0
\(195\) −6.98028 −0.499868
\(196\) 0 0
\(197\) −16.2809 −1.15996 −0.579982 0.814629i \(-0.696940\pi\)
−0.579982 + 0.814629i \(0.696940\pi\)
\(198\) 0 0
\(199\) −25.3988 −1.80047 −0.900237 0.435400i \(-0.856607\pi\)
−0.900237 + 0.435400i \(0.856607\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −16.2414 −1.13992
\(204\) 0 0
\(205\) 0.738851 0.0516036
\(206\) 0 0
\(207\) 0.280871 0.0195219
\(208\) 0 0
\(209\) −5.26115 −0.363921
\(210\) 0 0
\(211\) 27.3594 1.88350 0.941748 0.336318i \(-0.109182\pi\)
0.941748 + 0.336318i \(0.109182\pi\)
\(212\) 0 0
\(213\) −14.5223 −0.995051
\(214\) 0 0
\(215\) −5.54202 −0.377963
\(216\) 0 0
\(217\) 11.3594 0.771124
\(218\) 0 0
\(219\) −5.43826 −0.367483
\(220\) 0 0
\(221\) 1.96056 0.131881
\(222\) 0 0
\(223\) 4.56174 0.305477 0.152738 0.988267i \(-0.451191\pi\)
0.152738 + 0.988267i \(0.451191\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 20.7243 1.37552 0.687759 0.725939i \(-0.258594\pi\)
0.687759 + 0.725939i \(0.258594\pi\)
\(228\) 0 0
\(229\) −2.32031 −0.153331 −0.0766654 0.997057i \(-0.524427\pi\)
−0.0766654 + 0.997057i \(0.524427\pi\)
\(230\) 0 0
\(231\) 26.2020 1.72396
\(232\) 0 0
\(233\) 23.6063 1.54650 0.773251 0.634100i \(-0.218629\pi\)
0.773251 + 0.634100i \(0.218629\pi\)
\(234\) 0 0
\(235\) −4.28087 −0.279253
\(236\) 0 0
\(237\) 16.4829 1.07068
\(238\) 0 0
\(239\) −16.6994 −1.08019 −0.540097 0.841603i \(-0.681613\pi\)
−0.540097 + 0.841603i \(0.681613\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 17.8032 1.13740
\(246\) 0 0
\(247\) 6.98028 0.444144
\(248\) 0 0
\(249\) −10.8032 −0.684623
\(250\) 0 0
\(251\) −12.1377 −0.766123 −0.383061 0.923723i \(-0.625130\pi\)
−0.383061 + 0.923723i \(0.625130\pi\)
\(252\) 0 0
\(253\) −1.47770 −0.0929024
\(254\) 0 0
\(255\) −0.280871 −0.0175888
\(256\) 0 0
\(257\) 8.80317 0.549127 0.274563 0.961569i \(-0.411467\pi\)
0.274563 + 0.961569i \(0.411467\pi\)
\(258\) 0 0
\(259\) −14.8426 −0.922275
\(260\) 0 0
\(261\) −3.26115 −0.201860
\(262\) 0 0
\(263\) 13.1179 0.808887 0.404444 0.914563i \(-0.367465\pi\)
0.404444 + 0.914563i \(0.367465\pi\)
\(264\) 0 0
\(265\) 5.71913 0.351323
\(266\) 0 0
\(267\) −16.6600 −1.01957
\(268\) 0 0
\(269\) 13.2217 0.806142 0.403071 0.915169i \(-0.367943\pi\)
0.403071 + 0.915169i \(0.367943\pi\)
\(270\) 0 0
\(271\) 21.9606 1.33401 0.667004 0.745054i \(-0.267576\pi\)
0.667004 + 0.745054i \(0.267576\pi\)
\(272\) 0 0
\(273\) −34.7637 −2.10400
\(274\) 0 0
\(275\) −5.26115 −0.317259
\(276\) 0 0
\(277\) −19.6063 −1.17803 −0.589015 0.808122i \(-0.700484\pi\)
−0.589015 + 0.808122i \(0.700484\pi\)
\(278\) 0 0
\(279\) 2.28087 0.136552
\(280\) 0 0
\(281\) −5.78345 −0.345011 −0.172506 0.985009i \(-0.555186\pi\)
−0.172506 + 0.985009i \(0.555186\pi\)
\(282\) 0 0
\(283\) 0.418536 0.0248794 0.0124397 0.999923i \(-0.496040\pi\)
0.0124397 + 0.999923i \(0.496040\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) 3.67969 0.217205
\(288\) 0 0
\(289\) −16.9211 −0.995360
\(290\) 0 0
\(291\) −2.98028 −0.174707
\(292\) 0 0
\(293\) 24.2414 1.41620 0.708100 0.706113i \(-0.249553\pi\)
0.708100 + 0.706113i \(0.249553\pi\)
\(294\) 0 0
\(295\) 10.5223 0.612632
\(296\) 0 0
\(297\) 5.26115 0.305283
\(298\) 0 0
\(299\) 1.96056 0.113382
\(300\) 0 0
\(301\) −27.6008 −1.59088
\(302\) 0 0
\(303\) −13.0840 −0.751658
\(304\) 0 0
\(305\) −2.24143 −0.128344
\(306\) 0 0
\(307\) −9.39881 −0.536419 −0.268209 0.963361i \(-0.586432\pi\)
−0.268209 + 0.963361i \(0.586432\pi\)
\(308\) 0 0
\(309\) −4.56174 −0.259508
\(310\) 0 0
\(311\) −19.2217 −1.08996 −0.544981 0.838448i \(-0.683463\pi\)
−0.544981 + 0.838448i \(0.683463\pi\)
\(312\) 0 0
\(313\) −31.9606 −1.80652 −0.903259 0.429096i \(-0.858832\pi\)
−0.903259 + 0.429096i \(0.858832\pi\)
\(314\) 0 0
\(315\) 4.98028 0.280607
\(316\) 0 0
\(317\) −9.71913 −0.545881 −0.272940 0.962031i \(-0.587996\pi\)
−0.272940 + 0.962031i \(0.587996\pi\)
\(318\) 0 0
\(319\) 17.1574 0.960629
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 0.280871 0.0156281
\(324\) 0 0
\(325\) 6.98028 0.387196
\(326\) 0 0
\(327\) 4.52230 0.250084
\(328\) 0 0
\(329\) −21.3199 −1.17541
\(330\) 0 0
\(331\) −18.2020 −1.00047 −0.500236 0.865889i \(-0.666753\pi\)
−0.500236 + 0.865889i \(0.666753\pi\)
\(332\) 0 0
\(333\) −2.98028 −0.163318
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.37909 −0.456438 −0.228219 0.973610i \(-0.573290\pi\)
−0.228219 + 0.973610i \(0.573290\pi\)
\(338\) 0 0
\(339\) 0.241427 0.0131125
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 53.8028 2.90508
\(344\) 0 0
\(345\) −0.280871 −0.0151216
\(346\) 0 0
\(347\) −29.6008 −1.58905 −0.794527 0.607229i \(-0.792281\pi\)
−0.794527 + 0.607229i \(0.792281\pi\)
\(348\) 0 0
\(349\) 11.0446 0.591204 0.295602 0.955311i \(-0.404480\pi\)
0.295602 + 0.955311i \(0.404480\pi\)
\(350\) 0 0
\(351\) −6.98028 −0.372580
\(352\) 0 0
\(353\) −19.0446 −1.01364 −0.506821 0.862051i \(-0.669179\pi\)
−0.506821 + 0.862051i \(0.669179\pi\)
\(354\) 0 0
\(355\) 14.5223 0.770764
\(356\) 0 0
\(357\) −1.39881 −0.0740331
\(358\) 0 0
\(359\) 29.7440 1.56983 0.784914 0.619604i \(-0.212707\pi\)
0.784914 + 0.619604i \(0.212707\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −16.6797 −0.875456
\(364\) 0 0
\(365\) 5.43826 0.284651
\(366\) 0 0
\(367\) 27.5026 1.43562 0.717811 0.696238i \(-0.245144\pi\)
0.717811 + 0.696238i \(0.245144\pi\)
\(368\) 0 0
\(369\) 0.738851 0.0384631
\(370\) 0 0
\(371\) 28.4829 1.47876
\(372\) 0 0
\(373\) −16.4580 −0.852162 −0.426081 0.904685i \(-0.640106\pi\)
−0.426081 + 0.904685i \(0.640106\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −22.7637 −1.17239
\(378\) 0 0
\(379\) 2.84261 0.146015 0.0730076 0.997331i \(-0.476740\pi\)
0.0730076 + 0.997331i \(0.476740\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 15.4383 0.788858 0.394429 0.918926i \(-0.370942\pi\)
0.394429 + 0.918926i \(0.370942\pi\)
\(384\) 0 0
\(385\) −26.2020 −1.33538
\(386\) 0 0
\(387\) −5.54202 −0.281717
\(388\) 0 0
\(389\) −18.4829 −0.937118 −0.468559 0.883432i \(-0.655227\pi\)
−0.468559 + 0.883432i \(0.655227\pi\)
\(390\) 0 0
\(391\) 0.0788884 0.00398956
\(392\) 0 0
\(393\) 15.7834 0.796170
\(394\) 0 0
\(395\) −16.4829 −0.829342
\(396\) 0 0
\(397\) −34.9657 −1.75488 −0.877439 0.479688i \(-0.840750\pi\)
−0.877439 + 0.479688i \(0.840750\pi\)
\(398\) 0 0
\(399\) −4.98028 −0.249326
\(400\) 0 0
\(401\) 19.8229 0.989908 0.494954 0.868919i \(-0.335185\pi\)
0.494954 + 0.868919i \(0.335185\pi\)
\(402\) 0 0
\(403\) 15.9211 0.793087
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 15.6797 0.777213
\(408\) 0 0
\(409\) −9.64578 −0.476953 −0.238477 0.971148i \(-0.576648\pi\)
−0.238477 + 0.971148i \(0.576648\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) 52.4040 2.57863
\(414\) 0 0
\(415\) 10.8032 0.530307
\(416\) 0 0
\(417\) 10.5223 0.515279
\(418\) 0 0
\(419\) 6.65996 0.325360 0.162680 0.986679i \(-0.447986\pi\)
0.162680 + 0.986679i \(0.447986\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) −4.28087 −0.208143
\(424\) 0 0
\(425\) 0.280871 0.0136242
\(426\) 0 0
\(427\) −11.1629 −0.540212
\(428\) 0 0
\(429\) 36.7243 1.77306
\(430\) 0 0
\(431\) −20.8371 −1.00369 −0.501843 0.864959i \(-0.667345\pi\)
−0.501843 + 0.864959i \(0.667345\pi\)
\(432\) 0 0
\(433\) 9.58146 0.460456 0.230228 0.973137i \(-0.426053\pi\)
0.230228 + 0.973137i \(0.426053\pi\)
\(434\) 0 0
\(435\) 3.26115 0.156360
\(436\) 0 0
\(437\) 0.280871 0.0134359
\(438\) 0 0
\(439\) 5.04459 0.240765 0.120383 0.992728i \(-0.461588\pi\)
0.120383 + 0.992728i \(0.461588\pi\)
\(440\) 0 0
\(441\) 17.8032 0.847770
\(442\) 0 0
\(443\) 31.6402 1.50327 0.751637 0.659577i \(-0.229265\pi\)
0.751637 + 0.659577i \(0.229265\pi\)
\(444\) 0 0
\(445\) 16.6600 0.789758
\(446\) 0 0
\(447\) 5.43826 0.257221
\(448\) 0 0
\(449\) −7.82289 −0.369185 −0.184593 0.982815i \(-0.559097\pi\)
−0.184593 + 0.982815i \(0.559097\pi\)
\(450\) 0 0
\(451\) −3.88721 −0.183041
\(452\) 0 0
\(453\) 14.2020 0.667267
\(454\) 0 0
\(455\) 34.7637 1.62975
\(456\) 0 0
\(457\) −15.4777 −0.724016 −0.362008 0.932175i \(-0.617909\pi\)
−0.362008 + 0.932175i \(0.617909\pi\)
\(458\) 0 0
\(459\) −0.280871 −0.0131099
\(460\) 0 0
\(461\) 9.92111 0.462072 0.231036 0.972945i \(-0.425788\pi\)
0.231036 + 0.972945i \(0.425788\pi\)
\(462\) 0 0
\(463\) −21.4631 −0.997476 −0.498738 0.866753i \(-0.666203\pi\)
−0.498738 + 0.866753i \(0.666203\pi\)
\(464\) 0 0
\(465\) −2.28087 −0.105773
\(466\) 0 0
\(467\) −32.7637 −1.51612 −0.758062 0.652182i \(-0.773854\pi\)
−0.758062 + 0.652182i \(0.773854\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 15.3988 0.709540
\(472\) 0 0
\(473\) 29.1574 1.34066
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 5.71913 0.261861
\(478\) 0 0
\(479\) −27.7834 −1.26946 −0.634729 0.772735i \(-0.718888\pi\)
−0.634729 + 0.772735i \(0.718888\pi\)
\(480\) 0 0
\(481\) −20.8032 −0.948543
\(482\) 0 0
\(483\) −1.39881 −0.0636483
\(484\) 0 0
\(485\) 2.98028 0.135327
\(486\) 0 0
\(487\) −29.3199 −1.32861 −0.664306 0.747460i \(-0.731273\pi\)
−0.664306 + 0.747460i \(0.731273\pi\)
\(488\) 0 0
\(489\) 18.9408 0.856534
\(490\) 0 0
\(491\) 4.69941 0.212081 0.106041 0.994362i \(-0.466183\pi\)
0.106041 + 0.994362i \(0.466183\pi\)
\(492\) 0 0
\(493\) −0.915961 −0.0412528
\(494\) 0 0
\(495\) −5.26115 −0.236471
\(496\) 0 0
\(497\) 72.3251 3.24422
\(498\) 0 0
\(499\) 30.4434 1.36283 0.681417 0.731895i \(-0.261364\pi\)
0.681417 + 0.731895i \(0.261364\pi\)
\(500\) 0 0
\(501\) −11.6797 −0.521810
\(502\) 0 0
\(503\) −13.6797 −0.609947 −0.304974 0.952361i \(-0.598648\pi\)
−0.304974 + 0.952361i \(0.598648\pi\)
\(504\) 0 0
\(505\) 13.0840 0.582232
\(506\) 0 0
\(507\) −35.7243 −1.58657
\(508\) 0 0
\(509\) −17.7046 −0.784741 −0.392370 0.919807i \(-0.628345\pi\)
−0.392370 + 0.919807i \(0.628345\pi\)
\(510\) 0 0
\(511\) 27.0840 1.19813
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 4.56174 0.201014
\(516\) 0 0
\(517\) 22.5223 0.990530
\(518\) 0 0
\(519\) 16.2414 0.712919
\(520\) 0 0
\(521\) −9.30059 −0.407466 −0.203733 0.979026i \(-0.565308\pi\)
−0.203733 + 0.979026i \(0.565308\pi\)
\(522\) 0 0
\(523\) 7.64578 0.334327 0.167163 0.985929i \(-0.446539\pi\)
0.167163 + 0.985929i \(0.446539\pi\)
\(524\) 0 0
\(525\) −4.98028 −0.217357
\(526\) 0 0
\(527\) 0.640630 0.0279063
\(528\) 0 0
\(529\) −22.9211 −0.996570
\(530\) 0 0
\(531\) 10.5223 0.456629
\(532\) 0 0
\(533\) 5.15739 0.223391
\(534\) 0 0
\(535\) −16.0000 −0.691740
\(536\) 0 0
\(537\) 21.9606 0.947668
\(538\) 0 0
\(539\) −93.6651 −4.03444
\(540\) 0 0
\(541\) −12.8765 −0.553605 −0.276802 0.960927i \(-0.589275\pi\)
−0.276802 + 0.960927i \(0.589275\pi\)
\(542\) 0 0
\(543\) −19.9606 −0.856589
\(544\) 0 0
\(545\) −4.52230 −0.193714
\(546\) 0 0
\(547\) 13.4777 0.576265 0.288132 0.957591i \(-0.406966\pi\)
0.288132 + 0.957591i \(0.406966\pi\)
\(548\) 0 0
\(549\) −2.24143 −0.0956618
\(550\) 0 0
\(551\) −3.26115 −0.138930
\(552\) 0 0
\(553\) −82.0892 −3.49079
\(554\) 0 0
\(555\) 2.98028 0.126506
\(556\) 0 0
\(557\) −3.68522 −0.156148 −0.0780740 0.996948i \(-0.524877\pi\)
−0.0780740 + 0.996948i \(0.524877\pi\)
\(558\) 0 0
\(559\) −38.6848 −1.63619
\(560\) 0 0
\(561\) 1.47770 0.0623887
\(562\) 0 0
\(563\) 12.7243 0.536264 0.268132 0.963382i \(-0.413594\pi\)
0.268132 + 0.963382i \(0.413594\pi\)
\(564\) 0 0
\(565\) −0.241427 −0.0101569
\(566\) 0 0
\(567\) 4.98028 0.209152
\(568\) 0 0
\(569\) 25.3006 1.06066 0.530328 0.847793i \(-0.322069\pi\)
0.530328 + 0.847793i \(0.322069\pi\)
\(570\) 0 0
\(571\) 23.0052 0.962736 0.481368 0.876519i \(-0.340140\pi\)
0.481368 + 0.876519i \(0.340140\pi\)
\(572\) 0 0
\(573\) 19.2217 0.802998
\(574\) 0 0
\(575\) 0.280871 0.0117131
\(576\) 0 0
\(577\) −25.0840 −1.04426 −0.522131 0.852865i \(-0.674863\pi\)
−0.522131 + 0.852865i \(0.674863\pi\)
\(578\) 0 0
\(579\) 6.41854 0.266745
\(580\) 0 0
\(581\) 53.8028 2.23212
\(582\) 0 0
\(583\) −30.0892 −1.24617
\(584\) 0 0
\(585\) 6.98028 0.288599
\(586\) 0 0
\(587\) −28.6848 −1.18395 −0.591975 0.805956i \(-0.701652\pi\)
−0.591975 + 0.805956i \(0.701652\pi\)
\(588\) 0 0
\(589\) 2.28087 0.0939816
\(590\) 0 0
\(591\) 16.2809 0.669706
\(592\) 0 0
\(593\) 2.56174 0.105198 0.0525991 0.998616i \(-0.483249\pi\)
0.0525991 + 0.998616i \(0.483249\pi\)
\(594\) 0 0
\(595\) 1.39881 0.0573458
\(596\) 0 0
\(597\) 25.3988 1.03950
\(598\) 0 0
\(599\) 33.0446 1.35017 0.675083 0.737742i \(-0.264108\pi\)
0.675083 + 0.737742i \(0.264108\pi\)
\(600\) 0 0
\(601\) −22.8371 −0.931544 −0.465772 0.884905i \(-0.654223\pi\)
−0.465772 + 0.884905i \(0.654223\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 16.6797 0.678126
\(606\) 0 0
\(607\) −28.1286 −1.14171 −0.570853 0.821052i \(-0.693387\pi\)
−0.570853 + 0.821052i \(0.693387\pi\)
\(608\) 0 0
\(609\) 16.2414 0.658136
\(610\) 0 0
\(611\) −29.8817 −1.20888
\(612\) 0 0
\(613\) 15.9606 0.644641 0.322320 0.946631i \(-0.395537\pi\)
0.322320 + 0.946631i \(0.395537\pi\)
\(614\) 0 0
\(615\) −0.738851 −0.0297934
\(616\) 0 0
\(617\) −6.59565 −0.265531 −0.132765 0.991147i \(-0.542386\pi\)
−0.132765 + 0.991147i \(0.542386\pi\)
\(618\) 0 0
\(619\) 33.8817 1.36182 0.680910 0.732367i \(-0.261585\pi\)
0.680910 + 0.732367i \(0.261585\pi\)
\(620\) 0 0
\(621\) −0.280871 −0.0112710
\(622\) 0 0
\(623\) 82.9712 3.32417
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 5.26115 0.210110
\(628\) 0 0
\(629\) −0.837073 −0.0333763
\(630\) 0 0
\(631\) 26.7976 1.06680 0.533398 0.845864i \(-0.320915\pi\)
0.533398 + 0.845864i \(0.320915\pi\)
\(632\) 0 0
\(633\) −27.3594 −1.08744
\(634\) 0 0
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 124.271 4.92380
\(638\) 0 0
\(639\) 14.5223 0.574493
\(640\) 0 0
\(641\) 25.7834 1.01838 0.509192 0.860653i \(-0.329944\pi\)
0.509192 + 0.860653i \(0.329944\pi\)
\(642\) 0 0
\(643\) −19.4237 −0.765995 −0.382998 0.923749i \(-0.625108\pi\)
−0.382998 + 0.923749i \(0.625108\pi\)
\(644\) 0 0
\(645\) 5.54202 0.218217
\(646\) 0 0
\(647\) −8.55620 −0.336379 −0.168190 0.985755i \(-0.553792\pi\)
−0.168190 + 0.985755i \(0.553792\pi\)
\(648\) 0 0
\(649\) −55.3594 −2.17305
\(650\) 0 0
\(651\) −11.3594 −0.445209
\(652\) 0 0
\(653\) −0.842612 −0.0329740 −0.0164870 0.999864i \(-0.505248\pi\)
−0.0164870 + 0.999864i \(0.505248\pi\)
\(654\) 0 0
\(655\) −15.7834 −0.616710
\(656\) 0 0
\(657\) 5.43826 0.212167
\(658\) 0 0
\(659\) −3.43826 −0.133936 −0.0669678 0.997755i \(-0.521332\pi\)
−0.0669678 + 0.997755i \(0.521332\pi\)
\(660\) 0 0
\(661\) 9.08404 0.353328 0.176664 0.984271i \(-0.443469\pi\)
0.176664 + 0.984271i \(0.443469\pi\)
\(662\) 0 0
\(663\) −1.96056 −0.0761417
\(664\) 0 0
\(665\) 4.98028 0.193127
\(666\) 0 0
\(667\) −0.915961 −0.0354662
\(668\) 0 0
\(669\) −4.56174 −0.176367
\(670\) 0 0
\(671\) 11.7925 0.455244
\(672\) 0 0
\(673\) 2.98028 0.114881 0.0574406 0.998349i \(-0.481706\pi\)
0.0574406 + 0.998349i \(0.481706\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −48.9996 −1.88321 −0.941604 0.336722i \(-0.890682\pi\)
−0.941604 + 0.336722i \(0.890682\pi\)
\(678\) 0 0
\(679\) 14.8426 0.569607
\(680\) 0 0
\(681\) −20.7243 −0.794156
\(682\) 0 0
\(683\) 27.9211 1.06837 0.534186 0.845367i \(-0.320618\pi\)
0.534186 + 0.845367i \(0.320618\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 2.32031 0.0885255
\(688\) 0 0
\(689\) 39.9211 1.52087
\(690\) 0 0
\(691\) −9.96056 −0.378917 −0.189459 0.981889i \(-0.560673\pi\)
−0.189459 + 0.981889i \(0.560673\pi\)
\(692\) 0 0
\(693\) −26.2020 −0.995331
\(694\) 0 0
\(695\) −10.5223 −0.399133
\(696\) 0 0
\(697\) 0.207522 0.00786045
\(698\) 0 0
\(699\) −23.6063 −0.892874
\(700\) 0 0
\(701\) −10.4829 −0.395932 −0.197966 0.980209i \(-0.563434\pi\)
−0.197966 + 0.980209i \(0.563434\pi\)
\(702\) 0 0
\(703\) −2.98028 −0.112403
\(704\) 0 0
\(705\) 4.28087 0.161227
\(706\) 0 0
\(707\) 65.1621 2.45067
\(708\) 0 0
\(709\) 16.5562 0.621781 0.310891 0.950446i \(-0.399373\pi\)
0.310891 + 0.950446i \(0.399373\pi\)
\(710\) 0 0
\(711\) −16.4829 −0.618155
\(712\) 0 0
\(713\) 0.640630 0.0239918
\(714\) 0 0
\(715\) −36.7243 −1.37341
\(716\) 0 0
\(717\) 16.6994 0.623651
\(718\) 0 0
\(719\) −4.21655 −0.157251 −0.0786255 0.996904i \(-0.525053\pi\)
−0.0786255 + 0.996904i \(0.525053\pi\)
\(720\) 0 0
\(721\) 22.7187 0.846090
\(722\) 0 0
\(723\) 2.00000 0.0743808
\(724\) 0 0
\(725\) −3.26115 −0.121116
\(726\) 0 0
\(727\) 26.4580 0.981272 0.490636 0.871365i \(-0.336764\pi\)
0.490636 + 0.871365i \(0.336764\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.55659 −0.0575726
\(732\) 0 0
\(733\) −6.56174 −0.242363 −0.121182 0.992630i \(-0.538668\pi\)
−0.121182 + 0.992630i \(0.538668\pi\)
\(734\) 0 0
\(735\) −17.8032 −0.656680
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 44.9657 1.65409 0.827045 0.562136i \(-0.190020\pi\)
0.827045 + 0.562136i \(0.190020\pi\)
\(740\) 0 0
\(741\) −6.98028 −0.256427
\(742\) 0 0
\(743\) −24.4040 −0.895295 −0.447647 0.894210i \(-0.647738\pi\)
−0.447647 + 0.894210i \(0.647738\pi\)
\(744\) 0 0
\(745\) −5.43826 −0.199242
\(746\) 0 0
\(747\) 10.8032 0.395267
\(748\) 0 0
\(749\) −79.6844 −2.91161
\(750\) 0 0
\(751\) 5.23628 0.191074 0.0955372 0.995426i \(-0.469543\pi\)
0.0955372 + 0.995426i \(0.469543\pi\)
\(752\) 0 0
\(753\) 12.1377 0.442321
\(754\) 0 0
\(755\) −14.2020 −0.516863
\(756\) 0 0
\(757\) 20.5223 0.745896 0.372948 0.927852i \(-0.378347\pi\)
0.372948 + 0.927852i \(0.378347\pi\)
\(758\) 0 0
\(759\) 1.47770 0.0536372
\(760\) 0 0
\(761\) −13.5171 −0.489996 −0.244998 0.969524i \(-0.578787\pi\)
−0.244998 + 0.969524i \(0.578787\pi\)
\(762\) 0 0
\(763\) −22.5223 −0.815362
\(764\) 0 0
\(765\) 0.280871 0.0101549
\(766\) 0 0
\(767\) 73.4486 2.65207
\(768\) 0 0
\(769\) 33.9211 1.22323 0.611613 0.791157i \(-0.290521\pi\)
0.611613 + 0.791157i \(0.290521\pi\)
\(770\) 0 0
\(771\) −8.80317 −0.317038
\(772\) 0 0
\(773\) −5.79802 −0.208540 −0.104270 0.994549i \(-0.533251\pi\)
−0.104270 + 0.994549i \(0.533251\pi\)
\(774\) 0 0
\(775\) 2.28087 0.0819313
\(776\) 0 0
\(777\) 14.8426 0.532476
\(778\) 0 0
\(779\) 0.738851 0.0264721
\(780\) 0 0
\(781\) −76.4040 −2.73395
\(782\) 0 0
\(783\) 3.26115 0.116544
\(784\) 0 0
\(785\) −15.3988 −0.549607
\(786\) 0 0
\(787\) 11.5669 0.412315 0.206158 0.978519i \(-0.433904\pi\)
0.206158 + 0.978519i \(0.433904\pi\)
\(788\) 0 0
\(789\) −13.1179 −0.467011
\(790\) 0 0
\(791\) −1.20237 −0.0427514
\(792\) 0 0
\(793\) −15.6458 −0.555598
\(794\) 0 0
\(795\) −5.71913 −0.202837
\(796\) 0 0
\(797\) −17.3649 −0.615097 −0.307548 0.951532i \(-0.599509\pi\)
−0.307548 + 0.951532i \(0.599509\pi\)
\(798\) 0 0
\(799\) −1.20237 −0.0425368
\(800\) 0 0
\(801\) 16.6600 0.588651
\(802\) 0 0
\(803\) −28.6115 −1.00968
\(804\) 0 0
\(805\) 1.39881 0.0493017
\(806\) 0 0
\(807\) −13.2217 −0.465426
\(808\) 0 0
\(809\) 26.4829 0.931088 0.465544 0.885025i \(-0.345859\pi\)
0.465544 + 0.885025i \(0.345859\pi\)
\(810\) 0 0
\(811\) 35.8422 1.25859 0.629295 0.777166i \(-0.283344\pi\)
0.629295 + 0.777166i \(0.283344\pi\)
\(812\) 0 0
\(813\) −21.9606 −0.770190
\(814\) 0 0
\(815\) −18.9408 −0.663468
\(816\) 0 0
\(817\) −5.54202 −0.193891
\(818\) 0 0
\(819\) 34.7637 1.21474
\(820\) 0 0
\(821\) −12.5223 −0.437031 −0.218516 0.975833i \(-0.570121\pi\)
−0.218516 + 0.975833i \(0.570121\pi\)
\(822\) 0 0
\(823\) 4.06432 0.141673 0.0708366 0.997488i \(-0.477433\pi\)
0.0708366 + 0.997488i \(0.477433\pi\)
\(824\) 0 0
\(825\) 5.26115 0.183170
\(826\) 0 0
\(827\) −33.8478 −1.17700 −0.588501 0.808496i \(-0.700282\pi\)
−0.588501 + 0.808496i \(0.700282\pi\)
\(828\) 0 0
\(829\) −9.08404 −0.315502 −0.157751 0.987479i \(-0.550424\pi\)
−0.157751 + 0.987479i \(0.550424\pi\)
\(830\) 0 0
\(831\) 19.6063 0.680136
\(832\) 0 0
\(833\) 5.00039 0.173253
\(834\) 0 0
\(835\) 11.6797 0.404192
\(836\) 0 0
\(837\) −2.28087 −0.0788384
\(838\) 0 0
\(839\) 12.2753 0.423792 0.211896 0.977292i \(-0.432036\pi\)
0.211896 + 0.977292i \(0.432036\pi\)
\(840\) 0 0
\(841\) −18.3649 −0.633273
\(842\) 0 0
\(843\) 5.78345 0.199192
\(844\) 0 0
\(845\) 35.7243 1.22895
\(846\) 0 0
\(847\) 83.0695 2.85430
\(848\) 0 0
\(849\) −0.418536 −0.0143641
\(850\) 0 0
\(851\) −0.837073 −0.0286945
\(852\) 0 0
\(853\) −11.4777 −0.392989 −0.196495 0.980505i \(-0.562956\pi\)
−0.196495 + 0.980505i \(0.562956\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −46.7637 −1.59742 −0.798709 0.601717i \(-0.794483\pi\)
−0.798709 + 0.601717i \(0.794483\pi\)
\(858\) 0 0
\(859\) 30.7298 1.04849 0.524244 0.851568i \(-0.324348\pi\)
0.524244 + 0.851568i \(0.324348\pi\)
\(860\) 0 0
\(861\) −3.67969 −0.125403
\(862\) 0 0
\(863\) 44.0000 1.49778 0.748889 0.662696i \(-0.230588\pi\)
0.748889 + 0.662696i \(0.230588\pi\)
\(864\) 0 0
\(865\) −16.2414 −0.552225
\(866\) 0 0
\(867\) 16.9211 0.574671
\(868\) 0 0
\(869\) 86.7187 2.94173
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.98028 0.100867
\(874\) 0 0
\(875\) 4.98028 0.168364
\(876\) 0 0
\(877\) −31.1878 −1.05314 −0.526569 0.850133i \(-0.676522\pi\)
−0.526569 + 0.850133i \(0.676522\pi\)
\(878\) 0 0
\(879\) −24.2414 −0.817643
\(880\) 0 0
\(881\) 1.56689 0.0527899 0.0263950 0.999652i \(-0.491597\pi\)
0.0263950 + 0.999652i \(0.491597\pi\)
\(882\) 0 0
\(883\) −36.0643 −1.21366 −0.606830 0.794831i \(-0.707559\pi\)
−0.606830 + 0.794831i \(0.707559\pi\)
\(884\) 0 0
\(885\) −10.5223 −0.353703
\(886\) 0 0
\(887\) −9.12348 −0.306337 −0.153168 0.988200i \(-0.548948\pi\)
−0.153168 + 0.988200i \(0.548948\pi\)
\(888\) 0 0
\(889\) 19.9211 0.668133
\(890\) 0 0
\(891\) −5.26115 −0.176255
\(892\) 0 0
\(893\) −4.28087 −0.143254
\(894\) 0 0
\(895\) −21.9606 −0.734060
\(896\) 0 0
\(897\) −1.96056 −0.0654611
\(898\) 0 0
\(899\) −7.43826 −0.248080
\(900\) 0 0
\(901\) 1.60634 0.0535148
\(902\) 0 0
\(903\) 27.6008 0.918497
\(904\) 0 0
\(905\) 19.9606 0.663511
\(906\) 0 0
\(907\) −15.7136 −0.521761 −0.260881 0.965371i \(-0.584013\pi\)
−0.260881 + 0.965371i \(0.584013\pi\)
\(908\) 0 0
\(909\) 13.0840 0.433970
\(910\) 0 0
\(911\) −47.3594 −1.56909 −0.784543 0.620074i \(-0.787102\pi\)
−0.784543 + 0.620074i \(0.787102\pi\)
\(912\) 0 0
\(913\) −56.8371 −1.88103
\(914\) 0 0
\(915\) 2.24143 0.0740993
\(916\) 0 0
\(917\) −78.6059 −2.59580
\(918\) 0 0
\(919\) 24.4829 0.807615 0.403807 0.914844i \(-0.367687\pi\)
0.403807 + 0.914844i \(0.367687\pi\)
\(920\) 0 0
\(921\) 9.39881 0.309701
\(922\) 0 0
\(923\) 101.370 3.33662
\(924\) 0 0
\(925\) −2.98028 −0.0979909
\(926\) 0 0
\(927\) 4.56174 0.149827
\(928\) 0 0
\(929\) 51.3988 1.68634 0.843170 0.537647i \(-0.180687\pi\)
0.843170 + 0.537647i \(0.180687\pi\)
\(930\) 0 0
\(931\) 17.8032 0.583475
\(932\) 0 0
\(933\) 19.2217 0.629290
\(934\) 0 0
\(935\) −1.47770 −0.0483260
\(936\) 0 0
\(937\) 21.5669 0.704560 0.352280 0.935895i \(-0.385407\pi\)
0.352280 + 0.935895i \(0.385407\pi\)
\(938\) 0 0
\(939\) 31.9606 1.04299
\(940\) 0 0
\(941\) −28.8675 −0.941053 −0.470527 0.882386i \(-0.655936\pi\)
−0.470527 + 0.882386i \(0.655936\pi\)
\(942\) 0 0
\(943\) 0.207522 0.00675784
\(944\) 0 0
\(945\) −4.98028 −0.162008
\(946\) 0 0
\(947\) 26.2414 0.852732 0.426366 0.904551i \(-0.359794\pi\)
0.426366 + 0.904551i \(0.359794\pi\)
\(948\) 0 0
\(949\) 37.9606 1.23225
\(950\) 0 0
\(951\) 9.71913 0.315164
\(952\) 0 0
\(953\) −47.0391 −1.52374 −0.761872 0.647727i \(-0.775720\pi\)
−0.761872 + 0.647727i \(0.775720\pi\)
\(954\) 0 0
\(955\) −19.2217 −0.622000
\(956\) 0 0
\(957\) −17.1574 −0.554620
\(958\) 0 0
\(959\) −29.8817 −0.964929
\(960\) 0 0
\(961\) −25.7976 −0.832182
\(962\) 0 0
\(963\) −16.0000 −0.515593
\(964\) 0 0
\(965\) −6.41854 −0.206620
\(966\) 0 0
\(967\) −53.9460 −1.73479 −0.867393 0.497624i \(-0.834206\pi\)
−0.867393 + 0.497624i \(0.834206\pi\)
\(968\) 0 0
\(969\) −0.280871 −0.00902287
\(970\) 0 0
\(971\) −13.6852 −0.439180 −0.219590 0.975592i \(-0.570472\pi\)
−0.219590 + 0.975592i \(0.570472\pi\)
\(972\) 0 0
\(973\) −52.4040 −1.67999
\(974\) 0 0
\(975\) −6.98028 −0.223548
\(976\) 0 0
\(977\) 50.6848 1.62155 0.810776 0.585357i \(-0.199046\pi\)
0.810776 + 0.585357i \(0.199046\pi\)
\(978\) 0 0
\(979\) −87.6505 −2.80132
\(980\) 0 0
\(981\) −4.52230 −0.144386
\(982\) 0 0
\(983\) 3.19683 0.101963 0.0509816 0.998700i \(-0.483765\pi\)
0.0509816 + 0.998700i \(0.483765\pi\)
\(984\) 0 0
\(985\) −16.2809 −0.518752
\(986\) 0 0
\(987\) 21.3199 0.678621
\(988\) 0 0
\(989\) −1.55659 −0.0494967
\(990\) 0 0
\(991\) 5.04459 0.160247 0.0801234 0.996785i \(-0.474469\pi\)
0.0801234 + 0.996785i \(0.474469\pi\)
\(992\) 0 0
\(993\) 18.2020 0.577622
\(994\) 0 0
\(995\) −25.3988 −0.805197
\(996\) 0 0
\(997\) −1.64578 −0.0521224 −0.0260612 0.999660i \(-0.508296\pi\)
−0.0260612 + 0.999660i \(0.508296\pi\)
\(998\) 0 0
\(999\) 2.98028 0.0942918
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 1140.2.a.f.1.3 3
3.2 odd 2 3420.2.a.k.1.3 3
4.3 odd 2 4560.2.a.bu.1.1 3
5.2 odd 4 5700.2.f.p.3649.6 6
5.3 odd 4 5700.2.f.p.3649.1 6
5.4 even 2 5700.2.a.z.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.f.1.3 3 1.1 even 1 trivial
3420.2.a.k.1.3 3 3.2 odd 2
4560.2.a.bu.1.1 3 4.3 odd 2
5700.2.a.z.1.1 3 5.4 even 2
5700.2.f.p.3649.1 6 5.3 odd 4
5700.2.f.p.3649.6 6 5.2 odd 4