Properties

Label 572.2.a.d.1.2
Level $572$
Weight $2$
Character 572.1
Self dual yes
Analytic conductor $4.567$
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] = [572,2,Mod(1,572)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(572, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("572.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.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: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.79129\) of defining polynomial
Character \(\chi\) \(=\) 572.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.79129 q^{3} +2.00000 q^{5} +0.208712 q^{7} +4.79129 q^{9} +O(q^{10})\) \(q+2.79129 q^{3} +2.00000 q^{5} +0.208712 q^{7} +4.79129 q^{9} -1.00000 q^{11} -1.00000 q^{13} +5.58258 q^{15} -4.00000 q^{17} +1.20871 q^{19} +0.582576 q^{21} +0.208712 q^{23} -1.00000 q^{25} +5.00000 q^{27} -3.58258 q^{29} -2.79129 q^{33} +0.417424 q^{35} +4.00000 q^{37} -2.79129 q^{39} +1.79129 q^{41} -7.16515 q^{43} +9.58258 q^{45} +3.58258 q^{47} -6.95644 q^{49} -11.1652 q^{51} +4.37386 q^{53} -2.00000 q^{55} +3.37386 q^{57} -5.58258 q^{59} +2.41742 q^{61} +1.00000 q^{63} -2.00000 q^{65} +11.1652 q^{67} +0.582576 q^{69} -10.7913 q^{73} -2.79129 q^{75} -0.208712 q^{77} +12.7477 q^{79} -0.417424 q^{81} +0.626136 q^{83} -8.00000 q^{85} -10.0000 q^{87} +9.58258 q^{89} -0.208712 q^{91} +2.41742 q^{95} +18.7477 q^{97} -4.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 4 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 4 q^{5} + 5 q^{7} + 5 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{15} - 8 q^{17} + 7 q^{19} - 8 q^{21} + 5 q^{23} - 2 q^{25} + 10 q^{27} + 2 q^{29} - q^{33} + 10 q^{35} + 8 q^{37} - q^{39} - q^{41} + 4 q^{43} + 10 q^{45} - 2 q^{47} + 9 q^{49} - 4 q^{51} - 5 q^{53} - 4 q^{55} - 7 q^{57} - 2 q^{59} + 14 q^{61} + 2 q^{63} - 4 q^{65} + 4 q^{67} - 8 q^{69} - 17 q^{73} - q^{75} - 5 q^{77} - 2 q^{79} - 10 q^{81} + 15 q^{83} - 16 q^{85} - 20 q^{87} + 10 q^{89} - 5 q^{91} + 14 q^{95} + 10 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.79129 1.61155 0.805775 0.592221i \(-0.201749\pi\)
0.805775 + 0.592221i \(0.201749\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0.208712 0.0788858 0.0394429 0.999222i \(-0.487442\pi\)
0.0394429 + 0.999222i \(0.487442\pi\)
\(8\) 0 0
\(9\) 4.79129 1.59710
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 5.58258 1.44141
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 1.20871 0.277298 0.138649 0.990342i \(-0.455724\pi\)
0.138649 + 0.990342i \(0.455724\pi\)
\(20\) 0 0
\(21\) 0.582576 0.127128
\(22\) 0 0
\(23\) 0.208712 0.0435195 0.0217597 0.999763i \(-0.493073\pi\)
0.0217597 + 0.999763i \(0.493073\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −3.58258 −0.665268 −0.332634 0.943056i \(-0.607937\pi\)
−0.332634 + 0.943056i \(0.607937\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −2.79129 −0.485901
\(34\) 0 0
\(35\) 0.417424 0.0705576
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) −2.79129 −0.446964
\(40\) 0 0
\(41\) 1.79129 0.279752 0.139876 0.990169i \(-0.455330\pi\)
0.139876 + 0.990169i \(0.455330\pi\)
\(42\) 0 0
\(43\) −7.16515 −1.09268 −0.546338 0.837565i \(-0.683978\pi\)
−0.546338 + 0.837565i \(0.683978\pi\)
\(44\) 0 0
\(45\) 9.58258 1.42849
\(46\) 0 0
\(47\) 3.58258 0.522572 0.261286 0.965261i \(-0.415853\pi\)
0.261286 + 0.965261i \(0.415853\pi\)
\(48\) 0 0
\(49\) −6.95644 −0.993777
\(50\) 0 0
\(51\) −11.1652 −1.56343
\(52\) 0 0
\(53\) 4.37386 0.600796 0.300398 0.953814i \(-0.402880\pi\)
0.300398 + 0.953814i \(0.402880\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 3.37386 0.446879
\(58\) 0 0
\(59\) −5.58258 −0.726789 −0.363395 0.931635i \(-0.618382\pi\)
−0.363395 + 0.931635i \(0.618382\pi\)
\(60\) 0 0
\(61\) 2.41742 0.309519 0.154760 0.987952i \(-0.450540\pi\)
0.154760 + 0.987952i \(0.450540\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 11.1652 1.36404 0.682020 0.731333i \(-0.261102\pi\)
0.682020 + 0.731333i \(0.261102\pi\)
\(68\) 0 0
\(69\) 0.582576 0.0701339
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.7913 −1.26302 −0.631512 0.775366i \(-0.717565\pi\)
−0.631512 + 0.775366i \(0.717565\pi\)
\(74\) 0 0
\(75\) −2.79129 −0.322310
\(76\) 0 0
\(77\) −0.208712 −0.0237850
\(78\) 0 0
\(79\) 12.7477 1.43423 0.717116 0.696954i \(-0.245462\pi\)
0.717116 + 0.696954i \(0.245462\pi\)
\(80\) 0 0
\(81\) −0.417424 −0.0463805
\(82\) 0 0
\(83\) 0.626136 0.0687274 0.0343637 0.999409i \(-0.489060\pi\)
0.0343637 + 0.999409i \(0.489060\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) −10.0000 −1.07211
\(88\) 0 0
\(89\) 9.58258 1.01575 0.507875 0.861430i \(-0.330431\pi\)
0.507875 + 0.861430i \(0.330431\pi\)
\(90\) 0 0
\(91\) −0.208712 −0.0218790
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.41742 0.248023
\(96\) 0 0
\(97\) 18.7477 1.90354 0.951772 0.306807i \(-0.0992607\pi\)
0.951772 + 0.306807i \(0.0992607\pi\)
\(98\) 0 0
\(99\) −4.79129 −0.481543
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −0.208712 −0.0205650 −0.0102825 0.999947i \(-0.503273\pi\)
−0.0102825 + 0.999947i \(0.503273\pi\)
\(104\) 0 0
\(105\) 1.16515 0.113707
\(106\) 0 0
\(107\) 9.58258 0.926383 0.463191 0.886258i \(-0.346704\pi\)
0.463191 + 0.886258i \(0.346704\pi\)
\(108\) 0 0
\(109\) −1.20871 −0.115774 −0.0578868 0.998323i \(-0.518436\pi\)
−0.0578868 + 0.998323i \(0.518436\pi\)
\(110\) 0 0
\(111\) 11.1652 1.05975
\(112\) 0 0
\(113\) −21.1216 −1.98695 −0.993476 0.114041i \(-0.963621\pi\)
−0.993476 + 0.114041i \(0.963621\pi\)
\(114\) 0 0
\(115\) 0.417424 0.0389250
\(116\) 0 0
\(117\) −4.79129 −0.442955
\(118\) 0 0
\(119\) −0.834849 −0.0765304
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 5.00000 0.450835
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 17.5826 1.56020 0.780101 0.625654i \(-0.215168\pi\)
0.780101 + 0.625654i \(0.215168\pi\)
\(128\) 0 0
\(129\) −20.0000 −1.76090
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 0.252273 0.0218748
\(134\) 0 0
\(135\) 10.0000 0.860663
\(136\) 0 0
\(137\) −23.1652 −1.97913 −0.989566 0.144079i \(-0.953978\pi\)
−0.989566 + 0.144079i \(0.953978\pi\)
\(138\) 0 0
\(139\) −19.5826 −1.66097 −0.830486 0.557039i \(-0.811937\pi\)
−0.830486 + 0.557039i \(0.811937\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −7.16515 −0.595033
\(146\) 0 0
\(147\) −19.4174 −1.60152
\(148\) 0 0
\(149\) 1.37386 0.112551 0.0562756 0.998415i \(-0.482077\pi\)
0.0562756 + 0.998415i \(0.482077\pi\)
\(150\) 0 0
\(151\) 7.16515 0.583092 0.291546 0.956557i \(-0.405830\pi\)
0.291546 + 0.956557i \(0.405830\pi\)
\(152\) 0 0
\(153\) −19.1652 −1.54941
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.5390 −1.00072 −0.500361 0.865817i \(-0.666799\pi\)
−0.500361 + 0.865817i \(0.666799\pi\)
\(158\) 0 0
\(159\) 12.2087 0.968214
\(160\) 0 0
\(161\) 0.0435608 0.00343307
\(162\) 0 0
\(163\) −1.16515 −0.0912617 −0.0456309 0.998958i \(-0.514530\pi\)
−0.0456309 + 0.998958i \(0.514530\pi\)
\(164\) 0 0
\(165\) −5.58258 −0.434603
\(166\) 0 0
\(167\) 15.5390 1.20245 0.601223 0.799082i \(-0.294681\pi\)
0.601223 + 0.799082i \(0.294681\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 5.79129 0.442871
\(172\) 0 0
\(173\) −15.5826 −1.18472 −0.592361 0.805673i \(-0.701804\pi\)
−0.592361 + 0.805673i \(0.701804\pi\)
\(174\) 0 0
\(175\) −0.208712 −0.0157772
\(176\) 0 0
\(177\) −15.5826 −1.17126
\(178\) 0 0
\(179\) 3.16515 0.236575 0.118287 0.992979i \(-0.462260\pi\)
0.118287 + 0.992979i \(0.462260\pi\)
\(180\) 0 0
\(181\) 12.3739 0.919742 0.459871 0.887986i \(-0.347896\pi\)
0.459871 + 0.887986i \(0.347896\pi\)
\(182\) 0 0
\(183\) 6.74773 0.498806
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 1.04356 0.0759079
\(190\) 0 0
\(191\) −2.04356 −0.147867 −0.0739334 0.997263i \(-0.523555\pi\)
−0.0739334 + 0.997263i \(0.523555\pi\)
\(192\) 0 0
\(193\) 5.37386 0.386819 0.193410 0.981118i \(-0.438045\pi\)
0.193410 + 0.981118i \(0.438045\pi\)
\(194\) 0 0
\(195\) −5.58258 −0.399777
\(196\) 0 0
\(197\) −24.3739 −1.73657 −0.868283 0.496069i \(-0.834776\pi\)
−0.868283 + 0.496069i \(0.834776\pi\)
\(198\) 0 0
\(199\) 1.20871 0.0856833 0.0428417 0.999082i \(-0.486359\pi\)
0.0428417 + 0.999082i \(0.486359\pi\)
\(200\) 0 0
\(201\) 31.1652 2.19822
\(202\) 0 0
\(203\) −0.747727 −0.0524802
\(204\) 0 0
\(205\) 3.58258 0.250218
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −1.20871 −0.0836084
\(210\) 0 0
\(211\) 13.5826 0.935063 0.467532 0.883976i \(-0.345143\pi\)
0.467532 + 0.883976i \(0.345143\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −14.3303 −0.977319
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −30.1216 −2.03543
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 18.4174 1.23332 0.616661 0.787229i \(-0.288485\pi\)
0.616661 + 0.787229i \(0.288485\pi\)
\(224\) 0 0
\(225\) −4.79129 −0.319419
\(226\) 0 0
\(227\) −3.62614 −0.240675 −0.120338 0.992733i \(-0.538398\pi\)
−0.120338 + 0.992733i \(0.538398\pi\)
\(228\) 0 0
\(229\) 9.16515 0.605650 0.302825 0.953046i \(-0.402070\pi\)
0.302825 + 0.953046i \(0.402070\pi\)
\(230\) 0 0
\(231\) −0.582576 −0.0383307
\(232\) 0 0
\(233\) 6.74773 0.442058 0.221029 0.975267i \(-0.429058\pi\)
0.221029 + 0.975267i \(0.429058\pi\)
\(234\) 0 0
\(235\) 7.16515 0.467403
\(236\) 0 0
\(237\) 35.5826 2.31134
\(238\) 0 0
\(239\) −11.5390 −0.746397 −0.373198 0.927752i \(-0.621739\pi\)
−0.373198 + 0.927752i \(0.621739\pi\)
\(240\) 0 0
\(241\) −13.7913 −0.888375 −0.444187 0.895934i \(-0.646508\pi\)
−0.444187 + 0.895934i \(0.646508\pi\)
\(242\) 0 0
\(243\) −16.1652 −1.03699
\(244\) 0 0
\(245\) −13.9129 −0.888861
\(246\) 0 0
\(247\) −1.20871 −0.0769085
\(248\) 0 0
\(249\) 1.74773 0.110758
\(250\) 0 0
\(251\) −15.7913 −0.996737 −0.498369 0.866965i \(-0.666067\pi\)
−0.498369 + 0.866965i \(0.666067\pi\)
\(252\) 0 0
\(253\) −0.208712 −0.0131216
\(254\) 0 0
\(255\) −22.3303 −1.39838
\(256\) 0 0
\(257\) −9.79129 −0.610764 −0.305382 0.952230i \(-0.598784\pi\)
−0.305382 + 0.952230i \(0.598784\pi\)
\(258\) 0 0
\(259\) 0.834849 0.0518750
\(260\) 0 0
\(261\) −17.1652 −1.06250
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 8.74773 0.537369
\(266\) 0 0
\(267\) 26.7477 1.63693
\(268\) 0 0
\(269\) 23.7042 1.44527 0.722634 0.691231i \(-0.242931\pi\)
0.722634 + 0.691231i \(0.242931\pi\)
\(270\) 0 0
\(271\) 27.5390 1.67288 0.836438 0.548062i \(-0.184634\pi\)
0.836438 + 0.548062i \(0.184634\pi\)
\(272\) 0 0
\(273\) −0.582576 −0.0352591
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 20.0000 1.20168 0.600842 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 26.7042 1.59304 0.796519 0.604614i \(-0.206673\pi\)
0.796519 + 0.604614i \(0.206673\pi\)
\(282\) 0 0
\(283\) 27.4955 1.63444 0.817218 0.576329i \(-0.195515\pi\)
0.817218 + 0.576329i \(0.195515\pi\)
\(284\) 0 0
\(285\) 6.74773 0.399701
\(286\) 0 0
\(287\) 0.373864 0.0220685
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 52.3303 3.06766
\(292\) 0 0
\(293\) 17.1652 1.00280 0.501399 0.865216i \(-0.332819\pi\)
0.501399 + 0.865216i \(0.332819\pi\)
\(294\) 0 0
\(295\) −11.1652 −0.650060
\(296\) 0 0
\(297\) −5.00000 −0.290129
\(298\) 0 0
\(299\) −0.208712 −0.0120701
\(300\) 0 0
\(301\) −1.49545 −0.0861965
\(302\) 0 0
\(303\) −33.4955 −1.92426
\(304\) 0 0
\(305\) 4.83485 0.276843
\(306\) 0 0
\(307\) 18.3303 1.04617 0.523083 0.852282i \(-0.324782\pi\)
0.523083 + 0.852282i \(0.324782\pi\)
\(308\) 0 0
\(309\) −0.582576 −0.0331416
\(310\) 0 0
\(311\) 19.5390 1.10796 0.553978 0.832531i \(-0.313109\pi\)
0.553978 + 0.832531i \(0.313109\pi\)
\(312\) 0 0
\(313\) 2.20871 0.124844 0.0624219 0.998050i \(-0.480118\pi\)
0.0624219 + 0.998050i \(0.480118\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) 8.00000 0.449325 0.224662 0.974437i \(-0.427872\pi\)
0.224662 + 0.974437i \(0.427872\pi\)
\(318\) 0 0
\(319\) 3.58258 0.200586
\(320\) 0 0
\(321\) 26.7477 1.49291
\(322\) 0 0
\(323\) −4.83485 −0.269018
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) −3.37386 −0.186575
\(328\) 0 0
\(329\) 0.747727 0.0412235
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) 19.1652 1.05024
\(334\) 0 0
\(335\) 22.3303 1.22003
\(336\) 0 0
\(337\) −32.3303 −1.76114 −0.880572 0.473913i \(-0.842841\pi\)
−0.880572 + 0.473913i \(0.842841\pi\)
\(338\) 0 0
\(339\) −58.9564 −3.20207
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −2.91288 −0.157281
\(344\) 0 0
\(345\) 1.16515 0.0627296
\(346\) 0 0
\(347\) 14.8348 0.796376 0.398188 0.917304i \(-0.369639\pi\)
0.398188 + 0.917304i \(0.369639\pi\)
\(348\) 0 0
\(349\) 20.1216 1.07708 0.538542 0.842599i \(-0.318975\pi\)
0.538542 + 0.842599i \(0.318975\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) 10.4174 0.554464 0.277232 0.960803i \(-0.410583\pi\)
0.277232 + 0.960803i \(0.410583\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −2.33030 −0.123333
\(358\) 0 0
\(359\) −25.7042 −1.35661 −0.678307 0.734779i \(-0.737286\pi\)
−0.678307 + 0.734779i \(0.737286\pi\)
\(360\) 0 0
\(361\) −17.5390 −0.923106
\(362\) 0 0
\(363\) 2.79129 0.146505
\(364\) 0 0
\(365\) −21.5826 −1.12968
\(366\) 0 0
\(367\) 2.04356 0.106673 0.0533365 0.998577i \(-0.483014\pi\)
0.0533365 + 0.998577i \(0.483014\pi\)
\(368\) 0 0
\(369\) 8.58258 0.446791
\(370\) 0 0
\(371\) 0.912878 0.0473943
\(372\) 0 0
\(373\) 3.58258 0.185499 0.0927494 0.995689i \(-0.470434\pi\)
0.0927494 + 0.995689i \(0.470434\pi\)
\(374\) 0 0
\(375\) −33.4955 −1.72970
\(376\) 0 0
\(377\) 3.58258 0.184512
\(378\) 0 0
\(379\) −6.74773 −0.346607 −0.173304 0.984868i \(-0.555444\pi\)
−0.173304 + 0.984868i \(0.555444\pi\)
\(380\) 0 0
\(381\) 49.0780 2.51434
\(382\) 0 0
\(383\) −29.9129 −1.52848 −0.764238 0.644934i \(-0.776885\pi\)
−0.764238 + 0.644934i \(0.776885\pi\)
\(384\) 0 0
\(385\) −0.417424 −0.0212739
\(386\) 0 0
\(387\) −34.3303 −1.74511
\(388\) 0 0
\(389\) −26.7042 −1.35395 −0.676977 0.736004i \(-0.736711\pi\)
−0.676977 + 0.736004i \(0.736711\pi\)
\(390\) 0 0
\(391\) −0.834849 −0.0422201
\(392\) 0 0
\(393\) −16.7477 −0.844811
\(394\) 0 0
\(395\) 25.4955 1.28282
\(396\) 0 0
\(397\) 33.1652 1.66451 0.832256 0.554392i \(-0.187049\pi\)
0.832256 + 0.554392i \(0.187049\pi\)
\(398\) 0 0
\(399\) 0.704166 0.0352524
\(400\) 0 0
\(401\) 15.1652 0.757312 0.378656 0.925538i \(-0.376386\pi\)
0.378656 + 0.925538i \(0.376386\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.834849 −0.0414840
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 14.8348 0.733536 0.366768 0.930312i \(-0.380464\pi\)
0.366768 + 0.930312i \(0.380464\pi\)
\(410\) 0 0
\(411\) −64.6606 −3.18947
\(412\) 0 0
\(413\) −1.16515 −0.0573334
\(414\) 0 0
\(415\) 1.25227 0.0614717
\(416\) 0 0
\(417\) −54.6606 −2.67674
\(418\) 0 0
\(419\) 7.37386 0.360237 0.180118 0.983645i \(-0.442352\pi\)
0.180118 + 0.983645i \(0.442352\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) 17.1652 0.834598
\(424\) 0 0
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 0.504546 0.0244167
\(428\) 0 0
\(429\) 2.79129 0.134765
\(430\) 0 0
\(431\) −16.6261 −0.800853 −0.400426 0.916329i \(-0.631138\pi\)
−0.400426 + 0.916329i \(0.631138\pi\)
\(432\) 0 0
\(433\) −15.6261 −0.750944 −0.375472 0.926834i \(-0.622519\pi\)
−0.375472 + 0.926834i \(0.622519\pi\)
\(434\) 0 0
\(435\) −20.0000 −0.958927
\(436\) 0 0
\(437\) 0.252273 0.0120679
\(438\) 0 0
\(439\) −14.8348 −0.708029 −0.354014 0.935240i \(-0.615184\pi\)
−0.354014 + 0.935240i \(0.615184\pi\)
\(440\) 0 0
\(441\) −33.3303 −1.58716
\(442\) 0 0
\(443\) −36.4519 −1.73188 −0.865941 0.500146i \(-0.833280\pi\)
−0.865941 + 0.500146i \(0.833280\pi\)
\(444\) 0 0
\(445\) 19.1652 0.908515
\(446\) 0 0
\(447\) 3.83485 0.181382
\(448\) 0 0
\(449\) −9.16515 −0.432530 −0.216265 0.976335i \(-0.569388\pi\)
−0.216265 + 0.976335i \(0.569388\pi\)
\(450\) 0 0
\(451\) −1.79129 −0.0843485
\(452\) 0 0
\(453\) 20.0000 0.939682
\(454\) 0 0
\(455\) −0.417424 −0.0195692
\(456\) 0 0
\(457\) −7.62614 −0.356736 −0.178368 0.983964i \(-0.557082\pi\)
−0.178368 + 0.983964i \(0.557082\pi\)
\(458\) 0 0
\(459\) −20.0000 −0.933520
\(460\) 0 0
\(461\) 12.3739 0.576308 0.288154 0.957584i \(-0.406958\pi\)
0.288154 + 0.957584i \(0.406958\pi\)
\(462\) 0 0
\(463\) −39.1652 −1.82016 −0.910079 0.414434i \(-0.863980\pi\)
−0.910079 + 0.414434i \(0.863980\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.3303 1.03332 0.516662 0.856190i \(-0.327175\pi\)
0.516662 + 0.856190i \(0.327175\pi\)
\(468\) 0 0
\(469\) 2.33030 0.107603
\(470\) 0 0
\(471\) −35.0000 −1.61271
\(472\) 0 0
\(473\) 7.16515 0.329454
\(474\) 0 0
\(475\) −1.20871 −0.0554595
\(476\) 0 0
\(477\) 20.9564 0.959529
\(478\) 0 0
\(479\) −39.1652 −1.78950 −0.894751 0.446566i \(-0.852647\pi\)
−0.894751 + 0.446566i \(0.852647\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) 0.121591 0.00553257
\(484\) 0 0
\(485\) 37.4955 1.70258
\(486\) 0 0
\(487\) 17.5826 0.796743 0.398371 0.917224i \(-0.369576\pi\)
0.398371 + 0.917224i \(0.369576\pi\)
\(488\) 0 0
\(489\) −3.25227 −0.147073
\(490\) 0 0
\(491\) 21.1652 0.955170 0.477585 0.878586i \(-0.341512\pi\)
0.477585 + 0.878586i \(0.341512\pi\)
\(492\) 0 0
\(493\) 14.3303 0.645404
\(494\) 0 0
\(495\) −9.58258 −0.430705
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) 43.3739 1.93780
\(502\) 0 0
\(503\) 21.4955 0.958435 0.479217 0.877696i \(-0.340921\pi\)
0.479217 + 0.877696i \(0.340921\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) 2.79129 0.123965
\(508\) 0 0
\(509\) −19.5826 −0.867982 −0.433991 0.900917i \(-0.642895\pi\)
−0.433991 + 0.900917i \(0.642895\pi\)
\(510\) 0 0
\(511\) −2.25227 −0.0996347
\(512\) 0 0
\(513\) 6.04356 0.266830
\(514\) 0 0
\(515\) −0.417424 −0.0183939
\(516\) 0 0
\(517\) −3.58258 −0.157561
\(518\) 0 0
\(519\) −43.4955 −1.90924
\(520\) 0 0
\(521\) 2.62614 0.115053 0.0575266 0.998344i \(-0.481679\pi\)
0.0575266 + 0.998344i \(0.481679\pi\)
\(522\) 0 0
\(523\) 25.1652 1.10040 0.550198 0.835034i \(-0.314552\pi\)
0.550198 + 0.835034i \(0.314552\pi\)
\(524\) 0 0
\(525\) −0.582576 −0.0254257
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −22.9564 −0.998106
\(530\) 0 0
\(531\) −26.7477 −1.16075
\(532\) 0 0
\(533\) −1.79129 −0.0775893
\(534\) 0 0
\(535\) 19.1652 0.828582
\(536\) 0 0
\(537\) 8.83485 0.381252
\(538\) 0 0
\(539\) 6.95644 0.299635
\(540\) 0 0
\(541\) −8.53901 −0.367121 −0.183560 0.983008i \(-0.558762\pi\)
−0.183560 + 0.983008i \(0.558762\pi\)
\(542\) 0 0
\(543\) 34.5390 1.48221
\(544\) 0 0
\(545\) −2.41742 −0.103551
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 11.5826 0.494332
\(550\) 0 0
\(551\) −4.33030 −0.184477
\(552\) 0 0
\(553\) 2.66061 0.113140
\(554\) 0 0
\(555\) 22.3303 0.947869
\(556\) 0 0
\(557\) 38.8693 1.64695 0.823473 0.567356i \(-0.192033\pi\)
0.823473 + 0.567356i \(0.192033\pi\)
\(558\) 0 0
\(559\) 7.16515 0.303054
\(560\) 0 0
\(561\) 11.1652 0.471393
\(562\) 0 0
\(563\) −19.5826 −0.825307 −0.412654 0.910888i \(-0.635398\pi\)
−0.412654 + 0.910888i \(0.635398\pi\)
\(564\) 0 0
\(565\) −42.2432 −1.77718
\(566\) 0 0
\(567\) −0.0871215 −0.00365876
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 14.7477 0.617173 0.308587 0.951196i \(-0.400144\pi\)
0.308587 + 0.951196i \(0.400144\pi\)
\(572\) 0 0
\(573\) −5.70417 −0.238295
\(574\) 0 0
\(575\) −0.208712 −0.00870390
\(576\) 0 0
\(577\) −21.5826 −0.898494 −0.449247 0.893408i \(-0.648308\pi\)
−0.449247 + 0.893408i \(0.648308\pi\)
\(578\) 0 0
\(579\) 15.0000 0.623379
\(580\) 0 0
\(581\) 0.130682 0.00542161
\(582\) 0 0
\(583\) −4.37386 −0.181147
\(584\) 0 0
\(585\) −9.58258 −0.396191
\(586\) 0 0
\(587\) −28.4174 −1.17291 −0.586456 0.809981i \(-0.699477\pi\)
−0.586456 + 0.809981i \(0.699477\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −68.0345 −2.79856
\(592\) 0 0
\(593\) 29.3739 1.20624 0.603120 0.797650i \(-0.293924\pi\)
0.603120 + 0.797650i \(0.293924\pi\)
\(594\) 0 0
\(595\) −1.66970 −0.0684509
\(596\) 0 0
\(597\) 3.37386 0.138083
\(598\) 0 0
\(599\) 6.79129 0.277484 0.138742 0.990329i \(-0.455694\pi\)
0.138742 + 0.990329i \(0.455694\pi\)
\(600\) 0 0
\(601\) −20.4174 −0.832844 −0.416422 0.909171i \(-0.636716\pi\)
−0.416422 + 0.909171i \(0.636716\pi\)
\(602\) 0 0
\(603\) 53.4955 2.17850
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −5.25227 −0.213183 −0.106592 0.994303i \(-0.533994\pi\)
−0.106592 + 0.994303i \(0.533994\pi\)
\(608\) 0 0
\(609\) −2.08712 −0.0845744
\(610\) 0 0
\(611\) −3.58258 −0.144935
\(612\) 0 0
\(613\) −9.12159 −0.368418 −0.184209 0.982887i \(-0.558972\pi\)
−0.184209 + 0.982887i \(0.558972\pi\)
\(614\) 0 0
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) −22.7477 −0.915789 −0.457895 0.889007i \(-0.651396\pi\)
−0.457895 + 0.889007i \(0.651396\pi\)
\(618\) 0 0
\(619\) −0.834849 −0.0335554 −0.0167777 0.999859i \(-0.505341\pi\)
−0.0167777 + 0.999859i \(0.505341\pi\)
\(620\) 0 0
\(621\) 1.04356 0.0418767
\(622\) 0 0
\(623\) 2.00000 0.0801283
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −3.37386 −0.134739
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 12.0000 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) 0 0
\(633\) 37.9129 1.50690
\(634\) 0 0
\(635\) 35.1652 1.39549
\(636\) 0 0
\(637\) 6.95644 0.275624
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.9564 0.511749 0.255874 0.966710i \(-0.417637\pi\)
0.255874 + 0.966710i \(0.417637\pi\)
\(642\) 0 0
\(643\) −5.91288 −0.233181 −0.116591 0.993180i \(-0.537197\pi\)
−0.116591 + 0.993180i \(0.537197\pi\)
\(644\) 0 0
\(645\) −40.0000 −1.57500
\(646\) 0 0
\(647\) 14.0436 0.552109 0.276055 0.961142i \(-0.410973\pi\)
0.276055 + 0.961142i \(0.410973\pi\)
\(648\) 0 0
\(649\) 5.58258 0.219135
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 38.0000 1.48705 0.743527 0.668705i \(-0.233151\pi\)
0.743527 + 0.668705i \(0.233151\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) −51.7042 −2.01717
\(658\) 0 0
\(659\) −23.4955 −0.915253 −0.457626 0.889145i \(-0.651300\pi\)
−0.457626 + 0.889145i \(0.651300\pi\)
\(660\) 0 0
\(661\) −0.0871215 −0.00338863 −0.00169432 0.999999i \(-0.500539\pi\)
−0.00169432 + 0.999999i \(0.500539\pi\)
\(662\) 0 0
\(663\) 11.1652 0.433619
\(664\) 0 0
\(665\) 0.504546 0.0195654
\(666\) 0 0
\(667\) −0.747727 −0.0289521
\(668\) 0 0
\(669\) 51.4083 1.98756
\(670\) 0 0
\(671\) −2.41742 −0.0933236
\(672\) 0 0
\(673\) 37.4955 1.44534 0.722672 0.691191i \(-0.242914\pi\)
0.722672 + 0.691191i \(0.242914\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) −6.33030 −0.243293 −0.121647 0.992573i \(-0.538817\pi\)
−0.121647 + 0.992573i \(0.538817\pi\)
\(678\) 0 0
\(679\) 3.91288 0.150162
\(680\) 0 0
\(681\) −10.1216 −0.387860
\(682\) 0 0
\(683\) 38.0000 1.45403 0.727015 0.686622i \(-0.240907\pi\)
0.727015 + 0.686622i \(0.240907\pi\)
\(684\) 0 0
\(685\) −46.3303 −1.77019
\(686\) 0 0
\(687\) 25.5826 0.976036
\(688\) 0 0
\(689\) −4.37386 −0.166631
\(690\) 0 0
\(691\) −13.9129 −0.529271 −0.264635 0.964349i \(-0.585252\pi\)
−0.264635 + 0.964349i \(0.585252\pi\)
\(692\) 0 0
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) −39.1652 −1.48562
\(696\) 0 0
\(697\) −7.16515 −0.271399
\(698\) 0 0
\(699\) 18.8348 0.712399
\(700\) 0 0
\(701\) −29.9129 −1.12979 −0.564897 0.825161i \(-0.691084\pi\)
−0.564897 + 0.825161i \(0.691084\pi\)
\(702\) 0 0
\(703\) 4.83485 0.182350
\(704\) 0 0
\(705\) 20.0000 0.753244
\(706\) 0 0
\(707\) −2.50455 −0.0941931
\(708\) 0 0
\(709\) −32.0000 −1.20179 −0.600893 0.799330i \(-0.705188\pi\)
−0.600893 + 0.799330i \(0.705188\pi\)
\(710\) 0 0
\(711\) 61.0780 2.29061
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 0 0
\(717\) −32.2087 −1.20286
\(718\) 0 0
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) −0.0435608 −0.00162229
\(722\) 0 0
\(723\) −38.4955 −1.43166
\(724\) 0 0
\(725\) 3.58258 0.133054
\(726\) 0 0
\(727\) 38.1216 1.41385 0.706926 0.707288i \(-0.250081\pi\)
0.706926 + 0.707288i \(0.250081\pi\)
\(728\) 0 0
\(729\) −43.8693 −1.62479
\(730\) 0 0
\(731\) 28.6606 1.06005
\(732\) 0 0
\(733\) −15.0436 −0.555647 −0.277823 0.960632i \(-0.589613\pi\)
−0.277823 + 0.960632i \(0.589613\pi\)
\(734\) 0 0
\(735\) −38.8348 −1.43244
\(736\) 0 0
\(737\) −11.1652 −0.411274
\(738\) 0 0
\(739\) 34.9564 1.28589 0.642947 0.765911i \(-0.277712\pi\)
0.642947 + 0.765911i \(0.277712\pi\)
\(740\) 0 0
\(741\) −3.37386 −0.123942
\(742\) 0 0
\(743\) −15.1652 −0.556355 −0.278178 0.960530i \(-0.589730\pi\)
−0.278178 + 0.960530i \(0.589730\pi\)
\(744\) 0 0
\(745\) 2.74773 0.100669
\(746\) 0 0
\(747\) 3.00000 0.109764
\(748\) 0 0
\(749\) 2.00000 0.0730784
\(750\) 0 0
\(751\) −41.8693 −1.52783 −0.763917 0.645315i \(-0.776726\pi\)
−0.763917 + 0.645315i \(0.776726\pi\)
\(752\) 0 0
\(753\) −44.0780 −1.60629
\(754\) 0 0
\(755\) 14.3303 0.521533
\(756\) 0 0
\(757\) −0.956439 −0.0347624 −0.0173812 0.999849i \(-0.505533\pi\)
−0.0173812 + 0.999849i \(0.505533\pi\)
\(758\) 0 0
\(759\) −0.582576 −0.0211462
\(760\) 0 0
\(761\) 1.37386 0.0498025 0.0249013 0.999690i \(-0.492073\pi\)
0.0249013 + 0.999690i \(0.492073\pi\)
\(762\) 0 0
\(763\) −0.252273 −0.00913289
\(764\) 0 0
\(765\) −38.3303 −1.38584
\(766\) 0 0
\(767\) 5.58258 0.201575
\(768\) 0 0
\(769\) −46.4519 −1.67510 −0.837549 0.546362i \(-0.816012\pi\)
−0.837549 + 0.546362i \(0.816012\pi\)
\(770\) 0 0
\(771\) −27.3303 −0.984277
\(772\) 0 0
\(773\) −9.49545 −0.341528 −0.170764 0.985312i \(-0.554624\pi\)
−0.170764 + 0.985312i \(0.554624\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.33030 0.0835991
\(778\) 0 0
\(779\) 2.16515 0.0775746
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −17.9129 −0.640154
\(784\) 0 0
\(785\) −25.0780 −0.895073
\(786\) 0 0
\(787\) −17.7042 −0.631085 −0.315543 0.948911i \(-0.602187\pi\)
−0.315543 + 0.948911i \(0.602187\pi\)
\(788\) 0 0
\(789\) 50.2432 1.78870
\(790\) 0 0
\(791\) −4.40833 −0.156742
\(792\) 0 0
\(793\) −2.41742 −0.0858453
\(794\) 0 0
\(795\) 24.4174 0.865997
\(796\) 0 0
\(797\) −13.1652 −0.466334 −0.233167 0.972437i \(-0.574909\pi\)
−0.233167 + 0.972437i \(0.574909\pi\)
\(798\) 0 0
\(799\) −14.3303 −0.506970
\(800\) 0 0
\(801\) 45.9129 1.62225
\(802\) 0 0
\(803\) 10.7913 0.380816
\(804\) 0 0
\(805\) 0.0871215 0.00307063
\(806\) 0 0
\(807\) 66.1652 2.32912
\(808\) 0 0
\(809\) 36.0000 1.26569 0.632846 0.774277i \(-0.281886\pi\)
0.632846 + 0.774277i \(0.281886\pi\)
\(810\) 0 0
\(811\) 6.95644 0.244274 0.122137 0.992513i \(-0.461025\pi\)
0.122137 + 0.992513i \(0.461025\pi\)
\(812\) 0 0
\(813\) 76.8693 2.69592
\(814\) 0 0
\(815\) −2.33030 −0.0816269
\(816\) 0 0
\(817\) −8.66061 −0.302996
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) −51.4955 −1.79720 −0.898602 0.438765i \(-0.855416\pi\)
−0.898602 + 0.438765i \(0.855416\pi\)
\(822\) 0 0
\(823\) 5.87841 0.204908 0.102454 0.994738i \(-0.467330\pi\)
0.102454 + 0.994738i \(0.467330\pi\)
\(824\) 0 0
\(825\) 2.79129 0.0971802
\(826\) 0 0
\(827\) 12.2867 0.427252 0.213626 0.976916i \(-0.431473\pi\)
0.213626 + 0.976916i \(0.431473\pi\)
\(828\) 0 0
\(829\) 12.1216 0.421000 0.210500 0.977594i \(-0.432491\pi\)
0.210500 + 0.977594i \(0.432491\pi\)
\(830\) 0 0
\(831\) 55.8258 1.93657
\(832\) 0 0
\(833\) 27.8258 0.964105
\(834\) 0 0
\(835\) 31.0780 1.07550
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.16515 −0.0402255 −0.0201127 0.999798i \(-0.506403\pi\)
−0.0201127 + 0.999798i \(0.506403\pi\)
\(840\) 0 0
\(841\) −16.1652 −0.557419
\(842\) 0 0
\(843\) 74.5390 2.56726
\(844\) 0 0
\(845\) 2.00000 0.0688021
\(846\) 0 0
\(847\) 0.208712 0.00717143
\(848\) 0 0
\(849\) 76.7477 2.63398
\(850\) 0 0
\(851\) 0.834849 0.0286182
\(852\) 0 0
\(853\) −33.1216 −1.13406 −0.567031 0.823697i \(-0.691908\pi\)
−0.567031 + 0.823697i \(0.691908\pi\)
\(854\) 0 0
\(855\) 11.5826 0.396116
\(856\) 0 0
\(857\) 25.4955 0.870908 0.435454 0.900211i \(-0.356588\pi\)
0.435454 + 0.900211i \(0.356588\pi\)
\(858\) 0 0
\(859\) 19.6261 0.669635 0.334818 0.942283i \(-0.391325\pi\)
0.334818 + 0.942283i \(0.391325\pi\)
\(860\) 0 0
\(861\) 1.04356 0.0355645
\(862\) 0 0
\(863\) −30.6606 −1.04370 −0.521850 0.853038i \(-0.674758\pi\)
−0.521850 + 0.853038i \(0.674758\pi\)
\(864\) 0 0
\(865\) −31.1652 −1.05965
\(866\) 0 0
\(867\) −2.79129 −0.0947971
\(868\) 0 0
\(869\) −12.7477 −0.432437
\(870\) 0 0
\(871\) −11.1652 −0.378317
\(872\) 0 0
\(873\) 89.8258 3.04014
\(874\) 0 0
\(875\) −2.50455 −0.0846691
\(876\) 0 0
\(877\) 41.9564 1.41677 0.708384 0.705827i \(-0.249424\pi\)
0.708384 + 0.705827i \(0.249424\pi\)
\(878\) 0 0
\(879\) 47.9129 1.61606
\(880\) 0 0
\(881\) 14.0436 0.473140 0.236570 0.971614i \(-0.423977\pi\)
0.236570 + 0.971614i \(0.423977\pi\)
\(882\) 0 0
\(883\) 44.2867 1.49037 0.745184 0.666859i \(-0.232362\pi\)
0.745184 + 0.666859i \(0.232362\pi\)
\(884\) 0 0
\(885\) −31.1652 −1.04761
\(886\) 0 0
\(887\) −35.0780 −1.17780 −0.588902 0.808204i \(-0.700440\pi\)
−0.588902 + 0.808204i \(0.700440\pi\)
\(888\) 0 0
\(889\) 3.66970 0.123078
\(890\) 0 0
\(891\) 0.417424 0.0139842
\(892\) 0 0
\(893\) 4.33030 0.144908
\(894\) 0 0
\(895\) 6.33030 0.211599
\(896\) 0 0
\(897\) −0.582576 −0.0194516
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −17.4955 −0.582858
\(902\) 0 0
\(903\) −4.17424 −0.138910
\(904\) 0 0
\(905\) 24.7477 0.822642
\(906\) 0 0
\(907\) 12.8693 0.427319 0.213659 0.976908i \(-0.431462\pi\)
0.213659 + 0.976908i \(0.431462\pi\)
\(908\) 0 0
\(909\) −57.4955 −1.90700
\(910\) 0 0
\(911\) −32.6261 −1.08095 −0.540476 0.841359i \(-0.681756\pi\)
−0.540476 + 0.841359i \(0.681756\pi\)
\(912\) 0 0
\(913\) −0.626136 −0.0207221
\(914\) 0 0
\(915\) 13.4955 0.446146
\(916\) 0 0
\(917\) −1.25227 −0.0413537
\(918\) 0 0
\(919\) −35.4955 −1.17089 −0.585443 0.810713i \(-0.699080\pi\)
−0.585443 + 0.810713i \(0.699080\pi\)
\(920\) 0 0
\(921\) 51.1652 1.68595
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) −1.00000 −0.0328443
\(928\) 0 0
\(929\) 9.49545 0.311536 0.155768 0.987794i \(-0.450215\pi\)
0.155768 + 0.987794i \(0.450215\pi\)
\(930\) 0 0
\(931\) −8.40833 −0.275572
\(932\) 0 0
\(933\) 54.5390 1.78553
\(934\) 0 0
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) 57.8258 1.88909 0.944543 0.328389i \(-0.106506\pi\)
0.944543 + 0.328389i \(0.106506\pi\)
\(938\) 0 0
\(939\) 6.16515 0.201192
\(940\) 0 0
\(941\) −33.9564 −1.10695 −0.553474 0.832866i \(-0.686698\pi\)
−0.553474 + 0.832866i \(0.686698\pi\)
\(942\) 0 0
\(943\) 0.373864 0.0121747
\(944\) 0 0
\(945\) 2.08712 0.0678941
\(946\) 0 0
\(947\) 17.0780 0.554961 0.277481 0.960731i \(-0.410501\pi\)
0.277481 + 0.960731i \(0.410501\pi\)
\(948\) 0 0
\(949\) 10.7913 0.350300
\(950\) 0 0
\(951\) 22.3303 0.724110
\(952\) 0 0
\(953\) −23.0780 −0.747571 −0.373785 0.927515i \(-0.621940\pi\)
−0.373785 + 0.927515i \(0.621940\pi\)
\(954\) 0 0
\(955\) −4.08712 −0.132256
\(956\) 0 0
\(957\) 10.0000 0.323254
\(958\) 0 0
\(959\) −4.83485 −0.156125
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 45.9129 1.47952
\(964\) 0 0
\(965\) 10.7477 0.345982
\(966\) 0 0
\(967\) −2.79129 −0.0897618 −0.0448809 0.998992i \(-0.514291\pi\)
−0.0448809 + 0.998992i \(0.514291\pi\)
\(968\) 0 0
\(969\) −13.4955 −0.433536
\(970\) 0 0
\(971\) −35.4519 −1.13771 −0.568853 0.822439i \(-0.692613\pi\)
−0.568853 + 0.822439i \(0.692613\pi\)
\(972\) 0 0
\(973\) −4.08712 −0.131027
\(974\) 0 0
\(975\) 2.79129 0.0893928
\(976\) 0 0
\(977\) 5.66970 0.181390 0.0906948 0.995879i \(-0.471091\pi\)
0.0906948 + 0.995879i \(0.471091\pi\)
\(978\) 0 0
\(979\) −9.58258 −0.306260
\(980\) 0 0
\(981\) −5.79129 −0.184902
\(982\) 0 0
\(983\) −7.16515 −0.228533 −0.114266 0.993450i \(-0.536452\pi\)
−0.114266 + 0.993450i \(0.536452\pi\)
\(984\) 0 0
\(985\) −48.7477 −1.55323
\(986\) 0 0
\(987\) 2.08712 0.0664338
\(988\) 0 0
\(989\) −1.49545 −0.0475527
\(990\) 0 0
\(991\) −36.3739 −1.15545 −0.577727 0.816230i \(-0.696060\pi\)
−0.577727 + 0.816230i \(0.696060\pi\)
\(992\) 0 0
\(993\) 78.1561 2.48021
\(994\) 0 0
\(995\) 2.41742 0.0766375
\(996\) 0 0
\(997\) −13.5826 −0.430164 −0.215082 0.976596i \(-0.569002\pi\)
−0.215082 + 0.976596i \(0.569002\pi\)
\(998\) 0 0
\(999\) 20.0000 0.632772
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 572.2.a.d.1.2 2
3.2 odd 2 5148.2.a.g.1.1 2
4.3 odd 2 2288.2.a.n.1.1 2
8.3 odd 2 9152.2.a.bn.1.2 2
8.5 even 2 9152.2.a.bl.1.1 2
11.10 odd 2 6292.2.a.o.1.2 2
13.12 even 2 7436.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
572.2.a.d.1.2 2 1.1 even 1 trivial
2288.2.a.n.1.1 2 4.3 odd 2
5148.2.a.g.1.1 2 3.2 odd 2
6292.2.a.o.1.2 2 11.10 odd 2
7436.2.a.h.1.2 2 13.12 even 2
9152.2.a.bl.1.1 2 8.5 even 2
9152.2.a.bn.1.2 2 8.3 odd 2