Properties

Label 4080.2.a.bq.1.2
Level $4080$
Weight $2$
Character 4080.1
Self dual yes
Analytic conductor $32.579$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4080,2,Mod(1,4080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4080, 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("4080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4080.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: \(32.5789640247\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 510)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 4080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +4.89898 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +4.89898 q^{7} +1.00000 q^{9} +6.89898 q^{13} +1.00000 q^{15} -1.00000 q^{17} -4.00000 q^{19} +4.89898 q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -4.00000 q^{31} +4.89898 q^{35} +6.00000 q^{37} +6.89898 q^{39} -2.89898 q^{41} -8.89898 q^{43} +1.00000 q^{45} -9.79796 q^{47} +17.0000 q^{49} -1.00000 q^{51} -7.79796 q^{53} -4.00000 q^{57} -4.89898 q^{59} +11.7980 q^{61} +4.89898 q^{63} +6.89898 q^{65} -0.898979 q^{67} +4.00000 q^{69} +8.89898 q^{71} -10.8990 q^{73} +1.00000 q^{75} +5.79796 q^{79} +1.00000 q^{81} -13.7980 q^{83} -1.00000 q^{85} +6.00000 q^{87} -7.79796 q^{89} +33.7980 q^{91} -4.00000 q^{93} -4.00000 q^{95} -12.6969 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 2 q^{17} - 8 q^{19} + 8 q^{23} + 2 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{31} + 12 q^{37} + 4 q^{39} + 4 q^{41} - 8 q^{43} + 2 q^{45} + 34 q^{49} - 2 q^{51} + 4 q^{53} - 8 q^{57} + 4 q^{61} + 4 q^{65} + 8 q^{67} + 8 q^{69} + 8 q^{71} - 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} - 8 q^{83} - 2 q^{85} + 12 q^{87} + 4 q^{89} + 48 q^{91} - 8 q^{93} - 8 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.89898 1.85164 0.925820 0.377964i \(-0.123376\pi\)
0.925820 + 0.377964i \(0.123376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 6.89898 1.91343 0.956716 0.291022i \(-0.0939953\pi\)
0.956716 + 0.291022i \(0.0939953\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 4.89898 1.06904
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.89898 0.828079
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 6.89898 1.10472
\(40\) 0 0
\(41\) −2.89898 −0.452745 −0.226372 0.974041i \(-0.572687\pi\)
−0.226372 + 0.974041i \(0.572687\pi\)
\(42\) 0 0
\(43\) −8.89898 −1.35708 −0.678541 0.734563i \(-0.737387\pi\)
−0.678541 + 0.734563i \(0.737387\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −9.79796 −1.42918 −0.714590 0.699544i \(-0.753387\pi\)
−0.714590 + 0.699544i \(0.753387\pi\)
\(48\) 0 0
\(49\) 17.0000 2.42857
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) −7.79796 −1.07113 −0.535566 0.844493i \(-0.679902\pi\)
−0.535566 + 0.844493i \(0.679902\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −4.89898 −0.637793 −0.318896 0.947790i \(-0.603312\pi\)
−0.318896 + 0.947790i \(0.603312\pi\)
\(60\) 0 0
\(61\) 11.7980 1.51057 0.755287 0.655394i \(-0.227498\pi\)
0.755287 + 0.655394i \(0.227498\pi\)
\(62\) 0 0
\(63\) 4.89898 0.617213
\(64\) 0 0
\(65\) 6.89898 0.855713
\(66\) 0 0
\(67\) −0.898979 −0.109828 −0.0549139 0.998491i \(-0.517488\pi\)
−0.0549139 + 0.998491i \(0.517488\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 8.89898 1.05611 0.528057 0.849209i \(-0.322921\pi\)
0.528057 + 0.849209i \(0.322921\pi\)
\(72\) 0 0
\(73\) −10.8990 −1.27563 −0.637815 0.770190i \(-0.720161\pi\)
−0.637815 + 0.770190i \(0.720161\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.79796 0.652321 0.326161 0.945314i \(-0.394245\pi\)
0.326161 + 0.945314i \(0.394245\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.7980 −1.51452 −0.757261 0.653112i \(-0.773463\pi\)
−0.757261 + 0.653112i \(0.773463\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.108465
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −7.79796 −0.826582 −0.413291 0.910599i \(-0.635621\pi\)
−0.413291 + 0.910599i \(0.635621\pi\)
\(90\) 0 0
\(91\) 33.7980 3.54299
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −12.6969 −1.28918 −0.644589 0.764529i \(-0.722972\pi\)
−0.644589 + 0.764529i \(0.722972\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −18.8990 −1.88052 −0.940259 0.340459i \(-0.889418\pi\)
−0.940259 + 0.340459i \(0.889418\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 4.89898 0.478091
\(106\) 0 0
\(107\) −5.79796 −0.560510 −0.280255 0.959926i \(-0.590419\pi\)
−0.280255 + 0.959926i \(0.590419\pi\)
\(108\) 0 0
\(109\) 11.7980 1.13004 0.565020 0.825077i \(-0.308869\pi\)
0.565020 + 0.825077i \(0.308869\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 7.79796 0.733570 0.366785 0.930306i \(-0.380458\pi\)
0.366785 + 0.930306i \(0.380458\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 6.89898 0.637811
\(118\) 0 0
\(119\) −4.89898 −0.449089
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −2.89898 −0.261392
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) −8.89898 −0.783511
\(130\) 0 0
\(131\) −9.79796 −0.856052 −0.428026 0.903767i \(-0.640791\pi\)
−0.428026 + 0.903767i \(0.640791\pi\)
\(132\) 0 0
\(133\) −19.5959 −1.69918
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 14.0000 1.19610 0.598050 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 0 0
\(139\) −13.7980 −1.17033 −0.585164 0.810915i \(-0.698970\pi\)
−0.585164 + 0.810915i \(0.698970\pi\)
\(140\) 0 0
\(141\) −9.79796 −0.825137
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 17.0000 1.40214
\(148\) 0 0
\(149\) −18.8990 −1.54826 −0.774132 0.633024i \(-0.781814\pi\)
−0.774132 + 0.633024i \(0.781814\pi\)
\(150\) 0 0
\(151\) 9.79796 0.797347 0.398673 0.917093i \(-0.369471\pi\)
0.398673 + 0.917093i \(0.369471\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −1.10102 −0.0878710 −0.0439355 0.999034i \(-0.513990\pi\)
−0.0439355 + 0.999034i \(0.513990\pi\)
\(158\) 0 0
\(159\) −7.79796 −0.618418
\(160\) 0 0
\(161\) 19.5959 1.54437
\(162\) 0 0
\(163\) 2.20204 0.172477 0.0862386 0.996275i \(-0.472515\pi\)
0.0862386 + 0.996275i \(0.472515\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.20204 −0.170399 −0.0851995 0.996364i \(-0.527153\pi\)
−0.0851995 + 0.996364i \(0.527153\pi\)
\(168\) 0 0
\(169\) 34.5959 2.66122
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 4.89898 0.370328
\(176\) 0 0
\(177\) −4.89898 −0.368230
\(178\) 0 0
\(179\) 4.89898 0.366167 0.183083 0.983097i \(-0.441392\pi\)
0.183083 + 0.983097i \(0.441392\pi\)
\(180\) 0 0
\(181\) −4.20204 −0.312335 −0.156168 0.987731i \(-0.549914\pi\)
−0.156168 + 0.987731i \(0.549914\pi\)
\(182\) 0 0
\(183\) 11.7980 0.872130
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.89898 0.356348
\(190\) 0 0
\(191\) 9.79796 0.708955 0.354478 0.935064i \(-0.384659\pi\)
0.354478 + 0.935064i \(0.384659\pi\)
\(192\) 0 0
\(193\) −1.10102 −0.0792532 −0.0396266 0.999215i \(-0.512617\pi\)
−0.0396266 + 0.999215i \(0.512617\pi\)
\(194\) 0 0
\(195\) 6.89898 0.494046
\(196\) 0 0
\(197\) −13.5959 −0.968669 −0.484335 0.874883i \(-0.660938\pi\)
−0.484335 + 0.874883i \(0.660938\pi\)
\(198\) 0 0
\(199\) −15.5959 −1.10557 −0.552783 0.833325i \(-0.686434\pi\)
−0.552783 + 0.833325i \(0.686434\pi\)
\(200\) 0 0
\(201\) −0.898979 −0.0634091
\(202\) 0 0
\(203\) 29.3939 2.06305
\(204\) 0 0
\(205\) −2.89898 −0.202474
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 8.89898 0.609748
\(214\) 0 0
\(215\) −8.89898 −0.606905
\(216\) 0 0
\(217\) −19.5959 −1.33026
\(218\) 0 0
\(219\) −10.8990 −0.736485
\(220\) 0 0
\(221\) −6.89898 −0.464076
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 21.7980 1.44678 0.723391 0.690439i \(-0.242583\pi\)
0.723391 + 0.690439i \(0.242583\pi\)
\(228\) 0 0
\(229\) −13.5959 −0.898444 −0.449222 0.893420i \(-0.648299\pi\)
−0.449222 + 0.893420i \(0.648299\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) −9.79796 −0.639148
\(236\) 0 0
\(237\) 5.79796 0.376618
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 13.5959 0.875790 0.437895 0.899026i \(-0.355724\pi\)
0.437895 + 0.899026i \(0.355724\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 17.0000 1.08609
\(246\) 0 0
\(247\) −27.5959 −1.75589
\(248\) 0 0
\(249\) −13.7980 −0.874410
\(250\) 0 0
\(251\) −4.89898 −0.309221 −0.154610 0.987976i \(-0.549412\pi\)
−0.154610 + 0.987976i \(0.549412\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −1.00000 −0.0626224
\(256\) 0 0
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 29.3939 1.82645
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) −7.79796 −0.479025
\(266\) 0 0
\(267\) −7.79796 −0.477227
\(268\) 0 0
\(269\) 9.59592 0.585073 0.292537 0.956254i \(-0.405501\pi\)
0.292537 + 0.956254i \(0.405501\pi\)
\(270\) 0 0
\(271\) 1.79796 0.109218 0.0546091 0.998508i \(-0.482609\pi\)
0.0546091 + 0.998508i \(0.482609\pi\)
\(272\) 0 0
\(273\) 33.7980 2.04555
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.79796 0.468534 0.234267 0.972172i \(-0.424731\pi\)
0.234267 + 0.972172i \(0.424731\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 27.7980 1.65829 0.829144 0.559036i \(-0.188829\pi\)
0.829144 + 0.559036i \(0.188829\pi\)
\(282\) 0 0
\(283\) −23.5959 −1.40263 −0.701316 0.712851i \(-0.747404\pi\)
−0.701316 + 0.712851i \(0.747404\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −14.2020 −0.838320
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −12.6969 −0.744308
\(292\) 0 0
\(293\) 13.5959 0.794282 0.397141 0.917758i \(-0.370002\pi\)
0.397141 + 0.917758i \(0.370002\pi\)
\(294\) 0 0
\(295\) −4.89898 −0.285230
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 27.5959 1.59591
\(300\) 0 0
\(301\) −43.5959 −2.51283
\(302\) 0 0
\(303\) −18.8990 −1.08572
\(304\) 0 0
\(305\) 11.7980 0.675549
\(306\) 0 0
\(307\) −0.898979 −0.0513075 −0.0256537 0.999671i \(-0.508167\pi\)
−0.0256537 + 0.999671i \(0.508167\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 7.10102 0.402662 0.201331 0.979523i \(-0.435473\pi\)
0.201331 + 0.979523i \(0.435473\pi\)
\(312\) 0 0
\(313\) 6.89898 0.389953 0.194977 0.980808i \(-0.437537\pi\)
0.194977 + 0.980808i \(0.437537\pi\)
\(314\) 0 0
\(315\) 4.89898 0.276026
\(316\) 0 0
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −5.79796 −0.323611
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) 6.89898 0.382687
\(326\) 0 0
\(327\) 11.7980 0.652429
\(328\) 0 0
\(329\) −48.0000 −2.64633
\(330\) 0 0
\(331\) 5.79796 0.318685 0.159342 0.987223i \(-0.449063\pi\)
0.159342 + 0.987223i \(0.449063\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) −0.898979 −0.0491165
\(336\) 0 0
\(337\) 32.6969 1.78112 0.890558 0.454870i \(-0.150314\pi\)
0.890558 + 0.454870i \(0.150314\pi\)
\(338\) 0 0
\(339\) 7.79796 0.423527
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 48.9898 2.64520
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) −21.7980 −1.17018 −0.585088 0.810970i \(-0.698940\pi\)
−0.585088 + 0.810970i \(0.698940\pi\)
\(348\) 0 0
\(349\) 4.20204 0.224930 0.112465 0.993656i \(-0.464125\pi\)
0.112465 + 0.993656i \(0.464125\pi\)
\(350\) 0 0
\(351\) 6.89898 0.368240
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) 8.89898 0.472309
\(356\) 0 0
\(357\) −4.89898 −0.259281
\(358\) 0 0
\(359\) −37.3939 −1.97357 −0.986787 0.162025i \(-0.948198\pi\)
−0.986787 + 0.162025i \(0.948198\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) −10.8990 −0.570479
\(366\) 0 0
\(367\) 27.1010 1.41466 0.707331 0.706883i \(-0.249899\pi\)
0.707331 + 0.706883i \(0.249899\pi\)
\(368\) 0 0
\(369\) −2.89898 −0.150915
\(370\) 0 0
\(371\) −38.2020 −1.98335
\(372\) 0 0
\(373\) 24.6969 1.27876 0.639380 0.768891i \(-0.279191\pi\)
0.639380 + 0.768891i \(0.279191\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 41.3939 2.13189
\(378\) 0 0
\(379\) 29.7980 1.53062 0.765309 0.643663i \(-0.222586\pi\)
0.765309 + 0.643663i \(0.222586\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.89898 −0.452361
\(388\) 0 0
\(389\) −38.4949 −1.95177 −0.975884 0.218288i \(-0.929953\pi\)
−0.975884 + 0.218288i \(0.929953\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) −9.79796 −0.494242
\(394\) 0 0
\(395\) 5.79796 0.291727
\(396\) 0 0
\(397\) 1.59592 0.0800968 0.0400484 0.999198i \(-0.487249\pi\)
0.0400484 + 0.999198i \(0.487249\pi\)
\(398\) 0 0
\(399\) −19.5959 −0.981023
\(400\) 0 0
\(401\) 22.8990 1.14352 0.571760 0.820421i \(-0.306261\pi\)
0.571760 + 0.820421i \(0.306261\pi\)
\(402\) 0 0
\(403\) −27.5959 −1.37465
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −17.5959 −0.870062 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 0 0
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) −13.7980 −0.677315
\(416\) 0 0
\(417\) −13.7980 −0.675689
\(418\) 0 0
\(419\) 17.7980 0.869487 0.434744 0.900554i \(-0.356839\pi\)
0.434744 + 0.900554i \(0.356839\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 0 0
\(423\) −9.79796 −0.476393
\(424\) 0 0
\(425\) −1.00000 −0.0485071
\(426\) 0 0
\(427\) 57.7980 2.79704
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 23.1010 1.11274 0.556369 0.830936i \(-0.312194\pi\)
0.556369 + 0.830936i \(0.312194\pi\)
\(432\) 0 0
\(433\) −35.3939 −1.70092 −0.850461 0.526039i \(-0.823677\pi\)
−0.850461 + 0.526039i \(0.823677\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 0 0
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) −5.79796 −0.276721 −0.138361 0.990382i \(-0.544183\pi\)
−0.138361 + 0.990382i \(0.544183\pi\)
\(440\) 0 0
\(441\) 17.0000 0.809524
\(442\) 0 0
\(443\) −37.7980 −1.79584 −0.897918 0.440164i \(-0.854920\pi\)
−0.897918 + 0.440164i \(0.854920\pi\)
\(444\) 0 0
\(445\) −7.79796 −0.369659
\(446\) 0 0
\(447\) −18.8990 −0.893891
\(448\) 0 0
\(449\) 6.89898 0.325583 0.162791 0.986660i \(-0.447950\pi\)
0.162791 + 0.986660i \(0.447950\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 9.79796 0.460348
\(454\) 0 0
\(455\) 33.7980 1.58447
\(456\) 0 0
\(457\) 16.2020 0.757900 0.378950 0.925417i \(-0.376285\pi\)
0.378950 + 0.925417i \(0.376285\pi\)
\(458\) 0 0
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −28.6969 −1.33655 −0.668275 0.743914i \(-0.732967\pi\)
−0.668275 + 0.743914i \(0.732967\pi\)
\(462\) 0 0
\(463\) −7.59592 −0.353012 −0.176506 0.984300i \(-0.556480\pi\)
−0.176506 + 0.984300i \(0.556480\pi\)
\(464\) 0 0
\(465\) −4.00000 −0.185496
\(466\) 0 0
\(467\) 7.59592 0.351497 0.175749 0.984435i \(-0.443765\pi\)
0.175749 + 0.984435i \(0.443765\pi\)
\(468\) 0 0
\(469\) −4.40408 −0.203362
\(470\) 0 0
\(471\) −1.10102 −0.0507323
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −7.79796 −0.357044
\(478\) 0 0
\(479\) 32.8990 1.50319 0.751596 0.659623i \(-0.229284\pi\)
0.751596 + 0.659623i \(0.229284\pi\)
\(480\) 0 0
\(481\) 41.3939 1.88740
\(482\) 0 0
\(483\) 19.5959 0.891645
\(484\) 0 0
\(485\) −12.6969 −0.576538
\(486\) 0 0
\(487\) 20.8990 0.947023 0.473512 0.880788i \(-0.342986\pi\)
0.473512 + 0.880788i \(0.342986\pi\)
\(488\) 0 0
\(489\) 2.20204 0.0995797
\(490\) 0 0
\(491\) 1.30306 0.0588063 0.0294032 0.999568i \(-0.490639\pi\)
0.0294032 + 0.999568i \(0.490639\pi\)
\(492\) 0 0
\(493\) −6.00000 −0.270226
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 43.5959 1.95554
\(498\) 0 0
\(499\) 39.5959 1.77256 0.886278 0.463153i \(-0.153282\pi\)
0.886278 + 0.463153i \(0.153282\pi\)
\(500\) 0 0
\(501\) −2.20204 −0.0983799
\(502\) 0 0
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) −18.8990 −0.840994
\(506\) 0 0
\(507\) 34.5959 1.53646
\(508\) 0 0
\(509\) −15.3031 −0.678296 −0.339148 0.940733i \(-0.610139\pi\)
−0.339148 + 0.940733i \(0.610139\pi\)
\(510\) 0 0
\(511\) −53.3939 −2.36201
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −40.2929 −1.76526 −0.882631 0.470066i \(-0.844230\pi\)
−0.882631 + 0.470066i \(0.844230\pi\)
\(522\) 0 0
\(523\) −18.6969 −0.817560 −0.408780 0.912633i \(-0.634046\pi\)
−0.408780 + 0.912633i \(0.634046\pi\)
\(524\) 0 0
\(525\) 4.89898 0.213809
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.89898 −0.212598
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) 0 0
\(535\) −5.79796 −0.250668
\(536\) 0 0
\(537\) 4.89898 0.211407
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.79796 −0.335260 −0.167630 0.985850i \(-0.553611\pi\)
−0.167630 + 0.985850i \(0.553611\pi\)
\(542\) 0 0
\(543\) −4.20204 −0.180327
\(544\) 0 0
\(545\) 11.7980 0.505369
\(546\) 0 0
\(547\) −0.404082 −0.0172773 −0.00863865 0.999963i \(-0.502750\pi\)
−0.00863865 + 0.999963i \(0.502750\pi\)
\(548\) 0 0
\(549\) 11.7980 0.503525
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 28.4041 1.20786
\(554\) 0 0
\(555\) 6.00000 0.254686
\(556\) 0 0
\(557\) 19.7980 0.838866 0.419433 0.907786i \(-0.362229\pi\)
0.419433 + 0.907786i \(0.362229\pi\)
\(558\) 0 0
\(559\) −61.3939 −2.59668
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21.7980 −0.918674 −0.459337 0.888262i \(-0.651913\pi\)
−0.459337 + 0.888262i \(0.651913\pi\)
\(564\) 0 0
\(565\) 7.79796 0.328063
\(566\) 0 0
\(567\) 4.89898 0.205738
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) −29.7980 −1.24701 −0.623503 0.781821i \(-0.714291\pi\)
−0.623503 + 0.781821i \(0.714291\pi\)
\(572\) 0 0
\(573\) 9.79796 0.409316
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −27.3939 −1.14042 −0.570211 0.821498i \(-0.693139\pi\)
−0.570211 + 0.821498i \(0.693139\pi\)
\(578\) 0 0
\(579\) −1.10102 −0.0457569
\(580\) 0 0
\(581\) −67.5959 −2.80435
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 6.89898 0.285238
\(586\) 0 0
\(587\) 41.3939 1.70851 0.854254 0.519856i \(-0.174014\pi\)
0.854254 + 0.519856i \(0.174014\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) −13.5959 −0.559261
\(592\) 0 0
\(593\) 1.59592 0.0655365 0.0327682 0.999463i \(-0.489568\pi\)
0.0327682 + 0.999463i \(0.489568\pi\)
\(594\) 0 0
\(595\) −4.89898 −0.200839
\(596\) 0 0
\(597\) −15.5959 −0.638298
\(598\) 0 0
\(599\) −21.3939 −0.874130 −0.437065 0.899430i \(-0.643982\pi\)
−0.437065 + 0.899430i \(0.643982\pi\)
\(600\) 0 0
\(601\) 16.2020 0.660895 0.330448 0.943824i \(-0.392800\pi\)
0.330448 + 0.943824i \(0.392800\pi\)
\(602\) 0 0
\(603\) −0.898979 −0.0366093
\(604\) 0 0
\(605\) −11.0000 −0.447214
\(606\) 0 0
\(607\) 32.4949 1.31893 0.659464 0.751736i \(-0.270783\pi\)
0.659464 + 0.751736i \(0.270783\pi\)
\(608\) 0 0
\(609\) 29.3939 1.19110
\(610\) 0 0
\(611\) −67.5959 −2.73464
\(612\) 0 0
\(613\) −14.4949 −0.585443 −0.292722 0.956198i \(-0.594561\pi\)
−0.292722 + 0.956198i \(0.594561\pi\)
\(614\) 0 0
\(615\) −2.89898 −0.116898
\(616\) 0 0
\(617\) −37.5959 −1.51355 −0.756777 0.653673i \(-0.773227\pi\)
−0.756777 + 0.653673i \(0.773227\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) −38.2020 −1.53053
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) 117.283 4.64691
\(638\) 0 0
\(639\) 8.89898 0.352038
\(640\) 0 0
\(641\) −20.6969 −0.817480 −0.408740 0.912651i \(-0.634032\pi\)
−0.408740 + 0.912651i \(0.634032\pi\)
\(642\) 0 0
\(643\) −29.7980 −1.17512 −0.587558 0.809182i \(-0.699911\pi\)
−0.587558 + 0.809182i \(0.699911\pi\)
\(644\) 0 0
\(645\) −8.89898 −0.350397
\(646\) 0 0
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −19.5959 −0.768025
\(652\) 0 0
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 0 0
\(655\) −9.79796 −0.382838
\(656\) 0 0
\(657\) −10.8990 −0.425210
\(658\) 0 0
\(659\) −30.6969 −1.19578 −0.597891 0.801577i \(-0.703995\pi\)
−0.597891 + 0.801577i \(0.703995\pi\)
\(660\) 0 0
\(661\) −19.7980 −0.770051 −0.385026 0.922906i \(-0.625807\pi\)
−0.385026 + 0.922906i \(0.625807\pi\)
\(662\) 0 0
\(663\) −6.89898 −0.267934
\(664\) 0 0
\(665\) −19.5959 −0.759897
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −33.1010 −1.27595 −0.637975 0.770057i \(-0.720228\pi\)
−0.637975 + 0.770057i \(0.720228\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −13.5959 −0.522534 −0.261267 0.965267i \(-0.584140\pi\)
−0.261267 + 0.965267i \(0.584140\pi\)
\(678\) 0 0
\(679\) −62.2020 −2.38710
\(680\) 0 0
\(681\) 21.7980 0.835300
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) 0 0
\(687\) −13.5959 −0.518717
\(688\) 0 0
\(689\) −53.7980 −2.04954
\(690\) 0 0
\(691\) 27.1918 1.03443 0.517213 0.855857i \(-0.326969\pi\)
0.517213 + 0.855857i \(0.326969\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −13.7980 −0.523386
\(696\) 0 0
\(697\) 2.89898 0.109807
\(698\) 0 0
\(699\) 14.0000 0.529529
\(700\) 0 0
\(701\) 14.8990 0.562727 0.281363 0.959601i \(-0.409213\pi\)
0.281363 + 0.959601i \(0.409213\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) −9.79796 −0.369012
\(706\) 0 0
\(707\) −92.5857 −3.48204
\(708\) 0 0
\(709\) −31.7980 −1.19420 −0.597099 0.802168i \(-0.703680\pi\)
−0.597099 + 0.802168i \(0.703680\pi\)
\(710\) 0 0
\(711\) 5.79796 0.217440
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 12.4949 0.465981 0.232991 0.972479i \(-0.425149\pi\)
0.232991 + 0.972479i \(0.425149\pi\)
\(720\) 0 0
\(721\) −19.5959 −0.729790
\(722\) 0 0
\(723\) 13.5959 0.505638
\(724\) 0 0
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −0.404082 −0.0149866 −0.00749329 0.999972i \(-0.502385\pi\)
−0.00749329 + 0.999972i \(0.502385\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.89898 0.329141
\(732\) 0 0
\(733\) −46.4949 −1.71733 −0.858664 0.512539i \(-0.828705\pi\)
−0.858664 + 0.512539i \(0.828705\pi\)
\(734\) 0 0
\(735\) 17.0000 0.627054
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 9.39388 0.345559 0.172780 0.984960i \(-0.444725\pi\)
0.172780 + 0.984960i \(0.444725\pi\)
\(740\) 0 0
\(741\) −27.5959 −1.01376
\(742\) 0 0
\(743\) 9.39388 0.344628 0.172314 0.985042i \(-0.444876\pi\)
0.172314 + 0.985042i \(0.444876\pi\)
\(744\) 0 0
\(745\) −18.8990 −0.692405
\(746\) 0 0
\(747\) −13.7980 −0.504841
\(748\) 0 0
\(749\) −28.4041 −1.03786
\(750\) 0 0
\(751\) −21.7980 −0.795419 −0.397709 0.917511i \(-0.630195\pi\)
−0.397709 + 0.917511i \(0.630195\pi\)
\(752\) 0 0
\(753\) −4.89898 −0.178529
\(754\) 0 0
\(755\) 9.79796 0.356584
\(756\) 0 0
\(757\) −4.69694 −0.170713 −0.0853566 0.996350i \(-0.527203\pi\)
−0.0853566 + 0.996350i \(0.527203\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.59592 −0.0578520 −0.0289260 0.999582i \(-0.509209\pi\)
−0.0289260 + 0.999582i \(0.509209\pi\)
\(762\) 0 0
\(763\) 57.7980 2.09243
\(764\) 0 0
\(765\) −1.00000 −0.0361551
\(766\) 0 0
\(767\) −33.7980 −1.22037
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 0 0
\(773\) −35.3939 −1.27303 −0.636515 0.771265i \(-0.719625\pi\)
−0.636515 + 0.771265i \(0.719625\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 29.3939 1.05450
\(778\) 0 0
\(779\) 11.5959 0.415467
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) −1.10102 −0.0392971
\(786\) 0 0
\(787\) 45.7980 1.63252 0.816260 0.577684i \(-0.196043\pi\)
0.816260 + 0.577684i \(0.196043\pi\)
\(788\) 0 0
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) 38.2020 1.35831
\(792\) 0 0
\(793\) 81.3939 2.89038
\(794\) 0 0
\(795\) −7.79796 −0.276565
\(796\) 0 0
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) 0 0
\(799\) 9.79796 0.346627
\(800\) 0 0
\(801\) −7.79796 −0.275527
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 19.5959 0.690665
\(806\) 0 0
\(807\) 9.59592 0.337792
\(808\) 0 0
\(809\) 32.6969 1.14956 0.574782 0.818307i \(-0.305087\pi\)
0.574782 + 0.818307i \(0.305087\pi\)
\(810\) 0 0
\(811\) −25.3939 −0.891700 −0.445850 0.895108i \(-0.647098\pi\)
−0.445850 + 0.895108i \(0.647098\pi\)
\(812\) 0 0
\(813\) 1.79796 0.0630572
\(814\) 0 0
\(815\) 2.20204 0.0771341
\(816\) 0 0
\(817\) 35.5959 1.24534
\(818\) 0 0
\(819\) 33.7980 1.18100
\(820\) 0 0
\(821\) 15.7980 0.551353 0.275676 0.961251i \(-0.411098\pi\)
0.275676 + 0.961251i \(0.411098\pi\)
\(822\) 0 0
\(823\) −20.8990 −0.728493 −0.364246 0.931303i \(-0.618673\pi\)
−0.364246 + 0.931303i \(0.618673\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 0 0
\(829\) 3.39388 0.117874 0.0589371 0.998262i \(-0.481229\pi\)
0.0589371 + 0.998262i \(0.481229\pi\)
\(830\) 0 0
\(831\) 7.79796 0.270508
\(832\) 0 0
\(833\) −17.0000 −0.589015
\(834\) 0 0
\(835\) −2.20204 −0.0762048
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) 0 0
\(839\) 7.10102 0.245154 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 27.7980 0.957413
\(844\) 0 0
\(845\) 34.5959 1.19014
\(846\) 0 0
\(847\) −53.8888 −1.85164
\(848\) 0 0
\(849\) −23.5959 −0.809810
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) 11.3939 0.390119 0.195059 0.980791i \(-0.437510\pi\)
0.195059 + 0.980791i \(0.437510\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) −5.79796 −0.197824 −0.0989119 0.995096i \(-0.531536\pi\)
−0.0989119 + 0.995096i \(0.531536\pi\)
\(860\) 0 0
\(861\) −14.2020 −0.484004
\(862\) 0 0
\(863\) 45.3939 1.54523 0.772613 0.634878i \(-0.218949\pi\)
0.772613 + 0.634878i \(0.218949\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −6.20204 −0.210148
\(872\) 0 0
\(873\) −12.6969 −0.429726
\(874\) 0 0
\(875\) 4.89898 0.165616
\(876\) 0 0
\(877\) −31.3939 −1.06010 −0.530048 0.847968i \(-0.677826\pi\)
−0.530048 + 0.847968i \(0.677826\pi\)
\(878\) 0 0
\(879\) 13.5959 0.458579
\(880\) 0 0
\(881\) 34.4949 1.16216 0.581081 0.813846i \(-0.302630\pi\)
0.581081 + 0.813846i \(0.302630\pi\)
\(882\) 0 0
\(883\) −26.6969 −0.898424 −0.449212 0.893425i \(-0.648295\pi\)
−0.449212 + 0.893425i \(0.648295\pi\)
\(884\) 0 0
\(885\) −4.89898 −0.164677
\(886\) 0 0
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 58.7878 1.97168
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 39.1918 1.31150
\(894\) 0 0
\(895\) 4.89898 0.163755
\(896\) 0 0
\(897\) 27.5959 0.921401
\(898\) 0 0
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 7.79796 0.259788
\(902\) 0 0
\(903\) −43.5959 −1.45078
\(904\) 0 0
\(905\) −4.20204 −0.139681
\(906\) 0 0
\(907\) 33.3939 1.10883 0.554413 0.832242i \(-0.312943\pi\)
0.554413 + 0.832242i \(0.312943\pi\)
\(908\) 0 0
\(909\) −18.8990 −0.626840
\(910\) 0 0
\(911\) −23.1010 −0.765371 −0.382685 0.923879i \(-0.625001\pi\)
−0.382685 + 0.923879i \(0.625001\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 11.7980 0.390028
\(916\) 0 0
\(917\) −48.0000 −1.58510
\(918\) 0 0
\(919\) −1.79796 −0.0593092 −0.0296546 0.999560i \(-0.509441\pi\)
−0.0296546 + 0.999560i \(0.509441\pi\)
\(920\) 0 0
\(921\) −0.898979 −0.0296224
\(922\) 0 0
\(923\) 61.3939 2.02080
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) 36.2929 1.19073 0.595365 0.803455i \(-0.297007\pi\)
0.595365 + 0.803455i \(0.297007\pi\)
\(930\) 0 0
\(931\) −68.0000 −2.22861
\(932\) 0 0
\(933\) 7.10102 0.232477
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 39.3939 1.28694 0.643471 0.765471i \(-0.277494\pi\)
0.643471 + 0.765471i \(0.277494\pi\)
\(938\) 0 0
\(939\) 6.89898 0.225140
\(940\) 0 0
\(941\) −16.2020 −0.528171 −0.264086 0.964499i \(-0.585070\pi\)
−0.264086 + 0.964499i \(0.585070\pi\)
\(942\) 0 0
\(943\) −11.5959 −0.377615
\(944\) 0 0
\(945\) 4.89898 0.159364
\(946\) 0 0
\(947\) −21.7980 −0.708338 −0.354169 0.935181i \(-0.615236\pi\)
−0.354169 + 0.935181i \(0.615236\pi\)
\(948\) 0 0
\(949\) −75.1918 −2.44083
\(950\) 0 0
\(951\) 14.0000 0.453981
\(952\) 0 0
\(953\) −41.1918 −1.33433 −0.667167 0.744908i \(-0.732493\pi\)
−0.667167 + 0.744908i \(0.732493\pi\)
\(954\) 0 0
\(955\) 9.79796 0.317055
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 68.5857 2.21475
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −5.79796 −0.186837
\(964\) 0 0
\(965\) −1.10102 −0.0354431
\(966\) 0 0
\(967\) 41.3939 1.33114 0.665569 0.746337i \(-0.268189\pi\)
0.665569 + 0.746337i \(0.268189\pi\)
\(968\) 0 0
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) −38.6969 −1.24184 −0.620922 0.783872i \(-0.713242\pi\)
−0.620922 + 0.783872i \(0.713242\pi\)
\(972\) 0 0
\(973\) −67.5959 −2.16703
\(974\) 0 0
\(975\) 6.89898 0.220944
\(976\) 0 0
\(977\) −29.5959 −0.946857 −0.473429 0.880832i \(-0.656984\pi\)
−0.473429 + 0.880832i \(0.656984\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 11.7980 0.376680
\(982\) 0 0
\(983\) −9.39388 −0.299618 −0.149809 0.988715i \(-0.547866\pi\)
−0.149809 + 0.988715i \(0.547866\pi\)
\(984\) 0 0
\(985\) −13.5959 −0.433202
\(986\) 0 0
\(987\) −48.0000 −1.52786
\(988\) 0 0
\(989\) −35.5959 −1.13188
\(990\) 0 0
\(991\) 15.5959 0.495421 0.247710 0.968834i \(-0.420322\pi\)
0.247710 + 0.968834i \(0.420322\pi\)
\(992\) 0 0
\(993\) 5.79796 0.183993
\(994\) 0 0
\(995\) −15.5959 −0.494424
\(996\) 0 0
\(997\) −21.5959 −0.683950 −0.341975 0.939709i \(-0.611096\pi\)
−0.341975 + 0.939709i \(0.611096\pi\)
\(998\) 0 0
\(999\) 6.00000 0.189832
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 4080.2.a.bq.1.2 2
4.3 odd 2 510.2.a.h.1.1 2
12.11 even 2 1530.2.a.s.1.1 2
20.3 even 4 2550.2.d.u.2449.4 4
20.7 even 4 2550.2.d.u.2449.1 4
20.19 odd 2 2550.2.a.bl.1.2 2
60.59 even 2 7650.2.a.cu.1.2 2
68.67 odd 2 8670.2.a.be.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.h.1.1 2 4.3 odd 2
1530.2.a.s.1.1 2 12.11 even 2
2550.2.a.bl.1.2 2 20.19 odd 2
2550.2.d.u.2449.1 4 20.7 even 4
2550.2.d.u.2449.4 4 20.3 even 4
4080.2.a.bq.1.2 2 1.1 even 1 trivial
7650.2.a.cu.1.2 2 60.59 even 2
8670.2.a.be.1.2 2 68.67 odd 2