Properties

Label 6864.2.a.bm.1.1
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6864,2,Mod(1,6864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6864, 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("6864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.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: \(54.8093159474\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.732051 q^{5} +3.46410 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.732051 q^{5} +3.46410 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.732051 q^{15} -2.73205 q^{17} +2.00000 q^{19} +3.46410 q^{21} -6.00000 q^{23} -4.46410 q^{25} +1.00000 q^{27} -6.73205 q^{29} -5.26795 q^{31} -1.00000 q^{33} -2.53590 q^{35} -2.00000 q^{37} +1.00000 q^{39} -10.9282 q^{41} -3.26795 q^{43} -0.732051 q^{45} +1.46410 q^{47} +5.00000 q^{49} -2.73205 q^{51} -6.00000 q^{53} +0.732051 q^{55} +2.00000 q^{57} -5.46410 q^{59} +4.92820 q^{61} +3.46410 q^{63} -0.732051 q^{65} -8.19615 q^{67} -6.00000 q^{69} -4.00000 q^{71} -4.00000 q^{73} -4.46410 q^{75} -3.46410 q^{77} +3.26795 q^{79} +1.00000 q^{81} -10.9282 q^{83} +2.00000 q^{85} -6.73205 q^{87} +11.6603 q^{89} +3.46410 q^{91} -5.26795 q^{93} -1.46410 q^{95} +15.8564 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} - 2 q^{17} + 4 q^{19} - 12 q^{23} - 2 q^{25} + 2 q^{27} - 10 q^{29} - 14 q^{31} - 2 q^{33} - 12 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} - 10 q^{43} + 2 q^{45} - 4 q^{47} + 10 q^{49} - 2 q^{51} - 12 q^{53} - 2 q^{55} + 4 q^{57} - 4 q^{59} - 4 q^{61} + 2 q^{65} - 6 q^{67} - 12 q^{69} - 8 q^{71} - 8 q^{73} - 2 q^{75} + 10 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} - 10 q^{87} + 6 q^{89} - 14 q^{93} + 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(5\) −0.732051 −0.327383 −0.163692 0.986512i \(-0.552340\pi\)
−0.163692 + 0.986512i \(0.552340\pi\)
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −0.732051 −0.189015
\(16\) 0 0
\(17\) −2.73205 −0.662620 −0.331310 0.943522i \(-0.607491\pi\)
−0.331310 + 0.943522i \(0.607491\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −4.46410 −0.892820
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.73205 −1.25011 −0.625055 0.780581i \(-0.714924\pi\)
−0.625055 + 0.780581i \(0.714924\pi\)
\(30\) 0 0
\(31\) −5.26795 −0.946152 −0.473076 0.881022i \(-0.656856\pi\)
−0.473076 + 0.881022i \(0.656856\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −2.53590 −0.428645
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −10.9282 −1.70670 −0.853349 0.521340i \(-0.825432\pi\)
−0.853349 + 0.521340i \(0.825432\pi\)
\(42\) 0 0
\(43\) −3.26795 −0.498358 −0.249179 0.968458i \(-0.580161\pi\)
−0.249179 + 0.968458i \(0.580161\pi\)
\(44\) 0 0
\(45\) −0.732051 −0.109128
\(46\) 0 0
\(47\) 1.46410 0.213561 0.106781 0.994283i \(-0.465946\pi\)
0.106781 + 0.994283i \(0.465946\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) −2.73205 −0.382564
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0.732051 0.0987097
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −5.46410 −0.711365 −0.355683 0.934607i \(-0.615752\pi\)
−0.355683 + 0.934607i \(0.615752\pi\)
\(60\) 0 0
\(61\) 4.92820 0.630992 0.315496 0.948927i \(-0.397829\pi\)
0.315496 + 0.948927i \(0.397829\pi\)
\(62\) 0 0
\(63\) 3.46410 0.436436
\(64\) 0 0
\(65\) −0.732051 −0.0907997
\(66\) 0 0
\(67\) −8.19615 −1.00132 −0.500660 0.865644i \(-0.666909\pi\)
−0.500660 + 0.865644i \(0.666909\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) −4.46410 −0.515470
\(76\) 0 0
\(77\) −3.46410 −0.394771
\(78\) 0 0
\(79\) 3.26795 0.367673 0.183837 0.982957i \(-0.441148\pi\)
0.183837 + 0.982957i \(0.441148\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.9282 −1.19953 −0.599763 0.800178i \(-0.704739\pi\)
−0.599763 + 0.800178i \(0.704739\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) −6.73205 −0.721752
\(88\) 0 0
\(89\) 11.6603 1.23598 0.617992 0.786184i \(-0.287946\pi\)
0.617992 + 0.786184i \(0.287946\pi\)
\(90\) 0 0
\(91\) 3.46410 0.363137
\(92\) 0 0
\(93\) −5.26795 −0.546261
\(94\) 0 0
\(95\) −1.46410 −0.150214
\(96\) 0 0
\(97\) 15.8564 1.60997 0.804987 0.593292i \(-0.202172\pi\)
0.804987 + 0.593292i \(0.202172\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 10.7321 1.06788 0.533939 0.845523i \(-0.320711\pi\)
0.533939 + 0.845523i \(0.320711\pi\)
\(102\) 0 0
\(103\) 5.46410 0.538394 0.269197 0.963085i \(-0.413242\pi\)
0.269197 + 0.963085i \(0.413242\pi\)
\(104\) 0 0
\(105\) −2.53590 −0.247478
\(106\) 0 0
\(107\) −4.39230 −0.424620 −0.212310 0.977202i \(-0.568099\pi\)
−0.212310 + 0.977202i \(0.568099\pi\)
\(108\) 0 0
\(109\) −17.8564 −1.71033 −0.855167 0.518353i \(-0.826545\pi\)
−0.855167 + 0.518353i \(0.826545\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 4.39230 0.409585
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −9.46410 −0.867573
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −10.9282 −0.985363
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −10.5885 −0.939574 −0.469787 0.882780i \(-0.655669\pi\)
−0.469787 + 0.882780i \(0.655669\pi\)
\(128\) 0 0
\(129\) −3.26795 −0.287727
\(130\) 0 0
\(131\) −1.07180 −0.0936433 −0.0468217 0.998903i \(-0.514909\pi\)
−0.0468217 + 0.998903i \(0.514909\pi\)
\(132\) 0 0
\(133\) 6.92820 0.600751
\(134\) 0 0
\(135\) −0.732051 −0.0630049
\(136\) 0 0
\(137\) 1.80385 0.154113 0.0770565 0.997027i \(-0.475448\pi\)
0.0770565 + 0.997027i \(0.475448\pi\)
\(138\) 0 0
\(139\) 20.0526 1.70084 0.850418 0.526108i \(-0.176349\pi\)
0.850418 + 0.526108i \(0.176349\pi\)
\(140\) 0 0
\(141\) 1.46410 0.123300
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 4.92820 0.409265
\(146\) 0 0
\(147\) 5.00000 0.412393
\(148\) 0 0
\(149\) 2.53590 0.207749 0.103874 0.994590i \(-0.466876\pi\)
0.103874 + 0.994590i \(0.466876\pi\)
\(150\) 0 0
\(151\) −15.8564 −1.29038 −0.645188 0.764024i \(-0.723221\pi\)
−0.645188 + 0.764024i \(0.723221\pi\)
\(152\) 0 0
\(153\) −2.73205 −0.220873
\(154\) 0 0
\(155\) 3.85641 0.309754
\(156\) 0 0
\(157\) 23.3205 1.86118 0.930590 0.366064i \(-0.119295\pi\)
0.930590 + 0.366064i \(0.119295\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −20.7846 −1.63806
\(162\) 0 0
\(163\) 5.66025 0.443345 0.221673 0.975121i \(-0.428848\pi\)
0.221673 + 0.975121i \(0.428848\pi\)
\(164\) 0 0
\(165\) 0.732051 0.0569901
\(166\) 0 0
\(167\) −17.8564 −1.38177 −0.690885 0.722965i \(-0.742779\pi\)
−0.690885 + 0.722965i \(0.742779\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) 5.66025 0.430341 0.215171 0.976576i \(-0.430969\pi\)
0.215171 + 0.976576i \(0.430969\pi\)
\(174\) 0 0
\(175\) −15.4641 −1.16898
\(176\) 0 0
\(177\) −5.46410 −0.410707
\(178\) 0 0
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) 5.46410 0.406143 0.203072 0.979164i \(-0.434908\pi\)
0.203072 + 0.979164i \(0.434908\pi\)
\(182\) 0 0
\(183\) 4.92820 0.364303
\(184\) 0 0
\(185\) 1.46410 0.107643
\(186\) 0 0
\(187\) 2.73205 0.199787
\(188\) 0 0
\(189\) 3.46410 0.251976
\(190\) 0 0
\(191\) −26.9282 −1.94846 −0.974228 0.225565i \(-0.927577\pi\)
−0.974228 + 0.225565i \(0.927577\pi\)
\(192\) 0 0
\(193\) −10.9282 −0.786629 −0.393315 0.919404i \(-0.628672\pi\)
−0.393315 + 0.919404i \(0.628672\pi\)
\(194\) 0 0
\(195\) −0.732051 −0.0524232
\(196\) 0 0
\(197\) 19.3205 1.37653 0.688265 0.725460i \(-0.258373\pi\)
0.688265 + 0.725460i \(0.258373\pi\)
\(198\) 0 0
\(199\) 19.3205 1.36959 0.684797 0.728734i \(-0.259891\pi\)
0.684797 + 0.728734i \(0.259891\pi\)
\(200\) 0 0
\(201\) −8.19615 −0.578112
\(202\) 0 0
\(203\) −23.3205 −1.63678
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −9.80385 −0.674925 −0.337462 0.941339i \(-0.609569\pi\)
−0.337462 + 0.941339i \(0.609569\pi\)
\(212\) 0 0
\(213\) −4.00000 −0.274075
\(214\) 0 0
\(215\) 2.39230 0.163154
\(216\) 0 0
\(217\) −18.2487 −1.23880
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −2.73205 −0.183778
\(222\) 0 0
\(223\) −3.80385 −0.254724 −0.127362 0.991856i \(-0.540651\pi\)
−0.127362 + 0.991856i \(0.540651\pi\)
\(224\) 0 0
\(225\) −4.46410 −0.297607
\(226\) 0 0
\(227\) −0.535898 −0.0355688 −0.0177844 0.999842i \(-0.505661\pi\)
−0.0177844 + 0.999842i \(0.505661\pi\)
\(228\) 0 0
\(229\) −12.5359 −0.828395 −0.414198 0.910187i \(-0.635938\pi\)
−0.414198 + 0.910187i \(0.635938\pi\)
\(230\) 0 0
\(231\) −3.46410 −0.227921
\(232\) 0 0
\(233\) −11.1244 −0.728781 −0.364390 0.931246i \(-0.618723\pi\)
−0.364390 + 0.931246i \(0.618723\pi\)
\(234\) 0 0
\(235\) −1.07180 −0.0699163
\(236\) 0 0
\(237\) 3.26795 0.212276
\(238\) 0 0
\(239\) 10.3923 0.672222 0.336111 0.941822i \(-0.390888\pi\)
0.336111 + 0.941822i \(0.390888\pi\)
\(240\) 0 0
\(241\) 18.7846 1.21002 0.605012 0.796217i \(-0.293168\pi\)
0.605012 + 0.796217i \(0.293168\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −3.66025 −0.233845
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) −10.9282 −0.692547
\(250\) 0 0
\(251\) 1.07180 0.0676512 0.0338256 0.999428i \(-0.489231\pi\)
0.0338256 + 0.999428i \(0.489231\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 0 0
\(257\) 11.4641 0.715111 0.357556 0.933892i \(-0.383610\pi\)
0.357556 + 0.933892i \(0.383610\pi\)
\(258\) 0 0
\(259\) −6.92820 −0.430498
\(260\) 0 0
\(261\) −6.73205 −0.416703
\(262\) 0 0
\(263\) 23.3205 1.43800 0.719002 0.695008i \(-0.244599\pi\)
0.719002 + 0.695008i \(0.244599\pi\)
\(264\) 0 0
\(265\) 4.39230 0.269817
\(266\) 0 0
\(267\) 11.6603 0.713596
\(268\) 0 0
\(269\) −14.3923 −0.877514 −0.438757 0.898606i \(-0.644581\pi\)
−0.438757 + 0.898606i \(0.644581\pi\)
\(270\) 0 0
\(271\) −12.9282 −0.785332 −0.392666 0.919681i \(-0.628447\pi\)
−0.392666 + 0.919681i \(0.628447\pi\)
\(272\) 0 0
\(273\) 3.46410 0.209657
\(274\) 0 0
\(275\) 4.46410 0.269195
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) −5.26795 −0.315384
\(280\) 0 0
\(281\) 25.8564 1.54246 0.771232 0.636554i \(-0.219641\pi\)
0.771232 + 0.636554i \(0.219641\pi\)
\(282\) 0 0
\(283\) 1.80385 0.107228 0.0536138 0.998562i \(-0.482926\pi\)
0.0536138 + 0.998562i \(0.482926\pi\)
\(284\) 0 0
\(285\) −1.46410 −0.0867259
\(286\) 0 0
\(287\) −37.8564 −2.23459
\(288\) 0 0
\(289\) −9.53590 −0.560935
\(290\) 0 0
\(291\) 15.8564 0.929519
\(292\) 0 0
\(293\) 21.4641 1.25395 0.626973 0.779041i \(-0.284294\pi\)
0.626973 + 0.779041i \(0.284294\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −11.3205 −0.652503
\(302\) 0 0
\(303\) 10.7321 0.616540
\(304\) 0 0
\(305\) −3.60770 −0.206576
\(306\) 0 0
\(307\) 30.7846 1.75697 0.878485 0.477769i \(-0.158555\pi\)
0.878485 + 0.477769i \(0.158555\pi\)
\(308\) 0 0
\(309\) 5.46410 0.310842
\(310\) 0 0
\(311\) −12.9282 −0.733091 −0.366546 0.930400i \(-0.619460\pi\)
−0.366546 + 0.930400i \(0.619460\pi\)
\(312\) 0 0
\(313\) −1.46410 −0.0827559 −0.0413780 0.999144i \(-0.513175\pi\)
−0.0413780 + 0.999144i \(0.513175\pi\)
\(314\) 0 0
\(315\) −2.53590 −0.142882
\(316\) 0 0
\(317\) −3.26795 −0.183546 −0.0917732 0.995780i \(-0.529253\pi\)
−0.0917732 + 0.995780i \(0.529253\pi\)
\(318\) 0 0
\(319\) 6.73205 0.376922
\(320\) 0 0
\(321\) −4.39230 −0.245155
\(322\) 0 0
\(323\) −5.46410 −0.304031
\(324\) 0 0
\(325\) −4.46410 −0.247624
\(326\) 0 0
\(327\) −17.8564 −0.987462
\(328\) 0 0
\(329\) 5.07180 0.279617
\(330\) 0 0
\(331\) 18.7321 1.02961 0.514803 0.857308i \(-0.327865\pi\)
0.514803 + 0.857308i \(0.327865\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −27.1769 −1.48042 −0.740210 0.672375i \(-0.765274\pi\)
−0.740210 + 0.672375i \(0.765274\pi\)
\(338\) 0 0
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) 5.26795 0.285275
\(342\) 0 0
\(343\) −6.92820 −0.374088
\(344\) 0 0
\(345\) 4.39230 0.236474
\(346\) 0 0
\(347\) 4.39230 0.235791 0.117896 0.993026i \(-0.462385\pi\)
0.117896 + 0.993026i \(0.462385\pi\)
\(348\) 0 0
\(349\) 0.928203 0.0496856 0.0248428 0.999691i \(-0.492091\pi\)
0.0248428 + 0.999691i \(0.492091\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −26.9808 −1.43604 −0.718021 0.696022i \(-0.754952\pi\)
−0.718021 + 0.696022i \(0.754952\pi\)
\(354\) 0 0
\(355\) 2.92820 0.155413
\(356\) 0 0
\(357\) −9.46410 −0.500893
\(358\) 0 0
\(359\) −7.46410 −0.393940 −0.196970 0.980409i \(-0.563110\pi\)
−0.196970 + 0.980409i \(0.563110\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 2.92820 0.153269
\(366\) 0 0
\(367\) −28.7846 −1.50254 −0.751272 0.659992i \(-0.770559\pi\)
−0.751272 + 0.659992i \(0.770559\pi\)
\(368\) 0 0
\(369\) −10.9282 −0.568900
\(370\) 0 0
\(371\) −20.7846 −1.07908
\(372\) 0 0
\(373\) −11.4641 −0.593589 −0.296794 0.954941i \(-0.595918\pi\)
−0.296794 + 0.954941i \(0.595918\pi\)
\(374\) 0 0
\(375\) 6.92820 0.357771
\(376\) 0 0
\(377\) −6.73205 −0.346718
\(378\) 0 0
\(379\) −8.58846 −0.441159 −0.220580 0.975369i \(-0.570795\pi\)
−0.220580 + 0.975369i \(0.570795\pi\)
\(380\) 0 0
\(381\) −10.5885 −0.542463
\(382\) 0 0
\(383\) −36.3923 −1.85956 −0.929780 0.368116i \(-0.880003\pi\)
−0.929780 + 0.368116i \(0.880003\pi\)
\(384\) 0 0
\(385\) 2.53590 0.129241
\(386\) 0 0
\(387\) −3.26795 −0.166119
\(388\) 0 0
\(389\) 6.39230 0.324103 0.162051 0.986782i \(-0.448189\pi\)
0.162051 + 0.986782i \(0.448189\pi\)
\(390\) 0 0
\(391\) 16.3923 0.828994
\(392\) 0 0
\(393\) −1.07180 −0.0540650
\(394\) 0 0
\(395\) −2.39230 −0.120370
\(396\) 0 0
\(397\) 32.9282 1.65262 0.826310 0.563216i \(-0.190436\pi\)
0.826310 + 0.563216i \(0.190436\pi\)
\(398\) 0 0
\(399\) 6.92820 0.346844
\(400\) 0 0
\(401\) 20.7321 1.03531 0.517655 0.855590i \(-0.326805\pi\)
0.517655 + 0.855590i \(0.326805\pi\)
\(402\) 0 0
\(403\) −5.26795 −0.262415
\(404\) 0 0
\(405\) −0.732051 −0.0363759
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) 1.07180 0.0529969 0.0264985 0.999649i \(-0.491564\pi\)
0.0264985 + 0.999649i \(0.491564\pi\)
\(410\) 0 0
\(411\) 1.80385 0.0889772
\(412\) 0 0
\(413\) −18.9282 −0.931396
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) 20.0526 0.981978
\(418\) 0 0
\(419\) 17.7128 0.865328 0.432664 0.901555i \(-0.357574\pi\)
0.432664 + 0.901555i \(0.357574\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 1.46410 0.0711871
\(424\) 0 0
\(425\) 12.1962 0.591600
\(426\) 0 0
\(427\) 17.0718 0.826162
\(428\) 0 0
\(429\) −1.00000 −0.0482805
\(430\) 0 0
\(431\) −20.7846 −1.00116 −0.500580 0.865690i \(-0.666880\pi\)
−0.500580 + 0.865690i \(0.666880\pi\)
\(432\) 0 0
\(433\) −3.85641 −0.185327 −0.0926635 0.995697i \(-0.529538\pi\)
−0.0926635 + 0.995697i \(0.529538\pi\)
\(434\) 0 0
\(435\) 4.92820 0.236289
\(436\) 0 0
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) 4.73205 0.225848 0.112924 0.993604i \(-0.463978\pi\)
0.112924 + 0.993604i \(0.463978\pi\)
\(440\) 0 0
\(441\) 5.00000 0.238095
\(442\) 0 0
\(443\) 1.85641 0.0882005 0.0441003 0.999027i \(-0.485958\pi\)
0.0441003 + 0.999027i \(0.485958\pi\)
\(444\) 0 0
\(445\) −8.53590 −0.404640
\(446\) 0 0
\(447\) 2.53590 0.119944
\(448\) 0 0
\(449\) 14.1962 0.669958 0.334979 0.942226i \(-0.391271\pi\)
0.334979 + 0.942226i \(0.391271\pi\)
\(450\) 0 0
\(451\) 10.9282 0.514589
\(452\) 0 0
\(453\) −15.8564 −0.744999
\(454\) 0 0
\(455\) −2.53590 −0.118885
\(456\) 0 0
\(457\) 14.7846 0.691595 0.345797 0.938309i \(-0.387608\pi\)
0.345797 + 0.938309i \(0.387608\pi\)
\(458\) 0 0
\(459\) −2.73205 −0.127521
\(460\) 0 0
\(461\) 39.7128 1.84961 0.924805 0.380441i \(-0.124228\pi\)
0.924805 + 0.380441i \(0.124228\pi\)
\(462\) 0 0
\(463\) −22.0526 −1.02487 −0.512435 0.858726i \(-0.671256\pi\)
−0.512435 + 0.858726i \(0.671256\pi\)
\(464\) 0 0
\(465\) 3.85641 0.178837
\(466\) 0 0
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 0 0
\(469\) −28.3923 −1.31103
\(470\) 0 0
\(471\) 23.3205 1.07455
\(472\) 0 0
\(473\) 3.26795 0.150260
\(474\) 0 0
\(475\) −8.92820 −0.409654
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −5.32051 −0.243100 −0.121550 0.992585i \(-0.538787\pi\)
−0.121550 + 0.992585i \(0.538787\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) −20.7846 −0.945732
\(484\) 0 0
\(485\) −11.6077 −0.527078
\(486\) 0 0
\(487\) −34.4449 −1.56085 −0.780423 0.625252i \(-0.784996\pi\)
−0.780423 + 0.625252i \(0.784996\pi\)
\(488\) 0 0
\(489\) 5.66025 0.255966
\(490\) 0 0
\(491\) −1.46410 −0.0660740 −0.0330370 0.999454i \(-0.510518\pi\)
−0.0330370 + 0.999454i \(0.510518\pi\)
\(492\) 0 0
\(493\) 18.3923 0.828348
\(494\) 0 0
\(495\) 0.732051 0.0329032
\(496\) 0 0
\(497\) −13.8564 −0.621545
\(498\) 0 0
\(499\) 23.8038 1.06561 0.532803 0.846239i \(-0.321139\pi\)
0.532803 + 0.846239i \(0.321139\pi\)
\(500\) 0 0
\(501\) −17.8564 −0.797765
\(502\) 0 0
\(503\) −30.5359 −1.36153 −0.680764 0.732503i \(-0.738352\pi\)
−0.680764 + 0.732503i \(0.738352\pi\)
\(504\) 0 0
\(505\) −7.85641 −0.349605
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −18.5885 −0.823919 −0.411959 0.911202i \(-0.635155\pi\)
−0.411959 + 0.911202i \(0.635155\pi\)
\(510\) 0 0
\(511\) −13.8564 −0.612971
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −1.46410 −0.0643911
\(518\) 0 0
\(519\) 5.66025 0.248458
\(520\) 0 0
\(521\) 18.7846 0.822969 0.411484 0.911417i \(-0.365010\pi\)
0.411484 + 0.911417i \(0.365010\pi\)
\(522\) 0 0
\(523\) −6.58846 −0.288093 −0.144047 0.989571i \(-0.546012\pi\)
−0.144047 + 0.989571i \(0.546012\pi\)
\(524\) 0 0
\(525\) −15.4641 −0.674909
\(526\) 0 0
\(527\) 14.3923 0.626939
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −5.46410 −0.237122
\(532\) 0 0
\(533\) −10.9282 −0.473353
\(534\) 0 0
\(535\) 3.21539 0.139013
\(536\) 0 0
\(537\) −2.00000 −0.0863064
\(538\) 0 0
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −18.7846 −0.807613 −0.403807 0.914844i \(-0.632313\pi\)
−0.403807 + 0.914844i \(0.632313\pi\)
\(542\) 0 0
\(543\) 5.46410 0.234487
\(544\) 0 0
\(545\) 13.0718 0.559934
\(546\) 0 0
\(547\) 20.7321 0.886438 0.443219 0.896413i \(-0.353836\pi\)
0.443219 + 0.896413i \(0.353836\pi\)
\(548\) 0 0
\(549\) 4.92820 0.210331
\(550\) 0 0
\(551\) −13.4641 −0.573590
\(552\) 0 0
\(553\) 11.3205 0.481397
\(554\) 0 0
\(555\) 1.46410 0.0621477
\(556\) 0 0
\(557\) −7.60770 −0.322348 −0.161174 0.986926i \(-0.551528\pi\)
−0.161174 + 0.986926i \(0.551528\pi\)
\(558\) 0 0
\(559\) −3.26795 −0.138220
\(560\) 0 0
\(561\) 2.73205 0.115347
\(562\) 0 0
\(563\) −13.8564 −0.583978 −0.291989 0.956422i \(-0.594317\pi\)
−0.291989 + 0.956422i \(0.594317\pi\)
\(564\) 0 0
\(565\) −7.32051 −0.307976
\(566\) 0 0
\(567\) 3.46410 0.145479
\(568\) 0 0
\(569\) −40.9808 −1.71800 −0.859001 0.511973i \(-0.828915\pi\)
−0.859001 + 0.511973i \(0.828915\pi\)
\(570\) 0 0
\(571\) 23.2679 0.973733 0.486867 0.873476i \(-0.338140\pi\)
0.486867 + 0.873476i \(0.338140\pi\)
\(572\) 0 0
\(573\) −26.9282 −1.12494
\(574\) 0 0
\(575\) 26.7846 1.11700
\(576\) 0 0
\(577\) 2.78461 0.115925 0.0579624 0.998319i \(-0.481540\pi\)
0.0579624 + 0.998319i \(0.481540\pi\)
\(578\) 0 0
\(579\) −10.9282 −0.454161
\(580\) 0 0
\(581\) −37.8564 −1.57055
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) 0 0
\(585\) −0.732051 −0.0302666
\(586\) 0 0
\(587\) 19.6077 0.809296 0.404648 0.914472i \(-0.367394\pi\)
0.404648 + 0.914472i \(0.367394\pi\)
\(588\) 0 0
\(589\) −10.5359 −0.434124
\(590\) 0 0
\(591\) 19.3205 0.794740
\(592\) 0 0
\(593\) 33.4641 1.37421 0.687103 0.726560i \(-0.258882\pi\)
0.687103 + 0.726560i \(0.258882\pi\)
\(594\) 0 0
\(595\) 6.92820 0.284029
\(596\) 0 0
\(597\) 19.3205 0.790736
\(598\) 0 0
\(599\) 22.6410 0.925087 0.462543 0.886597i \(-0.346937\pi\)
0.462543 + 0.886597i \(0.346937\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) −8.19615 −0.333773
\(604\) 0 0
\(605\) −0.732051 −0.0297621
\(606\) 0 0
\(607\) −27.3731 −1.11104 −0.555519 0.831504i \(-0.687480\pi\)
−0.555519 + 0.831504i \(0.687480\pi\)
\(608\) 0 0
\(609\) −23.3205 −0.944995
\(610\) 0 0
\(611\) 1.46410 0.0592312
\(612\) 0 0
\(613\) 8.78461 0.354807 0.177404 0.984138i \(-0.443230\pi\)
0.177404 + 0.984138i \(0.443230\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) −8.73205 −0.351539 −0.175770 0.984431i \(-0.556241\pi\)
−0.175770 + 0.984431i \(0.556241\pi\)
\(618\) 0 0
\(619\) 18.3397 0.737137 0.368568 0.929601i \(-0.379848\pi\)
0.368568 + 0.929601i \(0.379848\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) 40.3923 1.61828
\(624\) 0 0
\(625\) 17.2487 0.689948
\(626\) 0 0
\(627\) −2.00000 −0.0798723
\(628\) 0 0
\(629\) 5.46410 0.217868
\(630\) 0 0
\(631\) 18.4449 0.734278 0.367139 0.930166i \(-0.380337\pi\)
0.367139 + 0.930166i \(0.380337\pi\)
\(632\) 0 0
\(633\) −9.80385 −0.389668
\(634\) 0 0
\(635\) 7.75129 0.307601
\(636\) 0 0
\(637\) 5.00000 0.198107
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) −36.9282 −1.45858 −0.729288 0.684207i \(-0.760149\pi\)
−0.729288 + 0.684207i \(0.760149\pi\)
\(642\) 0 0
\(643\) −6.73205 −0.265486 −0.132743 0.991150i \(-0.542379\pi\)
−0.132743 + 0.991150i \(0.542379\pi\)
\(644\) 0 0
\(645\) 2.39230 0.0941969
\(646\) 0 0
\(647\) 5.85641 0.230239 0.115120 0.993352i \(-0.463275\pi\)
0.115120 + 0.993352i \(0.463275\pi\)
\(648\) 0 0
\(649\) 5.46410 0.214485
\(650\) 0 0
\(651\) −18.2487 −0.715223
\(652\) 0 0
\(653\) 1.60770 0.0629140 0.0314570 0.999505i \(-0.489985\pi\)
0.0314570 + 0.999505i \(0.489985\pi\)
\(654\) 0 0
\(655\) 0.784610 0.0306572
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −6.14359 −0.239320 −0.119660 0.992815i \(-0.538181\pi\)
−0.119660 + 0.992815i \(0.538181\pi\)
\(660\) 0 0
\(661\) −44.2487 −1.72108 −0.860538 0.509387i \(-0.829872\pi\)
−0.860538 + 0.509387i \(0.829872\pi\)
\(662\) 0 0
\(663\) −2.73205 −0.106104
\(664\) 0 0
\(665\) −5.07180 −0.196676
\(666\) 0 0
\(667\) 40.3923 1.56400
\(668\) 0 0
\(669\) −3.80385 −0.147065
\(670\) 0 0
\(671\) −4.92820 −0.190251
\(672\) 0 0
\(673\) 9.60770 0.370349 0.185175 0.982706i \(-0.440715\pi\)
0.185175 + 0.982706i \(0.440715\pi\)
\(674\) 0 0
\(675\) −4.46410 −0.171823
\(676\) 0 0
\(677\) −41.2679 −1.58606 −0.793028 0.609185i \(-0.791497\pi\)
−0.793028 + 0.609185i \(0.791497\pi\)
\(678\) 0 0
\(679\) 54.9282 2.10795
\(680\) 0 0
\(681\) −0.535898 −0.0205357
\(682\) 0 0
\(683\) 20.7846 0.795301 0.397650 0.917537i \(-0.369826\pi\)
0.397650 + 0.917537i \(0.369826\pi\)
\(684\) 0 0
\(685\) −1.32051 −0.0504540
\(686\) 0 0
\(687\) −12.5359 −0.478274
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 45.6603 1.73700 0.868499 0.495691i \(-0.165085\pi\)
0.868499 + 0.495691i \(0.165085\pi\)
\(692\) 0 0
\(693\) −3.46410 −0.131590
\(694\) 0 0
\(695\) −14.6795 −0.556825
\(696\) 0 0
\(697\) 29.8564 1.13089
\(698\) 0 0
\(699\) −11.1244 −0.420762
\(700\) 0 0
\(701\) −44.1962 −1.66927 −0.834633 0.550807i \(-0.814320\pi\)
−0.834633 + 0.550807i \(0.814320\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) −1.07180 −0.0403662
\(706\) 0 0
\(707\) 37.1769 1.39818
\(708\) 0 0
\(709\) 44.2487 1.66180 0.830898 0.556425i \(-0.187827\pi\)
0.830898 + 0.556425i \(0.187827\pi\)
\(710\) 0 0
\(711\) 3.26795 0.122558
\(712\) 0 0
\(713\) 31.6077 1.18372
\(714\) 0 0
\(715\) 0.732051 0.0273771
\(716\) 0 0
\(717\) 10.3923 0.388108
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 18.9282 0.704923
\(722\) 0 0
\(723\) 18.7846 0.698607
\(724\) 0 0
\(725\) 30.0526 1.11612
\(726\) 0 0
\(727\) −39.3205 −1.45832 −0.729158 0.684345i \(-0.760088\pi\)
−0.729158 + 0.684345i \(0.760088\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.92820 0.330222
\(732\) 0 0
\(733\) 2.92820 0.108156 0.0540778 0.998537i \(-0.482778\pi\)
0.0540778 + 0.998537i \(0.482778\pi\)
\(734\) 0 0
\(735\) −3.66025 −0.135011
\(736\) 0 0
\(737\) 8.19615 0.301909
\(738\) 0 0
\(739\) −4.53590 −0.166856 −0.0834279 0.996514i \(-0.526587\pi\)
−0.0834279 + 0.996514i \(0.526587\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 21.0718 0.773049 0.386525 0.922279i \(-0.373675\pi\)
0.386525 + 0.922279i \(0.373675\pi\)
\(744\) 0 0
\(745\) −1.85641 −0.0680135
\(746\) 0 0
\(747\) −10.9282 −0.399842
\(748\) 0 0
\(749\) −15.2154 −0.555958
\(750\) 0 0
\(751\) 6.14359 0.224183 0.112091 0.993698i \(-0.464245\pi\)
0.112091 + 0.993698i \(0.464245\pi\)
\(752\) 0 0
\(753\) 1.07180 0.0390584
\(754\) 0 0
\(755\) 11.6077 0.422447
\(756\) 0 0
\(757\) −6.53590 −0.237551 −0.118776 0.992921i \(-0.537897\pi\)
−0.118776 + 0.992921i \(0.537897\pi\)
\(758\) 0 0
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) 27.3205 0.990368 0.495184 0.868788i \(-0.335101\pi\)
0.495184 + 0.868788i \(0.335101\pi\)
\(762\) 0 0
\(763\) −61.8564 −2.23935
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) −5.46410 −0.197297
\(768\) 0 0
\(769\) −14.7846 −0.533147 −0.266573 0.963815i \(-0.585891\pi\)
−0.266573 + 0.963815i \(0.585891\pi\)
\(770\) 0 0
\(771\) 11.4641 0.412870
\(772\) 0 0
\(773\) 30.5885 1.10019 0.550095 0.835102i \(-0.314591\pi\)
0.550095 + 0.835102i \(0.314591\pi\)
\(774\) 0 0
\(775\) 23.5167 0.844743
\(776\) 0 0
\(777\) −6.92820 −0.248548
\(778\) 0 0
\(779\) −21.8564 −0.783087
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) −6.73205 −0.240584
\(784\) 0 0
\(785\) −17.0718 −0.609319
\(786\) 0 0
\(787\) −4.53590 −0.161687 −0.0808437 0.996727i \(-0.525761\pi\)
−0.0808437 + 0.996727i \(0.525761\pi\)
\(788\) 0 0
\(789\) 23.3205 0.830232
\(790\) 0 0
\(791\) 34.6410 1.23169
\(792\) 0 0
\(793\) 4.92820 0.175006
\(794\) 0 0
\(795\) 4.39230 0.155779
\(796\) 0 0
\(797\) −34.3923 −1.21824 −0.609119 0.793079i \(-0.708477\pi\)
−0.609119 + 0.793079i \(0.708477\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 0 0
\(801\) 11.6603 0.411995
\(802\) 0 0
\(803\) 4.00000 0.141157
\(804\) 0 0
\(805\) 15.2154 0.536272
\(806\) 0 0
\(807\) −14.3923 −0.506633
\(808\) 0 0
\(809\) 19.5167 0.686169 0.343085 0.939304i \(-0.388528\pi\)
0.343085 + 0.939304i \(0.388528\pi\)
\(810\) 0 0
\(811\) 46.0000 1.61528 0.807639 0.589677i \(-0.200745\pi\)
0.807639 + 0.589677i \(0.200745\pi\)
\(812\) 0 0
\(813\) −12.9282 −0.453412
\(814\) 0 0
\(815\) −4.14359 −0.145144
\(816\) 0 0
\(817\) −6.53590 −0.228662
\(818\) 0 0
\(819\) 3.46410 0.121046
\(820\) 0 0
\(821\) 21.0718 0.735411 0.367705 0.929942i \(-0.380143\pi\)
0.367705 + 0.929942i \(0.380143\pi\)
\(822\) 0 0
\(823\) 26.2487 0.914973 0.457486 0.889217i \(-0.348750\pi\)
0.457486 + 0.889217i \(0.348750\pi\)
\(824\) 0 0
\(825\) 4.46410 0.155420
\(826\) 0 0
\(827\) −3.46410 −0.120459 −0.0602293 0.998185i \(-0.519183\pi\)
−0.0602293 + 0.998185i \(0.519183\pi\)
\(828\) 0 0
\(829\) −13.7128 −0.476266 −0.238133 0.971233i \(-0.576535\pi\)
−0.238133 + 0.971233i \(0.576535\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) −13.6603 −0.473300
\(834\) 0 0
\(835\) 13.0718 0.452368
\(836\) 0 0
\(837\) −5.26795 −0.182087
\(838\) 0 0
\(839\) 30.6410 1.05785 0.528923 0.848670i \(-0.322596\pi\)
0.528923 + 0.848670i \(0.322596\pi\)
\(840\) 0 0
\(841\) 16.3205 0.562776
\(842\) 0 0
\(843\) 25.8564 0.890542
\(844\) 0 0
\(845\) −0.732051 −0.0251833
\(846\) 0 0
\(847\) 3.46410 0.119028
\(848\) 0 0
\(849\) 1.80385 0.0619079
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) −22.7846 −0.780130 −0.390065 0.920787i \(-0.627547\pi\)
−0.390065 + 0.920787i \(0.627547\pi\)
\(854\) 0 0
\(855\) −1.46410 −0.0500712
\(856\) 0 0
\(857\) 42.8372 1.46329 0.731645 0.681686i \(-0.238753\pi\)
0.731645 + 0.681686i \(0.238753\pi\)
\(858\) 0 0
\(859\) 19.3205 0.659207 0.329604 0.944119i \(-0.393085\pi\)
0.329604 + 0.944119i \(0.393085\pi\)
\(860\) 0 0
\(861\) −37.8564 −1.29014
\(862\) 0 0
\(863\) −33.4641 −1.13913 −0.569566 0.821946i \(-0.692889\pi\)
−0.569566 + 0.821946i \(0.692889\pi\)
\(864\) 0 0
\(865\) −4.14359 −0.140886
\(866\) 0 0
\(867\) −9.53590 −0.323856
\(868\) 0 0
\(869\) −3.26795 −0.110858
\(870\) 0 0
\(871\) −8.19615 −0.277716
\(872\) 0 0
\(873\) 15.8564 0.536658
\(874\) 0 0
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) −20.9282 −0.706695 −0.353348 0.935492i \(-0.614957\pi\)
−0.353348 + 0.935492i \(0.614957\pi\)
\(878\) 0 0
\(879\) 21.4641 0.723966
\(880\) 0 0
\(881\) −27.8564 −0.938506 −0.469253 0.883064i \(-0.655477\pi\)
−0.469253 + 0.883064i \(0.655477\pi\)
\(882\) 0 0
\(883\) 22.5359 0.758393 0.379197 0.925316i \(-0.376200\pi\)
0.379197 + 0.925316i \(0.376200\pi\)
\(884\) 0 0
\(885\) 4.00000 0.134459
\(886\) 0 0
\(887\) −27.7128 −0.930505 −0.465253 0.885178i \(-0.654037\pi\)
−0.465253 + 0.885178i \(0.654037\pi\)
\(888\) 0 0
\(889\) −36.6795 −1.23019
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 2.92820 0.0979886
\(894\) 0 0
\(895\) 1.46410 0.0489395
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 0 0
\(899\) 35.4641 1.18279
\(900\) 0 0
\(901\) 16.3923 0.546107
\(902\) 0 0
\(903\) −11.3205 −0.376723
\(904\) 0 0
\(905\) −4.00000 −0.132964
\(906\) 0 0
\(907\) −43.7128 −1.45146 −0.725730 0.687980i \(-0.758498\pi\)
−0.725730 + 0.687980i \(0.758498\pi\)
\(908\) 0 0
\(909\) 10.7321 0.355960
\(910\) 0 0
\(911\) 37.8564 1.25424 0.627119 0.778923i \(-0.284234\pi\)
0.627119 + 0.778923i \(0.284234\pi\)
\(912\) 0 0
\(913\) 10.9282 0.361671
\(914\) 0 0
\(915\) −3.60770 −0.119267
\(916\) 0 0
\(917\) −3.71281 −0.122608
\(918\) 0 0
\(919\) −41.5167 −1.36951 −0.684754 0.728774i \(-0.740090\pi\)
−0.684754 + 0.728774i \(0.740090\pi\)
\(920\) 0 0
\(921\) 30.7846 1.01439
\(922\) 0 0
\(923\) −4.00000 −0.131662
\(924\) 0 0
\(925\) 8.92820 0.293558
\(926\) 0 0
\(927\) 5.46410 0.179465
\(928\) 0 0
\(929\) 15.6603 0.513796 0.256898 0.966438i \(-0.417300\pi\)
0.256898 + 0.966438i \(0.417300\pi\)
\(930\) 0 0
\(931\) 10.0000 0.327737
\(932\) 0 0
\(933\) −12.9282 −0.423250
\(934\) 0 0
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) −38.3923 −1.25422 −0.627111 0.778930i \(-0.715763\pi\)
−0.627111 + 0.778930i \(0.715763\pi\)
\(938\) 0 0
\(939\) −1.46410 −0.0477792
\(940\) 0 0
\(941\) −44.7846 −1.45994 −0.729968 0.683481i \(-0.760465\pi\)
−0.729968 + 0.683481i \(0.760465\pi\)
\(942\) 0 0
\(943\) 65.5692 2.13523
\(944\) 0 0
\(945\) −2.53590 −0.0824928
\(946\) 0 0
\(947\) −16.6795 −0.542011 −0.271005 0.962578i \(-0.587356\pi\)
−0.271005 + 0.962578i \(0.587356\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) −3.26795 −0.105971
\(952\) 0 0
\(953\) 23.9090 0.774487 0.387244 0.921977i \(-0.373427\pi\)
0.387244 + 0.921977i \(0.373427\pi\)
\(954\) 0 0
\(955\) 19.7128 0.637892
\(956\) 0 0
\(957\) 6.73205 0.217616
\(958\) 0 0
\(959\) 6.24871 0.201781
\(960\) 0 0
\(961\) −3.24871 −0.104797
\(962\) 0 0
\(963\) −4.39230 −0.141540
\(964\) 0 0
\(965\) 8.00000 0.257529
\(966\) 0 0
\(967\) 54.3923 1.74914 0.874569 0.484901i \(-0.161144\pi\)
0.874569 + 0.484901i \(0.161144\pi\)
\(968\) 0 0
\(969\) −5.46410 −0.175532
\(970\) 0 0
\(971\) 9.21539 0.295736 0.147868 0.989007i \(-0.452759\pi\)
0.147868 + 0.989007i \(0.452759\pi\)
\(972\) 0 0
\(973\) 69.4641 2.22692
\(974\) 0 0
\(975\) −4.46410 −0.142966
\(976\) 0 0
\(977\) −56.1577 −1.79664 −0.898322 0.439338i \(-0.855213\pi\)
−0.898322 + 0.439338i \(0.855213\pi\)
\(978\) 0 0
\(979\) −11.6603 −0.372663
\(980\) 0 0
\(981\) −17.8564 −0.570111
\(982\) 0 0
\(983\) 10.1436 0.323530 0.161765 0.986829i \(-0.448281\pi\)
0.161765 + 0.986829i \(0.448281\pi\)
\(984\) 0 0
\(985\) −14.1436 −0.450652
\(986\) 0 0
\(987\) 5.07180 0.161437
\(988\) 0 0
\(989\) 19.6077 0.623488
\(990\) 0 0
\(991\) −51.3205 −1.63025 −0.815125 0.579285i \(-0.803332\pi\)
−0.815125 + 0.579285i \(0.803332\pi\)
\(992\) 0 0
\(993\) 18.7321 0.594444
\(994\) 0 0
\(995\) −14.1436 −0.448382
\(996\) 0 0
\(997\) 13.6077 0.430960 0.215480 0.976508i \(-0.430868\pi\)
0.215480 + 0.976508i \(0.430868\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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 6864.2.a.bm.1.1 2
4.3 odd 2 3432.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.l.1.1 2 4.3 odd 2
6864.2.a.bm.1.1 2 1.1 even 1 trivial