Properties

Label 1728.2.d.e.865.2
Level $1728$
Weight $2$
Character 1728.865
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,2,Mod(865,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 865.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.865
Dual form 1728.2.d.e.865.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.46410i q^{5} +1.73205 q^{7} +O(q^{10})\) \(q-3.46410i q^{5} +1.73205 q^{7} +6.00000i q^{11} +5.19615i q^{13} -6.00000 q^{17} +5.00000i q^{19} -3.46410 q^{23} -7.00000 q^{25} +6.92820i q^{29} +3.46410 q^{31} -6.00000i q^{35} +1.73205i q^{37} +4.00000i q^{43} -3.46410 q^{47} -4.00000 q^{49} -6.92820i q^{53} +20.7846 q^{55} -6.00000i q^{59} +12.1244i q^{61} +18.0000 q^{65} -5.00000i q^{67} +13.8564 q^{71} -7.00000 q^{73} +10.3923i q^{77} +5.19615 q^{79} +20.7846i q^{85} +6.00000 q^{89} +9.00000i q^{91} +17.3205 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 24 q^{17} - 28 q^{25} - 16 q^{49} + 72 q^{65} - 28 q^{73} + 24 q^{89} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 3.46410i − 1.54919i −0.632456 0.774597i \(-0.717953\pi\)
0.632456 0.774597i \(-0.282047\pi\)
\(6\) 0 0
\(7\) 1.73205 0.654654 0.327327 0.944911i \(-0.393852\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000i 1.80907i 0.426401 + 0.904534i \(0.359781\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) 5.19615i 1.44115i 0.693375 + 0.720577i \(0.256123\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 5.00000i 1.14708i 0.819178 + 0.573539i \(0.194430\pi\)
−0.819178 + 0.573539i \(0.805570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) 3.46410 0.622171 0.311086 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 6.00000i − 1.01419i
\(36\) 0 0
\(37\) 1.73205i 0.284747i 0.989813 + 0.142374i \(0.0454735\pi\)
−0.989813 + 0.142374i \(0.954527\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.46410 −0.505291 −0.252646 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.92820i − 0.951662i −0.879537 0.475831i \(-0.842147\pi\)
0.879537 0.475831i \(-0.157853\pi\)
\(54\) 0 0
\(55\) 20.7846 2.80260
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 6.00000i − 0.781133i −0.920575 0.390567i \(-0.872279\pi\)
0.920575 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) 12.1244i 1.55236i 0.630509 + 0.776182i \(0.282846\pi\)
−0.630509 + 0.776182i \(0.717154\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 18.0000 2.23263
\(66\) 0 0
\(67\) − 5.00000i − 0.610847i −0.952217 0.305424i \(-0.901202\pi\)
0.952217 0.305424i \(-0.0987981\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3923i 1.18431i
\(78\) 0 0
\(79\) 5.19615 0.584613 0.292306 0.956325i \(-0.405577\pi\)
0.292306 + 0.956325i \(0.405577\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 20.7846i 2.25441i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 9.00000i 0.943456i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 17.3205 1.77705
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 6.92820i − 0.689382i −0.938716 0.344691i \(-0.887984\pi\)
0.938716 0.344691i \(-0.112016\pi\)
\(102\) 0 0
\(103\) −8.66025 −0.853320 −0.426660 0.904412i \(-0.640310\pi\)
−0.426660 + 0.904412i \(0.640310\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) − 13.8564i − 1.32720i −0.748086 0.663602i \(-0.769027\pi\)
0.748086 0.663602i \(-0.230973\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 12.0000i 1.11901i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −10.3923 −0.952661
\(120\) 0 0
\(121\) −25.0000 −2.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 17.3205 1.53695 0.768473 0.639882i \(-0.221017\pi\)
0.768473 + 0.639882i \(0.221017\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) 0 0
\(133\) 8.66025i 0.750939i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 11.0000i 0.933008i 0.884519 + 0.466504i \(0.154487\pi\)
−0.884519 + 0.466504i \(0.845513\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −31.1769 −2.60714
\(144\) 0 0
\(145\) 24.0000 1.99309
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 1.73205 0.140952 0.0704761 0.997513i \(-0.477548\pi\)
0.0704761 + 0.997513i \(0.477548\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 12.0000i − 0.963863i
\(156\) 0 0
\(157\) 13.8564i 1.10586i 0.833227 + 0.552931i \(0.186491\pi\)
−0.833227 + 0.552931i \(0.813509\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) 11.0000i 0.861586i 0.902451 + 0.430793i \(0.141766\pi\)
−0.902451 + 0.430793i \(0.858234\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.3205 −1.34030 −0.670151 0.742225i \(-0.733770\pi\)
−0.670151 + 0.742225i \(0.733770\pi\)
\(168\) 0 0
\(169\) −14.0000 −1.07692
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 13.8564i − 1.05348i −0.850026 0.526742i \(-0.823414\pi\)
0.850026 0.526742i \(-0.176586\pi\)
\(174\) 0 0
\(175\) −12.1244 −0.916515
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.0000i 1.79384i 0.442189 + 0.896922i \(0.354202\pi\)
−0.442189 + 0.896922i \(0.645798\pi\)
\(180\) 0 0
\(181\) − 12.1244i − 0.901196i −0.892727 0.450598i \(-0.851211\pi\)
0.892727 0.450598i \(-0.148789\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) − 36.0000i − 2.63258i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.2487 1.75458 0.877288 0.479965i \(-0.159351\pi\)
0.877288 + 0.479965i \(0.159351\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.3923i 0.740421i 0.928948 + 0.370211i \(0.120714\pi\)
−0.928948 + 0.370211i \(0.879286\pi\)
\(198\) 0 0
\(199\) 5.19615 0.368345 0.184173 0.982894i \(-0.441039\pi\)
0.184173 + 0.982894i \(0.441039\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.0000i 0.842235i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −30.0000 −2.07514
\(210\) 0 0
\(211\) − 23.0000i − 1.58339i −0.610920 0.791693i \(-0.709200\pi\)
0.610920 0.791693i \(-0.290800\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 13.8564 0.944999
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 31.1769i − 2.09719i
\(222\) 0 0
\(223\) −3.46410 −0.231973 −0.115987 0.993251i \(-0.537003\pi\)
−0.115987 + 0.993251i \(0.537003\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) − 6.92820i − 0.457829i −0.973447 0.228914i \(-0.926482\pi\)
0.973447 0.228914i \(-0.0735176\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) 12.0000i 0.782794i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.92820 −0.448148 −0.224074 0.974572i \(-0.571936\pi\)
−0.224074 + 0.974572i \(0.571936\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 13.8564i 0.885253i
\(246\) 0 0
\(247\) −25.9808 −1.65312
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000i 1.51487i 0.652913 + 0.757433i \(0.273547\pi\)
−0.652913 + 0.757433i \(0.726453\pi\)
\(252\) 0 0
\(253\) − 20.7846i − 1.30672i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 3.00000i 0.186411i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 20.7846 1.28163 0.640817 0.767694i \(-0.278596\pi\)
0.640817 + 0.767694i \(0.278596\pi\)
\(264\) 0 0
\(265\) −24.0000 −1.47431
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 10.3923i − 0.633630i −0.948487 0.316815i \(-0.897387\pi\)
0.948487 0.316815i \(-0.102613\pi\)
\(270\) 0 0
\(271\) 22.5167 1.36779 0.683895 0.729581i \(-0.260285\pi\)
0.683895 + 0.729581i \(0.260285\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 42.0000i − 2.53270i
\(276\) 0 0
\(277\) − 27.7128i − 1.66510i −0.553949 0.832551i \(-0.686880\pi\)
0.553949 0.832551i \(-0.313120\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.2487i 1.41662i 0.705899 + 0.708312i \(0.250543\pi\)
−0.705899 + 0.708312i \(0.749457\pi\)
\(294\) 0 0
\(295\) −20.7846 −1.21013
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 18.0000i − 1.04097i
\(300\) 0 0
\(301\) 6.92820i 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 42.0000 2.40491
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −17.3205 −0.982156 −0.491078 0.871116i \(-0.663397\pi\)
−0.491078 + 0.871116i \(0.663397\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −41.5692 −2.32743
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 30.0000i − 1.66924i
\(324\) 0 0
\(325\) − 36.3731i − 2.01761i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 11.0000i 0.604615i 0.953211 + 0.302307i \(0.0977569\pi\)
−0.953211 + 0.302307i \(0.902243\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −17.3205 −0.946320
\(336\) 0 0
\(337\) 31.0000 1.68868 0.844339 0.535810i \(-0.179994\pi\)
0.844339 + 0.535810i \(0.179994\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 20.7846i 1.12555i
\(342\) 0 0
\(343\) −19.0526 −1.02874
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) 1.73205i 0.0927146i 0.998925 + 0.0463573i \(0.0147613\pi\)
−0.998925 + 0.0463573i \(0.985239\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.0000 −0.638696 −0.319348 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 0 0
\(355\) − 48.0000i − 2.54758i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.46410 0.182828 0.0914141 0.995813i \(-0.470861\pi\)
0.0914141 + 0.995813i \(0.470861\pi\)
\(360\) 0 0
\(361\) −6.00000 −0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 24.2487i 1.26924i
\(366\) 0 0
\(367\) −29.4449 −1.53701 −0.768505 0.639844i \(-0.778999\pi\)
−0.768505 + 0.639844i \(0.778999\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 12.0000i − 0.623009i
\(372\) 0 0
\(373\) 15.5885i 0.807140i 0.914949 + 0.403570i \(0.132231\pi\)
−0.914949 + 0.403570i \(0.867769\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) − 29.0000i − 1.48963i −0.667271 0.744815i \(-0.732538\pi\)
0.667271 0.744815i \(-0.267462\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 34.6410 1.77007 0.885037 0.465521i \(-0.154133\pi\)
0.885037 + 0.465521i \(0.154133\pi\)
\(384\) 0 0
\(385\) 36.0000 1.83473
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.46410i 0.175637i 0.996136 + 0.0878185i \(0.0279895\pi\)
−0.996136 + 0.0878185i \(0.972010\pi\)
\(390\) 0 0
\(391\) 20.7846 1.05112
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 18.0000i − 0.905678i
\(396\) 0 0
\(397\) − 27.7128i − 1.39087i −0.718591 0.695433i \(-0.755213\pi\)
0.718591 0.695433i \(-0.244787\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 18.0000i 0.896644i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.3923 −0.515127
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 10.3923i − 0.511372i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 30.0000i 1.46560i 0.680446 + 0.732798i \(0.261786\pi\)
−0.680446 + 0.732798i \(0.738214\pi\)
\(420\) 0 0
\(421\) 12.1244i 0.590905i 0.955357 + 0.295452i \(0.0954704\pi\)
−0.955357 + 0.295452i \(0.904530\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 42.0000 2.03730
\(426\) 0 0
\(427\) 21.0000i 1.01626i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.46410 0.166860 0.0834300 0.996514i \(-0.473413\pi\)
0.0834300 + 0.996514i \(0.473413\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 17.3205i − 0.828552i
\(438\) 0 0
\(439\) −10.3923 −0.495998 −0.247999 0.968760i \(-0.579773\pi\)
−0.247999 + 0.968760i \(0.579773\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 36.0000i 1.71041i 0.518289 + 0.855206i \(0.326569\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(444\) 0 0
\(445\) − 20.7846i − 0.985285i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 31.1769 1.46160
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 31.1769i − 1.45205i −0.687666 0.726027i \(-0.741365\pi\)
0.687666 0.726027i \(-0.258635\pi\)
\(462\) 0 0
\(463\) −5.19615 −0.241486 −0.120743 0.992684i \(-0.538528\pi\)
−0.120743 + 0.992684i \(0.538528\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.0000i 0.832941i 0.909149 + 0.416470i \(0.136733\pi\)
−0.909149 + 0.416470i \(0.863267\pi\)
\(468\) 0 0
\(469\) − 8.66025i − 0.399893i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) − 35.0000i − 1.60591i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20.7846 −0.949673 −0.474837 0.880074i \(-0.657493\pi\)
−0.474837 + 0.880074i \(0.657493\pi\)
\(480\) 0 0
\(481\) −9.00000 −0.410365
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 24.2487i − 1.10108i
\(486\) 0 0
\(487\) −5.19615 −0.235460 −0.117730 0.993046i \(-0.537562\pi\)
−0.117730 + 0.993046i \(0.537562\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 6.00000i − 0.270776i −0.990793 0.135388i \(-0.956772\pi\)
0.990793 0.135388i \(-0.0432281\pi\)
\(492\) 0 0
\(493\) − 41.5692i − 1.87218i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) − 8.00000i − 0.358129i −0.983837 0.179065i \(-0.942693\pi\)
0.983837 0.179065i \(-0.0573071\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −24.2487 −1.08120 −0.540598 0.841281i \(-0.681802\pi\)
−0.540598 + 0.841281i \(0.681802\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 17.3205i − 0.767718i −0.923392 0.383859i \(-0.874595\pi\)
0.923392 0.383859i \(-0.125405\pi\)
\(510\) 0 0
\(511\) −12.1244 −0.536350
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 30.0000i 1.32196i
\(516\) 0 0
\(517\) − 20.7846i − 0.914106i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 7.00000i 0.306089i 0.988219 + 0.153044i \(0.0489077\pi\)
−0.988219 + 0.153044i \(0.951092\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20.7846 −0.905392
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −20.7846 −0.898597
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 24.0000i − 1.03375i
\(540\) 0 0
\(541\) 12.1244i 0.521267i 0.965438 + 0.260633i \(0.0839314\pi\)
−0.965438 + 0.260633i \(0.916069\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −48.0000 −2.05609
\(546\) 0 0
\(547\) − 1.00000i − 0.0427569i −0.999771 0.0213785i \(-0.993195\pi\)
0.999771 0.0213785i \(-0.00680549\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −34.6410 −1.47576
\(552\) 0 0
\(553\) 9.00000 0.382719
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 10.3923i − 0.440336i −0.975462 0.220168i \(-0.929339\pi\)
0.975462 0.220168i \(-0.0706606\pi\)
\(558\) 0 0
\(559\) −20.7846 −0.879095
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) 0 0
\(565\) 20.7846i 0.874415i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 5.00000i 0.209243i 0.994512 + 0.104622i \(0.0333632\pi\)
−0.994512 + 0.104622i \(0.966637\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 24.2487 1.01124
\(576\) 0 0
\(577\) −11.0000 −0.457936 −0.228968 0.973434i \(-0.573535\pi\)
−0.228968 + 0.973434i \(0.573535\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 41.5692 1.72162
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) 0 0
\(589\) 17.3205i 0.713679i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 0 0
\(595\) 36.0000i 1.47586i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.92820 0.283079 0.141539 0.989933i \(-0.454795\pi\)
0.141539 + 0.989933i \(0.454795\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 86.6025i 3.52089i
\(606\) 0 0
\(607\) 1.73205 0.0703018 0.0351509 0.999382i \(-0.488809\pi\)
0.0351509 + 0.999382i \(0.488809\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 18.0000i − 0.728202i
\(612\) 0 0
\(613\) 32.9090i 1.32918i 0.747208 + 0.664590i \(0.231394\pi\)
−0.747208 + 0.664590i \(0.768606\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) − 31.0000i − 1.24600i −0.782224 0.622998i \(-0.785915\pi\)
0.782224 0.622998i \(-0.214085\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.3923 0.416359
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 10.3923i − 0.414368i
\(630\) 0 0
\(631\) 29.4449 1.17218 0.586091 0.810245i \(-0.300666\pi\)
0.586091 + 0.810245i \(0.300666\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 60.0000i − 2.38103i
\(636\) 0 0
\(637\) − 20.7846i − 0.823516i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) − 28.0000i − 1.10421i −0.833774 0.552106i \(-0.813824\pi\)
0.833774 0.552106i \(-0.186176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.8564 0.544752 0.272376 0.962191i \(-0.412191\pi\)
0.272376 + 0.962191i \(0.412191\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 41.5692i 1.62673i 0.581754 + 0.813365i \(0.302367\pi\)
−0.581754 + 0.813365i \(0.697633\pi\)
\(654\) 0 0
\(655\) 41.5692 1.62424
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.0000i 0.467454i 0.972302 + 0.233727i \(0.0750921\pi\)
−0.972302 + 0.233727i \(0.924908\pi\)
\(660\) 0 0
\(661\) − 8.66025i − 0.336845i −0.985715 0.168422i \(-0.946133\pi\)
0.985715 0.168422i \(-0.0538673\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 30.0000 1.16335
\(666\) 0 0
\(667\) − 24.0000i − 0.929284i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −72.7461 −2.80833
\(672\) 0 0
\(673\) 25.0000 0.963679 0.481840 0.876259i \(-0.339969\pi\)
0.481840 + 0.876259i \(0.339969\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 24.2487i − 0.931954i −0.884797 0.465977i \(-0.845703\pi\)
0.884797 0.465977i \(-0.154297\pi\)
\(678\) 0 0
\(679\) 12.1244 0.465290
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 62.3538i 2.38242i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 44.0000i 1.67384i 0.547326 + 0.836919i \(0.315646\pi\)
−0.547326 + 0.836919i \(0.684354\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 38.1051 1.44541
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 31.1769i 1.17754i 0.808302 + 0.588768i \(0.200387\pi\)
−0.808302 + 0.588768i \(0.799613\pi\)
\(702\) 0 0
\(703\) −8.66025 −0.326628
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 12.0000i − 0.451306i
\(708\) 0 0
\(709\) 5.19615i 0.195146i 0.995228 + 0.0975728i \(0.0311079\pi\)
−0.995228 + 0.0975728i \(0.968892\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) 108.000i 4.03897i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 41.5692 1.55027 0.775135 0.631795i \(-0.217682\pi\)
0.775135 + 0.631795i \(0.217682\pi\)
\(720\) 0 0
\(721\) −15.0000 −0.558629
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 48.4974i − 1.80115i
\(726\) 0 0
\(727\) −31.1769 −1.15629 −0.578144 0.815935i \(-0.696223\pi\)
−0.578144 + 0.815935i \(0.696223\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 24.0000i − 0.887672i
\(732\) 0 0
\(733\) − 13.8564i − 0.511798i −0.966704 0.255899i \(-0.917629\pi\)
0.966704 0.255899i \(-0.0823715\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 30.0000 1.10506
\(738\) 0 0
\(739\) − 44.0000i − 1.61857i −0.587419 0.809283i \(-0.699856\pi\)
0.587419 0.809283i \(-0.300144\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.46410 0.127086 0.0635428 0.997979i \(-0.479760\pi\)
0.0635428 + 0.997979i \(0.479760\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 10.3923i − 0.379727i
\(750\) 0 0
\(751\) 53.6936 1.95931 0.979653 0.200698i \(-0.0643209\pi\)
0.979653 + 0.200698i \(0.0643209\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 6.00000i − 0.218362i
\(756\) 0 0
\(757\) − 22.5167i − 0.818382i −0.912449 0.409191i \(-0.865811\pi\)
0.912449 0.409191i \(-0.134189\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) − 24.0000i − 0.868858i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 31.1769 1.12573
\(768\) 0 0
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 27.7128i − 0.996761i −0.866959 0.498380i \(-0.833928\pi\)
0.866959 0.498380i \(-0.166072\pi\)
\(774\) 0 0
\(775\) −24.2487 −0.871039
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 83.1384i 2.97493i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 48.0000 1.71319
\(786\) 0 0
\(787\) 5.00000i 0.178231i 0.996021 + 0.0891154i \(0.0284040\pi\)
−0.996021 + 0.0891154i \(0.971596\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −10.3923 −0.369508
\(792\) 0 0
\(793\) −63.0000 −2.23720
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 34.6410i − 1.22705i −0.789676 0.613524i \(-0.789751\pi\)
0.789676 0.613524i \(-0.210249\pi\)
\(798\) 0 0
\(799\) 20.7846 0.735307
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 42.0000i − 1.48215i
\(804\) 0 0
\(805\) 20.7846i 0.732561i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) 52.0000i 1.82597i 0.407997 + 0.912983i \(0.366228\pi\)
−0.407997 + 0.912983i \(0.633772\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 38.1051 1.33476
\(816\) 0 0
\(817\) −20.0000 −0.699711
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 17.3205i 0.604490i 0.953230 + 0.302245i \(0.0977361\pi\)
−0.953230 + 0.302245i \(0.902264\pi\)
\(822\) 0 0
\(823\) 36.3731 1.26789 0.633943 0.773380i \(-0.281435\pi\)
0.633943 + 0.773380i \(0.281435\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.00000i 0.208640i 0.994544 + 0.104320i \(0.0332667\pi\)
−0.994544 + 0.104320i \(0.966733\pi\)
\(828\) 0 0
\(829\) 36.3731i 1.26329i 0.775258 + 0.631644i \(0.217620\pi\)
−0.775258 + 0.631644i \(0.782380\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 24.0000 0.831551
\(834\) 0 0
\(835\) 60.0000i 2.07639i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 6.92820 0.239188 0.119594 0.992823i \(-0.461841\pi\)
0.119594 + 0.992823i \(0.461841\pi\)
\(840\) 0 0
\(841\) −19.0000 −0.655172
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 48.4974i 1.66836i
\(846\) 0 0
\(847\) −43.3013 −1.48785
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 6.00000i − 0.205677i
\(852\) 0 0
\(853\) 8.66025i 0.296521i 0.988948 + 0.148261i \(0.0473675\pi\)
−0.988948 + 0.148261i \(0.952633\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −48.0000 −1.63965 −0.819824 0.572615i \(-0.805929\pi\)
−0.819824 + 0.572615i \(0.805929\pi\)
\(858\) 0 0
\(859\) 13.0000i 0.443554i 0.975097 + 0.221777i \(0.0711857\pi\)
−0.975097 + 0.221777i \(0.928814\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −17.3205 −0.589597 −0.294798 0.955559i \(-0.595253\pi\)
−0.294798 + 0.955559i \(0.595253\pi\)
\(864\) 0 0
\(865\) −48.0000 −1.63205
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 31.1769i 1.05760i
\(870\) 0 0
\(871\) 25.9808 0.880325
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.0000i 0.405674i
\(876\) 0 0
\(877\) − 19.0526i − 0.643359i −0.946849 0.321680i \(-0.895753\pi\)
0.946849 0.321680i \(-0.104247\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) − 25.0000i − 0.841317i −0.907219 0.420658i \(-0.861799\pi\)
0.907219 0.420658i \(-0.138201\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 41.5692 1.39576 0.697879 0.716216i \(-0.254127\pi\)
0.697879 + 0.716216i \(0.254127\pi\)
\(888\) 0 0
\(889\) 30.0000 1.00617
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 17.3205i − 0.579609i
\(894\) 0 0
\(895\) 83.1384 2.77901
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.0000i 0.800445i
\(900\) 0 0
\(901\) 41.5692i 1.38487i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −42.0000 −1.39613
\(906\) 0 0
\(907\) 7.00000i 0.232431i 0.993224 + 0.116216i \(0.0370764\pi\)
−0.993224 + 0.116216i \(0.962924\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −13.8564 −0.459083 −0.229542 0.973299i \(-0.573723\pi\)
−0.229542 + 0.973299i \(0.573723\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 20.7846i 0.686368i
\(918\) 0 0
\(919\) 51.9615 1.71405 0.857026 0.515273i \(-0.172309\pi\)
0.857026 + 0.515273i \(0.172309\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 72.0000i 2.36991i
\(924\) 0 0
\(925\) − 12.1244i − 0.398646i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 0 0
\(931\) − 20.0000i − 0.655474i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −124.708 −4.07838
\(936\) 0 0
\(937\) 1.00000 0.0326686 0.0163343 0.999867i \(-0.494800\pi\)
0.0163343 + 0.999867i \(0.494800\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 38.1051i 1.24219i 0.783735 + 0.621096i \(0.213312\pi\)
−0.783735 + 0.621096i \(0.786688\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 42.0000i − 1.36482i −0.730971 0.682408i \(-0.760933\pi\)
0.730971 0.682408i \(-0.239067\pi\)
\(948\) 0 0
\(949\) − 36.3731i − 1.18072i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) − 84.0000i − 2.71818i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −31.1769 −1.00676
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 38.1051i − 1.22665i
\(966\) 0 0
\(967\) −22.5167 −0.724087 −0.362043 0.932161i \(-0.617921\pi\)
−0.362043 + 0.932161i \(0.617921\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 42.0000i 1.34784i 0.738802 + 0.673922i \(0.235392\pi\)
−0.738802 + 0.673922i \(0.764608\pi\)
\(972\) 0 0
\(973\) 19.0526i 0.610797i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −48.0000 −1.53566 −0.767828 0.640656i \(-0.778662\pi\)
−0.767828 + 0.640656i \(0.778662\pi\)
\(978\) 0 0
\(979\) 36.0000i 1.15056i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 10.3923 0.331463 0.165732 0.986171i \(-0.447001\pi\)
0.165732 + 0.986171i \(0.447001\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 13.8564i − 0.440608i
\(990\) 0 0
\(991\) 12.1244 0.385143 0.192571 0.981283i \(-0.438317\pi\)
0.192571 + 0.981283i \(0.438317\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 18.0000i − 0.570638i
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1728.2.d.e.865.2 yes 4
3.2 odd 2 1728.2.d.j.865.4 yes 4
4.3 odd 2 inner 1728.2.d.e.865.1 4
8.3 odd 2 inner 1728.2.d.e.865.3 yes 4
8.5 even 2 inner 1728.2.d.e.865.4 yes 4
12.11 even 2 1728.2.d.j.865.3 yes 4
16.3 odd 4 6912.2.a.bg.1.1 2
16.5 even 4 6912.2.a.bg.1.2 2
16.11 odd 4 6912.2.a.bq.1.2 2
16.13 even 4 6912.2.a.bq.1.1 2
24.5 odd 2 1728.2.d.j.865.2 yes 4
24.11 even 2 1728.2.d.j.865.1 yes 4
48.5 odd 4 6912.2.a.br.1.1 2
48.11 even 4 6912.2.a.bh.1.1 2
48.29 odd 4 6912.2.a.bh.1.2 2
48.35 even 4 6912.2.a.br.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.2.d.e.865.1 4 4.3 odd 2 inner
1728.2.d.e.865.2 yes 4 1.1 even 1 trivial
1728.2.d.e.865.3 yes 4 8.3 odd 2 inner
1728.2.d.e.865.4 yes 4 8.5 even 2 inner
1728.2.d.j.865.1 yes 4 24.11 even 2
1728.2.d.j.865.2 yes 4 24.5 odd 2
1728.2.d.j.865.3 yes 4 12.11 even 2
1728.2.d.j.865.4 yes 4 3.2 odd 2
6912.2.a.bg.1.1 2 16.3 odd 4
6912.2.a.bg.1.2 2 16.5 even 4
6912.2.a.bh.1.1 2 48.11 even 4
6912.2.a.bh.1.2 2 48.29 odd 4
6912.2.a.bq.1.1 2 16.13 even 4
6912.2.a.bq.1.2 2 16.11 odd 4
6912.2.a.br.1.1 2 48.5 odd 4
6912.2.a.br.1.2 2 48.35 even 4