Properties

Label 2704.2.a.o.1.1
Level $2704$
Weight $2$
Character 2704.1
Self dual yes
Analytic conductor $21.592$
Analytic rank $1$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2704,2,Mod(1,2704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2704, 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("2704.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.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: \(21.5915487066\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2704.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} -1.73205 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} -1.73205 q^{5} +1.00000 q^{9} +3.46410 q^{15} +3.00000 q^{17} +3.46410 q^{19} -6.00000 q^{23} -2.00000 q^{25} +4.00000 q^{27} +3.00000 q^{29} -3.46410 q^{31} +8.66025 q^{37} +5.19615 q^{41} +8.00000 q^{43} -1.73205 q^{45} -3.46410 q^{47} -7.00000 q^{49} -6.00000 q^{51} -3.00000 q^{53} -6.92820 q^{57} +6.92820 q^{59} +1.00000 q^{61} +3.46410 q^{67} +12.0000 q^{69} -3.46410 q^{71} -1.73205 q^{73} +4.00000 q^{75} -4.00000 q^{79} -11.0000 q^{81} -13.8564 q^{83} -5.19615 q^{85} -6.00000 q^{87} -6.92820 q^{89} +6.92820 q^{93} -6.00000 q^{95} +6.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 2 q^{9} + 6 q^{17} - 12 q^{23} - 4 q^{25} + 8 q^{27} + 6 q^{29} + 16 q^{43} - 14 q^{49} - 12 q^{51} - 6 q^{53} + 2 q^{61} + 24 q^{69} + 8 q^{75} - 8 q^{79} - 22 q^{81} - 12 q^{87} - 12 q^{95}+O(q^{100}) \) 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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) −1.73205 −0.774597 −0.387298 0.921954i \(-0.626592\pi\)
−0.387298 + 0.921954i \(0.626592\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 3.46410 0.894427
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.66025 1.42374 0.711868 0.702313i \(-0.247849\pi\)
0.711868 + 0.702313i \(0.247849\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.19615 0.811503 0.405751 0.913984i \(-0.367010\pi\)
0.405751 + 0.913984i \(0.367010\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) −1.73205 −0.258199
\(46\) 0 0
\(47\) −3.46410 −0.505291 −0.252646 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.92820 −0.917663
\(58\) 0 0
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.46410 0.423207 0.211604 0.977356i \(-0.432131\pi\)
0.211604 + 0.977356i \(0.432131\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) −1.73205 −0.202721 −0.101361 0.994850i \(-0.532320\pi\)
−0.101361 + 0.994850i \(0.532320\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −13.8564 −1.52094 −0.760469 0.649374i \(-0.775031\pi\)
−0.760469 + 0.649374i \(0.775031\pi\)
\(84\) 0 0
\(85\) −5.19615 −0.563602
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −6.92820 −0.734388 −0.367194 0.930144i \(-0.619682\pi\)
−0.367194 + 0.930144i \(0.619682\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.92820 0.718421
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 6.92820 0.703452 0.351726 0.936103i \(-0.385595\pi\)
0.351726 + 0.936103i \(0.385595\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) −13.8564 −1.32720 −0.663602 0.748086i \(-0.730973\pi\)
−0.663602 + 0.748086i \(0.730973\pi\)
\(110\) 0 0
\(111\) −17.3205 −1.64399
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 10.3923 0.969087
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −10.3923 −0.937043
\(124\) 0 0
\(125\) 12.1244 1.08444
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) −16.0000 −1.40872
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −6.92820 −0.596285
\(136\) 0 0
\(137\) 15.5885 1.33181 0.665906 0.746036i \(-0.268045\pi\)
0.665906 + 0.746036i \(0.268045\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 6.92820 0.583460
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −5.19615 −0.431517
\(146\) 0 0
\(147\) 14.0000 1.15470
\(148\) 0 0
\(149\) −19.0526 −1.56085 −0.780423 0.625252i \(-0.784996\pi\)
−0.780423 + 0.625252i \(0.784996\pi\)
\(150\) 0 0
\(151\) 17.3205 1.40952 0.704761 0.709444i \(-0.251054\pi\)
0.704761 + 0.709444i \(0.251054\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.7846 1.62798 0.813988 0.580881i \(-0.197292\pi\)
0.813988 + 0.580881i \(0.197292\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.8564 1.07224 0.536120 0.844141i \(-0.319889\pi\)
0.536120 + 0.844141i \(0.319889\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 3.46410 0.264906
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −13.8564 −1.04151
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −11.0000 −0.817624 −0.408812 0.912619i \(-0.634057\pi\)
−0.408812 + 0.912619i \(0.634057\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −15.0000 −1.10282
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) −5.19615 −0.374027 −0.187014 0.982357i \(-0.559881\pi\)
−0.187014 + 0.982357i \(0.559881\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.8564 −0.987228 −0.493614 0.869681i \(-0.664324\pi\)
−0.493614 + 0.869681i \(0.664324\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) −6.92820 −0.488678
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) 6.92820 0.474713
\(214\) 0 0
\(215\) −13.8564 −0.944999
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3.46410 0.234082
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −10.3923 −0.695920 −0.347960 0.937509i \(-0.613126\pi\)
−0.347960 + 0.937509i \(0.613126\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −24.2487 −1.60944 −0.804722 0.593652i \(-0.797686\pi\)
−0.804722 + 0.593652i \(0.797686\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 20.7846 1.34444 0.672222 0.740349i \(-0.265340\pi\)
0.672222 + 0.740349i \(0.265340\pi\)
\(240\) 0 0
\(241\) 1.73205 0.111571 0.0557856 0.998443i \(-0.482234\pi\)
0.0557856 + 0.998443i \(0.482234\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 12.1244 0.774597
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 27.7128 1.75623
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 10.3923 0.650791
\(256\) 0 0
\(257\) −3.00000 −0.187135 −0.0935674 0.995613i \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 5.19615 0.319197
\(266\) 0 0
\(267\) 13.8564 0.847998
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −20.7846 −1.26258 −0.631288 0.775549i \(-0.717473\pi\)
−0.631288 + 0.775549i \(0.717473\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.00000 0.420589 0.210295 0.977638i \(-0.432558\pi\)
0.210295 + 0.977638i \(0.432558\pi\)
\(278\) 0 0
\(279\) −3.46410 −0.207390
\(280\) 0 0
\(281\) 22.5167 1.34323 0.671616 0.740900i \(-0.265601\pi\)
0.671616 + 0.740900i \(0.265601\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 12.0000 0.710819
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −13.8564 −0.812277
\(292\) 0 0
\(293\) −5.19615 −0.303562 −0.151781 0.988414i \(-0.548501\pi\)
−0.151781 + 0.988414i \(0.548501\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) −1.73205 −0.0991769
\(306\) 0 0
\(307\) −17.3205 −0.988534 −0.494267 0.869310i \(-0.664563\pi\)
−0.494267 + 0.869310i \(0.664563\pi\)
\(308\) 0 0
\(309\) 20.0000 1.13776
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.19615 −0.291845 −0.145922 0.989296i \(-0.546615\pi\)
−0.145922 + 0.989296i \(0.546615\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 10.3923 0.578243
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 27.7128 1.53252
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 27.7128 1.52323 0.761617 0.648027i \(-0.224406\pi\)
0.761617 + 0.648027i \(0.224406\pi\)
\(332\) 0 0
\(333\) 8.66025 0.474579
\(334\) 0 0
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) 30.0000 1.62938
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −20.7846 −1.11901
\(346\) 0 0
\(347\) 30.0000 1.61048 0.805242 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) 0 0
\(349\) 13.8564 0.741716 0.370858 0.928689i \(-0.379064\pi\)
0.370858 + 0.928689i \(0.379064\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −32.9090 −1.75157 −0.875784 0.482704i \(-0.839655\pi\)
−0.875784 + 0.482704i \(0.839655\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.92820 −0.365657 −0.182828 0.983145i \(-0.558525\pi\)
−0.182828 + 0.983145i \(0.558525\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) 0 0
\(363\) 22.0000 1.15470
\(364\) 0 0
\(365\) 3.00000 0.157027
\(366\) 0 0
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 0 0
\(369\) 5.19615 0.270501
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 19.0000 0.983783 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 0 0
\(375\) −24.2487 −1.25220
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −24.2487 −1.24557 −0.622786 0.782392i \(-0.713999\pi\)
−0.622786 + 0.782392i \(0.713999\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) 20.7846 1.06204 0.531022 0.847358i \(-0.321808\pi\)
0.531022 + 0.847358i \(0.321808\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000 0.406663
\(388\) 0 0
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 36.0000 1.81596
\(394\) 0 0
\(395\) 6.92820 0.348596
\(396\) 0 0
\(397\) 13.8564 0.695433 0.347717 0.937600i \(-0.386957\pi\)
0.347717 + 0.937600i \(0.386957\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.73205 −0.0864945 −0.0432472 0.999064i \(-0.513770\pi\)
−0.0432472 + 0.999064i \(0.513770\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 19.0526 0.946729
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 15.5885 0.770800 0.385400 0.922750i \(-0.374064\pi\)
0.385400 + 0.922750i \(0.374064\pi\)
\(410\) 0 0
\(411\) −31.1769 −1.53784
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) 15.5885 0.759735 0.379867 0.925041i \(-0.375970\pi\)
0.379867 + 0.925041i \(0.375970\pi\)
\(422\) 0 0
\(423\) −3.46410 −0.168430
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.92820 −0.333720 −0.166860 0.985981i \(-0.553363\pi\)
−0.166860 + 0.985981i \(0.553363\pi\)
\(432\) 0 0
\(433\) 17.0000 0.816968 0.408484 0.912766i \(-0.366058\pi\)
0.408484 + 0.912766i \(0.366058\pi\)
\(434\) 0 0
\(435\) 10.3923 0.498273
\(436\) 0 0
\(437\) −20.7846 −0.994263
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) 38.1051 1.80231
\(448\) 0 0
\(449\) 6.92820 0.326962 0.163481 0.986546i \(-0.447728\pi\)
0.163481 + 0.986546i \(0.447728\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −34.6410 −1.62758
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.73205 −0.0810219 −0.0405110 0.999179i \(-0.512899\pi\)
−0.0405110 + 0.999179i \(0.512899\pi\)
\(458\) 0 0
\(459\) 12.0000 0.560112
\(460\) 0 0
\(461\) 22.5167 1.04871 0.524353 0.851501i \(-0.324307\pi\)
0.524353 + 0.851501i \(0.324307\pi\)
\(462\) 0 0
\(463\) 13.8564 0.643962 0.321981 0.946746i \(-0.395651\pi\)
0.321981 + 0.946746i \(0.395651\pi\)
\(464\) 0 0
\(465\) −12.0000 −0.556487
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 26.0000 1.19802
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6.92820 −0.317888
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) 0 0
\(479\) −24.2487 −1.10795 −0.553976 0.832533i \(-0.686890\pi\)
−0.553976 + 0.832533i \(0.686890\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12.0000 −0.544892
\(486\) 0 0
\(487\) 6.92820 0.313947 0.156973 0.987603i \(-0.449826\pi\)
0.156973 + 0.987603i \(0.449826\pi\)
\(488\) 0 0
\(489\) −41.5692 −1.87983
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −31.1769 −1.39567 −0.697835 0.716258i \(-0.745853\pi\)
−0.697835 + 0.716258i \(0.745853\pi\)
\(500\) 0 0
\(501\) −27.7128 −1.23812
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) −5.19615 −0.231226
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −19.0526 −0.844490 −0.422245 0.906482i \(-0.638758\pi\)
−0.422245 + 0.906482i \(0.638758\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 13.8564 0.611775
\(514\) 0 0
\(515\) 17.3205 0.763233
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 9.00000 0.394297 0.197149 0.980374i \(-0.436832\pi\)
0.197149 + 0.980374i \(0.436832\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.3923 −0.452696
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 6.92820 0.300658
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 10.3923 0.449299
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −29.4449 −1.26593 −0.632967 0.774179i \(-0.718163\pi\)
−0.632967 + 0.774179i \(0.718163\pi\)
\(542\) 0 0
\(543\) 22.0000 0.944110
\(544\) 0 0
\(545\) 24.0000 1.02805
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) 0 0
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) 10.3923 0.442727
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 30.0000 1.27343
\(556\) 0 0
\(557\) −15.5885 −0.660504 −0.330252 0.943893i \(-0.607134\pi\)
−0.330252 + 0.943893i \(0.607134\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 25.9808 1.09302
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) 36.0000 1.50392
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −19.0526 −0.793168 −0.396584 0.917998i \(-0.629805\pi\)
−0.396584 + 0.917998i \(0.629805\pi\)
\(578\) 0 0
\(579\) 10.3923 0.431889
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.7846 −0.857873 −0.428936 0.903335i \(-0.641112\pi\)
−0.428936 + 0.903335i \(0.641112\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 27.7128 1.13995
\(592\) 0 0
\(593\) 25.9808 1.06690 0.533451 0.845831i \(-0.320895\pi\)
0.533451 + 0.845831i \(0.320895\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) 25.0000 1.01977 0.509886 0.860242i \(-0.329688\pi\)
0.509886 + 0.860242i \(0.329688\pi\)
\(602\) 0 0
\(603\) 3.46410 0.141069
\(604\) 0 0
\(605\) 19.0526 0.774597
\(606\) 0 0
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 12.1244 0.489698 0.244849 0.969561i \(-0.421262\pi\)
0.244849 + 0.969561i \(0.421262\pi\)
\(614\) 0 0
\(615\) 18.0000 0.725830
\(616\) 0 0
\(617\) −22.5167 −0.906487 −0.453243 0.891387i \(-0.649733\pi\)
−0.453243 + 0.891387i \(0.649733\pi\)
\(618\) 0 0
\(619\) −20.7846 −0.835404 −0.417702 0.908584i \(-0.637164\pi\)
−0.417702 + 0.908584i \(0.637164\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 25.9808 1.03592
\(630\) 0 0
\(631\) −48.4974 −1.93065 −0.965326 0.261048i \(-0.915932\pi\)
−0.965326 + 0.261048i \(0.915932\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 0 0
\(635\) 3.46410 0.137469
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −3.46410 −0.137038
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) −13.8564 −0.546443 −0.273222 0.961951i \(-0.588089\pi\)
−0.273222 + 0.961951i \(0.588089\pi\)
\(644\) 0 0
\(645\) 27.7128 1.09119
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 31.1769 1.21818
\(656\) 0 0
\(657\) −1.73205 −0.0675737
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −46.7654 −1.81896 −0.909481 0.415745i \(-0.863521\pi\)
−0.909481 + 0.415745i \(0.863521\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −18.0000 −0.696963
\(668\) 0 0
\(669\) 20.7846 0.803579
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 0 0
\(675\) −8.00000 −0.307920
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 48.4974 1.85843
\(682\) 0 0
\(683\) 24.2487 0.927851 0.463926 0.885874i \(-0.346441\pi\)
0.463926 + 0.885874i \(0.346441\pi\)
\(684\) 0 0
\(685\) −27.0000 −1.03162
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 13.8564 0.527123 0.263561 0.964643i \(-0.415103\pi\)
0.263561 + 0.964643i \(0.415103\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.92820 −0.262802
\(696\) 0 0
\(697\) 15.5885 0.590455
\(698\) 0 0
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 30.0000 1.13147
\(704\) 0 0
\(705\) −12.0000 −0.451946
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5.19615 0.195146 0.0975728 0.995228i \(-0.468892\pi\)
0.0975728 + 0.995228i \(0.468892\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 20.7846 0.778390
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −41.5692 −1.55243
\(718\) 0 0
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −3.46410 −0.128831
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 12.1244 0.447823 0.223912 0.974609i \(-0.428117\pi\)
0.223912 + 0.974609i \(0.428117\pi\)
\(734\) 0 0
\(735\) −24.2487 −0.894427
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.7846 0.764574 0.382287 0.924044i \(-0.375137\pi\)
0.382287 + 0.924044i \(0.375137\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34.6410 −1.27086 −0.635428 0.772160i \(-0.719176\pi\)
−0.635428 + 0.772160i \(0.719176\pi\)
\(744\) 0 0
\(745\) 33.0000 1.20903
\(746\) 0 0
\(747\) −13.8564 −0.506979
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 36.0000 1.31191
\(754\) 0 0
\(755\) −30.0000 −1.09181
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −34.6410 −1.25574 −0.627868 0.778320i \(-0.716072\pi\)
−0.627868 + 0.778320i \(0.716072\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −5.19615 −0.187867
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −6.92820 −0.249837 −0.124919 0.992167i \(-0.539867\pi\)
−0.124919 + 0.992167i \(0.539867\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) −34.6410 −1.24595 −0.622975 0.782241i \(-0.714076\pi\)
−0.622975 + 0.782241i \(0.714076\pi\)
\(774\) 0 0
\(775\) 6.92820 0.248868
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 12.0000 0.428845
\(784\) 0 0
\(785\) 22.5167 0.803654
\(786\) 0 0
\(787\) 38.1051 1.35830 0.679150 0.733999i \(-0.262348\pi\)
0.679150 + 0.733999i \(0.262348\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −10.3923 −0.368577
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −10.3923 −0.367653
\(800\) 0 0
\(801\) −6.92820 −0.244796
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) 0 0
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) 0 0
\(811\) 38.1051 1.33805 0.669026 0.743239i \(-0.266712\pi\)
0.669026 + 0.743239i \(0.266712\pi\)
\(812\) 0 0
\(813\) 41.5692 1.45790
\(814\) 0 0
\(815\) −36.0000 −1.26102
\(816\) 0 0
\(817\) 27.7128 0.969549
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 41.5692 1.45078 0.725388 0.688340i \(-0.241660\pi\)
0.725388 + 0.688340i \(0.241660\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.7846 0.722752 0.361376 0.932420i \(-0.382307\pi\)
0.361376 + 0.932420i \(0.382307\pi\)
\(828\) 0 0
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) −14.0000 −0.485655
\(832\) 0 0
\(833\) −21.0000 −0.727607
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) −13.8564 −0.478947
\(838\) 0 0
\(839\) 45.0333 1.55472 0.777361 0.629054i \(-0.216558\pi\)
0.777361 + 0.629054i \(0.216558\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −45.0333 −1.55103
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) −51.9615 −1.78122
\(852\) 0 0
\(853\) −25.9808 −0.889564 −0.444782 0.895639i \(-0.646719\pi\)
−0.444782 + 0.895639i \(0.646719\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) 0 0
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 27.7128 0.943355 0.471678 0.881771i \(-0.343649\pi\)
0.471678 + 0.881771i \(0.343649\pi\)
\(864\) 0 0
\(865\) 10.3923 0.353349
\(866\) 0 0
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 6.92820 0.234484
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 12.1244 0.409410 0.204705 0.978824i \(-0.434376\pi\)
0.204705 + 0.978824i \(0.434376\pi\)
\(878\) 0 0
\(879\) 10.3923 0.350524
\(880\) 0 0
\(881\) −27.0000 −0.909653 −0.454827 0.890580i \(-0.650299\pi\)
−0.454827 + 0.890580i \(0.650299\pi\)
\(882\) 0 0
\(883\) 10.0000 0.336527 0.168263 0.985742i \(-0.446184\pi\)
0.168263 + 0.985742i \(0.446184\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) 0 0
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −10.3923 −0.346603
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.0526 0.633328
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) 3.00000 0.0995037
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 3.46410 0.114520
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −22.0000 −0.725713 −0.362857 0.931845i \(-0.618198\pi\)
−0.362857 + 0.931845i \(0.618198\pi\)
\(920\) 0 0
\(921\) 34.6410 1.14146
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −17.3205 −0.569495
\(926\) 0 0
\(927\) −10.0000 −0.328443
\(928\) 0 0
\(929\) −46.7654 −1.53432 −0.767161 0.641455i \(-0.778331\pi\)
−0.767161 + 0.641455i \(0.778331\pi\)
\(930\) 0 0
\(931\) −24.2487 −0.794719
\(932\) 0 0
\(933\) 60.0000 1.96431
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) 20.7846 0.677559 0.338779 0.940866i \(-0.389986\pi\)
0.338779 + 0.940866i \(0.389986\pi\)
\(942\) 0 0
\(943\) −31.1769 −1.01526
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 17.3205 0.562841 0.281420 0.959585i \(-0.409194\pi\)
0.281420 + 0.959585i \(0.409194\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 10.3923 0.336994
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 31.1769 1.00886
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 0 0
\(965\) 9.00000 0.289720
\(966\) 0 0
\(967\) −58.8897 −1.89377 −0.946883 0.321578i \(-0.895787\pi\)
−0.946883 + 0.321578i \(0.895787\pi\)
\(968\) 0 0
\(969\) −20.7846 −0.667698
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −43.3013 −1.38533 −0.692665 0.721259i \(-0.743564\pi\)
−0.692665 + 0.721259i \(0.743564\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −13.8564 −0.442401
\(982\) 0 0
\(983\) 51.9615 1.65732 0.828658 0.559756i \(-0.189105\pi\)
0.828658 + 0.559756i \(0.189105\pi\)
\(984\) 0 0
\(985\) 24.0000 0.764704
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 0 0
\(993\) −55.4256 −1.75888
\(994\) 0 0
\(995\) 3.46410 0.109819
\(996\) 0 0
\(997\) 17.0000 0.538395 0.269198 0.963085i \(-0.413241\pi\)
0.269198 + 0.963085i \(0.413241\pi\)
\(998\) 0 0
\(999\) 34.6410 1.09599
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 2704.2.a.o.1.1 2
4.3 odd 2 169.2.a.a.1.2 2
12.11 even 2 1521.2.a.k.1.1 2
13.5 odd 4 2704.2.f.b.337.2 2
13.6 odd 12 208.2.w.b.49.1 2
13.8 odd 4 2704.2.f.b.337.1 2
13.11 odd 12 208.2.w.b.17.1 2
13.12 even 2 inner 2704.2.a.o.1.2 2
20.19 odd 2 4225.2.a.v.1.1 2
28.27 even 2 8281.2.a.q.1.2 2
39.11 even 12 1872.2.by.d.433.1 2
39.32 even 12 1872.2.by.d.1297.1 2
52.3 odd 6 169.2.c.a.22.1 4
52.7 even 12 169.2.e.a.23.1 2
52.11 even 12 13.2.e.a.4.1 2
52.15 even 12 169.2.e.a.147.1 2
52.19 even 12 13.2.e.a.10.1 yes 2
52.23 odd 6 169.2.c.a.22.2 4
52.31 even 4 169.2.b.a.168.1 2
52.35 odd 6 169.2.c.a.146.1 4
52.43 odd 6 169.2.c.a.146.2 4
52.47 even 4 169.2.b.a.168.2 2
52.51 odd 2 169.2.a.a.1.1 2
104.11 even 12 832.2.w.d.641.1 2
104.19 even 12 832.2.w.d.257.1 2
104.37 odd 12 832.2.w.a.641.1 2
104.45 odd 12 832.2.w.a.257.1 2
156.11 odd 12 117.2.q.c.82.1 2
156.47 odd 4 1521.2.b.a.1351.1 2
156.71 odd 12 117.2.q.c.10.1 2
156.83 odd 4 1521.2.b.a.1351.2 2
156.155 even 2 1521.2.a.k.1.2 2
260.19 even 12 325.2.n.a.101.1 2
260.63 odd 12 325.2.m.a.199.1 4
260.123 odd 12 325.2.m.a.49.2 4
260.167 odd 12 325.2.m.a.199.2 4
260.219 even 12 325.2.n.a.251.1 2
260.227 odd 12 325.2.m.a.49.1 4
260.259 odd 2 4225.2.a.v.1.2 2
364.11 even 12 637.2.u.c.30.1 2
364.19 odd 12 637.2.u.b.361.1 2
364.115 odd 12 637.2.u.b.30.1 2
364.123 even 12 637.2.k.a.569.1 2
364.167 odd 12 637.2.q.a.589.1 2
364.219 even 12 637.2.k.a.459.1 2
364.227 odd 12 637.2.k.c.569.1 2
364.271 odd 12 637.2.k.c.459.1 2
364.279 odd 12 637.2.q.a.491.1 2
364.331 even 12 637.2.u.c.361.1 2
364.363 even 2 8281.2.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 2 52.11 even 12
13.2.e.a.10.1 yes 2 52.19 even 12
117.2.q.c.10.1 2 156.71 odd 12
117.2.q.c.82.1 2 156.11 odd 12
169.2.a.a.1.1 2 52.51 odd 2
169.2.a.a.1.2 2 4.3 odd 2
169.2.b.a.168.1 2 52.31 even 4
169.2.b.a.168.2 2 52.47 even 4
169.2.c.a.22.1 4 52.3 odd 6
169.2.c.a.22.2 4 52.23 odd 6
169.2.c.a.146.1 4 52.35 odd 6
169.2.c.a.146.2 4 52.43 odd 6
169.2.e.a.23.1 2 52.7 even 12
169.2.e.a.147.1 2 52.15 even 12
208.2.w.b.17.1 2 13.11 odd 12
208.2.w.b.49.1 2 13.6 odd 12
325.2.m.a.49.1 4 260.227 odd 12
325.2.m.a.49.2 4 260.123 odd 12
325.2.m.a.199.1 4 260.63 odd 12
325.2.m.a.199.2 4 260.167 odd 12
325.2.n.a.101.1 2 260.19 even 12
325.2.n.a.251.1 2 260.219 even 12
637.2.k.a.459.1 2 364.219 even 12
637.2.k.a.569.1 2 364.123 even 12
637.2.k.c.459.1 2 364.271 odd 12
637.2.k.c.569.1 2 364.227 odd 12
637.2.q.a.491.1 2 364.279 odd 12
637.2.q.a.589.1 2 364.167 odd 12
637.2.u.b.30.1 2 364.115 odd 12
637.2.u.b.361.1 2 364.19 odd 12
637.2.u.c.30.1 2 364.11 even 12
637.2.u.c.361.1 2 364.331 even 12
832.2.w.a.257.1 2 104.45 odd 12
832.2.w.a.641.1 2 104.37 odd 12
832.2.w.d.257.1 2 104.19 even 12
832.2.w.d.641.1 2 104.11 even 12
1521.2.a.k.1.1 2 12.11 even 2
1521.2.a.k.1.2 2 156.155 even 2
1521.2.b.a.1351.1 2 156.47 odd 4
1521.2.b.a.1351.2 2 156.83 odd 4
1872.2.by.d.433.1 2 39.11 even 12
1872.2.by.d.1297.1 2 39.32 even 12
2704.2.a.o.1.1 2 1.1 even 1 trivial
2704.2.a.o.1.2 2 13.12 even 2 inner
2704.2.f.b.337.1 2 13.8 odd 4
2704.2.f.b.337.2 2 13.5 odd 4
4225.2.a.v.1.1 2 20.19 odd 2
4225.2.a.v.1.2 2 260.259 odd 2
8281.2.a.q.1.1 2 364.363 even 2
8281.2.a.q.1.2 2 28.27 even 2