Properties

Label 3724.2.a.i.1.1
Level $3724$
Weight $2$
Character 3724.1
Self dual yes
Analytic conductor $29.736$
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] = [3724,2,Mod(1,3724)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3724, 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("3724.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3724.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: \(29.7362897127\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.733.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 8 \) 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 532)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.69639\) of defining polynomial
Character \(\chi\) \(=\) 3724.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.69639 q^{3} -1.42586 q^{5} +4.27053 q^{9} +O(q^{10})\) \(q-2.69639 q^{3} -1.42586 q^{5} +4.27053 q^{9} -3.27053 q^{11} -3.42586 q^{13} +3.84468 q^{15} -5.27053 q^{17} +1.00000 q^{19} +0.574141 q^{23} -2.96693 q^{25} -3.42586 q^{27} -0.122252 q^{29} -2.12225 q^{31} +8.81864 q^{33} -3.96693 q^{37} +9.23746 q^{39} +4.66332 q^{41} -11.9339 q^{43} -6.08918 q^{45} -3.11521 q^{47} +14.2114 q^{51} -6.08918 q^{53} +4.66332 q^{55} -2.69639 q^{57} +1.72242 q^{59} -12.2114 q^{61} +4.88479 q^{65} -5.27053 q^{67} -1.54811 q^{69} -6.81864 q^{71} +11.5481 q^{73} +8.00000 q^{75} +11.0821 q^{79} -3.57414 q^{81} -5.23746 q^{83} +7.51504 q^{85} +0.329638 q^{87} +1.45893 q^{89} +5.72242 q^{93} -1.42586 q^{95} -17.6563 q^{97} -13.9669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} - 2 q^{5} + 6 q^{9} - 3 q^{11} - 8 q^{13} + 7 q^{15} - 9 q^{17} + 3 q^{19} + 4 q^{23} + 7 q^{25} - 8 q^{27} + 11 q^{29} + 5 q^{31} + 6 q^{33} + 4 q^{37} + 5 q^{39} - 11 q^{41} - 4 q^{43} + 9 q^{45} + 2 q^{47} + 4 q^{51} + 9 q^{53} - 11 q^{55} + q^{57} + 12 q^{59} + 2 q^{61} + 26 q^{65} - 9 q^{67} + 9 q^{69} + 21 q^{73} + 24 q^{75} + 6 q^{79} - 13 q^{81} + 7 q^{83} - 7 q^{85} + 26 q^{87} + 18 q^{89} + 24 q^{93} - 2 q^{95} - 28 q^{97} - 26 q^{99}+O(q^{100}) \) 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
\(3\) −2.69639 −1.55676 −0.778382 0.627791i \(-0.783959\pi\)
−0.778382 + 0.627791i \(0.783959\pi\)
\(4\) 0 0
\(5\) −1.42586 −0.637663 −0.318832 0.947811i \(-0.603290\pi\)
−0.318832 + 0.947811i \(0.603290\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 4.27053 1.42351
\(10\) 0 0
\(11\) −3.27053 −0.986103 −0.493052 0.870000i \(-0.664119\pi\)
−0.493052 + 0.870000i \(0.664119\pi\)
\(12\) 0 0
\(13\) −3.42586 −0.950162 −0.475081 0.879942i \(-0.657581\pi\)
−0.475081 + 0.879942i \(0.657581\pi\)
\(14\) 0 0
\(15\) 3.84468 0.992691
\(16\) 0 0
\(17\) −5.27053 −1.27829 −0.639146 0.769085i \(-0.720712\pi\)
−0.639146 + 0.769085i \(0.720712\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.574141 0.119717 0.0598584 0.998207i \(-0.480935\pi\)
0.0598584 + 0.998207i \(0.480935\pi\)
\(24\) 0 0
\(25\) −2.96693 −0.593385
\(26\) 0 0
\(27\) −3.42586 −0.659307
\(28\) 0 0
\(29\) −0.122252 −0.0227015 −0.0113508 0.999936i \(-0.503613\pi\)
−0.0113508 + 0.999936i \(0.503613\pi\)
\(30\) 0 0
\(31\) −2.12225 −0.381168 −0.190584 0.981671i \(-0.561038\pi\)
−0.190584 + 0.981671i \(0.561038\pi\)
\(32\) 0 0
\(33\) 8.81864 1.53513
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.96693 −0.652159 −0.326079 0.945342i \(-0.605728\pi\)
−0.326079 + 0.945342i \(0.605728\pi\)
\(38\) 0 0
\(39\) 9.23746 1.47918
\(40\) 0 0
\(41\) 4.66332 0.728288 0.364144 0.931343i \(-0.381362\pi\)
0.364144 + 0.931343i \(0.381362\pi\)
\(42\) 0 0
\(43\) −11.9339 −1.81990 −0.909948 0.414723i \(-0.863879\pi\)
−0.909948 + 0.414723i \(0.863879\pi\)
\(44\) 0 0
\(45\) −6.08918 −0.907721
\(46\) 0 0
\(47\) −3.11521 −0.454400 −0.227200 0.973848i \(-0.572957\pi\)
−0.227200 + 0.973848i \(0.572957\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 14.2114 1.99000
\(52\) 0 0
\(53\) −6.08918 −0.836413 −0.418206 0.908352i \(-0.637341\pi\)
−0.418206 + 0.908352i \(0.637341\pi\)
\(54\) 0 0
\(55\) 4.66332 0.628802
\(56\) 0 0
\(57\) −2.69639 −0.357146
\(58\) 0 0
\(59\) 1.72242 0.224240 0.112120 0.993695i \(-0.464236\pi\)
0.112120 + 0.993695i \(0.464236\pi\)
\(60\) 0 0
\(61\) −12.2114 −1.56351 −0.781757 0.623584i \(-0.785676\pi\)
−0.781757 + 0.623584i \(0.785676\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.88479 0.605884
\(66\) 0 0
\(67\) −5.27053 −0.643898 −0.321949 0.946757i \(-0.604338\pi\)
−0.321949 + 0.946757i \(0.604338\pi\)
\(68\) 0 0
\(69\) −1.54811 −0.186371
\(70\) 0 0
\(71\) −6.81864 −0.809224 −0.404612 0.914488i \(-0.632593\pi\)
−0.404612 + 0.914488i \(0.632593\pi\)
\(72\) 0 0
\(73\) 11.5481 1.35160 0.675802 0.737083i \(-0.263797\pi\)
0.675802 + 0.737083i \(0.263797\pi\)
\(74\) 0 0
\(75\) 8.00000 0.923760
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.0821 1.24684 0.623419 0.781888i \(-0.285743\pi\)
0.623419 + 0.781888i \(0.285743\pi\)
\(80\) 0 0
\(81\) −3.57414 −0.397127
\(82\) 0 0
\(83\) −5.23746 −0.574886 −0.287443 0.957798i \(-0.592805\pi\)
−0.287443 + 0.957798i \(0.592805\pi\)
\(84\) 0 0
\(85\) 7.51504 0.815120
\(86\) 0 0
\(87\) 0.329638 0.0353409
\(88\) 0 0
\(89\) 1.45893 0.154646 0.0773232 0.997006i \(-0.475363\pi\)
0.0773232 + 0.997006i \(0.475363\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 5.72242 0.593388
\(94\) 0 0
\(95\) −1.42586 −0.146290
\(96\) 0 0
\(97\) −17.6563 −1.79272 −0.896362 0.443324i \(-0.853799\pi\)
−0.896362 + 0.443324i \(0.853799\pi\)
\(98\) 0 0
\(99\) −13.9669 −1.40373
\(100\) 0 0
\(101\) −17.0491 −1.69645 −0.848223 0.529640i \(-0.822327\pi\)
−0.848223 + 0.529640i \(0.822327\pi\)
\(102\) 0 0
\(103\) 11.3597 1.11931 0.559653 0.828727i \(-0.310934\pi\)
0.559653 + 0.828727i \(0.310934\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.8186 1.04588 0.522939 0.852370i \(-0.324836\pi\)
0.522939 + 0.852370i \(0.324836\pi\)
\(108\) 0 0
\(109\) −12.2776 −1.17598 −0.587989 0.808869i \(-0.700080\pi\)
−0.587989 + 0.808869i \(0.700080\pi\)
\(110\) 0 0
\(111\) 10.6964 1.01526
\(112\) 0 0
\(113\) −0.696393 −0.0655111 −0.0327556 0.999463i \(-0.510428\pi\)
−0.0327556 + 0.999463i \(0.510428\pi\)
\(114\) 0 0
\(115\) −0.818644 −0.0763390
\(116\) 0 0
\(117\) −14.6302 −1.35257
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.303607 −0.0276007
\(122\) 0 0
\(123\) −12.5741 −1.13377
\(124\) 0 0
\(125\) 11.3597 1.01604
\(126\) 0 0
\(127\) −5.68935 −0.504848 −0.252424 0.967617i \(-0.581228\pi\)
−0.252424 + 0.967617i \(0.581228\pi\)
\(128\) 0 0
\(129\) 32.1784 2.83315
\(130\) 0 0
\(131\) 15.2044 1.32841 0.664207 0.747549i \(-0.268769\pi\)
0.664207 + 0.747549i \(0.268769\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 4.88479 0.420416
\(136\) 0 0
\(137\) 3.70344 0.316406 0.158203 0.987407i \(-0.449430\pi\)
0.158203 + 0.987407i \(0.449430\pi\)
\(138\) 0 0
\(139\) 11.6704 0.989867 0.494934 0.868931i \(-0.335192\pi\)
0.494934 + 0.868931i \(0.335192\pi\)
\(140\) 0 0
\(141\) 8.39983 0.707393
\(142\) 0 0
\(143\) 11.2044 0.936958
\(144\) 0 0
\(145\) 0.174313 0.0144759
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.8186 1.54168 0.770842 0.637027i \(-0.219836\pi\)
0.770842 + 0.637027i \(0.219836\pi\)
\(150\) 0 0
\(151\) −4.43290 −0.360744 −0.180372 0.983598i \(-0.557730\pi\)
−0.180372 + 0.983598i \(0.557730\pi\)
\(152\) 0 0
\(153\) −22.5080 −1.81966
\(154\) 0 0
\(155\) 3.02603 0.243057
\(156\) 0 0
\(157\) 9.82569 0.784175 0.392088 0.919928i \(-0.371753\pi\)
0.392088 + 0.919928i \(0.371753\pi\)
\(158\) 0 0
\(159\) 16.4188 1.30210
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −10.9409 −0.856957 −0.428479 0.903552i \(-0.640950\pi\)
−0.428479 + 0.903552i \(0.640950\pi\)
\(164\) 0 0
\(165\) −12.5741 −0.978896
\(166\) 0 0
\(167\) 12.7525 0.986818 0.493409 0.869797i \(-0.335751\pi\)
0.493409 + 0.869797i \(0.335751\pi\)
\(168\) 0 0
\(169\) −1.26349 −0.0971917
\(170\) 0 0
\(171\) 4.27053 0.326576
\(172\) 0 0
\(173\) −22.5411 −1.71377 −0.856883 0.515511i \(-0.827602\pi\)
−0.856883 + 0.515511i \(0.827602\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.64433 −0.349089
\(178\) 0 0
\(179\) 6.41882 0.479765 0.239882 0.970802i \(-0.422891\pi\)
0.239882 + 0.970802i \(0.422891\pi\)
\(180\) 0 0
\(181\) −9.81160 −0.729291 −0.364645 0.931146i \(-0.618810\pi\)
−0.364645 + 0.931146i \(0.618810\pi\)
\(182\) 0 0
\(183\) 32.9268 2.43402
\(184\) 0 0
\(185\) 5.65628 0.415858
\(186\) 0 0
\(187\) 17.2375 1.26053
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.2515 1.10356 0.551782 0.833989i \(-0.313948\pi\)
0.551782 + 0.833989i \(0.313948\pi\)
\(192\) 0 0
\(193\) −15.7455 −1.13338 −0.566691 0.823930i \(-0.691777\pi\)
−0.566691 + 0.823930i \(0.691777\pi\)
\(194\) 0 0
\(195\) −13.1713 −0.943217
\(196\) 0 0
\(197\) 24.8086 1.76754 0.883770 0.467922i \(-0.154997\pi\)
0.883770 + 0.467922i \(0.154997\pi\)
\(198\) 0 0
\(199\) 2.58823 0.183474 0.0917372 0.995783i \(-0.470758\pi\)
0.0917372 + 0.995783i \(0.470758\pi\)
\(200\) 0 0
\(201\) 14.2114 1.00240
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −6.64924 −0.464403
\(206\) 0 0
\(207\) 2.45189 0.170418
\(208\) 0 0
\(209\) −3.27053 −0.226228
\(210\) 0 0
\(211\) 21.7785 1.49930 0.749648 0.661837i \(-0.230223\pi\)
0.749648 + 0.661837i \(0.230223\pi\)
\(212\) 0 0
\(213\) 18.3857 1.25977
\(214\) 0 0
\(215\) 17.0160 1.16048
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −31.1382 −2.10413
\(220\) 0 0
\(221\) 18.0561 1.21459
\(222\) 0 0
\(223\) 9.90078 0.663005 0.331503 0.943454i \(-0.392444\pi\)
0.331503 + 0.943454i \(0.392444\pi\)
\(224\) 0 0
\(225\) −12.6704 −0.844691
\(226\) 0 0
\(227\) −2.71538 −0.180226 −0.0901131 0.995932i \(-0.528723\pi\)
−0.0901131 + 0.995932i \(0.528723\pi\)
\(228\) 0 0
\(229\) 8.29656 0.548252 0.274126 0.961694i \(-0.411611\pi\)
0.274126 + 0.961694i \(0.411611\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.7295 0.964959 0.482480 0.875907i \(-0.339736\pi\)
0.482480 + 0.875907i \(0.339736\pi\)
\(234\) 0 0
\(235\) 4.44185 0.289754
\(236\) 0 0
\(237\) −29.8818 −1.94103
\(238\) 0 0
\(239\) 28.9970 1.87566 0.937830 0.347095i \(-0.112832\pi\)
0.937830 + 0.347095i \(0.112832\pi\)
\(240\) 0 0
\(241\) 4.31065 0.277673 0.138837 0.990315i \(-0.455664\pi\)
0.138837 + 0.990315i \(0.455664\pi\)
\(242\) 0 0
\(243\) 19.9149 1.27754
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.42586 −0.217982
\(248\) 0 0
\(249\) 14.1223 0.894961
\(250\) 0 0
\(251\) 10.4329 0.658519 0.329259 0.944239i \(-0.393201\pi\)
0.329259 + 0.944239i \(0.393201\pi\)
\(252\) 0 0
\(253\) −1.87775 −0.118053
\(254\) 0 0
\(255\) −20.2635 −1.26895
\(256\) 0 0
\(257\) −23.7124 −1.47914 −0.739569 0.673081i \(-0.764971\pi\)
−0.739569 + 0.673081i \(0.764971\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −0.522079 −0.0323159
\(262\) 0 0
\(263\) 30.9409 1.90790 0.953949 0.299970i \(-0.0969766\pi\)
0.953949 + 0.299970i \(0.0969766\pi\)
\(264\) 0 0
\(265\) 8.68231 0.533350
\(266\) 0 0
\(267\) −3.93385 −0.240748
\(268\) 0 0
\(269\) 9.17131 0.559185 0.279592 0.960119i \(-0.409801\pi\)
0.279592 + 0.960119i \(0.409801\pi\)
\(270\) 0 0
\(271\) −20.0420 −1.21747 −0.608733 0.793375i \(-0.708322\pi\)
−0.608733 + 0.793375i \(0.708322\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.70344 0.585139
\(276\) 0 0
\(277\) −7.45893 −0.448164 −0.224082 0.974570i \(-0.571938\pi\)
−0.224082 + 0.974570i \(0.571938\pi\)
\(278\) 0 0
\(279\) −9.06315 −0.542596
\(280\) 0 0
\(281\) 12.2114 0.728473 0.364236 0.931307i \(-0.381330\pi\)
0.364236 + 0.931307i \(0.381330\pi\)
\(282\) 0 0
\(283\) 8.02303 0.476920 0.238460 0.971152i \(-0.423357\pi\)
0.238460 + 0.971152i \(0.423357\pi\)
\(284\) 0 0
\(285\) 3.84468 0.227739
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 10.7785 0.634031
\(290\) 0 0
\(291\) 47.6083 2.79085
\(292\) 0 0
\(293\) −14.0331 −0.819821 −0.409910 0.912126i \(-0.634440\pi\)
−0.409910 + 0.912126i \(0.634440\pi\)
\(294\) 0 0
\(295\) −2.45593 −0.142990
\(296\) 0 0
\(297\) 11.2044 0.650145
\(298\) 0 0
\(299\) −1.96693 −0.113750
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 45.9710 2.64096
\(304\) 0 0
\(305\) 17.4118 0.996995
\(306\) 0 0
\(307\) −16.0702 −0.917174 −0.458587 0.888649i \(-0.651644\pi\)
−0.458587 + 0.888649i \(0.651644\pi\)
\(308\) 0 0
\(309\) −30.6302 −1.74249
\(310\) 0 0
\(311\) 1.56710 0.0888620 0.0444310 0.999012i \(-0.485853\pi\)
0.0444310 + 0.999012i \(0.485853\pi\)
\(312\) 0 0
\(313\) −29.6232 −1.67440 −0.837201 0.546895i \(-0.815810\pi\)
−0.837201 + 0.546895i \(0.815810\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.4890 −1.26311 −0.631554 0.775332i \(-0.717583\pi\)
−0.631554 + 0.775332i \(0.717583\pi\)
\(318\) 0 0
\(319\) 0.399828 0.0223861
\(320\) 0 0
\(321\) −29.1713 −1.62818
\(322\) 0 0
\(323\) −5.27053 −0.293260
\(324\) 0 0
\(325\) 10.1643 0.563812
\(326\) 0 0
\(327\) 33.1052 1.83072
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 28.3857 1.56022 0.780111 0.625641i \(-0.215163\pi\)
0.780111 + 0.625641i \(0.215163\pi\)
\(332\) 0 0
\(333\) −16.9409 −0.928355
\(334\) 0 0
\(335\) 7.51504 0.410590
\(336\) 0 0
\(337\) 29.6272 1.61390 0.806950 0.590620i \(-0.201117\pi\)
0.806950 + 0.590620i \(0.201117\pi\)
\(338\) 0 0
\(339\) 1.87775 0.101985
\(340\) 0 0
\(341\) 6.94090 0.375871
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2.20739 0.118842
\(346\) 0 0
\(347\) −32.8607 −1.76405 −0.882026 0.471200i \(-0.843821\pi\)
−0.882026 + 0.471200i \(0.843821\pi\)
\(348\) 0 0
\(349\) 32.5641 1.74312 0.871558 0.490292i \(-0.163110\pi\)
0.871558 + 0.490292i \(0.163110\pi\)
\(350\) 0 0
\(351\) 11.7365 0.626448
\(352\) 0 0
\(353\) 30.6633 1.63204 0.816022 0.578021i \(-0.196175\pi\)
0.816022 + 0.578021i \(0.196175\pi\)
\(354\) 0 0
\(355\) 9.72242 0.516013
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.67740 −0.352420 −0.176210 0.984353i \(-0.556384\pi\)
−0.176210 + 0.984353i \(0.556384\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0.818644 0.0429677
\(364\) 0 0
\(365\) −16.4660 −0.861868
\(366\) 0 0
\(367\) −5.19544 −0.271200 −0.135600 0.990764i \(-0.543296\pi\)
−0.135600 + 0.990764i \(0.543296\pi\)
\(368\) 0 0
\(369\) 19.9149 1.03673
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.31769 −0.0682275 −0.0341137 0.999418i \(-0.510861\pi\)
−0.0341137 + 0.999418i \(0.510861\pi\)
\(374\) 0 0
\(375\) −30.6302 −1.58174
\(376\) 0 0
\(377\) 0.418817 0.0215701
\(378\) 0 0
\(379\) −15.6563 −0.804209 −0.402104 0.915594i \(-0.631721\pi\)
−0.402104 + 0.915594i \(0.631721\pi\)
\(380\) 0 0
\(381\) 15.3407 0.785929
\(382\) 0 0
\(383\) −29.0631 −1.48506 −0.742529 0.669814i \(-0.766374\pi\)
−0.742529 + 0.669814i \(0.766374\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −50.9639 −2.59064
\(388\) 0 0
\(389\) 10.7295 0.544006 0.272003 0.962296i \(-0.412314\pi\)
0.272003 + 0.962296i \(0.412314\pi\)
\(390\) 0 0
\(391\) −3.02603 −0.153033
\(392\) 0 0
\(393\) −40.9970 −2.06803
\(394\) 0 0
\(395\) −15.8016 −0.795063
\(396\) 0 0
\(397\) −10.7997 −0.542019 −0.271010 0.962577i \(-0.587358\pi\)
−0.271010 + 0.962577i \(0.587358\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.66332 −0.332750 −0.166375 0.986063i \(-0.553206\pi\)
−0.166375 + 0.986063i \(0.553206\pi\)
\(402\) 0 0
\(403\) 7.27053 0.362171
\(404\) 0 0
\(405\) 5.09622 0.253233
\(406\) 0 0
\(407\) 12.9740 0.643096
\(408\) 0 0
\(409\) 25.5481 1.26327 0.631636 0.775265i \(-0.282384\pi\)
0.631636 + 0.775265i \(0.282384\pi\)
\(410\) 0 0
\(411\) −9.98592 −0.492569
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7.46788 0.366584
\(416\) 0 0
\(417\) −31.4679 −1.54099
\(418\) 0 0
\(419\) −33.2415 −1.62395 −0.811977 0.583690i \(-0.801608\pi\)
−0.811977 + 0.583690i \(0.801608\pi\)
\(420\) 0 0
\(421\) 2.95094 0.143820 0.0719099 0.997411i \(-0.477091\pi\)
0.0719099 + 0.997411i \(0.477091\pi\)
\(422\) 0 0
\(423\) −13.3036 −0.646844
\(424\) 0 0
\(425\) 15.6373 0.758520
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −30.2114 −1.45862
\(430\) 0 0
\(431\) 4.29656 0.206958 0.103479 0.994632i \(-0.467003\pi\)
0.103479 + 0.994632i \(0.467003\pi\)
\(432\) 0 0
\(433\) −5.96693 −0.286752 −0.143376 0.989668i \(-0.545796\pi\)
−0.143376 + 0.989668i \(0.545796\pi\)
\(434\) 0 0
\(435\) −0.470017 −0.0225356
\(436\) 0 0
\(437\) 0.574141 0.0274649
\(438\) 0 0
\(439\) −37.5711 −1.79317 −0.896586 0.442869i \(-0.853961\pi\)
−0.896586 + 0.442869i \(0.853961\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −15.2515 −0.724623 −0.362311 0.932057i \(-0.618012\pi\)
−0.362311 + 0.932057i \(0.618012\pi\)
\(444\) 0 0
\(445\) −2.08023 −0.0986124
\(446\) 0 0
\(447\) −50.7425 −2.40004
\(448\) 0 0
\(449\) −35.7926 −1.68916 −0.844579 0.535431i \(-0.820149\pi\)
−0.844579 + 0.535431i \(0.820149\pi\)
\(450\) 0 0
\(451\) −15.2515 −0.718167
\(452\) 0 0
\(453\) 11.9528 0.561594
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.6443 0.872145 0.436073 0.899912i \(-0.356369\pi\)
0.436073 + 0.899912i \(0.356369\pi\)
\(458\) 0 0
\(459\) 18.0561 0.842787
\(460\) 0 0
\(461\) 11.8306 0.551006 0.275503 0.961300i \(-0.411156\pi\)
0.275503 + 0.961300i \(0.411156\pi\)
\(462\) 0 0
\(463\) 0.752498 0.0349715 0.0174858 0.999847i \(-0.494434\pi\)
0.0174858 + 0.999847i \(0.494434\pi\)
\(464\) 0 0
\(465\) −8.15937 −0.378382
\(466\) 0 0
\(467\) 13.8588 0.641307 0.320653 0.947197i \(-0.396098\pi\)
0.320653 + 0.947197i \(0.396098\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −26.4939 −1.22077
\(472\) 0 0
\(473\) 39.0301 1.79460
\(474\) 0 0
\(475\) −2.96693 −0.136132
\(476\) 0 0
\(477\) −26.0040 −1.19064
\(478\) 0 0
\(479\) 7.58118 0.346393 0.173197 0.984887i \(-0.444590\pi\)
0.173197 + 0.984887i \(0.444590\pi\)
\(480\) 0 0
\(481\) 13.5901 0.619657
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 25.1754 1.14315
\(486\) 0 0
\(487\) −10.6864 −0.484245 −0.242122 0.970246i \(-0.577844\pi\)
−0.242122 + 0.970246i \(0.577844\pi\)
\(488\) 0 0
\(489\) 29.5010 1.33408
\(490\) 0 0
\(491\) −14.1643 −0.639225 −0.319612 0.947548i \(-0.603553\pi\)
−0.319612 + 0.947548i \(0.603553\pi\)
\(492\) 0 0
\(493\) 0.644331 0.0290192
\(494\) 0 0
\(495\) 19.9149 0.895107
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −23.9108 −1.07040 −0.535198 0.844727i \(-0.679763\pi\)
−0.535198 + 0.844727i \(0.679763\pi\)
\(500\) 0 0
\(501\) −34.3857 −1.53624
\(502\) 0 0
\(503\) 37.4388 1.66932 0.834658 0.550769i \(-0.185665\pi\)
0.834658 + 0.550769i \(0.185665\pi\)
\(504\) 0 0
\(505\) 24.3096 1.08176
\(506\) 0 0
\(507\) 3.40687 0.151304
\(508\) 0 0
\(509\) −31.3126 −1.38790 −0.693952 0.720021i \(-0.744132\pi\)
−0.693952 + 0.720021i \(0.744132\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −3.42586 −0.151255
\(514\) 0 0
\(515\) −16.1973 −0.713740
\(516\) 0 0
\(517\) 10.1884 0.448085
\(518\) 0 0
\(519\) 60.7796 2.66793
\(520\) 0 0
\(521\) −21.8346 −0.956593 −0.478296 0.878199i \(-0.658746\pi\)
−0.478296 + 0.878199i \(0.658746\pi\)
\(522\) 0 0
\(523\) −9.37380 −0.409888 −0.204944 0.978774i \(-0.565701\pi\)
−0.204944 + 0.978774i \(0.565701\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.1854 0.487244
\(528\) 0 0
\(529\) −22.6704 −0.985668
\(530\) 0 0
\(531\) 7.35567 0.319209
\(532\) 0 0
\(533\) −15.9759 −0.691992
\(534\) 0 0
\(535\) −15.4259 −0.666918
\(536\) 0 0
\(537\) −17.3077 −0.746880
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −30.4608 −1.30961 −0.654807 0.755796i \(-0.727250\pi\)
−0.654807 + 0.755796i \(0.727250\pi\)
\(542\) 0 0
\(543\) 26.4559 1.13533
\(544\) 0 0
\(545\) 17.5061 0.749878
\(546\) 0 0
\(547\) 34.2675 1.46517 0.732587 0.680673i \(-0.238313\pi\)
0.732587 + 0.680673i \(0.238313\pi\)
\(548\) 0 0
\(549\) −52.1493 −2.22568
\(550\) 0 0
\(551\) −0.122252 −0.00520809
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −15.2515 −0.647392
\(556\) 0 0
\(557\) −1.44889 −0.0613915 −0.0306957 0.999529i \(-0.509772\pi\)
−0.0306957 + 0.999529i \(0.509772\pi\)
\(558\) 0 0
\(559\) 40.8837 1.72920
\(560\) 0 0
\(561\) −46.4790 −1.96234
\(562\) 0 0
\(563\) 10.8707 0.458146 0.229073 0.973409i \(-0.426431\pi\)
0.229073 + 0.973409i \(0.426431\pi\)
\(564\) 0 0
\(565\) 0.992958 0.0417740
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.0772 1.05129 0.525646 0.850703i \(-0.323824\pi\)
0.525646 + 0.850703i \(0.323824\pi\)
\(570\) 0 0
\(571\) −11.6042 −0.485621 −0.242811 0.970074i \(-0.578069\pi\)
−0.242811 + 0.970074i \(0.578069\pi\)
\(572\) 0 0
\(573\) −41.1242 −1.71799
\(574\) 0 0
\(575\) −1.70344 −0.0710382
\(576\) 0 0
\(577\) −26.5782 −1.10646 −0.553232 0.833027i \(-0.686606\pi\)
−0.553232 + 0.833027i \(0.686606\pi\)
\(578\) 0 0
\(579\) 42.4559 1.76441
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 19.9149 0.824789
\(584\) 0 0
\(585\) 20.8607 0.862482
\(586\) 0 0
\(587\) 16.5552 0.683304 0.341652 0.939826i \(-0.389014\pi\)
0.341652 + 0.939826i \(0.389014\pi\)
\(588\) 0 0
\(589\) −2.12225 −0.0874459
\(590\) 0 0
\(591\) −66.8937 −2.75164
\(592\) 0 0
\(593\) −26.6864 −1.09588 −0.547939 0.836519i \(-0.684587\pi\)
−0.547939 + 0.836519i \(0.684587\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.97887 −0.285626
\(598\) 0 0
\(599\) −19.6603 −0.803299 −0.401649 0.915793i \(-0.631563\pi\)
−0.401649 + 0.915793i \(0.631563\pi\)
\(600\) 0 0
\(601\) 23.1854 0.945752 0.472876 0.881129i \(-0.343216\pi\)
0.472876 + 0.881129i \(0.343216\pi\)
\(602\) 0 0
\(603\) −22.5080 −0.916596
\(604\) 0 0
\(605\) 0.432901 0.0175999
\(606\) 0 0
\(607\) −0.441848 −0.0179341 −0.00896704 0.999960i \(-0.502854\pi\)
−0.00896704 + 0.999960i \(0.502854\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 10.6723 0.431754
\(612\) 0 0
\(613\) −28.3478 −1.14496 −0.572478 0.819920i \(-0.694018\pi\)
−0.572478 + 0.819920i \(0.694018\pi\)
\(614\) 0 0
\(615\) 17.9289 0.722965
\(616\) 0 0
\(617\) 31.7455 1.27802 0.639012 0.769197i \(-0.279343\pi\)
0.639012 + 0.769197i \(0.279343\pi\)
\(618\) 0 0
\(619\) −22.2535 −0.894442 −0.447221 0.894424i \(-0.647586\pi\)
−0.447221 + 0.894424i \(0.647586\pi\)
\(620\) 0 0
\(621\) −1.96693 −0.0789301
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.36271 −0.0545085
\(626\) 0 0
\(627\) 8.81864 0.352183
\(628\) 0 0
\(629\) 20.9078 0.833649
\(630\) 0 0
\(631\) −19.4719 −0.775165 −0.387583 0.921835i \(-0.626690\pi\)
−0.387583 + 0.921835i \(0.626690\pi\)
\(632\) 0 0
\(633\) −58.7235 −2.33405
\(634\) 0 0
\(635\) 8.11221 0.321923
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −29.1193 −1.15194
\(640\) 0 0
\(641\) 40.2345 1.58917 0.794583 0.607156i \(-0.207690\pi\)
0.794583 + 0.607156i \(0.207690\pi\)
\(642\) 0 0
\(643\) 17.1011 0.674403 0.337201 0.941433i \(-0.390520\pi\)
0.337201 + 0.941433i \(0.390520\pi\)
\(644\) 0 0
\(645\) −45.8818 −1.80659
\(646\) 0 0
\(647\) 5.37380 0.211266 0.105633 0.994405i \(-0.466313\pi\)
0.105633 + 0.994405i \(0.466313\pi\)
\(648\) 0 0
\(649\) −5.63325 −0.221124
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −11.6183 −0.454659 −0.227330 0.973818i \(-0.572999\pi\)
−0.227330 + 0.973818i \(0.572999\pi\)
\(654\) 0 0
\(655\) −21.6793 −0.847081
\(656\) 0 0
\(657\) 49.3166 1.92402
\(658\) 0 0
\(659\) −41.8718 −1.63109 −0.815546 0.578692i \(-0.803563\pi\)
−0.815546 + 0.578692i \(0.803563\pi\)
\(660\) 0 0
\(661\) −6.71942 −0.261355 −0.130678 0.991425i \(-0.541715\pi\)
−0.130678 + 0.991425i \(0.541715\pi\)
\(662\) 0 0
\(663\) −48.6864 −1.89082
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −0.0701896 −0.00271775
\(668\) 0 0
\(669\) −26.6964 −1.03214
\(670\) 0 0
\(671\) 39.9379 1.54179
\(672\) 0 0
\(673\) 43.5280 1.67788 0.838941 0.544222i \(-0.183175\pi\)
0.838941 + 0.544222i \(0.183175\pi\)
\(674\) 0 0
\(675\) 10.1643 0.391223
\(676\) 0 0
\(677\) −18.1033 −0.695765 −0.347882 0.937538i \(-0.613099\pi\)
−0.347882 + 0.937538i \(0.613099\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 7.32173 0.280569
\(682\) 0 0
\(683\) 27.3456 1.04635 0.523176 0.852225i \(-0.324747\pi\)
0.523176 + 0.852225i \(0.324747\pi\)
\(684\) 0 0
\(685\) −5.28058 −0.201760
\(686\) 0 0
\(687\) −22.3708 −0.853499
\(688\) 0 0
\(689\) 20.8607 0.794728
\(690\) 0 0
\(691\) 30.8316 1.17289 0.586445 0.809989i \(-0.300527\pi\)
0.586445 + 0.809989i \(0.300527\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.6403 −0.631202
\(696\) 0 0
\(697\) −24.5782 −0.930965
\(698\) 0 0
\(699\) −39.7164 −1.50221
\(700\) 0 0
\(701\) −8.23042 −0.310859 −0.155429 0.987847i \(-0.549676\pi\)
−0.155429 + 0.987847i \(0.549676\pi\)
\(702\) 0 0
\(703\) −3.96693 −0.149615
\(704\) 0 0
\(705\) −11.9770 −0.451079
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −15.6183 −0.586558 −0.293279 0.956027i \(-0.594746\pi\)
−0.293279 + 0.956027i \(0.594746\pi\)
\(710\) 0 0
\(711\) 47.3266 1.77489
\(712\) 0 0
\(713\) −1.21847 −0.0456321
\(714\) 0 0
\(715\) −15.9759 −0.597464
\(716\) 0 0
\(717\) −78.1873 −2.91996
\(718\) 0 0
\(719\) −16.3567 −0.610002 −0.305001 0.952352i \(-0.598657\pi\)
−0.305001 + 0.952352i \(0.598657\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −11.6232 −0.432272
\(724\) 0 0
\(725\) 0.362711 0.0134708
\(726\) 0 0
\(727\) 17.6183 0.653427 0.326713 0.945123i \(-0.394059\pi\)
0.326713 + 0.945123i \(0.394059\pi\)
\(728\) 0 0
\(729\) −42.9759 −1.59170
\(730\) 0 0
\(731\) 62.8978 2.32636
\(732\) 0 0
\(733\) 42.8837 1.58395 0.791973 0.610556i \(-0.209054\pi\)
0.791973 + 0.610556i \(0.209054\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.2375 0.634950
\(738\) 0 0
\(739\) 19.4259 0.714592 0.357296 0.933991i \(-0.383699\pi\)
0.357296 + 0.933991i \(0.383699\pi\)
\(740\) 0 0
\(741\) 9.23746 0.339347
\(742\) 0 0
\(743\) 51.7305 1.89781 0.948904 0.315564i \(-0.102194\pi\)
0.948904 + 0.315564i \(0.102194\pi\)
\(744\) 0 0
\(745\) −26.8327 −0.983075
\(746\) 0 0
\(747\) −22.3668 −0.818357
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 23.9428 0.873685 0.436843 0.899538i \(-0.356097\pi\)
0.436843 + 0.899538i \(0.356097\pi\)
\(752\) 0 0
\(753\) −28.1312 −1.02516
\(754\) 0 0
\(755\) 6.32069 0.230033
\(756\) 0 0
\(757\) −2.48901 −0.0904645 −0.0452322 0.998976i \(-0.514403\pi\)
−0.0452322 + 0.998976i \(0.514403\pi\)
\(758\) 0 0
\(759\) 5.06315 0.183781
\(760\) 0 0
\(761\) 4.14528 0.150266 0.0751332 0.997174i \(-0.476062\pi\)
0.0751332 + 0.997174i \(0.476062\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 32.0932 1.16033
\(766\) 0 0
\(767\) −5.90078 −0.213065
\(768\) 0 0
\(769\) 11.0442 0.398263 0.199131 0.979973i \(-0.436188\pi\)
0.199131 + 0.979973i \(0.436188\pi\)
\(770\) 0 0
\(771\) 63.9379 2.30267
\(772\) 0 0
\(773\) 42.3426 1.52296 0.761479 0.648189i \(-0.224473\pi\)
0.761479 + 0.648189i \(0.224473\pi\)
\(774\) 0 0
\(775\) 6.29656 0.226179
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.66332 0.167081
\(780\) 0 0
\(781\) 22.3006 0.797979
\(782\) 0 0
\(783\) 0.418817 0.0149673
\(784\) 0 0
\(785\) −14.0100 −0.500040
\(786\) 0 0
\(787\) −49.5711 −1.76702 −0.883510 0.468412i \(-0.844826\pi\)
−0.883510 + 0.468412i \(0.844826\pi\)
\(788\) 0 0
\(789\) −83.4288 −2.97014
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 41.8346 1.48559
\(794\) 0 0
\(795\) −23.4109 −0.830300
\(796\) 0 0
\(797\) −38.8276 −1.37534 −0.687672 0.726022i \(-0.741367\pi\)
−0.687672 + 0.726022i \(0.741367\pi\)
\(798\) 0 0
\(799\) 16.4188 0.580856
\(800\) 0 0
\(801\) 6.23042 0.220141
\(802\) 0 0
\(803\) −37.7685 −1.33282
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −24.7295 −0.870518
\(808\) 0 0
\(809\) 13.4920 0.474354 0.237177 0.971466i \(-0.423778\pi\)
0.237177 + 0.971466i \(0.423778\pi\)
\(810\) 0 0
\(811\) −24.5552 −0.862248 −0.431124 0.902293i \(-0.641883\pi\)
−0.431124 + 0.902293i \(0.641883\pi\)
\(812\) 0 0
\(813\) 54.0412 1.89531
\(814\) 0 0
\(815\) 15.6002 0.546450
\(816\) 0 0
\(817\) −11.9339 −0.417513
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −37.9339 −1.32390 −0.661950 0.749548i \(-0.730271\pi\)
−0.661950 + 0.749548i \(0.730271\pi\)
\(822\) 0 0
\(823\) −10.0141 −0.349069 −0.174535 0.984651i \(-0.555842\pi\)
−0.174535 + 0.984651i \(0.555842\pi\)
\(824\) 0 0
\(825\) −26.1643 −0.910923
\(826\) 0 0
\(827\) −35.7685 −1.24379 −0.621896 0.783100i \(-0.713637\pi\)
−0.621896 + 0.783100i \(0.713637\pi\)
\(828\) 0 0
\(829\) −25.6753 −0.891739 −0.445869 0.895098i \(-0.647105\pi\)
−0.445869 + 0.895098i \(0.647105\pi\)
\(830\) 0 0
\(831\) 20.1122 0.697685
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −18.1833 −0.629258
\(836\) 0 0
\(837\) 7.27053 0.251306
\(838\) 0 0
\(839\) 42.7856 1.47712 0.738561 0.674187i \(-0.235506\pi\)
0.738561 + 0.674187i \(0.235506\pi\)
\(840\) 0 0
\(841\) −28.9851 −0.999485
\(842\) 0 0
\(843\) −32.9268 −1.13406
\(844\) 0 0
\(845\) 1.80156 0.0619756
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −21.6332 −0.742451
\(850\) 0 0
\(851\) −2.27758 −0.0780743
\(852\) 0 0
\(853\) −17.8447 −0.610990 −0.305495 0.952194i \(-0.598822\pi\)
−0.305495 + 0.952194i \(0.598822\pi\)
\(854\) 0 0
\(855\) −6.08918 −0.208246
\(856\) 0 0
\(857\) −37.8116 −1.29162 −0.645810 0.763498i \(-0.723480\pi\)
−0.645810 + 0.763498i \(0.723480\pi\)
\(858\) 0 0
\(859\) −22.8558 −0.779828 −0.389914 0.920851i \(-0.627495\pi\)
−0.389914 + 0.920851i \(0.627495\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 13.3177 0.453339 0.226670 0.973972i \(-0.427216\pi\)
0.226670 + 0.973972i \(0.427216\pi\)
\(864\) 0 0
\(865\) 32.1404 1.09281
\(866\) 0 0
\(867\) −29.0631 −0.987036
\(868\) 0 0
\(869\) −36.2445 −1.22951
\(870\) 0 0
\(871\) 18.0561 0.611808
\(872\) 0 0
\(873\) −75.4017 −2.55196
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −33.5571 −1.13314 −0.566571 0.824013i \(-0.691730\pi\)
−0.566571 + 0.824013i \(0.691730\pi\)
\(878\) 0 0
\(879\) 37.8387 1.27627
\(880\) 0 0
\(881\) 52.5641 1.77093 0.885465 0.464707i \(-0.153840\pi\)
0.885465 + 0.464707i \(0.153840\pi\)
\(882\) 0 0
\(883\) −5.68445 −0.191297 −0.0956484 0.995415i \(-0.530492\pi\)
−0.0956484 + 0.995415i \(0.530492\pi\)
\(884\) 0 0
\(885\) 6.62216 0.222601
\(886\) 0 0
\(887\) −26.3898 −0.886082 −0.443041 0.896501i \(-0.646100\pi\)
−0.443041 + 0.896501i \(0.646100\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 11.6894 0.391608
\(892\) 0 0
\(893\) −3.11521 −0.104247
\(894\) 0 0
\(895\) −9.15233 −0.305929
\(896\) 0 0
\(897\) 5.30361 0.177082
\(898\) 0 0
\(899\) 0.259449 0.00865309
\(900\) 0 0
\(901\) 32.0932 1.06918
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 13.9900 0.465042
\(906\) 0 0
\(907\) 0.211430 0.00702041 0.00351021 0.999994i \(-0.498883\pi\)
0.00351021 + 0.999994i \(0.498883\pi\)
\(908\) 0 0
\(909\) −72.8086 −2.41491
\(910\) 0 0
\(911\) −21.3787 −0.708308 −0.354154 0.935187i \(-0.615231\pi\)
−0.354154 + 0.935187i \(0.615231\pi\)
\(912\) 0 0
\(913\) 17.1293 0.566897
\(914\) 0 0
\(915\) −46.9490 −1.55209
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −16.8186 −0.554796 −0.277398 0.960755i \(-0.589472\pi\)
−0.277398 + 0.960755i \(0.589472\pi\)
\(920\) 0 0
\(921\) 43.3315 1.42782
\(922\) 0 0
\(923\) 23.3597 0.768894
\(924\) 0 0
\(925\) 11.7696 0.386981
\(926\) 0 0
\(927\) 48.5120 1.59334
\(928\) 0 0
\(929\) 35.3497 1.15979 0.579893 0.814693i \(-0.303095\pi\)
0.579893 + 0.814693i \(0.303095\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −4.22551 −0.138337
\(934\) 0 0
\(935\) −24.5782 −0.803793
\(936\) 0 0
\(937\) −4.78153 −0.156206 −0.0781029 0.996945i \(-0.524886\pi\)
−0.0781029 + 0.996945i \(0.524886\pi\)
\(938\) 0 0
\(939\) 79.8758 2.60665
\(940\) 0 0
\(941\) 13.6373 0.444563 0.222281 0.974983i \(-0.428650\pi\)
0.222281 + 0.974983i \(0.428650\pi\)
\(942\) 0 0
\(943\) 2.67740 0.0871883
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −59.3638 −1.92906 −0.964531 0.263968i \(-0.914969\pi\)
−0.964531 + 0.263968i \(0.914969\pi\)
\(948\) 0 0
\(949\) −39.5622 −1.28424
\(950\) 0 0
\(951\) 60.6392 1.96636
\(952\) 0 0
\(953\) −52.4790 −1.69996 −0.849980 0.526815i \(-0.823386\pi\)
−0.849980 + 0.526815i \(0.823386\pi\)
\(954\) 0 0
\(955\) −21.7465 −0.703702
\(956\) 0 0
\(957\) −1.07809 −0.0348498
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −26.4960 −0.854711
\(962\) 0 0
\(963\) 46.2014 1.48882
\(964\) 0 0
\(965\) 22.4508 0.722717
\(966\) 0 0
\(967\) 21.9900 0.707149 0.353575 0.935406i \(-0.384966\pi\)
0.353575 + 0.935406i \(0.384966\pi\)
\(968\) 0 0
\(969\) 14.2114 0.456537
\(970\) 0 0
\(971\) 40.9309 1.31353 0.656767 0.754094i \(-0.271924\pi\)
0.656767 + 0.754094i \(0.271924\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −27.4069 −0.877722
\(976\) 0 0
\(977\) 8.56924 0.274154 0.137077 0.990560i \(-0.456229\pi\)
0.137077 + 0.990560i \(0.456229\pi\)
\(978\) 0 0
\(979\) −4.77149 −0.152497
\(980\) 0 0
\(981\) −52.4318 −1.67402
\(982\) 0 0
\(983\) −35.2654 −1.12479 −0.562396 0.826868i \(-0.690120\pi\)
−0.562396 + 0.826868i \(0.690120\pi\)
\(984\) 0 0
\(985\) −35.3736 −1.12710
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.85172 −0.217872
\(990\) 0 0
\(991\) 1.78367 0.0566600 0.0283300 0.999599i \(-0.490981\pi\)
0.0283300 + 0.999599i \(0.490981\pi\)
\(992\) 0 0
\(993\) −76.5391 −2.42890
\(994\) 0 0
\(995\) −3.69044 −0.116995
\(996\) 0 0
\(997\) 37.4299 1.18542 0.592708 0.805417i \(-0.298059\pi\)
0.592708 + 0.805417i \(0.298059\pi\)
\(998\) 0 0
\(999\) 13.5901 0.429973
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 3724.2.a.i.1.1 3
7.6 odd 2 532.2.a.e.1.3 3
21.20 even 2 4788.2.a.o.1.2 3
28.27 even 2 2128.2.a.r.1.1 3
56.13 odd 2 8512.2.a.bn.1.1 3
56.27 even 2 8512.2.a.bl.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
532.2.a.e.1.3 3 7.6 odd 2
2128.2.a.r.1.1 3 28.27 even 2
3724.2.a.i.1.1 3 1.1 even 1 trivial
4788.2.a.o.1.2 3 21.20 even 2
8512.2.a.bl.1.3 3 56.27 even 2
8512.2.a.bn.1.1 3 56.13 odd 2