Properties

Label 6552.2.a.bk.1.1
Level $6552$
Weight $2$
Character 6552.1
Self dual yes
Analytic conductor $52.318$
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] = [6552,2,Mod(1,6552)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6552, 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("6552.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6552.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: \(52.3179834043\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2184)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 6552.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.561553 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-0.561553 q^{5} -1.00000 q^{7} +5.12311 q^{11} +1.00000 q^{13} -7.12311 q^{17} +2.56155 q^{19} +3.68466 q^{23} -4.68466 q^{25} -4.56155 q^{29} -1.43845 q^{31} +0.561553 q^{35} -7.12311 q^{37} +2.00000 q^{41} -5.43845 q^{43} -5.43845 q^{47} +1.00000 q^{49} +5.68466 q^{53} -2.87689 q^{55} +8.00000 q^{59} -2.00000 q^{61} -0.561553 q^{65} +1.12311 q^{67} -6.24621 q^{71} -12.5616 q^{73} -5.12311 q^{77} -3.68466 q^{79} +14.5616 q^{83} +4.00000 q^{85} +10.8078 q^{89} -1.00000 q^{91} -1.43845 q^{95} +5.68466 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{5} - 2 q^{7} + 2 q^{11} + 2 q^{13} - 6 q^{17} + q^{19} - 5 q^{23} + 3 q^{25} - 5 q^{29} - 7 q^{31} - 3 q^{35} - 6 q^{37} + 4 q^{41} - 15 q^{43} - 15 q^{47} + 2 q^{49} - q^{53} - 14 q^{55} + 16 q^{59} - 4 q^{61} + 3 q^{65} - 6 q^{67} + 4 q^{71} - 21 q^{73} - 2 q^{77} + 5 q^{79} + 25 q^{83} + 8 q^{85} + q^{89} - 2 q^{91} - 7 q^{95} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.561553 −0.251134 −0.125567 0.992085i \(-0.540075\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.12311 1.54467 0.772337 0.635213i \(-0.219088\pi\)
0.772337 + 0.635213i \(0.219088\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.12311 −1.72761 −0.863803 0.503829i \(-0.831924\pi\)
−0.863803 + 0.503829i \(0.831924\pi\)
\(18\) 0 0
\(19\) 2.56155 0.587661 0.293830 0.955858i \(-0.405070\pi\)
0.293830 + 0.955858i \(0.405070\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.68466 0.768304 0.384152 0.923270i \(-0.374494\pi\)
0.384152 + 0.923270i \(0.374494\pi\)
\(24\) 0 0
\(25\) −4.68466 −0.936932
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.56155 −0.847059 −0.423530 0.905882i \(-0.639209\pi\)
−0.423530 + 0.905882i \(0.639209\pi\)
\(30\) 0 0
\(31\) −1.43845 −0.258353 −0.129176 0.991622i \(-0.541233\pi\)
−0.129176 + 0.991622i \(0.541233\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.561553 0.0949197
\(36\) 0 0
\(37\) −7.12311 −1.17103 −0.585516 0.810661i \(-0.699108\pi\)
−0.585516 + 0.810661i \(0.699108\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −5.43845 −0.829355 −0.414678 0.909968i \(-0.636106\pi\)
−0.414678 + 0.909968i \(0.636106\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.43845 −0.793279 −0.396640 0.917974i \(-0.629824\pi\)
−0.396640 + 0.917974i \(0.629824\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.68466 0.780848 0.390424 0.920635i \(-0.372328\pi\)
0.390424 + 0.920635i \(0.372328\pi\)
\(54\) 0 0
\(55\) −2.87689 −0.387920
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.561553 −0.0696521
\(66\) 0 0
\(67\) 1.12311 0.137209 0.0686046 0.997644i \(-0.478145\pi\)
0.0686046 + 0.997644i \(0.478145\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.24621 −0.741289 −0.370644 0.928775i \(-0.620863\pi\)
−0.370644 + 0.928775i \(0.620863\pi\)
\(72\) 0 0
\(73\) −12.5616 −1.47022 −0.735109 0.677949i \(-0.762869\pi\)
−0.735109 + 0.677949i \(0.762869\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.12311 −0.583832
\(78\) 0 0
\(79\) −3.68466 −0.414556 −0.207278 0.978282i \(-0.566461\pi\)
−0.207278 + 0.978282i \(0.566461\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.5616 1.59834 0.799169 0.601106i \(-0.205273\pi\)
0.799169 + 0.601106i \(0.205273\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.8078 1.14562 0.572810 0.819688i \(-0.305853\pi\)
0.572810 + 0.819688i \(0.305853\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.43845 −0.147582
\(96\) 0 0
\(97\) 5.68466 0.577190 0.288595 0.957451i \(-0.406812\pi\)
0.288595 + 0.957451i \(0.406812\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.2462 1.21854 0.609272 0.792961i \(-0.291462\pi\)
0.609272 + 0.792961i \(0.291462\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.4924 −1.59438 −0.797191 0.603727i \(-0.793682\pi\)
−0.797191 + 0.603727i \(0.793682\pi\)
\(108\) 0 0
\(109\) 0.246211 0.0235828 0.0117914 0.999930i \(-0.496247\pi\)
0.0117914 + 0.999930i \(0.496247\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −13.6847 −1.28734 −0.643672 0.765301i \(-0.722590\pi\)
−0.643672 + 0.765301i \(0.722590\pi\)
\(114\) 0 0
\(115\) −2.06913 −0.192947
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 7.12311 0.652974
\(120\) 0 0
\(121\) 15.2462 1.38602
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.43845 0.486430
\(126\) 0 0
\(127\) −5.75379 −0.510566 −0.255283 0.966866i \(-0.582169\pi\)
−0.255283 + 0.966866i \(0.582169\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) −2.56155 −0.222115
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.87689 0.416661 0.208331 0.978058i \(-0.433197\pi\)
0.208331 + 0.978058i \(0.433197\pi\)
\(138\) 0 0
\(139\) −6.87689 −0.583291 −0.291645 0.956527i \(-0.594203\pi\)
−0.291645 + 0.956527i \(0.594203\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.12311 0.428416
\(144\) 0 0
\(145\) 2.56155 0.212725
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.24621 0.675556 0.337778 0.941226i \(-0.390325\pi\)
0.337778 + 0.941226i \(0.390325\pi\)
\(150\) 0 0
\(151\) −20.4924 −1.66765 −0.833825 0.552029i \(-0.813854\pi\)
−0.833825 + 0.552029i \(0.813854\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.807764 0.0648812
\(156\) 0 0
\(157\) 0.876894 0.0699838 0.0349919 0.999388i \(-0.488859\pi\)
0.0349919 + 0.999388i \(0.488859\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.68466 −0.290392
\(162\) 0 0
\(163\) −14.2462 −1.11585 −0.557925 0.829892i \(-0.688402\pi\)
−0.557925 + 0.829892i \(0.688402\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.315342 −0.0244019 −0.0122009 0.999926i \(-0.503884\pi\)
−0.0122009 + 0.999926i \(0.503884\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 17.3693 1.32056 0.660282 0.751017i \(-0.270437\pi\)
0.660282 + 0.751017i \(0.270437\pi\)
\(174\) 0 0
\(175\) 4.68466 0.354127
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.4384 1.00444 0.502218 0.864741i \(-0.332517\pi\)
0.502218 + 0.864741i \(0.332517\pi\)
\(180\) 0 0
\(181\) −4.24621 −0.315618 −0.157809 0.987470i \(-0.550443\pi\)
−0.157809 + 0.987470i \(0.550443\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) −36.4924 −2.66859
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.4924 −1.48278 −0.741390 0.671075i \(-0.765833\pi\)
−0.741390 + 0.671075i \(0.765833\pi\)
\(192\) 0 0
\(193\) −8.24621 −0.593575 −0.296788 0.954944i \(-0.595915\pi\)
−0.296788 + 0.954944i \(0.595915\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) −10.2462 −0.726335 −0.363167 0.931724i \(-0.618305\pi\)
−0.363167 + 0.931724i \(0.618305\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.56155 0.320158
\(204\) 0 0
\(205\) −1.12311 −0.0784411
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.1231 0.907744
\(210\) 0 0
\(211\) −15.6847 −1.07978 −0.539888 0.841737i \(-0.681533\pi\)
−0.539888 + 0.841737i \(0.681533\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.05398 0.208279
\(216\) 0 0
\(217\) 1.43845 0.0976482
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.12311 −0.479152
\(222\) 0 0
\(223\) −8.80776 −0.589812 −0.294906 0.955526i \(-0.595288\pi\)
−0.294906 + 0.955526i \(0.595288\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.75379 0.381892 0.190946 0.981601i \(-0.438844\pi\)
0.190946 + 0.981601i \(0.438844\pi\)
\(228\) 0 0
\(229\) 0.246211 0.0162701 0.00813505 0.999967i \(-0.497411\pi\)
0.00813505 + 0.999967i \(0.497411\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.3153 −0.937829 −0.468915 0.883244i \(-0.655355\pi\)
−0.468915 + 0.883244i \(0.655355\pi\)
\(234\) 0 0
\(235\) 3.05398 0.199219
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.49242 −0.549329 −0.274665 0.961540i \(-0.588567\pi\)
−0.274665 + 0.961540i \(0.588567\pi\)
\(240\) 0 0
\(241\) −14.8078 −0.953852 −0.476926 0.878943i \(-0.658249\pi\)
−0.476926 + 0.878943i \(0.658249\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.561553 −0.0358763
\(246\) 0 0
\(247\) 2.56155 0.162988
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.1231 1.08080 0.540400 0.841408i \(-0.318273\pi\)
0.540400 + 0.841408i \(0.318273\pi\)
\(252\) 0 0
\(253\) 18.8769 1.18678
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 7.12311 0.442608
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.43845 −0.582000 −0.291000 0.956723i \(-0.593988\pi\)
−0.291000 + 0.956723i \(0.593988\pi\)
\(264\) 0 0
\(265\) −3.19224 −0.196097
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −23.6155 −1.43986 −0.719932 0.694045i \(-0.755827\pi\)
−0.719932 + 0.694045i \(0.755827\pi\)
\(270\) 0 0
\(271\) 20.4924 1.24483 0.622413 0.782689i \(-0.286152\pi\)
0.622413 + 0.782689i \(0.286152\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −24.0000 −1.44725
\(276\) 0 0
\(277\) −21.0540 −1.26501 −0.632505 0.774556i \(-0.717973\pi\)
−0.632505 + 0.774556i \(0.717973\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.24621 −0.491928 −0.245964 0.969279i \(-0.579104\pi\)
−0.245964 + 0.969279i \(0.579104\pi\)
\(282\) 0 0
\(283\) 24.4924 1.45592 0.727962 0.685618i \(-0.240468\pi\)
0.727962 + 0.685618i \(0.240468\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) 33.7386 1.98463
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.43845 0.434559 0.217279 0.976109i \(-0.430282\pi\)
0.217279 + 0.976109i \(0.430282\pi\)
\(294\) 0 0
\(295\) −4.49242 −0.261559
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.68466 0.213089
\(300\) 0 0
\(301\) 5.43845 0.313467
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.12311 0.0643088
\(306\) 0 0
\(307\) 7.68466 0.438587 0.219293 0.975659i \(-0.429625\pi\)
0.219293 + 0.975659i \(0.429625\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −27.8617 −1.57989 −0.789947 0.613175i \(-0.789892\pi\)
−0.789947 + 0.613175i \(0.789892\pi\)
\(312\) 0 0
\(313\) −32.2462 −1.82266 −0.911332 0.411673i \(-0.864945\pi\)
−0.911332 + 0.411673i \(0.864945\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.36932 −0.526233 −0.263117 0.964764i \(-0.584750\pi\)
−0.263117 + 0.964764i \(0.584750\pi\)
\(318\) 0 0
\(319\) −23.3693 −1.30843
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −18.2462 −1.01525
\(324\) 0 0
\(325\) −4.68466 −0.259858
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.43845 0.299831
\(330\) 0 0
\(331\) −6.87689 −0.377988 −0.188994 0.981978i \(-0.560523\pi\)
−0.188994 + 0.981978i \(0.560523\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.630683 −0.0344579
\(336\) 0 0
\(337\) −9.05398 −0.493201 −0.246601 0.969117i \(-0.579314\pi\)
−0.246601 + 0.969117i \(0.579314\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7.36932 −0.399071
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −14.2462 −0.764777 −0.382388 0.924002i \(-0.624898\pi\)
−0.382388 + 0.924002i \(0.624898\pi\)
\(348\) 0 0
\(349\) 3.93087 0.210415 0.105207 0.994450i \(-0.466449\pi\)
0.105207 + 0.994450i \(0.466449\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.24621 0.226003 0.113002 0.993595i \(-0.463954\pi\)
0.113002 + 0.993595i \(0.463954\pi\)
\(354\) 0 0
\(355\) 3.50758 0.186163
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.49242 0.448213 0.224106 0.974565i \(-0.428054\pi\)
0.224106 + 0.974565i \(0.428054\pi\)
\(360\) 0 0
\(361\) −12.4384 −0.654655
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.05398 0.369222
\(366\) 0 0
\(367\) −33.6155 −1.75472 −0.877358 0.479836i \(-0.840696\pi\)
−0.877358 + 0.479836i \(0.840696\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.68466 −0.295133
\(372\) 0 0
\(373\) 1.50758 0.0780594 0.0390297 0.999238i \(-0.487573\pi\)
0.0390297 + 0.999238i \(0.487573\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.56155 −0.234932
\(378\) 0 0
\(379\) 11.3693 0.584003 0.292001 0.956418i \(-0.405679\pi\)
0.292001 + 0.956418i \(0.405679\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 0 0
\(385\) 2.87689 0.146620
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −38.9848 −1.97661 −0.988305 0.152490i \(-0.951271\pi\)
−0.988305 + 0.152490i \(0.951271\pi\)
\(390\) 0 0
\(391\) −26.2462 −1.32733
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.06913 0.104109
\(396\) 0 0
\(397\) 24.4233 1.22577 0.612885 0.790172i \(-0.290009\pi\)
0.612885 + 0.790172i \(0.290009\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.1231 −0.555461 −0.277731 0.960659i \(-0.589582\pi\)
−0.277731 + 0.960659i \(0.589582\pi\)
\(402\) 0 0
\(403\) −1.43845 −0.0716542
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −36.4924 −1.80886
\(408\) 0 0
\(409\) 19.4384 0.961169 0.480585 0.876948i \(-0.340424\pi\)
0.480585 + 0.876948i \(0.340424\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) −8.17708 −0.401397
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3.36932 0.164602 0.0823010 0.996608i \(-0.473773\pi\)
0.0823010 + 0.996608i \(0.473773\pi\)
\(420\) 0 0
\(421\) −15.1231 −0.737055 −0.368528 0.929617i \(-0.620138\pi\)
−0.368528 + 0.929617i \(0.620138\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 33.3693 1.61865
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3.36932 −0.162294 −0.0811471 0.996702i \(-0.525858\pi\)
−0.0811471 + 0.996702i \(0.525858\pi\)
\(432\) 0 0
\(433\) 19.6155 0.942662 0.471331 0.881956i \(-0.343774\pi\)
0.471331 + 0.881956i \(0.343774\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.43845 0.451502
\(438\) 0 0
\(439\) 41.6155 1.98620 0.993100 0.117267i \(-0.0374134\pi\)
0.993100 + 0.117267i \(0.0374134\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.19224 0.151668 0.0758339 0.997120i \(-0.475838\pi\)
0.0758339 + 0.997120i \(0.475838\pi\)
\(444\) 0 0
\(445\) −6.06913 −0.287704
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 29.8617 1.40926 0.704631 0.709574i \(-0.251112\pi\)
0.704631 + 0.709574i \(0.251112\pi\)
\(450\) 0 0
\(451\) 10.2462 0.482475
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.561553 0.0263260
\(456\) 0 0
\(457\) 35.6155 1.66602 0.833012 0.553255i \(-0.186614\pi\)
0.833012 + 0.553255i \(0.186614\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 19.7538 0.920026 0.460013 0.887912i \(-0.347845\pi\)
0.460013 + 0.887912i \(0.347845\pi\)
\(462\) 0 0
\(463\) 39.3693 1.82965 0.914824 0.403854i \(-0.132329\pi\)
0.914824 + 0.403854i \(0.132329\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.6307 −0.584478 −0.292239 0.956345i \(-0.594400\pi\)
−0.292239 + 0.956345i \(0.594400\pi\)
\(468\) 0 0
\(469\) −1.12311 −0.0518602
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −27.8617 −1.28108
\(474\) 0 0
\(475\) −12.0000 −0.550598
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −16.3153 −0.745467 −0.372733 0.927938i \(-0.621579\pi\)
−0.372733 + 0.927938i \(0.621579\pi\)
\(480\) 0 0
\(481\) −7.12311 −0.324786
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.19224 −0.144952
\(486\) 0 0
\(487\) −8.63068 −0.391094 −0.195547 0.980694i \(-0.562648\pi\)
−0.195547 + 0.980694i \(0.562648\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 24.4924 1.10533 0.552664 0.833404i \(-0.313611\pi\)
0.552664 + 0.833404i \(0.313611\pi\)
\(492\) 0 0
\(493\) 32.4924 1.46339
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.24621 0.280181
\(498\) 0 0
\(499\) −16.4924 −0.738302 −0.369151 0.929369i \(-0.620352\pi\)
−0.369151 + 0.929369i \(0.620352\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −23.3693 −1.04199 −0.520993 0.853561i \(-0.674438\pi\)
−0.520993 + 0.853561i \(0.674438\pi\)
\(504\) 0 0
\(505\) −6.87689 −0.306018
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8.56155 −0.379484 −0.189742 0.981834i \(-0.560765\pi\)
−0.189742 + 0.981834i \(0.560765\pi\)
\(510\) 0 0
\(511\) 12.5616 0.555690
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.49242 0.197960
\(516\) 0 0
\(517\) −27.8617 −1.22536
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −23.1231 −1.01304 −0.506521 0.862228i \(-0.669069\pi\)
−0.506521 + 0.862228i \(0.669069\pi\)
\(522\) 0 0
\(523\) −1.12311 −0.0491100 −0.0245550 0.999698i \(-0.507817\pi\)
−0.0245550 + 0.999698i \(0.507817\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.2462 0.446332
\(528\) 0 0
\(529\) −9.42329 −0.409708
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) 9.26137 0.400404
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.12311 0.220668
\(540\) 0 0
\(541\) −18.6307 −0.800996 −0.400498 0.916298i \(-0.631163\pi\)
−0.400498 + 0.916298i \(0.631163\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.138261 −0.00592243
\(546\) 0 0
\(547\) 0.946025 0.0404491 0.0202245 0.999795i \(-0.493562\pi\)
0.0202245 + 0.999795i \(0.493562\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −11.6847 −0.497783
\(552\) 0 0
\(553\) 3.68466 0.156688
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −17.3693 −0.735962 −0.367981 0.929833i \(-0.619951\pi\)
−0.367981 + 0.929833i \(0.619951\pi\)
\(558\) 0 0
\(559\) −5.43845 −0.230022
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.49242 0.357913 0.178956 0.983857i \(-0.442728\pi\)
0.178956 + 0.983857i \(0.442728\pi\)
\(564\) 0 0
\(565\) 7.68466 0.323296
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 38.1771 1.60047 0.800233 0.599689i \(-0.204709\pi\)
0.800233 + 0.599689i \(0.204709\pi\)
\(570\) 0 0
\(571\) 7.05398 0.295200 0.147600 0.989047i \(-0.452845\pi\)
0.147600 + 0.989047i \(0.452845\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −17.2614 −0.719849
\(576\) 0 0
\(577\) 4.24621 0.176772 0.0883860 0.996086i \(-0.471829\pi\)
0.0883860 + 0.996086i \(0.471829\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −14.5616 −0.604115
\(582\) 0 0
\(583\) 29.1231 1.20616
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.9309 −0.574989 −0.287494 0.957782i \(-0.592822\pi\)
−0.287494 + 0.957782i \(0.592822\pi\)
\(588\) 0 0
\(589\) −3.68466 −0.151824
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 31.9309 1.31124 0.655622 0.755089i \(-0.272407\pi\)
0.655622 + 0.755089i \(0.272407\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.1922 0.620738 0.310369 0.950616i \(-0.399547\pi\)
0.310369 + 0.950616i \(0.399547\pi\)
\(600\) 0 0
\(601\) 14.4924 0.591158 0.295579 0.955318i \(-0.404487\pi\)
0.295579 + 0.955318i \(0.404487\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8.56155 −0.348077
\(606\) 0 0
\(607\) 2.87689 0.116770 0.0583848 0.998294i \(-0.481405\pi\)
0.0583848 + 0.998294i \(0.481405\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5.43845 −0.220016
\(612\) 0 0
\(613\) −23.1231 −0.933933 −0.466967 0.884275i \(-0.654653\pi\)
−0.466967 + 0.884275i \(0.654653\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −24.4924 −0.984434 −0.492217 0.870473i \(-0.663813\pi\)
−0.492217 + 0.870473i \(0.663813\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.8078 −0.433004
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 50.7386 2.02308
\(630\) 0 0
\(631\) 48.3542 1.92495 0.962474 0.271373i \(-0.0874775\pi\)
0.962474 + 0.271373i \(0.0874775\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.23106 0.128221
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 41.0540 1.62153 0.810767 0.585369i \(-0.199050\pi\)
0.810767 + 0.585369i \(0.199050\pi\)
\(642\) 0 0
\(643\) −26.7386 −1.05447 −0.527234 0.849720i \(-0.676771\pi\)
−0.527234 + 0.849720i \(0.676771\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 47.3693 1.86228 0.931140 0.364662i \(-0.118815\pi\)
0.931140 + 0.364662i \(0.118815\pi\)
\(648\) 0 0
\(649\) 40.9848 1.60880
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19.7538 −0.773025 −0.386513 0.922284i \(-0.626320\pi\)
−0.386513 + 0.922284i \(0.626320\pi\)
\(654\) 0 0
\(655\) 2.24621 0.0877667
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 47.0540 1.83296 0.916481 0.400077i \(-0.131017\pi\)
0.916481 + 0.400077i \(0.131017\pi\)
\(660\) 0 0
\(661\) −34.8078 −1.35387 −0.676933 0.736045i \(-0.736691\pi\)
−0.676933 + 0.736045i \(0.736691\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.43845 0.0557806
\(666\) 0 0
\(667\) −16.8078 −0.650799
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −10.2462 −0.395551
\(672\) 0 0
\(673\) 31.9309 1.23084 0.615422 0.788198i \(-0.288985\pi\)
0.615422 + 0.788198i \(0.288985\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10.4924 −0.403257 −0.201628 0.979462i \(-0.564623\pi\)
−0.201628 + 0.979462i \(0.564623\pi\)
\(678\) 0 0
\(679\) −5.68466 −0.218157
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −32.9848 −1.26213 −0.631065 0.775730i \(-0.717382\pi\)
−0.631065 + 0.775730i \(0.717382\pi\)
\(684\) 0 0
\(685\) −2.73863 −0.104638
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 5.68466 0.216568
\(690\) 0 0
\(691\) 13.4384 0.511223 0.255611 0.966780i \(-0.417723\pi\)
0.255611 + 0.966780i \(0.417723\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.86174 0.146484
\(696\) 0 0
\(697\) −14.2462 −0.539614
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15.3002 0.577880 0.288940 0.957347i \(-0.406697\pi\)
0.288940 + 0.957347i \(0.406697\pi\)
\(702\) 0 0
\(703\) −18.2462 −0.688169
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.2462 −0.460566
\(708\) 0 0
\(709\) 34.4924 1.29539 0.647695 0.761900i \(-0.275733\pi\)
0.647695 + 0.761900i \(0.275733\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.30019 −0.198494
\(714\) 0 0
\(715\) −2.87689 −0.107590
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −47.3693 −1.76658 −0.883289 0.468829i \(-0.844676\pi\)
−0.883289 + 0.468829i \(0.844676\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 21.3693 0.793637
\(726\) 0 0
\(727\) 22.1080 0.819939 0.409969 0.912099i \(-0.365539\pi\)
0.409969 + 0.912099i \(0.365539\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 38.7386 1.43280
\(732\) 0 0
\(733\) −45.0540 −1.66411 −0.832053 0.554696i \(-0.812835\pi\)
−0.832053 + 0.554696i \(0.812835\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.75379 0.211944
\(738\) 0 0
\(739\) −51.3693 −1.88965 −0.944825 0.327574i \(-0.893769\pi\)
−0.944825 + 0.327574i \(0.893769\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −17.7538 −0.651323 −0.325662 0.945486i \(-0.605587\pi\)
−0.325662 + 0.945486i \(0.605587\pi\)
\(744\) 0 0
\(745\) −4.63068 −0.169655
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 16.4924 0.602620
\(750\) 0 0
\(751\) −21.9309 −0.800269 −0.400134 0.916456i \(-0.631037\pi\)
−0.400134 + 0.916456i \(0.631037\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 11.5076 0.418804
\(756\) 0 0
\(757\) 15.4384 0.561120 0.280560 0.959837i \(-0.409480\pi\)
0.280560 + 0.959837i \(0.409480\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −19.3002 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(762\) 0 0
\(763\) −0.246211 −0.00891345
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 6.31534 0.227737 0.113869 0.993496i \(-0.463676\pi\)
0.113869 + 0.993496i \(0.463676\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −13.5076 −0.485834 −0.242917 0.970047i \(-0.578104\pi\)
−0.242917 + 0.970047i \(0.578104\pi\)
\(774\) 0 0
\(775\) 6.73863 0.242059
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.12311 0.183554
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.492423 −0.0175753
\(786\) 0 0
\(787\) −14.4233 −0.514135 −0.257067 0.966393i \(-0.582756\pi\)
−0.257067 + 0.966393i \(0.582756\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 13.6847 0.486570
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 38.7386 1.37047
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −64.3542 −2.27101
\(804\) 0 0
\(805\) 2.06913 0.0729273
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 50.0388 1.75927 0.879636 0.475648i \(-0.157786\pi\)
0.879636 + 0.475648i \(0.157786\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) −13.9309 −0.487379
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −53.2311 −1.85778 −0.928888 0.370360i \(-0.879234\pi\)
−0.928888 + 0.370360i \(0.879234\pi\)
\(822\) 0 0
\(823\) 40.9848 1.42864 0.714321 0.699818i \(-0.246736\pi\)
0.714321 + 0.699818i \(0.246736\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −19.5076 −0.678345 −0.339172 0.940724i \(-0.610147\pi\)
−0.339172 + 0.940724i \(0.610147\pi\)
\(828\) 0 0
\(829\) 3.75379 0.130374 0.0651872 0.997873i \(-0.479236\pi\)
0.0651872 + 0.997873i \(0.479236\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −7.12311 −0.246801
\(834\) 0 0
\(835\) 0.177081 0.00612814
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26.7386 −0.923120 −0.461560 0.887109i \(-0.652710\pi\)
−0.461560 + 0.887109i \(0.652710\pi\)
\(840\) 0 0
\(841\) −8.19224 −0.282491
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.561553 −0.0193180
\(846\) 0 0
\(847\) −15.2462 −0.523866
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −26.2462 −0.899709
\(852\) 0 0
\(853\) −46.6695 −1.59793 −0.798967 0.601375i \(-0.794620\pi\)
−0.798967 + 0.601375i \(0.794620\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.2462 1.10151 0.550755 0.834667i \(-0.314340\pi\)
0.550755 + 0.834667i \(0.314340\pi\)
\(858\) 0 0
\(859\) −30.8769 −1.05351 −0.526753 0.850018i \(-0.676591\pi\)
−0.526753 + 0.850018i \(0.676591\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −20.9848 −0.714332 −0.357166 0.934041i \(-0.616257\pi\)
−0.357166 + 0.934041i \(0.616257\pi\)
\(864\) 0 0
\(865\) −9.75379 −0.331639
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −18.8769 −0.640355
\(870\) 0 0
\(871\) 1.12311 0.0380550
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.43845 −0.183853
\(876\) 0 0
\(877\) −24.1080 −0.814068 −0.407034 0.913413i \(-0.633437\pi\)
−0.407034 + 0.913413i \(0.633437\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 40.4924 1.36268 0.681339 0.731968i \(-0.261398\pi\)
0.681339 + 0.731968i \(0.261398\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −50.2462 −1.68710 −0.843551 0.537049i \(-0.819539\pi\)
−0.843551 + 0.537049i \(0.819539\pi\)
\(888\) 0 0
\(889\) 5.75379 0.192976
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −13.9309 −0.466179
\(894\) 0 0
\(895\) −7.54640 −0.252248
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.56155 0.218840
\(900\) 0 0
\(901\) −40.4924 −1.34900
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.38447 0.0792625
\(906\) 0 0
\(907\) −13.4384 −0.446216 −0.223108 0.974794i \(-0.571620\pi\)
−0.223108 + 0.974794i \(0.571620\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 43.0540 1.42644 0.713221 0.700939i \(-0.247236\pi\)
0.713221 + 0.700939i \(0.247236\pi\)
\(912\) 0 0
\(913\) 74.6004 2.46891
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) 28.4924 0.939878 0.469939 0.882699i \(-0.344276\pi\)
0.469939 + 0.882699i \(0.344276\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −6.24621 −0.205597
\(924\) 0 0
\(925\) 33.3693 1.09718
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.4384 1.16270 0.581349 0.813654i \(-0.302525\pi\)
0.581349 + 0.813654i \(0.302525\pi\)
\(930\) 0 0
\(931\) 2.56155 0.0839515
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 20.4924 0.670174
\(936\) 0 0
\(937\) 27.6155 0.902160 0.451080 0.892484i \(-0.351039\pi\)
0.451080 + 0.892484i \(0.351039\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 53.5464 1.74556 0.872781 0.488111i \(-0.162314\pi\)
0.872781 + 0.488111i \(0.162314\pi\)
\(942\) 0 0
\(943\) 7.36932 0.239978
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19.8617 −0.645420 −0.322710 0.946498i \(-0.604594\pi\)
−0.322710 + 0.946498i \(0.604594\pi\)
\(948\) 0 0
\(949\) −12.5616 −0.407765
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −40.9157 −1.32539 −0.662695 0.748889i \(-0.730587\pi\)
−0.662695 + 0.748889i \(0.730587\pi\)
\(954\) 0 0
\(955\) 11.5076 0.372376
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.87689 −0.157483
\(960\) 0 0
\(961\) −28.9309 −0.933254
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.63068 0.149067
\(966\) 0 0
\(967\) −38.1080 −1.22547 −0.612735 0.790289i \(-0.709931\pi\)
−0.612735 + 0.790289i \(0.709931\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −36.3542 −1.16666 −0.583330 0.812235i \(-0.698251\pi\)
−0.583330 + 0.812235i \(0.698251\pi\)
\(972\) 0 0
\(973\) 6.87689 0.220463
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −50.4924 −1.61540 −0.807698 0.589597i \(-0.799287\pi\)
−0.807698 + 0.589597i \(0.799287\pi\)
\(978\) 0 0
\(979\) 55.3693 1.76961
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −27.5464 −0.878594 −0.439297 0.898342i \(-0.644772\pi\)
−0.439297 + 0.898342i \(0.644772\pi\)
\(984\) 0 0
\(985\) 5.61553 0.178926
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −20.0388 −0.637197
\(990\) 0 0
\(991\) −38.7386 −1.23057 −0.615287 0.788303i \(-0.710960\pi\)
−0.615287 + 0.788303i \(0.710960\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5.75379 0.182407
\(996\) 0 0
\(997\) 48.6004 1.53919 0.769595 0.638533i \(-0.220458\pi\)
0.769595 + 0.638533i \(0.220458\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6552.2.a.bk.1.1 2
3.2 odd 2 2184.2.a.n.1.2 2
12.11 even 2 4368.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2184.2.a.n.1.2 2 3.2 odd 2
4368.2.a.bf.1.2 2 12.11 even 2
6552.2.a.bk.1.1 2 1.1 even 1 trivial