Properties

Label 1568.2.a.t.1.1
Level $1568$
Weight $2$
Character 1568.1
Self dual yes
Analytic conductor $12.521$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1568,2,Mod(1,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, 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("1568.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.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: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 1568.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.64575 q^{3} +3.00000 q^{5} +4.00000 q^{9} +O(q^{10})\) \(q-2.64575 q^{3} +3.00000 q^{5} +4.00000 q^{9} -2.64575 q^{11} +4.00000 q^{13} -7.93725 q^{15} -1.00000 q^{17} +7.93725 q^{19} -2.64575 q^{23} +4.00000 q^{25} -2.64575 q^{27} -4.00000 q^{29} +2.64575 q^{31} +7.00000 q^{33} -5.00000 q^{37} -10.5830 q^{39} -8.00000 q^{41} +10.5830 q^{43} +12.0000 q^{45} -2.64575 q^{47} +2.64575 q^{51} +7.00000 q^{53} -7.93725 q^{55} -21.0000 q^{57} -2.64575 q^{59} +5.00000 q^{61} +12.0000 q^{65} +2.64575 q^{67} +7.00000 q^{69} +9.00000 q^{73} -10.5830 q^{75} -2.64575 q^{79} -5.00000 q^{81} +10.5830 q^{83} -3.00000 q^{85} +10.5830 q^{87} +9.00000 q^{89} -7.00000 q^{93} +23.8118 q^{95} +8.00000 q^{97} -10.5830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{5} + 8 q^{9} + 8 q^{13} - 2 q^{17} + 8 q^{25} - 8 q^{29} + 14 q^{33} - 10 q^{37} - 16 q^{41} + 24 q^{45} + 14 q^{53} - 42 q^{57} + 10 q^{61} + 24 q^{65} + 14 q^{69} + 18 q^{73} - 10 q^{81} - 6 q^{85} + 18 q^{89} - 14 q^{93} + 16 q^{97}+O(q^{100}) \) 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
\(3\) −2.64575 −1.52753 −0.763763 0.645497i \(-0.776650\pi\)
−0.763763 + 0.645497i \(0.776650\pi\)
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 4.00000 1.33333
\(10\) 0 0
\(11\) −2.64575 −0.797724 −0.398862 0.917011i \(-0.630595\pi\)
−0.398862 + 0.917011i \(0.630595\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) −7.93725 −2.04939
\(16\) 0 0
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) 7.93725 1.82093 0.910465 0.413585i \(-0.135724\pi\)
0.910465 + 0.413585i \(0.135724\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.64575 −0.551677 −0.275839 0.961204i \(-0.588956\pi\)
−0.275839 + 0.961204i \(0.588956\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −2.64575 −0.509175
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 2.64575 0.475191 0.237595 0.971364i \(-0.423641\pi\)
0.237595 + 0.971364i \(0.423641\pi\)
\(32\) 0 0
\(33\) 7.00000 1.21854
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −10.5830 −1.69464
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 10.5830 1.61389 0.806947 0.590624i \(-0.201119\pi\)
0.806947 + 0.590624i \(0.201119\pi\)
\(44\) 0 0
\(45\) 12.0000 1.78885
\(46\) 0 0
\(47\) −2.64575 −0.385922 −0.192961 0.981206i \(-0.561809\pi\)
−0.192961 + 0.981206i \(0.561809\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.64575 0.370479
\(52\) 0 0
\(53\) 7.00000 0.961524 0.480762 0.876851i \(-0.340360\pi\)
0.480762 + 0.876851i \(0.340360\pi\)
\(54\) 0 0
\(55\) −7.93725 −1.07026
\(56\) 0 0
\(57\) −21.0000 −2.78152
\(58\) 0 0
\(59\) −2.64575 −0.344447 −0.172224 0.985058i \(-0.555095\pi\)
−0.172224 + 0.985058i \(0.555095\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 2.64575 0.323230 0.161615 0.986854i \(-0.448330\pi\)
0.161615 + 0.986854i \(0.448330\pi\)
\(68\) 0 0
\(69\) 7.00000 0.842701
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) −10.5830 −1.22202
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.64575 −0.297670 −0.148835 0.988862i \(-0.547552\pi\)
−0.148835 + 0.988862i \(0.547552\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 10.5830 1.16164 0.580818 0.814034i \(-0.302733\pi\)
0.580818 + 0.814034i \(0.302733\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) 10.5830 1.13462
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −7.00000 −0.725866
\(94\) 0 0
\(95\) 23.8118 2.44304
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) −10.5830 −1.06363
\(100\) 0 0
\(101\) 15.0000 1.49256 0.746278 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 0 0
\(103\) 2.64575 0.260694 0.130347 0.991468i \(-0.458391\pi\)
0.130347 + 0.991468i \(0.458391\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.2288 1.27887 0.639436 0.768844i \(-0.279168\pi\)
0.639436 + 0.768844i \(0.279168\pi\)
\(108\) 0 0
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 0 0
\(111\) 13.2288 1.25562
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −7.93725 −0.740153
\(116\) 0 0
\(117\) 16.0000 1.47920
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.00000 −0.363636
\(122\) 0 0
\(123\) 21.1660 1.90847
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 21.1660 1.87818 0.939090 0.343672i \(-0.111671\pi\)
0.939090 + 0.343672i \(0.111671\pi\)
\(128\) 0 0
\(129\) −28.0000 −2.46526
\(130\) 0 0
\(131\) −7.93725 −0.693481 −0.346741 0.937961i \(-0.612712\pi\)
−0.346741 + 0.937961i \(0.612712\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −7.93725 −0.683130
\(136\) 0 0
\(137\) 1.00000 0.0854358 0.0427179 0.999087i \(-0.486398\pi\)
0.0427179 + 0.999087i \(0.486398\pi\)
\(138\) 0 0
\(139\) 10.5830 0.897639 0.448819 0.893622i \(-0.351845\pi\)
0.448819 + 0.893622i \(0.351845\pi\)
\(140\) 0 0
\(141\) 7.00000 0.589506
\(142\) 0 0
\(143\) −10.5830 −0.884995
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 21.0000 1.72039 0.860194 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 0 0
\(151\) −2.64575 −0.215308 −0.107654 0.994188i \(-0.534334\pi\)
−0.107654 + 0.994188i \(0.534334\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) 7.93725 0.637536
\(156\) 0 0
\(157\) 9.00000 0.718278 0.359139 0.933284i \(-0.383070\pi\)
0.359139 + 0.933284i \(0.383070\pi\)
\(158\) 0 0
\(159\) −18.5203 −1.46875
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −13.2288 −1.03616 −0.518078 0.855333i \(-0.673352\pi\)
−0.518078 + 0.855333i \(0.673352\pi\)
\(164\) 0 0
\(165\) 21.0000 1.63485
\(166\) 0 0
\(167\) −5.29150 −0.409469 −0.204734 0.978818i \(-0.565633\pi\)
−0.204734 + 0.978818i \(0.565633\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 31.7490 2.42791
\(172\) 0 0
\(173\) −3.00000 −0.228086 −0.114043 0.993476i \(-0.536380\pi\)
−0.114043 + 0.993476i \(0.536380\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 7.00000 0.526152
\(178\) 0 0
\(179\) −23.8118 −1.77977 −0.889887 0.456180i \(-0.849217\pi\)
−0.889887 + 0.456180i \(0.849217\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −13.2288 −0.977898
\(184\) 0 0
\(185\) −15.0000 −1.10282
\(186\) 0 0
\(187\) 2.64575 0.193476
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 23.8118 1.72296 0.861479 0.507793i \(-0.169538\pi\)
0.861479 + 0.507793i \(0.169538\pi\)
\(192\) 0 0
\(193\) −7.00000 −0.503871 −0.251936 0.967744i \(-0.581067\pi\)
−0.251936 + 0.967744i \(0.581067\pi\)
\(194\) 0 0
\(195\) −31.7490 −2.27359
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −18.5203 −1.31287 −0.656433 0.754384i \(-0.727936\pi\)
−0.656433 + 0.754384i \(0.727936\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −24.0000 −1.67623
\(206\) 0 0
\(207\) −10.5830 −0.735570
\(208\) 0 0
\(209\) −21.0000 −1.45260
\(210\) 0 0
\(211\) −10.5830 −0.728564 −0.364282 0.931289i \(-0.618686\pi\)
−0.364282 + 0.931289i \(0.618686\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 31.7490 2.16526
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −23.8118 −1.60905
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 16.0000 1.06667
\(226\) 0 0
\(227\) 2.64575 0.175605 0.0878023 0.996138i \(-0.472016\pi\)
0.0878023 + 0.996138i \(0.472016\pi\)
\(228\) 0 0
\(229\) 11.0000 0.726900 0.363450 0.931614i \(-0.381599\pi\)
0.363450 + 0.931614i \(0.381599\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 0 0
\(235\) −7.93725 −0.517769
\(236\) 0 0
\(237\) 7.00000 0.454699
\(238\) 0 0
\(239\) 15.8745 1.02684 0.513418 0.858138i \(-0.328379\pi\)
0.513418 + 0.858138i \(0.328379\pi\)
\(240\) 0 0
\(241\) 1.00000 0.0644157 0.0322078 0.999481i \(-0.489746\pi\)
0.0322078 + 0.999481i \(0.489746\pi\)
\(242\) 0 0
\(243\) 21.1660 1.35780
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 31.7490 2.02014
\(248\) 0 0
\(249\) −28.0000 −1.77443
\(250\) 0 0
\(251\) −5.29150 −0.333997 −0.166998 0.985957i \(-0.553407\pi\)
−0.166998 + 0.985957i \(0.553407\pi\)
\(252\) 0 0
\(253\) 7.00000 0.440086
\(254\) 0 0
\(255\) 7.93725 0.497050
\(256\) 0 0
\(257\) −17.0000 −1.06043 −0.530215 0.847863i \(-0.677889\pi\)
−0.530215 + 0.847863i \(0.677889\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −16.0000 −0.990375
\(262\) 0 0
\(263\) −18.5203 −1.14201 −0.571004 0.820947i \(-0.693446\pi\)
−0.571004 + 0.820947i \(0.693446\pi\)
\(264\) 0 0
\(265\) 21.0000 1.29002
\(266\) 0 0
\(267\) −23.8118 −1.45726
\(268\) 0 0
\(269\) −15.0000 −0.914566 −0.457283 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(270\) 0 0
\(271\) 2.64575 0.160718 0.0803590 0.996766i \(-0.474393\pi\)
0.0803590 + 0.996766i \(0.474393\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10.5830 −0.638179
\(276\) 0 0
\(277\) −25.0000 −1.50210 −0.751052 0.660243i \(-0.770453\pi\)
−0.751052 + 0.660243i \(0.770453\pi\)
\(278\) 0 0
\(279\) 10.5830 0.633588
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 0 0
\(283\) −18.5203 −1.10092 −0.550458 0.834863i \(-0.685547\pi\)
−0.550458 + 0.834863i \(0.685547\pi\)
\(284\) 0 0
\(285\) −63.0000 −3.73180
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −21.1660 −1.24077
\(292\) 0 0
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) −7.93725 −0.462125
\(296\) 0 0
\(297\) 7.00000 0.406181
\(298\) 0 0
\(299\) −10.5830 −0.612031
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −39.6863 −2.27992
\(304\) 0 0
\(305\) 15.0000 0.858898
\(306\) 0 0
\(307\) −10.5830 −0.604004 −0.302002 0.953307i \(-0.597655\pi\)
−0.302002 + 0.953307i \(0.597655\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) −23.8118 −1.35024 −0.675121 0.737707i \(-0.735908\pi\)
−0.675121 + 0.737707i \(0.735908\pi\)
\(312\) 0 0
\(313\) 17.0000 0.960897 0.480448 0.877023i \(-0.340474\pi\)
0.480448 + 0.877023i \(0.340474\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.0000 −0.730153 −0.365076 0.930978i \(-0.618957\pi\)
−0.365076 + 0.930978i \(0.618957\pi\)
\(318\) 0 0
\(319\) 10.5830 0.592535
\(320\) 0 0
\(321\) −35.0000 −1.95351
\(322\) 0 0
\(323\) −7.93725 −0.441641
\(324\) 0 0
\(325\) 16.0000 0.887520
\(326\) 0 0
\(327\) −2.64575 −0.146310
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7.93725 0.436271 0.218135 0.975918i \(-0.430003\pi\)
0.218135 + 0.975918i \(0.430003\pi\)
\(332\) 0 0
\(333\) −20.0000 −1.09599
\(334\) 0 0
\(335\) 7.93725 0.433659
\(336\) 0 0
\(337\) 24.0000 1.30736 0.653682 0.756770i \(-0.273224\pi\)
0.653682 + 0.756770i \(0.273224\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7.00000 −0.379071
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 21.0000 1.13060
\(346\) 0 0
\(347\) 23.8118 1.27828 0.639141 0.769089i \(-0.279290\pi\)
0.639141 + 0.769089i \(0.279290\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) −10.5830 −0.564879
\(352\) 0 0
\(353\) −9.00000 −0.479022 −0.239511 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.5203 −0.977462 −0.488731 0.872435i \(-0.662540\pi\)
−0.488731 + 0.872435i \(0.662540\pi\)
\(360\) 0 0
\(361\) 44.0000 2.31579
\(362\) 0 0
\(363\) 10.5830 0.555464
\(364\) 0 0
\(365\) 27.0000 1.41324
\(366\) 0 0
\(367\) 18.5203 0.966750 0.483375 0.875413i \(-0.339411\pi\)
0.483375 + 0.875413i \(0.339411\pi\)
\(368\) 0 0
\(369\) −32.0000 −1.66585
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −21.0000 −1.08734 −0.543669 0.839299i \(-0.682965\pi\)
−0.543669 + 0.839299i \(0.682965\pi\)
\(374\) 0 0
\(375\) 7.93725 0.409878
\(376\) 0 0
\(377\) −16.0000 −0.824042
\(378\) 0 0
\(379\) −5.29150 −0.271806 −0.135903 0.990722i \(-0.543394\pi\)
−0.135903 + 0.990722i \(0.543394\pi\)
\(380\) 0 0
\(381\) −56.0000 −2.86897
\(382\) 0 0
\(383\) 18.5203 0.946341 0.473171 0.880971i \(-0.343109\pi\)
0.473171 + 0.880971i \(0.343109\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 42.3320 2.15186
\(388\) 0 0
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 2.64575 0.133801
\(392\) 0 0
\(393\) 21.0000 1.05931
\(394\) 0 0
\(395\) −7.93725 −0.399367
\(396\) 0 0
\(397\) −5.00000 −0.250943 −0.125471 0.992097i \(-0.540044\pi\)
−0.125471 + 0.992097i \(0.540044\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −23.0000 −1.14857 −0.574283 0.818657i \(-0.694719\pi\)
−0.574283 + 0.818657i \(0.694719\pi\)
\(402\) 0 0
\(403\) 10.5830 0.527177
\(404\) 0 0
\(405\) −15.0000 −0.745356
\(406\) 0 0
\(407\) 13.2288 0.655725
\(408\) 0 0
\(409\) 23.0000 1.13728 0.568638 0.822588i \(-0.307470\pi\)
0.568638 + 0.822588i \(0.307470\pi\)
\(410\) 0 0
\(411\) −2.64575 −0.130505
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 31.7490 1.55850
\(416\) 0 0
\(417\) −28.0000 −1.37117
\(418\) 0 0
\(419\) 31.7490 1.55104 0.775520 0.631322i \(-0.217488\pi\)
0.775520 + 0.631322i \(0.217488\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) 0 0
\(423\) −10.5830 −0.514563
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 28.0000 1.35185
\(430\) 0 0
\(431\) −2.64575 −0.127441 −0.0637207 0.997968i \(-0.520297\pi\)
−0.0637207 + 0.997968i \(0.520297\pi\)
\(432\) 0 0
\(433\) 32.0000 1.53782 0.768911 0.639356i \(-0.220799\pi\)
0.768911 + 0.639356i \(0.220799\pi\)
\(434\) 0 0
\(435\) 31.7490 1.52225
\(436\) 0 0
\(437\) −21.0000 −1.00457
\(438\) 0 0
\(439\) −23.8118 −1.13647 −0.568237 0.822865i \(-0.692374\pi\)
−0.568237 + 0.822865i \(0.692374\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.93725 −0.377110 −0.188555 0.982063i \(-0.560380\pi\)
−0.188555 + 0.982063i \(0.560380\pi\)
\(444\) 0 0
\(445\) 27.0000 1.27992
\(446\) 0 0
\(447\) −55.5608 −2.62793
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 21.1660 0.996669
\(452\) 0 0
\(453\) 7.00000 0.328889
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7.00000 0.327446 0.163723 0.986506i \(-0.447650\pi\)
0.163723 + 0.986506i \(0.447650\pi\)
\(458\) 0 0
\(459\) 2.64575 0.123493
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) −21.0000 −0.973852
\(466\) 0 0
\(467\) −7.93725 −0.367292 −0.183646 0.982992i \(-0.558790\pi\)
−0.183646 + 0.982992i \(0.558790\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −23.8118 −1.09719
\(472\) 0 0
\(473\) −28.0000 −1.28744
\(474\) 0 0
\(475\) 31.7490 1.45674
\(476\) 0 0
\(477\) 28.0000 1.28203
\(478\) 0 0
\(479\) 39.6863 1.81331 0.906656 0.421871i \(-0.138626\pi\)
0.906656 + 0.421871i \(0.138626\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 24.0000 1.08978
\(486\) 0 0
\(487\) 23.8118 1.07901 0.539507 0.841981i \(-0.318611\pi\)
0.539507 + 0.841981i \(0.318611\pi\)
\(488\) 0 0
\(489\) 35.0000 1.58275
\(490\) 0 0
\(491\) 37.0405 1.67162 0.835808 0.549022i \(-0.185000\pi\)
0.835808 + 0.549022i \(0.185000\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) 0 0
\(495\) −31.7490 −1.42701
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −13.2288 −0.592200 −0.296100 0.955157i \(-0.595686\pi\)
−0.296100 + 0.955157i \(0.595686\pi\)
\(500\) 0 0
\(501\) 14.0000 0.625474
\(502\) 0 0
\(503\) −21.1660 −0.943746 −0.471873 0.881667i \(-0.656422\pi\)
−0.471873 + 0.881667i \(0.656422\pi\)
\(504\) 0 0
\(505\) 45.0000 2.00247
\(506\) 0 0
\(507\) −7.93725 −0.352506
\(508\) 0 0
\(509\) 19.0000 0.842160 0.421080 0.907023i \(-0.361651\pi\)
0.421080 + 0.907023i \(0.361651\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −21.0000 −0.927173
\(514\) 0 0
\(515\) 7.93725 0.349757
\(516\) 0 0
\(517\) 7.00000 0.307860
\(518\) 0 0
\(519\) 7.93725 0.348407
\(520\) 0 0
\(521\) −23.0000 −1.00765 −0.503824 0.863806i \(-0.668074\pi\)
−0.503824 + 0.863806i \(0.668074\pi\)
\(522\) 0 0
\(523\) −34.3948 −1.50398 −0.751989 0.659176i \(-0.770905\pi\)
−0.751989 + 0.659176i \(0.770905\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.64575 −0.115251
\(528\) 0 0
\(529\) −16.0000 −0.695652
\(530\) 0 0
\(531\) −10.5830 −0.459263
\(532\) 0 0
\(533\) −32.0000 −1.38607
\(534\) 0 0
\(535\) 39.6863 1.71579
\(536\) 0 0
\(537\) 63.0000 2.71865
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 21.0000 0.902861 0.451430 0.892306i \(-0.350914\pi\)
0.451430 + 0.892306i \(0.350914\pi\)
\(542\) 0 0
\(543\) −15.8745 −0.681240
\(544\) 0 0
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) 26.4575 1.13124 0.565621 0.824665i \(-0.308637\pi\)
0.565621 + 0.824665i \(0.308637\pi\)
\(548\) 0 0
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) −31.7490 −1.35255
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 39.6863 1.68459
\(556\) 0 0
\(557\) −39.0000 −1.65248 −0.826242 0.563316i \(-0.809525\pi\)
−0.826242 + 0.563316i \(0.809525\pi\)
\(558\) 0 0
\(559\) 42.3320 1.79045
\(560\) 0 0
\(561\) −7.00000 −0.295540
\(562\) 0 0
\(563\) −2.64575 −0.111505 −0.0557526 0.998445i \(-0.517756\pi\)
−0.0557526 + 0.998445i \(0.517756\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −33.0000 −1.38343 −0.691716 0.722170i \(-0.743145\pi\)
−0.691716 + 0.722170i \(0.743145\pi\)
\(570\) 0 0
\(571\) −2.64575 −0.110721 −0.0553606 0.998466i \(-0.517631\pi\)
−0.0553606 + 0.998466i \(0.517631\pi\)
\(572\) 0 0
\(573\) −63.0000 −2.63186
\(574\) 0 0
\(575\) −10.5830 −0.441342
\(576\) 0 0
\(577\) 41.0000 1.70685 0.853426 0.521214i \(-0.174521\pi\)
0.853426 + 0.521214i \(0.174521\pi\)
\(578\) 0 0
\(579\) 18.5203 0.769676
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −18.5203 −0.767031
\(584\) 0 0
\(585\) 48.0000 1.98456
\(586\) 0 0
\(587\) −31.7490 −1.31042 −0.655211 0.755446i \(-0.727420\pi\)
−0.655211 + 0.755446i \(0.727420\pi\)
\(588\) 0 0
\(589\) 21.0000 0.865290
\(590\) 0 0
\(591\) −31.7490 −1.30598
\(592\) 0 0
\(593\) −33.0000 −1.35515 −0.677574 0.735455i \(-0.736969\pi\)
−0.677574 + 0.735455i \(0.736969\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 49.0000 2.00544
\(598\) 0 0
\(599\) 23.8118 0.972922 0.486461 0.873702i \(-0.338288\pi\)
0.486461 + 0.873702i \(0.338288\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) 10.5830 0.430973
\(604\) 0 0
\(605\) −12.0000 −0.487869
\(606\) 0 0
\(607\) 39.6863 1.61082 0.805408 0.592721i \(-0.201946\pi\)
0.805408 + 0.592721i \(0.201946\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10.5830 −0.428143
\(612\) 0 0
\(613\) 41.0000 1.65597 0.827987 0.560747i \(-0.189486\pi\)
0.827987 + 0.560747i \(0.189486\pi\)
\(614\) 0 0
\(615\) 63.4980 2.56049
\(616\) 0 0
\(617\) −32.0000 −1.28827 −0.644136 0.764911i \(-0.722783\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 0 0
\(619\) −44.9778 −1.80781 −0.903905 0.427734i \(-0.859312\pi\)
−0.903905 + 0.427734i \(0.859312\pi\)
\(620\) 0 0
\(621\) 7.00000 0.280900
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 55.5608 2.21888
\(628\) 0 0
\(629\) 5.00000 0.199363
\(630\) 0 0
\(631\) −42.3320 −1.68521 −0.842606 0.538531i \(-0.818979\pi\)
−0.842606 + 0.538531i \(0.818979\pi\)
\(632\) 0 0
\(633\) 28.0000 1.11290
\(634\) 0 0
\(635\) 63.4980 2.51984
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.00000 −0.0394976 −0.0197488 0.999805i \(-0.506287\pi\)
−0.0197488 + 0.999805i \(0.506287\pi\)
\(642\) 0 0
\(643\) 10.5830 0.417353 0.208676 0.977985i \(-0.433084\pi\)
0.208676 + 0.977985i \(0.433084\pi\)
\(644\) 0 0
\(645\) −84.0000 −3.30750
\(646\) 0 0
\(647\) 39.6863 1.56023 0.780114 0.625637i \(-0.215161\pi\)
0.780114 + 0.625637i \(0.215161\pi\)
\(648\) 0 0
\(649\) 7.00000 0.274774
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.0000 0.743527 0.371764 0.928327i \(-0.378753\pi\)
0.371764 + 0.928327i \(0.378753\pi\)
\(654\) 0 0
\(655\) −23.8118 −0.930403
\(656\) 0 0
\(657\) 36.0000 1.40449
\(658\) 0 0
\(659\) −31.7490 −1.23677 −0.618383 0.785877i \(-0.712212\pi\)
−0.618383 + 0.785877i \(0.712212\pi\)
\(660\) 0 0
\(661\) 33.0000 1.28355 0.641776 0.766892i \(-0.278198\pi\)
0.641776 + 0.766892i \(0.278198\pi\)
\(662\) 0 0
\(663\) 10.5830 0.411010
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.5830 0.409776
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −13.2288 −0.510690
\(672\) 0 0
\(673\) −24.0000 −0.925132 −0.462566 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 0 0
\(675\) −10.5830 −0.407340
\(676\) 0 0
\(677\) 3.00000 0.115299 0.0576497 0.998337i \(-0.481639\pi\)
0.0576497 + 0.998337i \(0.481639\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −7.00000 −0.268241
\(682\) 0 0
\(683\) −23.8118 −0.911132 −0.455566 0.890202i \(-0.650563\pi\)
−0.455566 + 0.890202i \(0.650563\pi\)
\(684\) 0 0
\(685\) 3.00000 0.114624
\(686\) 0 0
\(687\) −29.1033 −1.11036
\(688\) 0 0
\(689\) 28.0000 1.06672
\(690\) 0 0
\(691\) −29.1033 −1.10714 −0.553570 0.832803i \(-0.686735\pi\)
−0.553570 + 0.832803i \(0.686735\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 31.7490 1.20431
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 0 0
\(699\) −23.8118 −0.900644
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −39.6863 −1.49680
\(704\) 0 0
\(705\) 21.0000 0.790906
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −37.0000 −1.38956 −0.694782 0.719220i \(-0.744499\pi\)
−0.694782 + 0.719220i \(0.744499\pi\)
\(710\) 0 0
\(711\) −10.5830 −0.396894
\(712\) 0 0
\(713\) −7.00000 −0.262152
\(714\) 0 0
\(715\) −31.7490 −1.18735
\(716\) 0 0
\(717\) −42.0000 −1.56852
\(718\) 0 0
\(719\) −39.6863 −1.48005 −0.740024 0.672580i \(-0.765186\pi\)
−0.740024 + 0.672580i \(0.765186\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −2.64575 −0.0983965
\(724\) 0 0
\(725\) −16.0000 −0.594225
\(726\) 0 0
\(727\) −42.3320 −1.57001 −0.785004 0.619491i \(-0.787339\pi\)
−0.785004 + 0.619491i \(0.787339\pi\)
\(728\) 0 0
\(729\) −41.0000 −1.51852
\(730\) 0 0
\(731\) −10.5830 −0.391427
\(732\) 0 0
\(733\) −11.0000 −0.406294 −0.203147 0.979148i \(-0.565117\pi\)
−0.203147 + 0.979148i \(0.565117\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.00000 −0.257848
\(738\) 0 0
\(739\) −44.9778 −1.65453 −0.827267 0.561809i \(-0.810106\pi\)
−0.827267 + 0.561809i \(0.810106\pi\)
\(740\) 0 0
\(741\) −84.0000 −3.08582
\(742\) 0 0
\(743\) −21.1660 −0.776506 −0.388253 0.921553i \(-0.626921\pi\)
−0.388253 + 0.921553i \(0.626921\pi\)
\(744\) 0 0
\(745\) 63.0000 2.30814
\(746\) 0 0
\(747\) 42.3320 1.54885
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −39.6863 −1.44817 −0.724086 0.689709i \(-0.757738\pi\)
−0.724086 + 0.689709i \(0.757738\pi\)
\(752\) 0 0
\(753\) 14.0000 0.510188
\(754\) 0 0
\(755\) −7.93725 −0.288866
\(756\) 0 0
\(757\) 4.00000 0.145382 0.0726912 0.997354i \(-0.476841\pi\)
0.0726912 + 0.997354i \(0.476841\pi\)
\(758\) 0 0
\(759\) −18.5203 −0.672243
\(760\) 0 0
\(761\) −9.00000 −0.326250 −0.163125 0.986605i \(-0.552157\pi\)
−0.163125 + 0.986605i \(0.552157\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −12.0000 −0.433861
\(766\) 0 0
\(767\) −10.5830 −0.382130
\(768\) 0 0
\(769\) 24.0000 0.865462 0.432731 0.901523i \(-0.357550\pi\)
0.432731 + 0.901523i \(0.357550\pi\)
\(770\) 0 0
\(771\) 44.9778 1.61983
\(772\) 0 0
\(773\) −33.0000 −1.18693 −0.593464 0.804861i \(-0.702240\pi\)
−0.593464 + 0.804861i \(0.702240\pi\)
\(774\) 0 0
\(775\) 10.5830 0.380153
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −63.4980 −2.27505
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 10.5830 0.378206
\(784\) 0 0
\(785\) 27.0000 0.963671
\(786\) 0 0
\(787\) 18.5203 0.660176 0.330088 0.943950i \(-0.392922\pi\)
0.330088 + 0.943950i \(0.392922\pi\)
\(788\) 0 0
\(789\) 49.0000 1.74445
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) 0 0
\(795\) −55.5608 −1.97054
\(796\) 0 0
\(797\) 52.0000 1.84193 0.920967 0.389640i \(-0.127401\pi\)
0.920967 + 0.389640i \(0.127401\pi\)
\(798\) 0 0
\(799\) 2.64575 0.0936000
\(800\) 0 0
\(801\) 36.0000 1.27200
\(802\) 0 0
\(803\) −23.8118 −0.840299
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 39.6863 1.39702
\(808\) 0 0
\(809\) 31.0000 1.08990 0.544951 0.838468i \(-0.316548\pi\)
0.544951 + 0.838468i \(0.316548\pi\)
\(810\) 0 0
\(811\) −10.5830 −0.371620 −0.185810 0.982586i \(-0.559491\pi\)
−0.185810 + 0.982586i \(0.559491\pi\)
\(812\) 0 0
\(813\) −7.00000 −0.245501
\(814\) 0 0
\(815\) −39.6863 −1.39015
\(816\) 0 0
\(817\) 84.0000 2.93879
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.0000 0.453703 0.226852 0.973929i \(-0.427157\pi\)
0.226852 + 0.973929i \(0.427157\pi\)
\(822\) 0 0
\(823\) −18.5203 −0.645576 −0.322788 0.946471i \(-0.604620\pi\)
−0.322788 + 0.946471i \(0.604620\pi\)
\(824\) 0 0
\(825\) 28.0000 0.974835
\(826\) 0 0
\(827\) −31.7490 −1.10402 −0.552011 0.833837i \(-0.686139\pi\)
−0.552011 + 0.833837i \(0.686139\pi\)
\(828\) 0 0
\(829\) 15.0000 0.520972 0.260486 0.965478i \(-0.416117\pi\)
0.260486 + 0.965478i \(0.416117\pi\)
\(830\) 0 0
\(831\) 66.1438 2.29450
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −15.8745 −0.549360
\(836\) 0 0
\(837\) −7.00000 −0.241955
\(838\) 0 0
\(839\) −21.1660 −0.730732 −0.365366 0.930864i \(-0.619056\pi\)
−0.365366 + 0.930864i \(0.619056\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 63.4980 2.18699
\(844\) 0 0
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 49.0000 1.68168
\(850\) 0 0
\(851\) 13.2288 0.453476
\(852\) 0 0
\(853\) 36.0000 1.23262 0.616308 0.787505i \(-0.288628\pi\)
0.616308 + 0.787505i \(0.288628\pi\)
\(854\) 0 0
\(855\) 95.2470 3.25738
\(856\) 0 0
\(857\) −15.0000 −0.512390 −0.256195 0.966625i \(-0.582469\pi\)
−0.256195 + 0.966625i \(0.582469\pi\)
\(858\) 0 0
\(859\) 29.1033 0.992991 0.496495 0.868039i \(-0.334620\pi\)
0.496495 + 0.868039i \(0.334620\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 39.6863 1.35094 0.675468 0.737389i \(-0.263942\pi\)
0.675468 + 0.737389i \(0.263942\pi\)
\(864\) 0 0
\(865\) −9.00000 −0.306009
\(866\) 0 0
\(867\) 42.3320 1.43767
\(868\) 0 0
\(869\) 7.00000 0.237459
\(870\) 0 0
\(871\) 10.5830 0.358591
\(872\) 0 0
\(873\) 32.0000 1.08304
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 37.0000 1.24940 0.624701 0.780864i \(-0.285221\pi\)
0.624701 + 0.780864i \(0.285221\pi\)
\(878\) 0 0
\(879\) −26.4575 −0.892390
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) 0 0
\(883\) −10.5830 −0.356146 −0.178073 0.984017i \(-0.556986\pi\)
−0.178073 + 0.984017i \(0.556986\pi\)
\(884\) 0 0
\(885\) 21.0000 0.705907
\(886\) 0 0
\(887\) −2.64575 −0.0888356 −0.0444178 0.999013i \(-0.514143\pi\)
−0.0444178 + 0.999013i \(0.514143\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 13.2288 0.443180
\(892\) 0 0
\(893\) −21.0000 −0.702738
\(894\) 0 0
\(895\) −71.4353 −2.38782
\(896\) 0 0
\(897\) 28.0000 0.934893
\(898\) 0 0
\(899\) −10.5830 −0.352963
\(900\) 0 0
\(901\) −7.00000 −0.233204
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.0000 0.598340
\(906\) 0 0
\(907\) −18.5203 −0.614955 −0.307478 0.951555i \(-0.599485\pi\)
−0.307478 + 0.951555i \(0.599485\pi\)
\(908\) 0 0
\(909\) 60.0000 1.99007
\(910\) 0 0
\(911\) −21.1660 −0.701261 −0.350631 0.936514i \(-0.614033\pi\)
−0.350631 + 0.936514i \(0.614033\pi\)
\(912\) 0 0
\(913\) −28.0000 −0.926665
\(914\) 0 0
\(915\) −39.6863 −1.31199
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −18.5203 −0.610927 −0.305463 0.952204i \(-0.598811\pi\)
−0.305463 + 0.952204i \(0.598811\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −20.0000 −0.657596
\(926\) 0 0
\(927\) 10.5830 0.347591
\(928\) 0 0
\(929\) 9.00000 0.295280 0.147640 0.989041i \(-0.452832\pi\)
0.147640 + 0.989041i \(0.452832\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 63.0000 2.06253
\(934\) 0 0
\(935\) 7.93725 0.259576
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) −44.9778 −1.46779
\(940\) 0 0
\(941\) 1.00000 0.0325991 0.0162995 0.999867i \(-0.494811\pi\)
0.0162995 + 0.999867i \(0.494811\pi\)
\(942\) 0 0
\(943\) 21.1660 0.689260
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 29.1033 0.945729 0.472865 0.881135i \(-0.343220\pi\)
0.472865 + 0.881135i \(0.343220\pi\)
\(948\) 0 0
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) 34.3948 1.11533
\(952\) 0 0
\(953\) −16.0000 −0.518291 −0.259145 0.965838i \(-0.583441\pi\)
−0.259145 + 0.965838i \(0.583441\pi\)
\(954\) 0 0
\(955\) 71.4353 2.31159
\(956\) 0 0
\(957\) −28.0000 −0.905111
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −24.0000 −0.774194
\(962\) 0 0
\(963\) 52.9150 1.70516
\(964\) 0 0
\(965\) −21.0000 −0.676014
\(966\) 0 0
\(967\) 42.3320 1.36131 0.680653 0.732606i \(-0.261696\pi\)
0.680653 + 0.732606i \(0.261696\pi\)
\(968\) 0 0
\(969\) 21.0000 0.674617
\(970\) 0 0
\(971\) −7.93725 −0.254719 −0.127359 0.991857i \(-0.540650\pi\)
−0.127359 + 0.991857i \(0.540650\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −42.3320 −1.35571
\(976\) 0 0
\(977\) 41.0000 1.31171 0.655853 0.754889i \(-0.272309\pi\)
0.655853 + 0.754889i \(0.272309\pi\)
\(978\) 0 0
\(979\) −23.8118 −0.761027
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) 23.8118 0.759477 0.379739 0.925094i \(-0.376014\pi\)
0.379739 + 0.925094i \(0.376014\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −28.0000 −0.890348
\(990\) 0 0
\(991\) 44.9778 1.42877 0.714383 0.699755i \(-0.246707\pi\)
0.714383 + 0.699755i \(0.246707\pi\)
\(992\) 0 0
\(993\) −21.0000 −0.666415
\(994\) 0 0
\(995\) −55.5608 −1.76140
\(996\) 0 0
\(997\) −23.0000 −0.728417 −0.364209 0.931317i \(-0.618661\pi\)
−0.364209 + 0.931317i \(0.618661\pi\)
\(998\) 0 0
\(999\) 13.2288 0.418539
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 1568.2.a.t.1.1 2
4.3 odd 2 inner 1568.2.a.t.1.2 2
7.2 even 3 1568.2.i.p.1537.2 4
7.3 odd 6 224.2.i.c.65.1 4
7.4 even 3 1568.2.i.p.961.2 4
7.5 odd 6 224.2.i.c.193.1 yes 4
7.6 odd 2 1568.2.a.m.1.2 2
8.3 odd 2 3136.2.a.bg.1.1 2
8.5 even 2 3136.2.a.bg.1.2 2
21.5 even 6 2016.2.s.o.865.1 4
21.17 even 6 2016.2.s.o.289.1 4
28.3 even 6 224.2.i.c.65.2 yes 4
28.11 odd 6 1568.2.i.p.961.1 4
28.19 even 6 224.2.i.c.193.2 yes 4
28.23 odd 6 1568.2.i.p.1537.1 4
28.27 even 2 1568.2.a.m.1.1 2
56.3 even 6 448.2.i.h.65.1 4
56.5 odd 6 448.2.i.h.193.2 4
56.13 odd 2 3136.2.a.bv.1.1 2
56.19 even 6 448.2.i.h.193.1 4
56.27 even 2 3136.2.a.bv.1.2 2
56.45 odd 6 448.2.i.h.65.2 4
84.47 odd 6 2016.2.s.o.865.2 4
84.59 odd 6 2016.2.s.o.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.i.c.65.1 4 7.3 odd 6
224.2.i.c.65.2 yes 4 28.3 even 6
224.2.i.c.193.1 yes 4 7.5 odd 6
224.2.i.c.193.2 yes 4 28.19 even 6
448.2.i.h.65.1 4 56.3 even 6
448.2.i.h.65.2 4 56.45 odd 6
448.2.i.h.193.1 4 56.19 even 6
448.2.i.h.193.2 4 56.5 odd 6
1568.2.a.m.1.1 2 28.27 even 2
1568.2.a.m.1.2 2 7.6 odd 2
1568.2.a.t.1.1 2 1.1 even 1 trivial
1568.2.a.t.1.2 2 4.3 odd 2 inner
1568.2.i.p.961.1 4 28.11 odd 6
1568.2.i.p.961.2 4 7.4 even 3
1568.2.i.p.1537.1 4 28.23 odd 6
1568.2.i.p.1537.2 4 7.2 even 3
2016.2.s.o.289.1 4 21.17 even 6
2016.2.s.o.289.2 4 84.59 odd 6
2016.2.s.o.865.1 4 21.5 even 6
2016.2.s.o.865.2 4 84.47 odd 6
3136.2.a.bg.1.1 2 8.3 odd 2
3136.2.a.bg.1.2 2 8.5 even 2
3136.2.a.bv.1.1 2 56.13 odd 2
3136.2.a.bv.1.2 2 56.27 even 2