Properties

Label 228.2.a.c.1.1
Level $228$
Weight $2$
Character 228.1
Self dual yes
Analytic conductor $1.821$
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] = [228,2,Mod(1,228)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(228, 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("228.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 228.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: \(1.82058916609\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 228.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.37228 q^{5} +3.37228 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.37228 q^{5} +3.37228 q^{7} +1.00000 q^{9} +1.37228 q^{11} +2.00000 q^{13} -1.37228 q^{15} +1.37228 q^{17} +1.00000 q^{19} +3.37228 q^{21} -8.74456 q^{23} -3.11684 q^{25} +1.00000 q^{27} +2.74456 q^{29} -6.74456 q^{31} +1.37228 q^{33} -4.62772 q^{35} +4.74456 q^{37} +2.00000 q^{39} +3.37228 q^{43} -1.37228 q^{45} -13.3723 q^{47} +4.37228 q^{49} +1.37228 q^{51} -2.74456 q^{53} -1.88316 q^{55} +1.00000 q^{57} -2.62772 q^{61} +3.37228 q^{63} -2.74456 q^{65} -9.48913 q^{67} -8.74456 q^{69} -12.0000 q^{71} -5.37228 q^{73} -3.11684 q^{75} +4.62772 q^{77} +8.00000 q^{79} +1.00000 q^{81} +8.74456 q^{83} -1.88316 q^{85} +2.74456 q^{87} +14.7446 q^{89} +6.74456 q^{91} -6.74456 q^{93} -1.37228 q^{95} +14.0000 q^{97} +1.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 3 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 3 q^{5} + q^{7} + 2 q^{9} - 3 q^{11} + 4 q^{13} + 3 q^{15} - 3 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 11 q^{25} + 2 q^{27} - 6 q^{29} - 2 q^{31} - 3 q^{33} - 15 q^{35} - 2 q^{37} + 4 q^{39} + q^{43} + 3 q^{45} - 21 q^{47} + 3 q^{49} - 3 q^{51} + 6 q^{53} - 21 q^{55} + 2 q^{57} - 11 q^{61} + q^{63} + 6 q^{65} + 4 q^{67} - 6 q^{69} - 24 q^{71} - 5 q^{73} + 11 q^{75} + 15 q^{77} + 16 q^{79} + 2 q^{81} + 6 q^{83} - 21 q^{85} - 6 q^{87} + 18 q^{89} + 2 q^{91} - 2 q^{93} + 3 q^{95} + 28 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.37228 −0.613703 −0.306851 0.951757i \(-0.599275\pi\)
−0.306851 + 0.951757i \(0.599275\pi\)
\(6\) 0 0
\(7\) 3.37228 1.27460 0.637301 0.770615i \(-0.280051\pi\)
0.637301 + 0.770615i \(0.280051\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.37228 0.413758 0.206879 0.978366i \(-0.433669\pi\)
0.206879 + 0.978366i \(0.433669\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −1.37228 −0.354322
\(16\) 0 0
\(17\) 1.37228 0.332827 0.166414 0.986056i \(-0.446781\pi\)
0.166414 + 0.986056i \(0.446781\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.37228 0.735892
\(22\) 0 0
\(23\) −8.74456 −1.82337 −0.911684 0.410893i \(-0.865217\pi\)
−0.911684 + 0.410893i \(0.865217\pi\)
\(24\) 0 0
\(25\) −3.11684 −0.623369
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.74456 0.509652 0.254826 0.966987i \(-0.417982\pi\)
0.254826 + 0.966987i \(0.417982\pi\)
\(30\) 0 0
\(31\) −6.74456 −1.21136 −0.605680 0.795709i \(-0.707099\pi\)
−0.605680 + 0.795709i \(0.707099\pi\)
\(32\) 0 0
\(33\) 1.37228 0.238884
\(34\) 0 0
\(35\) −4.62772 −0.782227
\(36\) 0 0
\(37\) 4.74456 0.780001 0.390001 0.920815i \(-0.372475\pi\)
0.390001 + 0.920815i \(0.372475\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 3.37228 0.514268 0.257134 0.966376i \(-0.417222\pi\)
0.257134 + 0.966376i \(0.417222\pi\)
\(44\) 0 0
\(45\) −1.37228 −0.204568
\(46\) 0 0
\(47\) −13.3723 −1.95055 −0.975274 0.221000i \(-0.929068\pi\)
−0.975274 + 0.221000i \(0.929068\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 0 0
\(51\) 1.37228 0.192158
\(52\) 0 0
\(53\) −2.74456 −0.376995 −0.188497 0.982074i \(-0.560362\pi\)
−0.188497 + 0.982074i \(0.560362\pi\)
\(54\) 0 0
\(55\) −1.88316 −0.253925
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.62772 −0.336445 −0.168222 0.985749i \(-0.553803\pi\)
−0.168222 + 0.985749i \(0.553803\pi\)
\(62\) 0 0
\(63\) 3.37228 0.424868
\(64\) 0 0
\(65\) −2.74456 −0.340421
\(66\) 0 0
\(67\) −9.48913 −1.15928 −0.579641 0.814872i \(-0.696807\pi\)
−0.579641 + 0.814872i \(0.696807\pi\)
\(68\) 0 0
\(69\) −8.74456 −1.05272
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −5.37228 −0.628778 −0.314389 0.949294i \(-0.601800\pi\)
−0.314389 + 0.949294i \(0.601800\pi\)
\(74\) 0 0
\(75\) −3.11684 −0.359902
\(76\) 0 0
\(77\) 4.62772 0.527377
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.74456 0.959840 0.479920 0.877312i \(-0.340666\pi\)
0.479920 + 0.877312i \(0.340666\pi\)
\(84\) 0 0
\(85\) −1.88316 −0.204257
\(86\) 0 0
\(87\) 2.74456 0.294248
\(88\) 0 0
\(89\) 14.7446 1.56292 0.781460 0.623955i \(-0.214475\pi\)
0.781460 + 0.623955i \(0.214475\pi\)
\(90\) 0 0
\(91\) 6.74456 0.707022
\(92\) 0 0
\(93\) −6.74456 −0.699379
\(94\) 0 0
\(95\) −1.37228 −0.140793
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 1.37228 0.137919
\(100\) 0 0
\(101\) 11.4891 1.14321 0.571605 0.820529i \(-0.306321\pi\)
0.571605 + 0.820529i \(0.306321\pi\)
\(102\) 0 0
\(103\) −1.25544 −0.123702 −0.0618510 0.998085i \(-0.519700\pi\)
−0.0618510 + 0.998085i \(0.519700\pi\)
\(104\) 0 0
\(105\) −4.62772 −0.451619
\(106\) 0 0
\(107\) 14.7446 1.42541 0.712705 0.701464i \(-0.247470\pi\)
0.712705 + 0.701464i \(0.247470\pi\)
\(108\) 0 0
\(109\) −18.2337 −1.74647 −0.873235 0.487299i \(-0.837982\pi\)
−0.873235 + 0.487299i \(0.837982\pi\)
\(110\) 0 0
\(111\) 4.74456 0.450334
\(112\) 0 0
\(113\) 14.7446 1.38705 0.693526 0.720432i \(-0.256056\pi\)
0.693526 + 0.720432i \(0.256056\pi\)
\(114\) 0 0
\(115\) 12.0000 1.11901
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 4.62772 0.424222
\(120\) 0 0
\(121\) −9.11684 −0.828804
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1386 0.996266
\(126\) 0 0
\(127\) −12.2337 −1.08556 −0.542782 0.839874i \(-0.682629\pi\)
−0.542782 + 0.839874i \(0.682629\pi\)
\(128\) 0 0
\(129\) 3.37228 0.296913
\(130\) 0 0
\(131\) −18.8614 −1.64793 −0.823964 0.566642i \(-0.808242\pi\)
−0.823964 + 0.566642i \(0.808242\pi\)
\(132\) 0 0
\(133\) 3.37228 0.292414
\(134\) 0 0
\(135\) −1.37228 −0.118107
\(136\) 0 0
\(137\) 1.37228 0.117242 0.0586210 0.998280i \(-0.481330\pi\)
0.0586210 + 0.998280i \(0.481330\pi\)
\(138\) 0 0
\(139\) 15.3723 1.30386 0.651930 0.758279i \(-0.273960\pi\)
0.651930 + 0.758279i \(0.273960\pi\)
\(140\) 0 0
\(141\) −13.3723 −1.12615
\(142\) 0 0
\(143\) 2.74456 0.229512
\(144\) 0 0
\(145\) −3.76631 −0.312775
\(146\) 0 0
\(147\) 4.37228 0.360620
\(148\) 0 0
\(149\) 1.37228 0.112422 0.0562108 0.998419i \(-0.482098\pi\)
0.0562108 + 0.998419i \(0.482098\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 1.37228 0.110942
\(154\) 0 0
\(155\) 9.25544 0.743415
\(156\) 0 0
\(157\) 7.48913 0.597697 0.298849 0.954301i \(-0.403397\pi\)
0.298849 + 0.954301i \(0.403397\pi\)
\(158\) 0 0
\(159\) −2.74456 −0.217658
\(160\) 0 0
\(161\) −29.4891 −2.32407
\(162\) 0 0
\(163\) −9.48913 −0.743246 −0.371623 0.928384i \(-0.621199\pi\)
−0.371623 + 0.928384i \(0.621199\pi\)
\(164\) 0 0
\(165\) −1.88316 −0.146603
\(166\) 0 0
\(167\) 20.2337 1.56573 0.782865 0.622192i \(-0.213758\pi\)
0.782865 + 0.622192i \(0.213758\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 9.25544 0.703678 0.351839 0.936061i \(-0.385556\pi\)
0.351839 + 0.936061i \(0.385556\pi\)
\(174\) 0 0
\(175\) −10.5109 −0.794547
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.7446 1.10206 0.551030 0.834485i \(-0.314235\pi\)
0.551030 + 0.834485i \(0.314235\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) −2.62772 −0.194247
\(184\) 0 0
\(185\) −6.51087 −0.478689
\(186\) 0 0
\(187\) 1.88316 0.137710
\(188\) 0 0
\(189\) 3.37228 0.245297
\(190\) 0 0
\(191\) −10.6277 −0.768995 −0.384497 0.923126i \(-0.625625\pi\)
−0.384497 + 0.923126i \(0.625625\pi\)
\(192\) 0 0
\(193\) 16.7446 1.20530 0.602650 0.798006i \(-0.294112\pi\)
0.602650 + 0.798006i \(0.294112\pi\)
\(194\) 0 0
\(195\) −2.74456 −0.196542
\(196\) 0 0
\(197\) −11.4891 −0.818566 −0.409283 0.912407i \(-0.634221\pi\)
−0.409283 + 0.912407i \(0.634221\pi\)
\(198\) 0 0
\(199\) −14.1168 −1.00072 −0.500358 0.865818i \(-0.666798\pi\)
−0.500358 + 0.865818i \(0.666798\pi\)
\(200\) 0 0
\(201\) −9.48913 −0.669311
\(202\) 0 0
\(203\) 9.25544 0.649604
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −8.74456 −0.607789
\(208\) 0 0
\(209\) 1.37228 0.0949227
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) −4.62772 −0.315608
\(216\) 0 0
\(217\) −22.7446 −1.54400
\(218\) 0 0
\(219\) −5.37228 −0.363025
\(220\) 0 0
\(221\) 2.74456 0.184619
\(222\) 0 0
\(223\) 13.4891 0.903299 0.451649 0.892196i \(-0.350836\pi\)
0.451649 + 0.892196i \(0.350836\pi\)
\(224\) 0 0
\(225\) −3.11684 −0.207790
\(226\) 0 0
\(227\) −8.23369 −0.546489 −0.273245 0.961945i \(-0.588097\pi\)
−0.273245 + 0.961945i \(0.588097\pi\)
\(228\) 0 0
\(229\) −5.37228 −0.355010 −0.177505 0.984120i \(-0.556803\pi\)
−0.177505 + 0.984120i \(0.556803\pi\)
\(230\) 0 0
\(231\) 4.62772 0.304482
\(232\) 0 0
\(233\) −1.37228 −0.0899011 −0.0449506 0.998989i \(-0.514313\pi\)
−0.0449506 + 0.998989i \(0.514313\pi\)
\(234\) 0 0
\(235\) 18.3505 1.19706
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 18.8614 1.22004 0.610021 0.792385i \(-0.291161\pi\)
0.610021 + 0.792385i \(0.291161\pi\)
\(240\) 0 0
\(241\) −0.744563 −0.0479615 −0.0239807 0.999712i \(-0.507634\pi\)
−0.0239807 + 0.999712i \(0.507634\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) 8.74456 0.554164
\(250\) 0 0
\(251\) −1.37228 −0.0866176 −0.0433088 0.999062i \(-0.513790\pi\)
−0.0433088 + 0.999062i \(0.513790\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 0 0
\(255\) −1.88316 −0.117928
\(256\) 0 0
\(257\) −17.4891 −1.09094 −0.545471 0.838130i \(-0.683649\pi\)
−0.545471 + 0.838130i \(0.683649\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 2.74456 0.169884
\(262\) 0 0
\(263\) −30.8614 −1.90300 −0.951498 0.307655i \(-0.900456\pi\)
−0.951498 + 0.307655i \(0.900456\pi\)
\(264\) 0 0
\(265\) 3.76631 0.231363
\(266\) 0 0
\(267\) 14.7446 0.902353
\(268\) 0 0
\(269\) 14.7446 0.898992 0.449496 0.893282i \(-0.351604\pi\)
0.449496 + 0.893282i \(0.351604\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) 6.74456 0.408200
\(274\) 0 0
\(275\) −4.27719 −0.257924
\(276\) 0 0
\(277\) 24.1168 1.44904 0.724520 0.689253i \(-0.242061\pi\)
0.724520 + 0.689253i \(0.242061\pi\)
\(278\) 0 0
\(279\) −6.74456 −0.403786
\(280\) 0 0
\(281\) −26.7446 −1.59545 −0.797723 0.603023i \(-0.793963\pi\)
−0.797723 + 0.603023i \(0.793963\pi\)
\(282\) 0 0
\(283\) −26.1168 −1.55249 −0.776243 0.630434i \(-0.782877\pi\)
−0.776243 + 0.630434i \(0.782877\pi\)
\(284\) 0 0
\(285\) −1.37228 −0.0812869
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.1168 −0.889226
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 0 0
\(293\) 29.4891 1.72277 0.861387 0.507950i \(-0.169597\pi\)
0.861387 + 0.507950i \(0.169597\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.37228 0.0796278
\(298\) 0 0
\(299\) −17.4891 −1.01142
\(300\) 0 0
\(301\) 11.3723 0.655487
\(302\) 0 0
\(303\) 11.4891 0.660033
\(304\) 0 0
\(305\) 3.60597 0.206477
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −1.25544 −0.0714193
\(310\) 0 0
\(311\) −18.8614 −1.06953 −0.534766 0.845000i \(-0.679600\pi\)
−0.534766 + 0.845000i \(0.679600\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) −4.62772 −0.260742
\(316\) 0 0
\(317\) 29.4891 1.65627 0.828137 0.560526i \(-0.189401\pi\)
0.828137 + 0.560526i \(0.189401\pi\)
\(318\) 0 0
\(319\) 3.76631 0.210873
\(320\) 0 0
\(321\) 14.7446 0.822961
\(322\) 0 0
\(323\) 1.37228 0.0763558
\(324\) 0 0
\(325\) −6.23369 −0.345783
\(326\) 0 0
\(327\) −18.2337 −1.00833
\(328\) 0 0
\(329\) −45.0951 −2.48617
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 4.74456 0.260000
\(334\) 0 0
\(335\) 13.0217 0.711454
\(336\) 0 0
\(337\) 19.4891 1.06164 0.530820 0.847484i \(-0.321884\pi\)
0.530820 + 0.847484i \(0.321884\pi\)
\(338\) 0 0
\(339\) 14.7446 0.800815
\(340\) 0 0
\(341\) −9.25544 −0.501210
\(342\) 0 0
\(343\) −8.86141 −0.478471
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) 16.1168 0.865198 0.432599 0.901587i \(-0.357597\pi\)
0.432599 + 0.901587i \(0.357597\pi\)
\(348\) 0 0
\(349\) 18.6277 0.997119 0.498559 0.866856i \(-0.333863\pi\)
0.498559 + 0.866856i \(0.333863\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) 16.4674 0.873998
\(356\) 0 0
\(357\) 4.62772 0.244925
\(358\) 0 0
\(359\) −5.13859 −0.271205 −0.135602 0.990763i \(-0.543297\pi\)
−0.135602 + 0.990763i \(0.543297\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −9.11684 −0.478510
\(364\) 0 0
\(365\) 7.37228 0.385883
\(366\) 0 0
\(367\) 25.4891 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −9.25544 −0.480518
\(372\) 0 0
\(373\) −30.2337 −1.56544 −0.782721 0.622373i \(-0.786169\pi\)
−0.782721 + 0.622373i \(0.786169\pi\)
\(374\) 0 0
\(375\) 11.1386 0.575194
\(376\) 0 0
\(377\) 5.48913 0.282704
\(378\) 0 0
\(379\) 21.7228 1.11583 0.557913 0.829899i \(-0.311602\pi\)
0.557913 + 0.829899i \(0.311602\pi\)
\(380\) 0 0
\(381\) −12.2337 −0.626751
\(382\) 0 0
\(383\) 8.23369 0.420722 0.210361 0.977624i \(-0.432536\pi\)
0.210361 + 0.977624i \(0.432536\pi\)
\(384\) 0 0
\(385\) −6.35053 −0.323653
\(386\) 0 0
\(387\) 3.37228 0.171423
\(388\) 0 0
\(389\) −25.3723 −1.28643 −0.643213 0.765687i \(-0.722399\pi\)
−0.643213 + 0.765687i \(0.722399\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) −18.8614 −0.951432
\(394\) 0 0
\(395\) −10.9783 −0.552376
\(396\) 0 0
\(397\) 29.6060 1.48588 0.742941 0.669357i \(-0.233431\pi\)
0.742941 + 0.669357i \(0.233431\pi\)
\(398\) 0 0
\(399\) 3.37228 0.168825
\(400\) 0 0
\(401\) 5.48913 0.274114 0.137057 0.990563i \(-0.456236\pi\)
0.137057 + 0.990563i \(0.456236\pi\)
\(402\) 0 0
\(403\) −13.4891 −0.671941
\(404\) 0 0
\(405\) −1.37228 −0.0681892
\(406\) 0 0
\(407\) 6.51087 0.322732
\(408\) 0 0
\(409\) 30.4674 1.50651 0.753257 0.657726i \(-0.228481\pi\)
0.753257 + 0.657726i \(0.228481\pi\)
\(410\) 0 0
\(411\) 1.37228 0.0676896
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 15.3723 0.752784
\(418\) 0 0
\(419\) −19.7228 −0.963522 −0.481761 0.876303i \(-0.660003\pi\)
−0.481761 + 0.876303i \(0.660003\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) −13.3723 −0.650183
\(424\) 0 0
\(425\) −4.27719 −0.207474
\(426\) 0 0
\(427\) −8.86141 −0.428834
\(428\) 0 0
\(429\) 2.74456 0.132509
\(430\) 0 0
\(431\) −2.74456 −0.132201 −0.0661005 0.997813i \(-0.521056\pi\)
−0.0661005 + 0.997813i \(0.521056\pi\)
\(432\) 0 0
\(433\) 7.48913 0.359904 0.179952 0.983675i \(-0.442406\pi\)
0.179952 + 0.983675i \(0.442406\pi\)
\(434\) 0 0
\(435\) −3.76631 −0.180581
\(436\) 0 0
\(437\) −8.74456 −0.418309
\(438\) 0 0
\(439\) 4.23369 0.202063 0.101031 0.994883i \(-0.467786\pi\)
0.101031 + 0.994883i \(0.467786\pi\)
\(440\) 0 0
\(441\) 4.37228 0.208204
\(442\) 0 0
\(443\) −13.3723 −0.635336 −0.317668 0.948202i \(-0.602900\pi\)
−0.317668 + 0.948202i \(0.602900\pi\)
\(444\) 0 0
\(445\) −20.2337 −0.959169
\(446\) 0 0
\(447\) 1.37228 0.0649067
\(448\) 0 0
\(449\) −22.9783 −1.08441 −0.542205 0.840246i \(-0.682411\pi\)
−0.542205 + 0.840246i \(0.682411\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 8.00000 0.375873
\(454\) 0 0
\(455\) −9.25544 −0.433902
\(456\) 0 0
\(457\) 14.8614 0.695187 0.347594 0.937645i \(-0.386999\pi\)
0.347594 + 0.937645i \(0.386999\pi\)
\(458\) 0 0
\(459\) 1.37228 0.0640526
\(460\) 0 0
\(461\) 12.3505 0.575222 0.287611 0.957747i \(-0.407139\pi\)
0.287611 + 0.957747i \(0.407139\pi\)
\(462\) 0 0
\(463\) −34.3505 −1.59640 −0.798202 0.602389i \(-0.794215\pi\)
−0.798202 + 0.602389i \(0.794215\pi\)
\(464\) 0 0
\(465\) 9.25544 0.429211
\(466\) 0 0
\(467\) 4.11684 0.190505 0.0952524 0.995453i \(-0.469634\pi\)
0.0952524 + 0.995453i \(0.469634\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) 7.48913 0.345081
\(472\) 0 0
\(473\) 4.62772 0.212783
\(474\) 0 0
\(475\) −3.11684 −0.143011
\(476\) 0 0
\(477\) −2.74456 −0.125665
\(478\) 0 0
\(479\) 38.2337 1.74694 0.873471 0.486876i \(-0.161864\pi\)
0.873471 + 0.486876i \(0.161864\pi\)
\(480\) 0 0
\(481\) 9.48913 0.432667
\(482\) 0 0
\(483\) −29.4891 −1.34180
\(484\) 0 0
\(485\) −19.2119 −0.872369
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −9.48913 −0.429113
\(490\) 0 0
\(491\) −27.2554 −1.23002 −0.615010 0.788519i \(-0.710848\pi\)
−0.615010 + 0.788519i \(0.710848\pi\)
\(492\) 0 0
\(493\) 3.76631 0.169626
\(494\) 0 0
\(495\) −1.88316 −0.0846416
\(496\) 0 0
\(497\) −40.4674 −1.81521
\(498\) 0 0
\(499\) −34.3505 −1.53774 −0.768870 0.639405i \(-0.779181\pi\)
−0.768870 + 0.639405i \(0.779181\pi\)
\(500\) 0 0
\(501\) 20.2337 0.903975
\(502\) 0 0
\(503\) 3.25544 0.145153 0.0725764 0.997363i \(-0.476878\pi\)
0.0725764 + 0.997363i \(0.476878\pi\)
\(504\) 0 0
\(505\) −15.7663 −0.701592
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −14.7446 −0.653541 −0.326771 0.945104i \(-0.605960\pi\)
−0.326771 + 0.945104i \(0.605960\pi\)
\(510\) 0 0
\(511\) −18.1168 −0.801442
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 1.72281 0.0759162
\(516\) 0 0
\(517\) −18.3505 −0.807055
\(518\) 0 0
\(519\) 9.25544 0.406269
\(520\) 0 0
\(521\) 5.48913 0.240483 0.120241 0.992745i \(-0.461633\pi\)
0.120241 + 0.992745i \(0.461633\pi\)
\(522\) 0 0
\(523\) −18.7446 −0.819642 −0.409821 0.912166i \(-0.634409\pi\)
−0.409821 + 0.912166i \(0.634409\pi\)
\(524\) 0 0
\(525\) −10.5109 −0.458732
\(526\) 0 0
\(527\) −9.25544 −0.403173
\(528\) 0 0
\(529\) 53.4674 2.32467
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −20.2337 −0.874779
\(536\) 0 0
\(537\) 14.7446 0.636275
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 26.8614 1.15486 0.577431 0.816439i \(-0.304055\pi\)
0.577431 + 0.816439i \(0.304055\pi\)
\(542\) 0 0
\(543\) −22.0000 −0.944110
\(544\) 0 0
\(545\) 25.0217 1.07181
\(546\) 0 0
\(547\) −18.7446 −0.801460 −0.400730 0.916196i \(-0.631243\pi\)
−0.400730 + 0.916196i \(0.631243\pi\)
\(548\) 0 0
\(549\) −2.62772 −0.112148
\(550\) 0 0
\(551\) 2.74456 0.116922
\(552\) 0 0
\(553\) 26.9783 1.14723
\(554\) 0 0
\(555\) −6.51087 −0.276371
\(556\) 0 0
\(557\) 6.86141 0.290727 0.145364 0.989378i \(-0.453565\pi\)
0.145364 + 0.989378i \(0.453565\pi\)
\(558\) 0 0
\(559\) 6.74456 0.285265
\(560\) 0 0
\(561\) 1.88316 0.0795069
\(562\) 0 0
\(563\) 46.9783 1.97990 0.989949 0.141428i \(-0.0451692\pi\)
0.989949 + 0.141428i \(0.0451692\pi\)
\(564\) 0 0
\(565\) −20.2337 −0.851238
\(566\) 0 0
\(567\) 3.37228 0.141623
\(568\) 0 0
\(569\) −21.2554 −0.891074 −0.445537 0.895263i \(-0.646987\pi\)
−0.445537 + 0.895263i \(0.646987\pi\)
\(570\) 0 0
\(571\) 18.9783 0.794215 0.397108 0.917772i \(-0.370014\pi\)
0.397108 + 0.917772i \(0.370014\pi\)
\(572\) 0 0
\(573\) −10.6277 −0.443979
\(574\) 0 0
\(575\) 27.2554 1.13663
\(576\) 0 0
\(577\) 3.88316 0.161658 0.0808290 0.996728i \(-0.474243\pi\)
0.0808290 + 0.996728i \(0.474243\pi\)
\(578\) 0 0
\(579\) 16.7446 0.695880
\(580\) 0 0
\(581\) 29.4891 1.22342
\(582\) 0 0
\(583\) −3.76631 −0.155985
\(584\) 0 0
\(585\) −2.74456 −0.113474
\(586\) 0 0
\(587\) 33.6060 1.38707 0.693533 0.720424i \(-0.256053\pi\)
0.693533 + 0.720424i \(0.256053\pi\)
\(588\) 0 0
\(589\) −6.74456 −0.277905
\(590\) 0 0
\(591\) −11.4891 −0.472599
\(592\) 0 0
\(593\) 28.9783 1.18999 0.594997 0.803728i \(-0.297153\pi\)
0.594997 + 0.803728i \(0.297153\pi\)
\(594\) 0 0
\(595\) −6.35053 −0.260346
\(596\) 0 0
\(597\) −14.1168 −0.577764
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −12.7446 −0.519862 −0.259931 0.965627i \(-0.583700\pi\)
−0.259931 + 0.965627i \(0.583700\pi\)
\(602\) 0 0
\(603\) −9.48913 −0.386427
\(604\) 0 0
\(605\) 12.5109 0.508639
\(606\) 0 0
\(607\) −30.7446 −1.24788 −0.623942 0.781471i \(-0.714470\pi\)
−0.623942 + 0.781471i \(0.714470\pi\)
\(608\) 0 0
\(609\) 9.25544 0.375049
\(610\) 0 0
\(611\) −26.7446 −1.08197
\(612\) 0 0
\(613\) −11.8832 −0.479956 −0.239978 0.970778i \(-0.577140\pi\)
−0.239978 + 0.970778i \(0.577140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −28.1168 −1.13194 −0.565971 0.824425i \(-0.691498\pi\)
−0.565971 + 0.824425i \(0.691498\pi\)
\(618\) 0 0
\(619\) −38.9783 −1.56667 −0.783334 0.621601i \(-0.786483\pi\)
−0.783334 + 0.621601i \(0.786483\pi\)
\(620\) 0 0
\(621\) −8.74456 −0.350907
\(622\) 0 0
\(623\) 49.7228 1.99210
\(624\) 0 0
\(625\) 0.298936 0.0119574
\(626\) 0 0
\(627\) 1.37228 0.0548036
\(628\) 0 0
\(629\) 6.51087 0.259606
\(630\) 0 0
\(631\) 32.8614 1.30819 0.654096 0.756412i \(-0.273049\pi\)
0.654096 + 0.756412i \(0.273049\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) 16.7881 0.666214
\(636\) 0 0
\(637\) 8.74456 0.346472
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −20.2337 −0.799183 −0.399591 0.916693i \(-0.630848\pi\)
−0.399591 + 0.916693i \(0.630848\pi\)
\(642\) 0 0
\(643\) 14.3505 0.565930 0.282965 0.959130i \(-0.408682\pi\)
0.282965 + 0.959130i \(0.408682\pi\)
\(644\) 0 0
\(645\) −4.62772 −0.182216
\(646\) 0 0
\(647\) 6.86141 0.269750 0.134875 0.990863i \(-0.456937\pi\)
0.134875 + 0.990863i \(0.456937\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −22.7446 −0.891430
\(652\) 0 0
\(653\) −9.60597 −0.375911 −0.187955 0.982178i \(-0.560186\pi\)
−0.187955 + 0.982178i \(0.560186\pi\)
\(654\) 0 0
\(655\) 25.8832 1.01134
\(656\) 0 0
\(657\) −5.37228 −0.209593
\(658\) 0 0
\(659\) 2.74456 0.106913 0.0534565 0.998570i \(-0.482976\pi\)
0.0534565 + 0.998570i \(0.482976\pi\)
\(660\) 0 0
\(661\) 10.2337 0.398044 0.199022 0.979995i \(-0.436223\pi\)
0.199022 + 0.979995i \(0.436223\pi\)
\(662\) 0 0
\(663\) 2.74456 0.106590
\(664\) 0 0
\(665\) −4.62772 −0.179455
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 13.4891 0.521520
\(670\) 0 0
\(671\) −3.60597 −0.139207
\(672\) 0 0
\(673\) −1.76631 −0.0680863 −0.0340432 0.999420i \(-0.510838\pi\)
−0.0340432 + 0.999420i \(0.510838\pi\)
\(674\) 0 0
\(675\) −3.11684 −0.119967
\(676\) 0 0
\(677\) −2.74456 −0.105482 −0.0527411 0.998608i \(-0.516796\pi\)
−0.0527411 + 0.998608i \(0.516796\pi\)
\(678\) 0 0
\(679\) 47.2119 1.81183
\(680\) 0 0
\(681\) −8.23369 −0.315516
\(682\) 0 0
\(683\) 9.25544 0.354149 0.177075 0.984197i \(-0.443337\pi\)
0.177075 + 0.984197i \(0.443337\pi\)
\(684\) 0 0
\(685\) −1.88316 −0.0719517
\(686\) 0 0
\(687\) −5.37228 −0.204965
\(688\) 0 0
\(689\) −5.48913 −0.209119
\(690\) 0 0
\(691\) −22.3505 −0.850254 −0.425127 0.905134i \(-0.639771\pi\)
−0.425127 + 0.905134i \(0.639771\pi\)
\(692\) 0 0
\(693\) 4.62772 0.175792
\(694\) 0 0
\(695\) −21.0951 −0.800183
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −1.37228 −0.0519044
\(700\) 0 0
\(701\) 35.4891 1.34041 0.670203 0.742178i \(-0.266207\pi\)
0.670203 + 0.742178i \(0.266207\pi\)
\(702\) 0 0
\(703\) 4.74456 0.178945
\(704\) 0 0
\(705\) 18.3505 0.691121
\(706\) 0 0
\(707\) 38.7446 1.45714
\(708\) 0 0
\(709\) 18.4674 0.693557 0.346778 0.937947i \(-0.387276\pi\)
0.346778 + 0.937947i \(0.387276\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 58.9783 2.20875
\(714\) 0 0
\(715\) −3.76631 −0.140852
\(716\) 0 0
\(717\) 18.8614 0.704392
\(718\) 0 0
\(719\) −28.1168 −1.04858 −0.524291 0.851539i \(-0.675669\pi\)
−0.524291 + 0.851539i \(0.675669\pi\)
\(720\) 0 0
\(721\) −4.23369 −0.157671
\(722\) 0 0
\(723\) −0.744563 −0.0276906
\(724\) 0 0
\(725\) −8.55437 −0.317701
\(726\) 0 0
\(727\) 46.5842 1.72771 0.863857 0.503738i \(-0.168042\pi\)
0.863857 + 0.503738i \(0.168042\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.62772 0.171162
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) −13.0217 −0.479662
\(738\) 0 0
\(739\) 4.39403 0.161637 0.0808185 0.996729i \(-0.474247\pi\)
0.0808185 + 0.996729i \(0.474247\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) −28.4674 −1.04437 −0.522183 0.852833i \(-0.674882\pi\)
−0.522183 + 0.852833i \(0.674882\pi\)
\(744\) 0 0
\(745\) −1.88316 −0.0689935
\(746\) 0 0
\(747\) 8.74456 0.319947
\(748\) 0 0
\(749\) 49.7228 1.81683
\(750\) 0 0
\(751\) 42.9783 1.56830 0.784149 0.620572i \(-0.213100\pi\)
0.784149 + 0.620572i \(0.213100\pi\)
\(752\) 0 0
\(753\) −1.37228 −0.0500087
\(754\) 0 0
\(755\) −10.9783 −0.399539
\(756\) 0 0
\(757\) −31.0951 −1.13017 −0.565085 0.825033i \(-0.691157\pi\)
−0.565085 + 0.825033i \(0.691157\pi\)
\(758\) 0 0
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −44.5842 −1.61618 −0.808088 0.589061i \(-0.799498\pi\)
−0.808088 + 0.589061i \(0.799498\pi\)
\(762\) 0 0
\(763\) −61.4891 −2.22606
\(764\) 0 0
\(765\) −1.88316 −0.0680856
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 3.88316 0.140030 0.0700151 0.997546i \(-0.477695\pi\)
0.0700151 + 0.997546i \(0.477695\pi\)
\(770\) 0 0
\(771\) −17.4891 −0.629855
\(772\) 0 0
\(773\) 18.5109 0.665790 0.332895 0.942964i \(-0.391975\pi\)
0.332895 + 0.942964i \(0.391975\pi\)
\(774\) 0 0
\(775\) 21.0217 0.755124
\(776\) 0 0
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −16.4674 −0.589249
\(782\) 0 0
\(783\) 2.74456 0.0980827
\(784\) 0 0
\(785\) −10.2772 −0.366809
\(786\) 0 0
\(787\) 1.48913 0.0530816 0.0265408 0.999648i \(-0.491551\pi\)
0.0265408 + 0.999648i \(0.491551\pi\)
\(788\) 0 0
\(789\) −30.8614 −1.09870
\(790\) 0 0
\(791\) 49.7228 1.76794
\(792\) 0 0
\(793\) −5.25544 −0.186626
\(794\) 0 0
\(795\) 3.76631 0.133577
\(796\) 0 0
\(797\) −29.4891 −1.04456 −0.522279 0.852775i \(-0.674918\pi\)
−0.522279 + 0.852775i \(0.674918\pi\)
\(798\) 0 0
\(799\) −18.3505 −0.649195
\(800\) 0 0
\(801\) 14.7446 0.520974
\(802\) 0 0
\(803\) −7.37228 −0.260162
\(804\) 0 0
\(805\) 40.4674 1.42629
\(806\) 0 0
\(807\) 14.7446 0.519033
\(808\) 0 0
\(809\) −25.3723 −0.892042 −0.446021 0.895023i \(-0.647159\pi\)
−0.446021 + 0.895023i \(0.647159\pi\)
\(810\) 0 0
\(811\) −0.233688 −0.00820589 −0.00410295 0.999992i \(-0.501306\pi\)
−0.00410295 + 0.999992i \(0.501306\pi\)
\(812\) 0 0
\(813\) −4.00000 −0.140286
\(814\) 0 0
\(815\) 13.0217 0.456132
\(816\) 0 0
\(817\) 3.37228 0.117981
\(818\) 0 0
\(819\) 6.74456 0.235674
\(820\) 0 0
\(821\) −44.5842 −1.55600 −0.778000 0.628264i \(-0.783766\pi\)
−0.778000 + 0.628264i \(0.783766\pi\)
\(822\) 0 0
\(823\) 14.3505 0.500228 0.250114 0.968216i \(-0.419532\pi\)
0.250114 + 0.968216i \(0.419532\pi\)
\(824\) 0 0
\(825\) −4.27719 −0.148913
\(826\) 0 0
\(827\) 5.48913 0.190876 0.0954378 0.995435i \(-0.469575\pi\)
0.0954378 + 0.995435i \(0.469575\pi\)
\(828\) 0 0
\(829\) −42.2337 −1.46684 −0.733418 0.679778i \(-0.762076\pi\)
−0.733418 + 0.679778i \(0.762076\pi\)
\(830\) 0 0
\(831\) 24.1168 0.836604
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −27.7663 −0.960893
\(836\) 0 0
\(837\) −6.74456 −0.233126
\(838\) 0 0
\(839\) −2.74456 −0.0947528 −0.0473764 0.998877i \(-0.515086\pi\)
−0.0473764 + 0.998877i \(0.515086\pi\)
\(840\) 0 0
\(841\) −21.4674 −0.740254
\(842\) 0 0
\(843\) −26.7446 −0.921132
\(844\) 0 0
\(845\) 12.3505 0.424871
\(846\) 0 0
\(847\) −30.7446 −1.05640
\(848\) 0 0
\(849\) −26.1168 −0.896328
\(850\) 0 0
\(851\) −41.4891 −1.42223
\(852\) 0 0
\(853\) 8.51087 0.291407 0.145703 0.989328i \(-0.453455\pi\)
0.145703 + 0.989328i \(0.453455\pi\)
\(854\) 0 0
\(855\) −1.37228 −0.0469310
\(856\) 0 0
\(857\) −34.9783 −1.19483 −0.597417 0.801931i \(-0.703806\pi\)
−0.597417 + 0.801931i \(0.703806\pi\)
\(858\) 0 0
\(859\) 4.39403 0.149922 0.0749612 0.997186i \(-0.476117\pi\)
0.0749612 + 0.997186i \(0.476117\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 46.9783 1.59916 0.799579 0.600561i \(-0.205056\pi\)
0.799579 + 0.600561i \(0.205056\pi\)
\(864\) 0 0
\(865\) −12.7011 −0.431849
\(866\) 0 0
\(867\) −15.1168 −0.513395
\(868\) 0 0
\(869\) 10.9783 0.372412
\(870\) 0 0
\(871\) −18.9783 −0.643053
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 37.5625 1.26984
\(876\) 0 0
\(877\) −39.4891 −1.33345 −0.666727 0.745302i \(-0.732305\pi\)
−0.666727 + 0.745302i \(0.732305\pi\)
\(878\) 0 0
\(879\) 29.4891 0.994644
\(880\) 0 0
\(881\) 17.8397 0.601033 0.300517 0.953777i \(-0.402841\pi\)
0.300517 + 0.953777i \(0.402841\pi\)
\(882\) 0 0
\(883\) 50.3505 1.69443 0.847215 0.531250i \(-0.178277\pi\)
0.847215 + 0.531250i \(0.178277\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18.5109 −0.621534 −0.310767 0.950486i \(-0.600586\pi\)
−0.310767 + 0.950486i \(0.600586\pi\)
\(888\) 0 0
\(889\) −41.2554 −1.38366
\(890\) 0 0
\(891\) 1.37228 0.0459732
\(892\) 0 0
\(893\) −13.3723 −0.447486
\(894\) 0 0
\(895\) −20.2337 −0.676338
\(896\) 0 0
\(897\) −17.4891 −0.583945
\(898\) 0 0
\(899\) −18.5109 −0.617372
\(900\) 0 0
\(901\) −3.76631 −0.125474
\(902\) 0 0
\(903\) 11.3723 0.378446
\(904\) 0 0
\(905\) 30.1902 1.00356
\(906\) 0 0
\(907\) 40.2337 1.33594 0.667969 0.744189i \(-0.267164\pi\)
0.667969 + 0.744189i \(0.267164\pi\)
\(908\) 0 0
\(909\) 11.4891 0.381070
\(910\) 0 0
\(911\) −3.76631 −0.124783 −0.0623917 0.998052i \(-0.519873\pi\)
−0.0623917 + 0.998052i \(0.519873\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 3.60597 0.119210
\(916\) 0 0
\(917\) −63.6060 −2.10045
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −14.7881 −0.486228
\(926\) 0 0
\(927\) −1.25544 −0.0412340
\(928\) 0 0
\(929\) −11.4891 −0.376946 −0.188473 0.982078i \(-0.560354\pi\)
−0.188473 + 0.982078i \(0.560354\pi\)
\(930\) 0 0
\(931\) 4.37228 0.143296
\(932\) 0 0
\(933\) −18.8614 −0.617495
\(934\) 0 0
\(935\) −2.58422 −0.0845130
\(936\) 0 0
\(937\) −57.8397 −1.88954 −0.944770 0.327735i \(-0.893715\pi\)
−0.944770 + 0.327735i \(0.893715\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) 38.7446 1.26304 0.631518 0.775361i \(-0.282432\pi\)
0.631518 + 0.775361i \(0.282432\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −4.62772 −0.150540
\(946\) 0 0
\(947\) −43.7228 −1.42080 −0.710400 0.703798i \(-0.751486\pi\)
−0.710400 + 0.703798i \(0.751486\pi\)
\(948\) 0 0
\(949\) −10.7446 −0.348783
\(950\) 0 0
\(951\) 29.4891 0.956250
\(952\) 0 0
\(953\) −52.4674 −1.69959 −0.849793 0.527117i \(-0.823273\pi\)
−0.849793 + 0.527117i \(0.823273\pi\)
\(954\) 0 0
\(955\) 14.5842 0.471934
\(956\) 0 0
\(957\) 3.76631 0.121748
\(958\) 0 0
\(959\) 4.62772 0.149437
\(960\) 0 0
\(961\) 14.4891 0.467391
\(962\) 0 0
\(963\) 14.7446 0.475137
\(964\) 0 0
\(965\) −22.9783 −0.739696
\(966\) 0 0
\(967\) −5.02175 −0.161489 −0.0807443 0.996735i \(-0.525730\pi\)
−0.0807443 + 0.996735i \(0.525730\pi\)
\(968\) 0 0
\(969\) 1.37228 0.0440840
\(970\) 0 0
\(971\) −46.9783 −1.50760 −0.753802 0.657102i \(-0.771782\pi\)
−0.753802 + 0.657102i \(0.771782\pi\)
\(972\) 0 0
\(973\) 51.8397 1.66190
\(974\) 0 0
\(975\) −6.23369 −0.199638
\(976\) 0 0
\(977\) −49.7228 −1.59077 −0.795387 0.606102i \(-0.792732\pi\)
−0.795387 + 0.606102i \(0.792732\pi\)
\(978\) 0 0
\(979\) 20.2337 0.646671
\(980\) 0 0
\(981\) −18.2337 −0.582157
\(982\) 0 0
\(983\) −29.4891 −0.940557 −0.470279 0.882518i \(-0.655847\pi\)
−0.470279 + 0.882518i \(0.655847\pi\)
\(984\) 0 0
\(985\) 15.7663 0.502356
\(986\) 0 0
\(987\) −45.0951 −1.43539
\(988\) 0 0
\(989\) −29.4891 −0.937700
\(990\) 0 0
\(991\) 53.9565 1.71398 0.856992 0.515329i \(-0.172330\pi\)
0.856992 + 0.515329i \(0.172330\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 19.3723 0.614143
\(996\) 0 0
\(997\) 36.1168 1.14383 0.571916 0.820312i \(-0.306200\pi\)
0.571916 + 0.820312i \(0.306200\pi\)
\(998\) 0 0
\(999\) 4.74456 0.150111
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 228.2.a.c.1.1 2
3.2 odd 2 684.2.a.d.1.2 2
4.3 odd 2 912.2.a.n.1.1 2
5.2 odd 4 5700.2.f.m.3649.2 4
5.3 odd 4 5700.2.f.m.3649.3 4
5.4 even 2 5700.2.a.t.1.1 2
8.3 odd 2 3648.2.a.bq.1.2 2
8.5 even 2 3648.2.a.bk.1.2 2
12.11 even 2 2736.2.a.y.1.2 2
19.18 odd 2 4332.2.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
228.2.a.c.1.1 2 1.1 even 1 trivial
684.2.a.d.1.2 2 3.2 odd 2
912.2.a.n.1.1 2 4.3 odd 2
2736.2.a.y.1.2 2 12.11 even 2
3648.2.a.bk.1.2 2 8.5 even 2
3648.2.a.bq.1.2 2 8.3 odd 2
4332.2.a.i.1.1 2 19.18 odd 2
5700.2.a.t.1.1 2 5.4 even 2
5700.2.f.m.3649.2 4 5.2 odd 4
5700.2.f.m.3649.3 4 5.3 odd 4