Properties

Label 1440.2.f.j.289.6
Level $1440$
Weight $2$
Character 1440.289
Analytic conductor $11.498$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1440,2,Mod(289,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1440.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.f (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: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.6
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1440.289
Dual form 1440.2.f.j.289.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.41421 - 1.73205i) q^{5} +2.82843i q^{7} +O(q^{10})\) \(q+(1.41421 - 1.73205i) q^{5} +2.82843i q^{7} -4.89898 q^{11} +4.89898i q^{13} +3.46410i q^{17} -6.92820 q^{19} -4.00000i q^{23} +(-1.00000 - 4.89898i) q^{25} -8.48528 q^{29} -6.92820 q^{31} +(4.89898 + 4.00000i) q^{35} -4.89898i q^{37} +5.65685 q^{41} +11.3137i q^{43} +4.00000i q^{47} -1.00000 q^{49} -3.46410i q^{53} +(-6.92820 + 8.48528i) q^{55} +4.89898 q^{59} -6.00000 q^{61} +(8.48528 + 6.92820i) q^{65} +5.65685i q^{67} +9.79796 q^{71} -13.8564i q^{77} -6.92820 q^{79} -16.0000i q^{83} +(6.00000 + 4.89898i) q^{85} -13.8564 q^{91} +(-9.79796 + 12.0000i) q^{95} +9.79796i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{25} - 8 q^{49} - 48 q^{61} + 48 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) 1.41421 1.73205i 0.632456 0.774597i
\(6\) 0 0
\(7\) 2.82843i 1.06904i 0.845154 + 0.534522i \(0.179509\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 0 0
\(13\) 4.89898i 1.35873i 0.733799 + 0.679366i \(0.237745\pi\)
−0.733799 + 0.679366i \(0.762255\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 0 0
\(19\) −6.92820 −1.58944 −0.794719 0.606977i \(-0.792382\pi\)
−0.794719 + 0.606977i \(0.792382\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) −1.00000 4.89898i −0.200000 0.979796i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.48528 −1.57568 −0.787839 0.615882i \(-0.788800\pi\)
−0.787839 + 0.615882i \(0.788800\pi\)
\(30\) 0 0
\(31\) −6.92820 −1.24434 −0.622171 0.782881i \(-0.713749\pi\)
−0.622171 + 0.782881i \(0.713749\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.89898 + 4.00000i 0.828079 + 0.676123i
\(36\) 0 0
\(37\) 4.89898i 0.805387i −0.915335 0.402694i \(-0.868074\pi\)
0.915335 0.402694i \(-0.131926\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.65685 0.883452 0.441726 0.897150i \(-0.354366\pi\)
0.441726 + 0.897150i \(0.354366\pi\)
\(42\) 0 0
\(43\) 11.3137i 1.72532i 0.505781 + 0.862662i \(0.331205\pi\)
−0.505781 + 0.862662i \(0.668795\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.46410i 0.475831i −0.971286 0.237915i \(-0.923536\pi\)
0.971286 0.237915i \(-0.0764641\pi\)
\(54\) 0 0
\(55\) −6.92820 + 8.48528i −0.934199 + 1.14416i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.89898 0.637793 0.318896 0.947790i \(-0.396688\pi\)
0.318896 + 0.947790i \(0.396688\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.48528 + 6.92820i 1.05247 + 0.859338i
\(66\) 0 0
\(67\) 5.65685i 0.691095i 0.938401 + 0.345547i \(0.112307\pi\)
−0.938401 + 0.345547i \(0.887693\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.79796 1.16280 0.581402 0.813617i \(-0.302504\pi\)
0.581402 + 0.813617i \(0.302504\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 13.8564i 1.57908i
\(78\) 0 0
\(79\) −6.92820 −0.779484 −0.389742 0.920924i \(-0.627436\pi\)
−0.389742 + 0.920924i \(0.627436\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.0000i 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) 0 0
\(85\) 6.00000 + 4.89898i 0.650791 + 0.531369i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −13.8564 −1.45255
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −9.79796 + 12.0000i −1.00525 + 1.23117i
\(96\) 0 0
\(97\) 9.79796i 0.994832i 0.867512 + 0.497416i \(0.165718\pi\)
−0.867512 + 0.497416i \(0.834282\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.48528 −0.844317 −0.422159 0.906522i \(-0.638727\pi\)
−0.422159 + 0.906522i \(0.638727\pi\)
\(102\) 0 0
\(103\) 2.82843i 0.278693i −0.990244 0.139347i \(-0.955500\pi\)
0.990244 0.139347i \(-0.0445002\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 17.3205i 1.62938i 0.579899 + 0.814688i \(0.303092\pi\)
−0.579899 + 0.814688i \(0.696908\pi\)
\(114\) 0 0
\(115\) −6.92820 5.65685i −0.646058 0.527504i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.79796 −0.898177
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.89949 5.19615i −0.885438 0.464758i
\(126\) 0 0
\(127\) 14.1421i 1.25491i 0.778652 + 0.627456i \(0.215904\pi\)
−0.778652 + 0.627456i \(0.784096\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.89898 0.428026 0.214013 0.976831i \(-0.431347\pi\)
0.214013 + 0.976831i \(0.431347\pi\)
\(132\) 0 0
\(133\) 19.5959i 1.69918i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.3923i 0.887875i 0.896058 + 0.443937i \(0.146419\pi\)
−0.896058 + 0.443937i \(0.853581\pi\)
\(138\) 0 0
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 24.0000i 2.00698i
\(144\) 0 0
\(145\) −12.0000 + 14.6969i −0.996546 + 1.22051i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.82843 0.231714 0.115857 0.993266i \(-0.463039\pi\)
0.115857 + 0.993266i \(0.463039\pi\)
\(150\) 0 0
\(151\) 6.92820 0.563809 0.281905 0.959442i \(-0.409034\pi\)
0.281905 + 0.959442i \(0.409034\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −9.79796 + 12.0000i −0.786991 + 0.963863i
\(156\) 0 0
\(157\) 4.89898i 0.390981i 0.980706 + 0.195491i \(0.0626299\pi\)
−0.980706 + 0.195491i \(0.937370\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 11.3137 0.891645
\(162\) 0 0
\(163\) 5.65685i 0.443079i 0.975151 + 0.221540i \(0.0711082\pi\)
−0.975151 + 0.221540i \(0.928892\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.00000i 0.309529i 0.987951 + 0.154765i \(0.0494619\pi\)
−0.987951 + 0.154765i \(0.950538\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 24.2487i 1.84360i −0.387671 0.921798i \(-0.626720\pi\)
0.387671 0.921798i \(-0.373280\pi\)
\(174\) 0 0
\(175\) 13.8564 2.82843i 1.04745 0.213809i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −14.6969 −1.09850 −0.549250 0.835658i \(-0.685087\pi\)
−0.549250 + 0.835658i \(0.685087\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.48528 6.92820i −0.623850 0.509372i
\(186\) 0 0
\(187\) 16.9706i 1.24101i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.79796 −0.708955 −0.354478 0.935064i \(-0.615341\pi\)
−0.354478 + 0.935064i \(0.615341\pi\)
\(192\) 0 0
\(193\) 9.79796i 0.705273i −0.935760 0.352636i \(-0.885285\pi\)
0.935760 0.352636i \(-0.114715\pi\)
\(194\) 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
\(199\) −6.92820 −0.491127 −0.245564 0.969380i \(-0.578973\pi\)
−0.245564 + 0.969380i \(0.578973\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 24.0000i 1.68447i
\(204\) 0 0
\(205\) 8.00000 9.79796i 0.558744 0.684319i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 33.9411 2.34776
\(210\) 0 0
\(211\) 20.7846 1.43087 0.715436 0.698679i \(-0.246228\pi\)
0.715436 + 0.698679i \(0.246228\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 19.5959 + 16.0000i 1.33643 + 1.09119i
\(216\) 0 0
\(217\) 19.5959i 1.33026i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −16.9706 −1.14156
\(222\) 0 0
\(223\) 14.1421i 0.947027i −0.880786 0.473514i \(-0.842985\pi\)
0.880786 0.473514i \(-0.157015\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.0000i 1.06196i 0.847385 + 0.530979i \(0.178176\pi\)
−0.847385 + 0.530979i \(0.821824\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.46410i 0.226941i −0.993541 0.113470i \(-0.963803\pi\)
0.993541 0.113470i \(-0.0361967\pi\)
\(234\) 0 0
\(235\) 6.92820 + 5.65685i 0.451946 + 0.369012i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 29.3939 1.90133 0.950666 0.310217i \(-0.100402\pi\)
0.950666 + 0.310217i \(0.100402\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.41421 + 1.73205i −0.0903508 + 0.110657i
\(246\) 0 0
\(247\) 33.9411i 2.15962i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.89898 0.309221 0.154610 0.987976i \(-0.450588\pi\)
0.154610 + 0.987976i \(0.450588\pi\)
\(252\) 0 0
\(253\) 19.5959i 1.23198i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.3923i 0.648254i 0.946014 + 0.324127i \(0.105071\pi\)
−0.946014 + 0.324127i \(0.894929\pi\)
\(258\) 0 0
\(259\) 13.8564 0.860995
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.0000i 1.72655i −0.504730 0.863277i \(-0.668408\pi\)
0.504730 0.863277i \(-0.331592\pi\)
\(264\) 0 0
\(265\) −6.00000 4.89898i −0.368577 0.300942i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.82843 −0.172452 −0.0862261 0.996276i \(-0.527481\pi\)
−0.0862261 + 0.996276i \(0.527481\pi\)
\(270\) 0 0
\(271\) 6.92820 0.420858 0.210429 0.977609i \(-0.432514\pi\)
0.210429 + 0.977609i \(0.432514\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.89898 + 24.0000i 0.295420 + 1.44725i
\(276\) 0 0
\(277\) 24.4949i 1.47176i 0.677114 + 0.735878i \(0.263230\pi\)
−0.677114 + 0.735878i \(0.736770\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −22.6274 −1.34984 −0.674919 0.737892i \(-0.735822\pi\)
−0.674919 + 0.737892i \(0.735822\pi\)
\(282\) 0 0
\(283\) 28.2843i 1.68133i −0.541559 0.840663i \(-0.682166\pi\)
0.541559 0.840663i \(-0.317834\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.0000i 0.944450i
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.3205i 1.01187i 0.862570 + 0.505937i \(0.168853\pi\)
−0.862570 + 0.505937i \(0.831147\pi\)
\(294\) 0 0
\(295\) 6.92820 8.48528i 0.403376 0.494032i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 19.5959 1.13326
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.48528 + 10.3923i −0.485866 + 0.595062i
\(306\) 0 0
\(307\) 11.3137i 0.645707i 0.946449 + 0.322854i \(0.104642\pi\)
−0.946449 + 0.322854i \(0.895358\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −29.3939 −1.66677 −0.833387 0.552690i \(-0.813601\pi\)
−0.833387 + 0.552690i \(0.813601\pi\)
\(312\) 0 0
\(313\) 29.3939i 1.66144i 0.556690 + 0.830720i \(0.312071\pi\)
−0.556690 + 0.830720i \(0.687929\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.2487i 1.36194i −0.732310 0.680972i \(-0.761558\pi\)
0.732310 0.680972i \(-0.238442\pi\)
\(318\) 0 0
\(319\) 41.5692 2.32743
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000i 1.33540i
\(324\) 0 0
\(325\) 24.0000 4.89898i 1.33128 0.271746i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.3137 −0.623745
\(330\) 0 0
\(331\) 20.7846 1.14243 0.571213 0.820802i \(-0.306473\pi\)
0.571213 + 0.820802i \(0.306473\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.79796 + 8.00000i 0.535320 + 0.437087i
\(336\) 0 0
\(337\) 19.5959i 1.06746i −0.845656 0.533729i \(-0.820790\pi\)
0.845656 0.533729i \(-0.179210\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 33.9411 1.83801
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.00000i 0.429463i −0.976673 0.214731i \(-0.931112\pi\)
0.976673 0.214731i \(-0.0688876\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.3923i 0.553127i 0.960996 + 0.276563i \(0.0891955\pi\)
−0.960996 + 0.276563i \(0.910804\pi\)
\(354\) 0 0
\(355\) 13.8564 16.9706i 0.735422 0.900704i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −19.5959 −1.03423 −0.517116 0.855915i \(-0.672995\pi\)
−0.517116 + 0.855915i \(0.672995\pi\)
\(360\) 0 0
\(361\) 29.0000 1.52632
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.82843i 0.147643i 0.997271 + 0.0738213i \(0.0235195\pi\)
−0.997271 + 0.0738213i \(0.976481\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9.79796 0.508685
\(372\) 0 0
\(373\) 14.6969i 0.760979i 0.924785 + 0.380489i \(0.124244\pi\)
−0.924785 + 0.380489i \(0.875756\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 41.5692i 2.14092i
\(378\) 0 0
\(379\) −6.92820 −0.355878 −0.177939 0.984042i \(-0.556943\pi\)
−0.177939 + 0.984042i \(0.556943\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 20.0000i 1.02195i −0.859595 0.510976i \(-0.829284\pi\)
0.859595 0.510976i \(-0.170716\pi\)
\(384\) 0 0
\(385\) −24.0000 19.5959i −1.22315 0.998700i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 14.1421 0.717035 0.358517 0.933523i \(-0.383282\pi\)
0.358517 + 0.933523i \(0.383282\pi\)
\(390\) 0 0
\(391\) 13.8564 0.700749
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.79796 + 12.0000i −0.492989 + 0.603786i
\(396\) 0 0
\(397\) 24.4949i 1.22936i −0.788775 0.614682i \(-0.789284\pi\)
0.788775 0.614682i \(-0.210716\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 16.9706 0.847469 0.423735 0.905786i \(-0.360719\pi\)
0.423735 + 0.905786i \(0.360719\pi\)
\(402\) 0 0
\(403\) 33.9411i 1.69073i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000i 1.18964i
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.8564i 0.681829i
\(414\) 0 0
\(415\) −27.7128 22.6274i −1.36037 1.11074i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −24.4949 −1.19665 −0.598327 0.801252i \(-0.704168\pi\)
−0.598327 + 0.801252i \(0.704168\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 16.9706 3.46410i 0.823193 0.168034i
\(426\) 0 0
\(427\) 16.9706i 0.821263i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 9.79796i 0.470860i −0.971891 0.235430i \(-0.924350\pi\)
0.971891 0.235430i \(-0.0756498\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 27.7128i 1.32568i
\(438\) 0 0
\(439\) 20.7846 0.991995 0.495998 0.868324i \(-0.334802\pi\)
0.495998 + 0.868324i \(0.334802\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 32.0000i 1.52037i −0.649709 0.760183i \(-0.725109\pi\)
0.649709 0.760183i \(-0.274891\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −39.5980 −1.86874 −0.934372 0.356299i \(-0.884039\pi\)
−0.934372 + 0.356299i \(0.884039\pi\)
\(450\) 0 0
\(451\) −27.7128 −1.30495
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −19.5959 + 24.0000i −0.918671 + 1.12514i
\(456\) 0 0
\(457\) 29.3939i 1.37499i 0.726190 + 0.687494i \(0.241289\pi\)
−0.726190 + 0.687494i \(0.758711\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −25.4558 −1.18560 −0.592798 0.805351i \(-0.701977\pi\)
−0.592798 + 0.805351i \(0.701977\pi\)
\(462\) 0 0
\(463\) 2.82843i 0.131448i 0.997838 + 0.0657241i \(0.0209357\pi\)
−0.997838 + 0.0657241i \(0.979064\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.00000i 0.370196i 0.982720 + 0.185098i \(0.0592602\pi\)
−0.982720 + 0.185098i \(0.940740\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 55.4256i 2.54847i
\(474\) 0 0
\(475\) 6.92820 + 33.9411i 0.317888 + 1.55733i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9.79796 −0.447680 −0.223840 0.974626i \(-0.571859\pi\)
−0.223840 + 0.974626i \(0.571859\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.9706 + 13.8564i 0.770594 + 0.629187i
\(486\) 0 0
\(487\) 2.82843i 0.128168i −0.997944 0.0640841i \(-0.979587\pi\)
0.997944 0.0640841i \(-0.0204126\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.89898 0.221088 0.110544 0.993871i \(-0.464741\pi\)
0.110544 + 0.993871i \(0.464741\pi\)
\(492\) 0 0
\(493\) 29.3939i 1.32383i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 27.7128i 1.24309i
\(498\) 0 0
\(499\) 6.92820 0.310149 0.155074 0.987903i \(-0.450438\pi\)
0.155074 + 0.987903i \(0.450438\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 20.0000i 0.891756i −0.895094 0.445878i \(-0.852892\pi\)
0.895094 0.445878i \(-0.147108\pi\)
\(504\) 0 0
\(505\) −12.0000 + 14.6969i −0.533993 + 0.654005i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8.48528 −0.376103 −0.188052 0.982159i \(-0.560217\pi\)
−0.188052 + 0.982159i \(0.560217\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.89898 4.00000i −0.215875 0.176261i
\(516\) 0 0
\(517\) 19.5959i 0.861827i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −16.9706 −0.743494 −0.371747 0.928334i \(-0.621241\pi\)
−0.371747 + 0.928334i \(0.621241\pi\)
\(522\) 0 0
\(523\) 28.2843i 1.23678i 0.785869 + 0.618392i \(0.212216\pi\)
−0.785869 + 0.618392i \(0.787784\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24.0000i 1.04546i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 27.7128i 1.20038i
\(534\) 0 0
\(535\) 13.8564 + 11.3137i 0.599065 + 0.489134i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.89898 0.211014
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.48528 10.3923i 0.363470 0.445157i
\(546\) 0 0
\(547\) 11.3137i 0.483739i 0.970309 + 0.241870i \(0.0777606\pi\)
−0.970309 + 0.241870i \(0.922239\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 58.7878 2.50444
\(552\) 0 0
\(553\) 19.5959i 0.833303i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.1769i 1.32101i 0.750822 + 0.660504i \(0.229657\pi\)
−0.750822 + 0.660504i \(0.770343\pi\)
\(558\) 0 0
\(559\) −55.4256 −2.34425
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.00000i 0.337160i 0.985688 + 0.168580i \(0.0539181\pi\)
−0.985688 + 0.168580i \(0.946082\pi\)
\(564\) 0 0
\(565\) 30.0000 + 24.4949i 1.26211 + 1.03051i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −39.5980 −1.66003 −0.830017 0.557738i \(-0.811669\pi\)
−0.830017 + 0.557738i \(0.811669\pi\)
\(570\) 0 0
\(571\) 20.7846 0.869809 0.434904 0.900477i \(-0.356782\pi\)
0.434904 + 0.900477i \(0.356782\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −19.5959 + 4.00000i −0.817206 + 0.166812i
\(576\) 0 0
\(577\) 39.1918i 1.63158i 0.578350 + 0.815789i \(0.303697\pi\)
−0.578350 + 0.815789i \(0.696303\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 45.2548 1.87749
\(582\) 0 0
\(583\) 16.9706i 0.702849i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 40.0000i 1.65098i 0.564419 + 0.825488i \(0.309100\pi\)
−0.564419 + 0.825488i \(0.690900\pi\)
\(588\) 0 0
\(589\) 48.0000 1.97781
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 10.3923i 0.426761i −0.976969 0.213380i \(-0.931553\pi\)
0.976969 0.213380i \(-0.0684474\pi\)
\(594\) 0 0
\(595\) −13.8564 + 16.9706i −0.568057 + 0.695725i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 19.5959 0.800668 0.400334 0.916369i \(-0.368894\pi\)
0.400334 + 0.916369i \(0.368894\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 18.3848 22.5167i 0.747447 0.915432i
\(606\) 0 0
\(607\) 2.82843i 0.114802i −0.998351 0.0574012i \(-0.981719\pi\)
0.998351 0.0574012i \(-0.0182814\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −19.5959 −0.792766
\(612\) 0 0
\(613\) 4.89898i 0.197868i 0.995094 + 0.0989340i \(0.0315433\pi\)
−0.995094 + 0.0989340i \(0.968457\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.2487i 0.976216i −0.872783 0.488108i \(-0.837687\pi\)
0.872783 0.488108i \(-0.162313\pi\)
\(618\) 0 0
\(619\) −20.7846 −0.835404 −0.417702 0.908584i \(-0.637164\pi\)
−0.417702 + 0.908584i \(0.637164\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −23.0000 + 9.79796i −0.920000 + 0.391918i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.9706 0.676661
\(630\) 0 0
\(631\) −48.4974 −1.93065 −0.965326 0.261048i \(-0.915932\pi\)
−0.965326 + 0.261048i \(0.915932\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 24.4949 + 20.0000i 0.972050 + 0.793676i
\(636\) 0 0
\(637\) 4.89898i 0.194105i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.65685 0.223432 0.111716 0.993740i \(-0.464365\pi\)
0.111716 + 0.993740i \(0.464365\pi\)
\(642\) 0 0
\(643\) 5.65685i 0.223085i −0.993760 0.111542i \(-0.964421\pi\)
0.993760 0.111542i \(-0.0355790\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.00000i 0.157256i −0.996904 0.0786281i \(-0.974946\pi\)
0.996904 0.0786281i \(-0.0250540\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.3205i 0.677804i −0.940822 0.338902i \(-0.889945\pi\)
0.940822 0.338902i \(-0.110055\pi\)
\(654\) 0 0
\(655\) 6.92820 8.48528i 0.270707 0.331547i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 34.2929 1.33586 0.667930 0.744224i \(-0.267181\pi\)
0.667930 + 0.744224i \(0.267181\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −33.9411 27.7128i −1.31618 1.07466i
\(666\) 0 0
\(667\) 33.9411i 1.31421i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 29.3939 1.13474
\(672\) 0 0
\(673\) 9.79796i 0.377684i −0.982008 0.188842i \(-0.939527\pi\)
0.982008 0.188842i \(-0.0604733\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.46410i 0.133136i 0.997782 + 0.0665681i \(0.0212050\pi\)
−0.997782 + 0.0665681i \(0.978795\pi\)
\(678\) 0 0
\(679\) −27.7128 −1.06352
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 8.00000i 0.306111i −0.988218 0.153056i \(-0.951089\pi\)
0.988218 0.153056i \(-0.0489114\pi\)
\(684\) 0 0
\(685\) 18.0000 + 14.6969i 0.687745 + 0.561541i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 16.9706 0.646527
\(690\) 0 0
\(691\) −34.6410 −1.31781 −0.658903 0.752228i \(-0.728979\pi\)
−0.658903 + 0.752228i \(0.728979\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.79796 12.0000i 0.371658 0.455186i
\(696\) 0 0
\(697\) 19.5959i 0.742248i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 31.1127 1.17511 0.587555 0.809184i \(-0.300091\pi\)
0.587555 + 0.809184i \(0.300091\pi\)
\(702\) 0 0
\(703\) 33.9411i 1.28011i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.0000i 0.902613i
\(708\) 0 0
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 27.7128i 1.03785i
\(714\) 0 0
\(715\) −41.5692 33.9411i −1.55460 1.26933i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.5959 −0.730804 −0.365402 0.930850i \(-0.619069\pi\)
−0.365402 + 0.930850i \(0.619069\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 8.48528 + 41.5692i 0.315135 + 1.54384i
\(726\) 0 0
\(727\) 31.1127i 1.15391i 0.816777 + 0.576953i \(0.195758\pi\)
−0.816777 + 0.576953i \(0.804242\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −39.1918 −1.44956
\(732\) 0 0
\(733\) 24.4949i 0.904740i −0.891830 0.452370i \(-0.850579\pi\)
0.891830 0.452370i \(-0.149421\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 27.7128i 1.02081i
\(738\) 0 0
\(739\) −20.7846 −0.764574 −0.382287 0.924044i \(-0.624863\pi\)
−0.382287 + 0.924044i \(0.624863\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.00000i 0.146746i 0.997305 + 0.0733729i \(0.0233763\pi\)
−0.997305 + 0.0733729i \(0.976624\pi\)
\(744\) 0 0
\(745\) 4.00000 4.89898i 0.146549 0.179485i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −22.6274 −0.826788
\(750\) 0 0
\(751\) 20.7846 0.758441 0.379221 0.925306i \(-0.376192\pi\)
0.379221 + 0.925306i \(0.376192\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 9.79796 12.0000i 0.356584 0.436725i
\(756\) 0 0
\(757\) 4.89898i 0.178056i 0.996029 + 0.0890282i \(0.0283761\pi\)
−0.996029 + 0.0890282i \(0.971624\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −50.9117 −1.84555 −0.922774 0.385342i \(-0.874083\pi\)
−0.922774 + 0.385342i \(0.874083\pi\)
\(762\) 0 0
\(763\) 16.9706i 0.614376i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 38.1051i 1.37055i 0.728286 + 0.685273i \(0.240317\pi\)
−0.728286 + 0.685273i \(0.759683\pi\)
\(774\) 0 0
\(775\) 6.92820 + 33.9411i 0.248868 + 1.21920i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −39.1918 −1.40419
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 8.48528 + 6.92820i 0.302853 + 0.247278i
\(786\) 0 0
\(787\) 28.2843i 1.00823i 0.863638 + 0.504113i \(0.168180\pi\)
−0.863638 + 0.504113i \(0.831820\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −48.9898 −1.74188
\(792\) 0 0
\(793\) 29.3939i 1.04381i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.46410i 0.122705i −0.998116 0.0613524i \(-0.980459\pi\)
0.998116 0.0613524i \(-0.0195413\pi\)
\(798\) 0 0
\(799\) −13.8564 −0.490204
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 16.0000 19.5959i 0.563926 0.690665i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 16.9706 0.596653 0.298327 0.954464i \(-0.403572\pi\)
0.298327 + 0.954464i \(0.403572\pi\)
\(810\) 0 0
\(811\) −34.6410 −1.21641 −0.608205 0.793780i \(-0.708110\pi\)
−0.608205 + 0.793780i \(0.708110\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.79796 + 8.00000i 0.343208 + 0.280228i
\(816\) 0 0
\(817\) 78.3837i 2.74230i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −36.7696 −1.28327 −0.641633 0.767012i \(-0.721743\pi\)
−0.641633 + 0.767012i \(0.721743\pi\)
\(822\) 0 0
\(823\) 19.7990i 0.690149i −0.938575 0.345075i \(-0.887854\pi\)
0.938575 0.345075i \(-0.112146\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.0000i 0.556375i −0.960527 0.278187i \(-0.910266\pi\)
0.960527 0.278187i \(-0.0897336\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.46410i 0.120024i
\(834\) 0 0
\(835\) 6.92820 + 5.65685i 0.239760 + 0.195764i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 43.0000 1.48276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −15.5563 + 19.0526i −0.535155 + 0.655428i
\(846\) 0 0
\(847\) 36.7696i 1.26342i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −19.5959 −0.671739
\(852\) 0 0
\(853\) 44.0908i 1.50964i 0.655932 + 0.754820i \(0.272276\pi\)
−0.655932 + 0.754820i \(0.727724\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38.1051i 1.30165i −0.759229 0.650823i \(-0.774424\pi\)
0.759229 0.650823i \(-0.225576\pi\)
\(858\) 0 0
\(859\) 20.7846 0.709162 0.354581 0.935025i \(-0.384624\pi\)
0.354581 + 0.935025i \(0.384624\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 28.0000i 0.953131i 0.879139 + 0.476566i \(0.158119\pi\)
−0.879139 + 0.476566i \(0.841881\pi\)
\(864\) 0 0
\(865\) −42.0000 34.2929i −1.42804 1.16599i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 33.9411 1.15137
\(870\) 0 0
\(871\) −27.7128 −0.939013
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 14.6969 28.0000i 0.496847 0.946573i
\(876\) 0 0
\(877\) 44.0908i 1.48884i 0.667711 + 0.744421i \(0.267274\pi\)
−0.667711 + 0.744421i \(0.732726\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −11.3137 −0.381169 −0.190584 0.981671i \(-0.561038\pi\)
−0.190584 + 0.981671i \(0.561038\pi\)
\(882\) 0 0
\(883\) 39.5980i 1.33258i 0.745694 + 0.666289i \(0.232118\pi\)
−0.745694 + 0.666289i \(0.767882\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.0000i 0.940148i −0.882627 0.470074i \(-0.844227\pi\)
0.882627 0.470074i \(-0.155773\pi\)
\(888\) 0 0
\(889\) −40.0000 −1.34156
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 27.7128i 0.927374i
\(894\) 0 0
\(895\) −20.7846 + 25.4558i −0.694753 + 0.850895i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 58.7878 1.96068
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −19.7990 + 24.2487i −0.658141 + 0.806054i
\(906\) 0 0
\(907\) 45.2548i 1.50266i −0.659925 0.751331i \(-0.729412\pi\)
0.659925 0.751331i \(-0.270588\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −39.1918 −1.29848 −0.649242 0.760582i \(-0.724914\pi\)
−0.649242 + 0.760582i \(0.724914\pi\)
\(912\) 0 0
\(913\) 78.3837i 2.59412i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.8564i 0.457579i
\(918\) 0 0
\(919\) 6.92820 0.228540 0.114270 0.993450i \(-0.463547\pi\)
0.114270 + 0.993450i \(0.463547\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 48.0000i 1.57994i
\(924\) 0 0
\(925\) −24.0000 + 4.89898i −0.789115 + 0.161077i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 39.5980 1.29917 0.649584 0.760290i \(-0.274943\pi\)
0.649584 + 0.760290i \(0.274943\pi\)
\(930\) 0 0
\(931\) 6.92820 0.227063
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −29.3939 24.0000i −0.961283 0.784884i
\(936\) 0 0
\(937\) 19.5959i 0.640171i −0.947389 0.320085i \(-0.896288\pi\)
0.947389 0.320085i \(-0.103712\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8.48528 0.276612 0.138306 0.990390i \(-0.455834\pi\)
0.138306 + 0.990390i \(0.455834\pi\)
\(942\) 0 0
\(943\) 22.6274i 0.736850i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 8.00000i 0.259965i −0.991516 0.129983i \(-0.958508\pi\)
0.991516 0.129983i \(-0.0414921\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 38.1051i 1.23435i −0.786828 0.617173i \(-0.788278\pi\)
0.786828 0.617173i \(-0.211722\pi\)
\(954\) 0 0
\(955\) −13.8564 + 16.9706i −0.448383 + 0.549155i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −29.3939 −0.949178
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −16.9706 13.8564i −0.546302 0.446054i
\(966\) 0 0
\(967\) 31.1127i 1.00052i −0.865876 0.500258i \(-0.833238\pi\)
0.865876 0.500258i \(-0.166762\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 24.4949 0.786079 0.393039 0.919522i \(-0.371424\pi\)
0.393039 + 0.919522i \(0.371424\pi\)
\(972\) 0 0
\(973\) 19.5959i 0.628216i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 31.1769i 0.997438i 0.866764 + 0.498719i \(0.166196\pi\)
−0.866764 + 0.498719i \(0.833804\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.00000i 0.127580i 0.997963 + 0.0637901i \(0.0203188\pi\)
−0.997963 + 0.0637901i \(0.979681\pi\)
\(984\) 0 0
\(985\) 18.0000 + 14.6969i 0.573528 + 0.468283i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 45.2548 1.43902
\(990\) 0 0
\(991\) −6.92820 −0.220082 −0.110041 0.993927i \(-0.535098\pi\)
−0.110041 + 0.993927i \(0.535098\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −9.79796 + 12.0000i −0.310616 + 0.380426i
\(996\) 0 0
\(997\) 14.6969i 0.465457i −0.972542 0.232728i \(-0.925235\pi\)
0.972542 0.232728i \(-0.0747653\pi\)
\(998\) 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 1440.2.f.j.289.6 yes 8
3.2 odd 2 inner 1440.2.f.j.289.4 yes 8
4.3 odd 2 inner 1440.2.f.j.289.5 yes 8
5.2 odd 4 7200.2.a.cs.1.1 4
5.3 odd 4 7200.2.a.ct.1.3 4
5.4 even 2 inner 1440.2.f.j.289.7 yes 8
8.3 odd 2 2880.2.f.x.1729.3 8
8.5 even 2 2880.2.f.x.1729.4 8
12.11 even 2 inner 1440.2.f.j.289.3 yes 8
15.2 even 4 7200.2.a.ct.1.2 4
15.8 even 4 7200.2.a.cs.1.4 4
15.14 odd 2 inner 1440.2.f.j.289.1 8
20.3 even 4 7200.2.a.cs.1.2 4
20.7 even 4 7200.2.a.ct.1.4 4
20.19 odd 2 inner 1440.2.f.j.289.8 yes 8
24.5 odd 2 2880.2.f.x.1729.6 8
24.11 even 2 2880.2.f.x.1729.5 8
40.19 odd 2 2880.2.f.x.1729.2 8
40.29 even 2 2880.2.f.x.1729.1 8
60.23 odd 4 7200.2.a.ct.1.1 4
60.47 odd 4 7200.2.a.cs.1.3 4
60.59 even 2 inner 1440.2.f.j.289.2 yes 8
120.29 odd 2 2880.2.f.x.1729.7 8
120.59 even 2 2880.2.f.x.1729.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1440.2.f.j.289.1 8 15.14 odd 2 inner
1440.2.f.j.289.2 yes 8 60.59 even 2 inner
1440.2.f.j.289.3 yes 8 12.11 even 2 inner
1440.2.f.j.289.4 yes 8 3.2 odd 2 inner
1440.2.f.j.289.5 yes 8 4.3 odd 2 inner
1440.2.f.j.289.6 yes 8 1.1 even 1 trivial
1440.2.f.j.289.7 yes 8 5.4 even 2 inner
1440.2.f.j.289.8 yes 8 20.19 odd 2 inner
2880.2.f.x.1729.1 8 40.29 even 2
2880.2.f.x.1729.2 8 40.19 odd 2
2880.2.f.x.1729.3 8 8.3 odd 2
2880.2.f.x.1729.4 8 8.5 even 2
2880.2.f.x.1729.5 8 24.11 even 2
2880.2.f.x.1729.6 8 24.5 odd 2
2880.2.f.x.1729.7 8 120.29 odd 2
2880.2.f.x.1729.8 8 120.59 even 2
7200.2.a.cs.1.1 4 5.2 odd 4
7200.2.a.cs.1.2 4 20.3 even 4
7200.2.a.cs.1.3 4 60.47 odd 4
7200.2.a.cs.1.4 4 15.8 even 4
7200.2.a.ct.1.1 4 60.23 odd 4
7200.2.a.ct.1.2 4 15.2 even 4
7200.2.a.ct.1.3 4 5.3 odd 4
7200.2.a.ct.1.4 4 20.7 even 4