Properties

Label 1728.2.d.e.865.4
Level $1728$
Weight $2$
Character 1728.865
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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

$q$-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 \(0.282047\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 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 \(-0.640219\pi\)
0.426401 0.904534i \(-0.359781\pi\)
\(12\) 0 0
\(13\) − 5.19615i − 1.44115i −0.693375 0.720577i \(-0.743877\pi\)
0.693375 0.720577i \(-0.256123\pi\)
\(14\) 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 \(-0.805570\pi\)
0.819178 0.573539i \(-0.194430\pi\)
\(20\) 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 \(-0.777578\pi\)
0.765641 0.643268i \(-0.222422\pi\)
\(30\) 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 \(-0.954527\pi\)
0.989813 0.142374i \(-0.0454735\pi\)
\(38\) 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 \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 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 \(0.157853\pi\)
−0.879537 + 0.475831i \(0.842147\pi\)
\(54\) 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 \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) − 12.1244i − 1.55236i −0.630509 0.776182i \(-0.717154\pi\)
0.630509 0.776182i \(-0.282846\pi\)
\(62\) 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 \(0.0987981\pi\)
−0.952217 + 0.305424i \(0.901202\pi\)
\(68\) 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 \(0.112016\pi\)
−0.938716 + 0.344691i \(0.887984\pi\)
\(102\) 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 \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) 0 0
\(109\) 13.8564i 1.32720i 0.748086 + 0.663602i \(0.230973\pi\)
−0.748086 + 0.663602i \(0.769027\pi\)
\(110\) 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 \(-0.824356\pi\)
0.851581 0.524222i \(-0.175644\pi\)
\(132\) 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 \(-0.845513\pi\)
0.884519 0.466504i \(-0.154487\pi\)
\(140\) 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 \(-0.813509\pi\)
0.833227 0.552931i \(-0.186491\pi\)
\(158\) 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 \(-0.858234\pi\)
0.902451 0.430793i \(-0.141766\pi\)
\(164\) 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 \(0.176586\pi\)
−0.850026 + 0.526742i \(0.823414\pi\)
\(174\) 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 \(-0.645798\pi\)
0.442189 0.896922i \(-0.354202\pi\)
\(180\) 0 0
\(181\) 12.1244i 0.901196i 0.892727 + 0.450598i \(0.148789\pi\)
−0.892727 + 0.450598i \(0.851211\pi\)
\(182\) 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 \(-0.879286\pi\)
0.928948 0.370211i \(-0.120714\pi\)
\(198\) 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 \(0.290800\pi\)
−0.610920 + 0.791693i \(0.709200\pi\)
\(212\) 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 \(0.0735176\pi\)
−0.973447 + 0.228914i \(0.926482\pi\)
\(230\) 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 \(-0.726453\pi\)
0.652913 0.757433i \(-0.273547\pi\)
\(252\) 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 \(0.102613\pi\)
−0.948487 + 0.316815i \(0.897387\pi\)
\(270\) 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 \(0.313120\pi\)
−0.553949 + 0.832551i \(0.686880\pi\)
\(278\) 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 \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 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 \(-0.749457\pi\)
0.705899 0.708312i \(-0.250543\pi\)
\(294\) 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 \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 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 \(-0.902243\pi\)
0.953211 0.302307i \(-0.0977569\pi\)
\(332\) 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 \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) − 1.73205i − 0.0927146i −0.998925 0.0463573i \(-0.985239\pi\)
0.998925 0.0463573i \(-0.0147613\pi\)
\(350\) 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 \(-0.867769\pi\)
0.914949 0.403570i \(-0.132231\pi\)
\(374\) 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 \(0.267462\pi\)
−0.667271 + 0.744815i \(0.732538\pi\)
\(380\) 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 \(-0.972010\pi\)
0.996136 0.0878185i \(-0.0279895\pi\)
\(390\) 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 \(0.244787\pi\)
−0.718591 + 0.695433i \(0.755213\pi\)
\(398\) 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 \(-0.738214\pi\)
0.680446 0.732798i \(-0.261786\pi\)
\(420\) 0 0
\(421\) − 12.1244i − 0.590905i −0.955357 0.295452i \(-0.904530\pi\)
0.955357 0.295452i \(-0.0954704\pi\)
\(422\) 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 \(-0.673431\pi\)
0.518289 0.855206i \(-0.326569\pi\)
\(444\) 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 \(0.258635\pi\)
−0.687666 + 0.726027i \(0.741365\pi\)
\(462\) 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 \(-0.863267\pi\)
0.909149 0.416470i \(-0.136733\pi\)
\(468\) 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 \(0.0432281\pi\)
−0.990793 + 0.135388i \(0.956772\pi\)
\(492\) 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 \(0.0573071\pi\)
−0.983837 + 0.179065i \(0.942693\pi\)
\(500\) 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 \(0.125405\pi\)
−0.923392 + 0.383859i \(0.874595\pi\)
\(510\) 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 \(-0.951092\pi\)
0.988219 0.153044i \(-0.0489077\pi\)
\(524\) 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 \(-0.916069\pi\)
0.965438 0.260633i \(-0.0839314\pi\)
\(542\) 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 \(0.00680549\pi\)
−0.999771 + 0.0213785i \(0.993195\pi\)
\(548\) 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 \(0.0706606\pi\)
−0.975462 + 0.220168i \(0.929339\pi\)
\(558\) 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 \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 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 \(-0.966637\pi\)
0.994512 0.104622i \(-0.0333632\pi\)
\(572\) 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 \(-0.878854\pi\)
0.928445 0.371470i \(-0.121146\pi\)
\(588\) 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 \(-0.768606\pi\)
0.747208 0.664590i \(-0.231394\pi\)
\(614\) 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 \(0.214085\pi\)
−0.782224 + 0.622998i \(0.785915\pi\)
\(620\) 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 \(0.186176\pi\)
−0.833774 + 0.552106i \(0.813824\pi\)
\(644\) 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 \(-0.697633\pi\)
0.581754 0.813365i \(-0.302367\pi\)
\(654\) 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 \(-0.924908\pi\)
0.972302 0.233727i \(-0.0750921\pi\)
\(660\) 0 0
\(661\) 8.66025i 0.336845i 0.985715 + 0.168422i \(0.0538673\pi\)
−0.985715 + 0.168422i \(0.946133\pi\)
\(662\) 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 \(0.154297\pi\)
−0.884797 + 0.465977i \(0.845703\pi\)
\(678\) 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 \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 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 \(-0.684354\pi\)
0.547326 0.836919i \(-0.315646\pi\)
\(692\) 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 \(-0.799613\pi\)
0.808302 0.588768i \(-0.200387\pi\)
\(702\) 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 \(-0.968892\pi\)
0.995228 0.0975728i \(-0.0311079\pi\)
\(710\) 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 \(0.0823715\pi\)
−0.966704 + 0.255899i \(0.917629\pi\)
\(734\) 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 \(0.300144\pi\)
−0.587419 + 0.809283i \(0.699856\pi\)
\(740\) 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 \(0.134189\pi\)
−0.912449 + 0.409191i \(0.865811\pi\)
\(758\) 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 \(0.166072\pi\)
−0.866959 + 0.498380i \(0.833928\pi\)
\(774\) 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 \(-0.971596\pi\)
0.996021 0.0891154i \(-0.0284040\pi\)
\(788\) 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 \(0.210249\pi\)
−0.789676 + 0.613524i \(0.789751\pi\)
\(798\) 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 \(-0.633772\pi\)
0.407997 0.912983i \(-0.366228\pi\)
\(812\) 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 \(-0.902264\pi\)
0.953230 0.302245i \(-0.0977361\pi\)
\(822\) 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 \(-0.966733\pi\)
0.994544 0.104320i \(-0.0332667\pi\)
\(828\) 0 0
\(829\) − 36.3731i − 1.26329i −0.775258 0.631644i \(-0.782380\pi\)
0.775258 0.631644i \(-0.217620\pi\)
\(830\) 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 \(-0.952633\pi\)
0.988948 0.148261i \(-0.0473675\pi\)
\(854\) 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 \(-0.928814\pi\)
0.975097 0.221777i \(-0.0711857\pi\)
\(860\) 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 \(0.104247\pi\)
−0.946849 + 0.321680i \(0.895753\pi\)
\(878\) 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 \(0.138201\pi\)
−0.907219 + 0.420658i \(0.861799\pi\)
\(884\) 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 \(-0.962924\pi\)
0.993224 0.116216i \(-0.0370764\pi\)
\(908\) 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 \(-0.786688\pi\)
0.783735 0.621096i \(-0.213312\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42.0000i 1.36482i 0.730971 + 0.682408i \(0.239067\pi\)
−0.730971 + 0.682408i \(0.760933\pi\)
\(948\) 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 \(-0.764608\pi\)
0.738802 0.673922i \(-0.235392\pi\)
\(972\) 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.865.4 yes 4
3.2 odd 2 1728.2.d.j.865.2 yes 4
4.3 odd 2 inner 1728.2.d.e.865.3 yes 4
8.3 odd 2 inner 1728.2.d.e.865.1 4
8.5 even 2 inner 1728.2.d.e.865.2 yes 4
12.11 even 2 1728.2.d.j.865.1 yes 4
16.3 odd 4 6912.2.a.bq.1.2 2
16.5 even 4 6912.2.a.bq.1.1 2
16.11 odd 4 6912.2.a.bg.1.1 2
16.13 even 4 6912.2.a.bg.1.2 2
24.5 odd 2 1728.2.d.j.865.4 yes 4
24.11 even 2 1728.2.d.j.865.3 yes 4
48.5 odd 4 6912.2.a.bh.1.2 2
48.11 even 4 6912.2.a.br.1.2 2
48.29 odd 4 6912.2.a.br.1.1 2
48.35 even 4 6912.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.2.d.e.865.1 4 8.3 odd 2 inner
1728.2.d.e.865.2 yes 4 8.5 even 2 inner
1728.2.d.e.865.3 yes 4 4.3 odd 2 inner
1728.2.d.e.865.4 yes 4 1.1 even 1 trivial
1728.2.d.j.865.1 yes 4 12.11 even 2
1728.2.d.j.865.2 yes 4 3.2 odd 2
1728.2.d.j.865.3 yes 4 24.11 even 2
1728.2.d.j.865.4 yes 4 24.5 odd 2
6912.2.a.bg.1.1 2 16.11 odd 4
6912.2.a.bg.1.2 2 16.13 even 4
6912.2.a.bh.1.1 2 48.35 even 4
6912.2.a.bh.1.2 2 48.5 odd 4
6912.2.a.bq.1.1 2 16.5 even 4
6912.2.a.bq.1.2 2 16.3 odd 4
6912.2.a.br.1.1 2 48.29 odd 4
6912.2.a.br.1.2 2 48.11 even 4