Properties

Label 4032.2.j.f.2591.3
Level $4032$
Weight $2$
Character 4032.2591
Analytic conductor $32.196$
Analytic rank $0$
Dimension $16$
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] = [4032,2,Mod(2591,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("4032.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.j (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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.11007531417600000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.3
Root \(-0.418778 + 1.56290i\) of defining polynomial
Character \(\chi\) \(=\) 4032.2591
Dual form 4032.2.j.f.2591.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.96336 q^{5} +1.00000i q^{7} +O(q^{10})\) \(q-3.96336 q^{5} +1.00000i q^{7} +6.41285i q^{11} -5.60503i q^{13} -6.86474i q^{17} +3.46410 q^{19} -2.62210 q^{23} +10.7082 q^{25} -2.44949 q^{29} -2.00000i q^{31} -3.96336i q^{35} +5.60503i q^{37} -6.86474i q^{41} -7.74597 q^{43} -1.00000 q^{49} +0.578246 q^{53} -25.4164i q^{55} +9.79796i q^{59} -4.28187i q^{61} +22.2148i q^{65} +5.60503 q^{67} +11.1074 q^{71} +7.70820 q^{73} -6.41285 q^{77} -11.7082i q^{79} +12.8257i q^{83} +27.2074i q^{85} +6.86474i q^{89} +5.60503 q^{91} -13.7295 q^{95} +4.29180 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 64 q^{25} - 16 q^{49} + 16 q^{73} + 176 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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
\(5\) −3.96336 −1.77247 −0.886234 0.463238i \(-0.846687\pi\)
−0.886234 + 0.463238i \(0.846687\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.41285i 1.93355i 0.255636 + 0.966773i \(0.417715\pi\)
−0.255636 + 0.966773i \(0.582285\pi\)
\(12\) 0 0
\(13\) − 5.60503i − 1.55456i −0.629157 0.777278i \(-0.716600\pi\)
0.629157 0.777278i \(-0.283400\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.86474i − 1.66494i −0.554068 0.832472i \(-0.686925\pi\)
0.554068 0.832472i \(-0.313075\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.62210 −0.546745 −0.273372 0.961908i \(-0.588139\pi\)
−0.273372 + 0.961908i \(0.588139\pi\)
\(24\) 0 0
\(25\) 10.7082 2.14164
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.44949 −0.454859 −0.227429 0.973795i \(-0.573032\pi\)
−0.227429 + 0.973795i \(0.573032\pi\)
\(30\) 0 0
\(31\) − 2.00000i − 0.359211i −0.983739 0.179605i \(-0.942518\pi\)
0.983739 0.179605i \(-0.0574821\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 3.96336i − 0.669930i
\(36\) 0 0
\(37\) 5.60503i 0.921462i 0.887540 + 0.460731i \(0.152413\pi\)
−0.887540 + 0.460731i \(0.847587\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 6.86474i − 1.07209i −0.844189 0.536046i \(-0.819917\pi\)
0.844189 0.536046i \(-0.180083\pi\)
\(42\) 0 0
\(43\) −7.74597 −1.18125 −0.590624 0.806947i \(-0.701119\pi\)
−0.590624 + 0.806947i \(0.701119\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.578246 0.0794282 0.0397141 0.999211i \(-0.487355\pi\)
0.0397141 + 0.999211i \(0.487355\pi\)
\(54\) 0 0
\(55\) − 25.4164i − 3.42715i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.79796i 1.27559i 0.770208 + 0.637793i \(0.220152\pi\)
−0.770208 + 0.637793i \(0.779848\pi\)
\(60\) 0 0
\(61\) − 4.28187i − 0.548237i −0.961696 0.274118i \(-0.911614\pi\)
0.961696 0.274118i \(-0.0883860\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 22.2148i 2.75540i
\(66\) 0 0
\(67\) 5.60503 0.684764 0.342382 0.939561i \(-0.388766\pi\)
0.342382 + 0.939561i \(0.388766\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.1074 1.31820 0.659102 0.752054i \(-0.270937\pi\)
0.659102 + 0.752054i \(0.270937\pi\)
\(72\) 0 0
\(73\) 7.70820 0.902177 0.451089 0.892479i \(-0.351036\pi\)
0.451089 + 0.892479i \(0.351036\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.41285 −0.730812
\(78\) 0 0
\(79\) − 11.7082i − 1.31728i −0.752460 0.658638i \(-0.771133\pi\)
0.752460 0.658638i \(-0.228867\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.8257i 1.40780i 0.710298 + 0.703901i \(0.248560\pi\)
−0.710298 + 0.703901i \(0.751440\pi\)
\(84\) 0 0
\(85\) 27.2074i 2.95106i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.86474i 0.727661i 0.931465 + 0.363830i \(0.118531\pi\)
−0.931465 + 0.363830i \(0.881469\pi\)
\(90\) 0 0
\(91\) 5.60503 0.587567
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −13.7295 −1.40861
\(96\) 0 0
\(97\) 4.29180 0.435766 0.217883 0.975975i \(-0.430085\pi\)
0.217883 + 0.975975i \(0.430085\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.86234 −0.881836 −0.440918 0.897548i \(-0.645347\pi\)
−0.440918 + 0.897548i \(0.645347\pi\)
\(102\) 0 0
\(103\) 17.4164i 1.71609i 0.513575 + 0.858045i \(0.328321\pi\)
−0.513575 + 0.858045i \(0.671679\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.38511i 0.327251i 0.986523 + 0.163626i \(0.0523189\pi\)
−0.986523 + 0.163626i \(0.947681\pi\)
\(108\) 0 0
\(109\) − 7.74597i − 0.741929i −0.928647 0.370965i \(-0.879027\pi\)
0.928647 0.370965i \(-0.120973\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.7279i 1.19734i 0.800995 + 0.598671i \(0.204304\pi\)
−0.800995 + 0.598671i \(0.795696\pi\)
\(114\) 0 0
\(115\) 10.3923 0.969087
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.86474 0.629289
\(120\) 0 0
\(121\) −30.1246 −2.73860
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −22.6237 −2.02352
\(126\) 0 0
\(127\) 11.7082i 1.03894i 0.854490 + 0.519468i \(0.173870\pi\)
−0.854490 + 0.519468i \(0.826130\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.87124i 0.163491i 0.996653 + 0.0817457i \(0.0260495\pi\)
−0.996653 + 0.0817457i \(0.973950\pi\)
\(132\) 0 0
\(133\) 3.46410i 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.24264i 0.362473i 0.983440 + 0.181237i \(0.0580100\pi\)
−0.983440 + 0.181237i \(0.941990\pi\)
\(138\) 0 0
\(139\) −11.2101 −0.950826 −0.475413 0.879763i \(-0.657701\pi\)
−0.475413 + 0.879763i \(0.657701\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 35.9442 3.00581
\(144\) 0 0
\(145\) 9.70820 0.806222
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.4039 1.09809 0.549047 0.835792i \(-0.314991\pi\)
0.549047 + 0.835792i \(0.314991\pi\)
\(150\) 0 0
\(151\) 15.4164i 1.25457i 0.778790 + 0.627285i \(0.215834\pi\)
−0.778790 + 0.627285i \(0.784166\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.92672i 0.636689i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 2.62210i − 0.206650i
\(162\) 0 0
\(163\) 8.25137 0.646297 0.323149 0.946348i \(-0.395259\pi\)
0.323149 + 0.946348i \(0.395259\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.7295 −1.06242 −0.531209 0.847241i \(-0.678262\pi\)
−0.531209 + 0.847241i \(0.678262\pi\)
\(168\) 0 0
\(169\) −18.4164 −1.41665
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.83460 −0.443597 −0.221798 0.975093i \(-0.571193\pi\)
−0.221798 + 0.975093i \(0.571193\pi\)
\(174\) 0 0
\(175\) 10.7082i 0.809464i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 1.51387i − 0.113152i −0.998398 0.0565759i \(-0.981982\pi\)
0.998398 0.0565759i \(-0.0180183\pi\)
\(180\) 0 0
\(181\) 23.7433i 1.76483i 0.470476 + 0.882413i \(0.344082\pi\)
−0.470476 + 0.882413i \(0.655918\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 22.2148i − 1.63326i
\(186\) 0 0
\(187\) 44.0225 3.21924
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.86319 0.424245 0.212123 0.977243i \(-0.431962\pi\)
0.212123 + 0.977243i \(0.431962\pi\)
\(192\) 0 0
\(193\) 23.7082 1.70655 0.853277 0.521458i \(-0.174612\pi\)
0.853277 + 0.521458i \(0.174612\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.29300 −0.0921223 −0.0460611 0.998939i \(-0.514667\pi\)
−0.0460611 + 0.998939i \(0.514667\pi\)
\(198\) 0 0
\(199\) 11.4164i 0.809288i 0.914474 + 0.404644i \(0.132605\pi\)
−0.914474 + 0.404644i \(0.867395\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 2.44949i − 0.171920i
\(204\) 0 0
\(205\) 27.2074i 1.90025i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 22.2148i 1.53663i
\(210\) 0 0
\(211\) 18.9560 1.30499 0.652494 0.757794i \(-0.273723\pi\)
0.652494 + 0.757794i \(0.273723\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 30.7000 2.09373
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −38.4771 −2.58825
\(222\) 0 0
\(223\) − 4.00000i − 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 4.18423i − 0.277717i −0.990312 0.138858i \(-0.955657\pi\)
0.990312 0.138858i \(-0.0443433\pi\)
\(228\) 0 0
\(229\) − 1.32317i − 0.0874375i −0.999044 0.0437187i \(-0.986079\pi\)
0.999044 0.0437187i \(-0.0139205\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.7310i 0.965062i 0.875879 + 0.482531i \(0.160282\pi\)
−0.875879 + 0.482531i \(0.839718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −11.1074 −0.718477 −0.359238 0.933246i \(-0.616963\pi\)
−0.359238 + 0.933246i \(0.616963\pi\)
\(240\) 0 0
\(241\) 19.7082 1.26952 0.634759 0.772711i \(-0.281100\pi\)
0.634759 + 0.772711i \(0.281100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.96336 0.253210
\(246\) 0 0
\(247\) − 19.4164i − 1.23544i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 18.8812i − 1.19177i −0.803070 0.595884i \(-0.796802\pi\)
0.803070 0.595884i \(-0.203198\pi\)
\(252\) 0 0
\(253\) − 16.8151i − 1.05716i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 3.62365i − 0.226037i −0.993593 0.113018i \(-0.963948\pi\)
0.993593 0.113018i \(-0.0360519\pi\)
\(258\) 0 0
\(259\) −5.60503 −0.348280
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 30.0810 1.85488 0.927438 0.373976i \(-0.122006\pi\)
0.927438 + 0.373976i \(0.122006\pi\)
\(264\) 0 0
\(265\) −2.29180 −0.140784
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 12.6048 0.768530 0.384265 0.923223i \(-0.374455\pi\)
0.384265 + 0.923223i \(0.374455\pi\)
\(270\) 0 0
\(271\) 2.00000i 0.121491i 0.998153 + 0.0607457i \(0.0193479\pi\)
−0.998153 + 0.0607457i \(0.980652\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 68.6701i 4.14096i
\(276\) 0 0
\(277\) 29.8537i 1.79374i 0.442297 + 0.896869i \(0.354164\pi\)
−0.442297 + 0.896869i \(0.645836\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.48683i 0.565937i 0.959129 + 0.282969i \(0.0913192\pi\)
−0.959129 + 0.282969i \(0.908681\pi\)
\(282\) 0 0
\(283\) 28.5306 1.69597 0.847983 0.530023i \(-0.177817\pi\)
0.847983 + 0.530023i \(0.177817\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.86474 0.405213
\(288\) 0 0
\(289\) −30.1246 −1.77204
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.09211 0.122223 0.0611113 0.998131i \(-0.480536\pi\)
0.0611113 + 0.998131i \(0.480536\pi\)
\(294\) 0 0
\(295\) − 38.8328i − 2.26093i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 14.6969i 0.849946i
\(300\) 0 0
\(301\) − 7.74597i − 0.446470i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 16.9706i 0.971732i
\(306\) 0 0
\(307\) −25.8842 −1.47729 −0.738646 0.674094i \(-0.764534\pi\)
−0.738646 + 0.674094i \(0.764534\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.4884 −0.594742 −0.297371 0.954762i \(-0.596110\pi\)
−0.297371 + 0.954762i \(0.596110\pi\)
\(312\) 0 0
\(313\) 17.4164 0.984434 0.492217 0.870473i \(-0.336187\pi\)
0.492217 + 0.870473i \(0.336187\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.3029 1.02799 0.513997 0.857792i \(-0.328164\pi\)
0.513997 + 0.857792i \(0.328164\pi\)
\(318\) 0 0
\(319\) − 15.7082i − 0.879491i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 23.7801i − 1.32316i
\(324\) 0 0
\(325\) − 60.0198i − 3.32930i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −16.3097 −0.896462 −0.448231 0.893918i \(-0.647946\pi\)
−0.448231 + 0.893918i \(0.647946\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −22.2148 −1.21372
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 12.8257 0.694550
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 16.2108i − 0.870242i −0.900372 0.435121i \(-0.856706\pi\)
0.900372 0.435121i \(-0.143294\pi\)
\(348\) 0 0
\(349\) 25.0665i 1.34178i 0.741558 + 0.670889i \(0.234087\pi\)
−0.741558 + 0.670889i \(0.765913\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.3500i 0.816999i 0.912759 + 0.408500i \(0.133948\pi\)
−0.912759 + 0.408500i \(0.866052\pi\)
\(354\) 0 0
\(355\) −44.0225 −2.33647
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −33.3221 −1.75867 −0.879337 0.476199i \(-0.842014\pi\)
−0.879337 + 0.476199i \(0.842014\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −30.5504 −1.59908
\(366\) 0 0
\(367\) − 0.583592i − 0.0304633i −0.999884 0.0152316i \(-0.995151\pi\)
0.999884 0.0152316i \(-0.00484857\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.578246i 0.0300210i
\(372\) 0 0
\(373\) − 13.8564i − 0.717458i −0.933442 0.358729i \(-0.883210\pi\)
0.933442 0.358729i \(-0.116790\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 13.7295i 0.707104i
\(378\) 0 0
\(379\) −21.6024 −1.10964 −0.554820 0.831971i \(-0.687213\pi\)
−0.554820 + 0.831971i \(0.687213\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.7295 0.701543 0.350772 0.936461i \(-0.385919\pi\)
0.350772 + 0.936461i \(0.385919\pi\)
\(384\) 0 0
\(385\) 25.4164 1.29534
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.3762 −0.526094 −0.263047 0.964783i \(-0.584728\pi\)
−0.263047 + 0.964783i \(0.584728\pi\)
\(390\) 0 0
\(391\) 18.0000i 0.910299i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 46.4038i 2.33483i
\(396\) 0 0
\(397\) 15.4919i 0.777518i 0.921340 + 0.388759i \(0.127096\pi\)
−0.921340 + 0.388759i \(0.872904\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 31.7016i − 1.58310i −0.611103 0.791551i \(-0.709274\pi\)
0.611103 0.791551i \(-0.290726\pi\)
\(402\) 0 0
\(403\) −11.2101 −0.558413
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −35.9442 −1.78169
\(408\) 0 0
\(409\) −16.2918 −0.805577 −0.402789 0.915293i \(-0.631959\pi\)
−0.402789 + 0.915293i \(0.631959\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −9.79796 −0.482126
\(414\) 0 0
\(415\) − 50.8328i − 2.49528i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 32.8634i 1.60548i 0.596329 + 0.802740i \(0.296625\pi\)
−0.596329 + 0.802740i \(0.703375\pi\)
\(420\) 0 0
\(421\) 7.43361i 0.362292i 0.983456 + 0.181146i \(0.0579806\pi\)
−0.983456 + 0.181146i \(0.942019\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 73.5090i − 3.56571i
\(426\) 0 0
\(427\) 4.28187 0.207214
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.62210 −0.126302 −0.0631510 0.998004i \(-0.520115\pi\)
−0.0631510 + 0.998004i \(0.520115\pi\)
\(432\) 0 0
\(433\) −9.41641 −0.452524 −0.226262 0.974067i \(-0.572651\pi\)
−0.226262 + 0.974067i \(0.572651\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −9.08321 −0.434509
\(438\) 0 0
\(439\) − 20.0000i − 0.954548i −0.878755 0.477274i \(-0.841625\pi\)
0.878755 0.477274i \(-0.158375\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 23.4228i − 1.11285i −0.830898 0.556425i \(-0.812173\pi\)
0.830898 0.556425i \(-0.187827\pi\)
\(444\) 0 0
\(445\) − 27.2074i − 1.28975i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 21.2132i 1.00111i 0.865704 + 0.500556i \(0.166871\pi\)
−0.865704 + 0.500556i \(0.833129\pi\)
\(450\) 0 0
\(451\) 44.0225 2.07294
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −22.2148 −1.04144
\(456\) 0 0
\(457\) 11.4164 0.534037 0.267019 0.963691i \(-0.413962\pi\)
0.267019 + 0.963691i \(0.413962\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.9178 0.694792 0.347396 0.937719i \(-0.387066\pi\)
0.347396 + 0.937719i \(0.387066\pi\)
\(462\) 0 0
\(463\) 4.29180i 0.199457i 0.995015 + 0.0997283i \(0.0317974\pi\)
−0.995015 + 0.0997283i \(0.968203\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.02774i 0.140107i 0.997543 + 0.0700535i \(0.0223170\pi\)
−0.997543 + 0.0700535i \(0.977683\pi\)
\(468\) 0 0
\(469\) 5.60503i 0.258816i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 49.6737i − 2.28400i
\(474\) 0 0
\(475\) 37.0943 1.70200
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 25.4558 1.16311 0.581554 0.813508i \(-0.302445\pi\)
0.581554 + 0.813508i \(0.302445\pi\)
\(480\) 0 0
\(481\) 31.4164 1.43246
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −17.0099 −0.772381
\(486\) 0 0
\(487\) 32.0000i 1.45006i 0.688718 + 0.725029i \(0.258174\pi\)
−0.688718 + 0.725029i \(0.741826\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.69810i 0.257151i 0.991700 + 0.128576i \(0.0410405\pi\)
−0.991700 + 0.128576i \(0.958959\pi\)
\(492\) 0 0
\(493\) 16.8151i 0.757314i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 11.1074i 0.498234i
\(498\) 0 0
\(499\) −28.5306 −1.27720 −0.638602 0.769537i \(-0.720487\pi\)
−0.638602 + 0.769537i \(0.720487\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 22.2148 0.990507 0.495253 0.868749i \(-0.335075\pi\)
0.495253 + 0.868749i \(0.335075\pi\)
\(504\) 0 0
\(505\) 35.1246 1.56302
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8.14759 −0.361135 −0.180568 0.983563i \(-0.557793\pi\)
−0.180568 + 0.983563i \(0.557793\pi\)
\(510\) 0 0
\(511\) 7.70820i 0.340991i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 69.0275i − 3.04171i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 27.0764i − 1.18624i −0.805115 0.593119i \(-0.797896\pi\)
0.805115 0.593119i \(-0.202104\pi\)
\(522\) 0 0
\(523\) −11.2101 −0.490182 −0.245091 0.969500i \(-0.578818\pi\)
−0.245091 + 0.969500i \(0.578818\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −13.7295 −0.598065
\(528\) 0 0
\(529\) −16.1246 −0.701070
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −38.4771 −1.66663
\(534\) 0 0
\(535\) − 13.4164i − 0.580042i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 6.41285i − 0.276221i
\(540\) 0 0
\(541\) − 0.505406i − 0.0217291i −0.999941 0.0108645i \(-0.996542\pi\)
0.999941 0.0108645i \(-0.00345836\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 30.7000i 1.31505i
\(546\) 0 0
\(547\) 21.0970 0.902041 0.451021 0.892514i \(-0.351060\pi\)
0.451021 + 0.892514i \(0.351060\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.48528 −0.361485
\(552\) 0 0
\(553\) 11.7082 0.497883
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.5604 0.616945 0.308473 0.951233i \(-0.400182\pi\)
0.308473 + 0.951233i \(0.400182\pi\)
\(558\) 0 0
\(559\) 43.4164i 1.83632i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 2.31298i − 0.0974807i −0.998811 0.0487403i \(-0.984479\pi\)
0.998811 0.0487403i \(-0.0155207\pi\)
\(564\) 0 0
\(565\) − 50.4453i − 2.12225i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 21.2132i − 0.889304i −0.895703 0.444652i \(-0.853327\pi\)
0.895703 0.444652i \(-0.146673\pi\)
\(570\) 0 0
\(571\) −10.8977 −0.456055 −0.228027 0.973655i \(-0.573228\pi\)
−0.228027 + 0.973655i \(0.573228\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −28.0779 −1.17093
\(576\) 0 0
\(577\) 5.41641 0.225488 0.112744 0.993624i \(-0.464036\pi\)
0.112744 + 0.993624i \(0.464036\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.8257 −0.532099
\(582\) 0 0
\(583\) 3.70820i 0.153578i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 21.4672i − 0.886045i −0.896511 0.443022i \(-0.853906\pi\)
0.896511 0.443022i \(-0.146094\pi\)
\(588\) 0 0
\(589\) − 6.92820i − 0.285472i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.3237i 1.40950i 0.709453 + 0.704752i \(0.248942\pi\)
−0.709453 + 0.704752i \(0.751058\pi\)
\(594\) 0 0
\(595\) −27.2074 −1.11539
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −0.618993 −0.0252914 −0.0126457 0.999920i \(-0.504025\pi\)
−0.0126457 + 0.999920i \(0.504025\pi\)
\(600\) 0 0
\(601\) −29.4164 −1.19992 −0.599960 0.800030i \(-0.704817\pi\)
−0.599960 + 0.800030i \(0.704817\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 119.395 4.85408
\(606\) 0 0
\(607\) 8.00000i 0.324710i 0.986732 + 0.162355i \(0.0519090\pi\)
−0.986732 + 0.162355i \(0.948091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 29.3483i 1.18537i 0.805435 + 0.592684i \(0.201932\pi\)
−0.805435 + 0.592684i \(0.798068\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 12.7279i − 0.512407i −0.966623 0.256203i \(-0.917528\pi\)
0.966623 0.256203i \(-0.0824717\pi\)
\(618\) 0 0
\(619\) −1.63553 −0.0657374 −0.0328687 0.999460i \(-0.510464\pi\)
−0.0328687 + 0.999460i \(0.510464\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.86474 −0.275030
\(624\) 0 0
\(625\) 36.1246 1.44498
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 38.4771 1.53418
\(630\) 0 0
\(631\) 38.5410i 1.53429i 0.641471 + 0.767147i \(0.278324\pi\)
−0.641471 + 0.767147i \(0.721676\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 46.4038i − 1.84148i
\(636\) 0 0
\(637\) 5.60503i 0.222080i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.9690i 0.630738i 0.948969 + 0.315369i \(0.102128\pi\)
−0.948969 + 0.315369i \(0.897872\pi\)
\(642\) 0 0
\(643\) 7.74597 0.305471 0.152736 0.988267i \(-0.451192\pi\)
0.152736 + 0.988267i \(0.451192\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.2179 0.952102 0.476051 0.879418i \(-0.342068\pi\)
0.476051 + 0.879418i \(0.342068\pi\)
\(648\) 0 0
\(649\) −62.8328 −2.46640
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 37.8988 1.48310 0.741548 0.670900i \(-0.234092\pi\)
0.741548 + 0.670900i \(0.234092\pi\)
\(654\) 0 0
\(655\) − 7.41641i − 0.289783i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 45.6047i − 1.77651i −0.459354 0.888253i \(-0.651919\pi\)
0.459354 0.888253i \(-0.348081\pi\)
\(660\) 0 0
\(661\) − 48.4974i − 1.88633i −0.332323 0.943166i \(-0.607832\pi\)
0.332323 0.943166i \(-0.392168\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 13.7295i − 0.532406i
\(666\) 0 0
\(667\) 6.42280 0.248692
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 27.4589 1.06004
\(672\) 0 0
\(673\) 24.2918 0.936380 0.468190 0.883628i \(-0.344906\pi\)
0.468190 + 0.883628i \(0.344906\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −25.8723 −0.994352 −0.497176 0.867650i \(-0.665630\pi\)
−0.497176 + 0.867650i \(0.665630\pi\)
\(678\) 0 0
\(679\) 4.29180i 0.164704i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 23.4228i 0.896247i 0.893972 + 0.448124i \(0.147908\pi\)
−0.893972 + 0.448124i \(0.852092\pi\)
\(684\) 0 0
\(685\) − 16.8151i − 0.642472i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 3.24109i − 0.123476i
\(690\) 0 0
\(691\) 23.4309 0.891355 0.445678 0.895194i \(-0.352963\pi\)
0.445678 + 0.895194i \(0.352963\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 44.4295 1.68531
\(696\) 0 0
\(697\) −47.1246 −1.78497
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0.578246 0.0218401 0.0109200 0.999940i \(-0.496524\pi\)
0.0109200 + 0.999940i \(0.496524\pi\)
\(702\) 0 0
\(703\) 19.4164i 0.732304i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 8.86234i − 0.333302i
\(708\) 0 0
\(709\) − 35.4588i − 1.33168i −0.746093 0.665841i \(-0.768073\pi\)
0.746093 0.665841i \(-0.231927\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.24419i 0.196397i
\(714\) 0 0
\(715\) −142.460 −5.32770
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −47.6706 −1.77781 −0.888907 0.458088i \(-0.848534\pi\)
−0.888907 + 0.458088i \(0.848534\pi\)
\(720\) 0 0
\(721\) −17.4164 −0.648621
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −26.2296 −0.974144
\(726\) 0 0
\(727\) 28.8328i 1.06935i 0.845058 + 0.534675i \(0.179566\pi\)
−0.845058 + 0.534675i \(0.820434\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 53.1740i 1.96671i
\(732\) 0 0
\(733\) − 1.32317i − 0.0488724i −0.999701 0.0244362i \(-0.992221\pi\)
0.999701 0.0244362i \(-0.00777905\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 35.9442i 1.32402i
\(738\) 0 0
\(739\) 11.5224 0.423859 0.211930 0.977285i \(-0.432025\pi\)
0.211930 + 0.977285i \(0.432025\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.86319 −0.215099 −0.107550 0.994200i \(-0.534300\pi\)
−0.107550 + 0.994200i \(0.534300\pi\)
\(744\) 0 0
\(745\) −53.1246 −1.94634
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3.38511 −0.123689
\(750\) 0 0
\(751\) 11.4164i 0.416591i 0.978066 + 0.208295i \(0.0667915\pi\)
−0.978066 + 0.208295i \(0.933208\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 61.1007i − 2.22368i
\(756\) 0 0
\(757\) 18.9560i 0.688969i 0.938792 + 0.344484i \(0.111946\pi\)
−0.938792 + 0.344484i \(0.888054\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 36.3268i 1.31685i 0.752649 + 0.658423i \(0.228776\pi\)
−0.752649 + 0.658423i \(0.771224\pi\)
\(762\) 0 0
\(763\) 7.74597 0.280423
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 54.9179 1.98297
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 32.2007 1.15818 0.579090 0.815264i \(-0.303408\pi\)
0.579090 + 0.815264i \(0.303408\pi\)
\(774\) 0 0
\(775\) − 21.4164i − 0.769300i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 23.7801i − 0.852012i
\(780\) 0 0
\(781\) 71.2299i 2.54881i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −4.28187 −0.152632 −0.0763160 0.997084i \(-0.524316\pi\)
−0.0763160 + 0.997084i \(0.524316\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.7279 −0.452553
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 13.7613 0.487451 0.243725 0.969844i \(-0.421630\pi\)
0.243725 + 0.969844i \(0.421630\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 49.4315i 1.74440i
\(804\) 0 0
\(805\) 10.3923i 0.366281i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 47.4342i 1.66770i 0.551994 + 0.833848i \(0.313867\pi\)
−0.551994 + 0.833848i \(0.686133\pi\)
\(810\) 0 0
\(811\) −7.93901 −0.278777 −0.139388 0.990238i \(-0.544514\pi\)
−0.139388 + 0.990238i \(0.544514\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −32.7031 −1.14554
\(816\) 0 0
\(817\) −26.8328 −0.938761
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.16424 0.110433 0.0552164 0.998474i \(-0.482415\pi\)
0.0552164 + 0.998474i \(0.482415\pi\)
\(822\) 0 0
\(823\) 11.7082i 0.408122i 0.978958 + 0.204061i \(0.0654141\pi\)
−0.978958 + 0.204061i \(0.934586\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 15.0543i − 0.523490i −0.965137 0.261745i \(-0.915702\pi\)
0.965137 0.261745i \(-0.0842979\pi\)
\(828\) 0 0
\(829\) 15.8043i 0.548906i 0.961601 + 0.274453i \(0.0884967\pi\)
−0.961601 + 0.274453i \(0.911503\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6.86474i 0.237849i
\(834\) 0 0
\(835\) 54.4148 1.88310
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −39.1853 −1.35283 −0.676414 0.736522i \(-0.736467\pi\)
−0.676414 + 0.736522i \(0.736467\pi\)
\(840\) 0 0
\(841\) −23.0000 −0.793103
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 72.9908 2.51096
\(846\) 0 0
\(847\) − 30.1246i − 1.03509i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 14.6969i − 0.503805i
\(852\) 0 0
\(853\) − 5.91739i − 0.202608i −0.994856 0.101304i \(-0.967699\pi\)
0.994856 0.101304i \(-0.0323014\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 35.5617i − 1.21476i −0.794410 0.607382i \(-0.792220\pi\)
0.794410 0.607382i \(-0.207780\pi\)
\(858\) 0 0
\(859\) 31.8016 1.08506 0.542529 0.840037i \(-0.317467\pi\)
0.542529 + 0.840037i \(0.317467\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 22.8337 0.777270 0.388635 0.921392i \(-0.372947\pi\)
0.388635 + 0.921392i \(0.372947\pi\)
\(864\) 0 0
\(865\) 23.1246 0.786260
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 75.0829 2.54701
\(870\) 0 0
\(871\) − 31.4164i − 1.06450i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 22.6237i − 0.764819i
\(876\) 0 0
\(877\) − 50.1329i − 1.69287i −0.532493 0.846435i \(-0.678745\pi\)
0.532493 0.846435i \(-0.321255\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 14.1120i − 0.475446i −0.971333 0.237723i \(-0.923599\pi\)
0.971333 0.237723i \(-0.0764011\pi\)
\(882\) 0 0
\(883\) 43.0117 1.44746 0.723729 0.690084i \(-0.242426\pi\)
0.723729 + 0.690084i \(0.242426\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 37.1822 1.24846 0.624228 0.781242i \(-0.285414\pi\)
0.624228 + 0.781242i \(0.285414\pi\)
\(888\) 0 0
\(889\) −11.7082 −0.392681
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 6.00000i 0.200558i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.89898i 0.163390i
\(900\) 0 0
\(901\) − 3.96951i − 0.132243i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 94.1032i − 3.12810i
\(906\) 0 0
\(907\) −34.4480 −1.14383 −0.571913 0.820314i \(-0.693799\pi\)
−0.571913 + 0.820314i \(0.693799\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.86008 −0.127890 −0.0639451 0.997953i \(-0.520368\pi\)
−0.0639451 + 0.997953i \(0.520368\pi\)
\(912\) 0 0
\(913\) −82.2492 −2.72205
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.87124 −0.0617939
\(918\) 0 0
\(919\) 27.4164i 0.904384i 0.891921 + 0.452192i \(0.149358\pi\)
−0.891921 + 0.452192i \(0.850642\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 62.2572i − 2.04922i
\(924\) 0 0
\(925\) 60.0198i 1.97344i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 51.2942i − 1.68291i −0.540327 0.841455i \(-0.681700\pi\)
0.540327 0.841455i \(-0.318300\pi\)
\(930\) 0 0
\(931\) −3.46410 −0.113531
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −174.477 −5.70601
\(936\) 0 0
\(937\) 2.58359 0.0844023 0.0422011 0.999109i \(-0.486563\pi\)
0.0422011 + 0.999109i \(0.486563\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 6.54935 0.213503 0.106751 0.994286i \(-0.465955\pi\)
0.106751 + 0.994286i \(0.465955\pi\)
\(942\) 0 0
\(943\) 18.0000i 0.586161i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1.51387i − 0.0491941i −0.999697 0.0245970i \(-0.992170\pi\)
0.999697 0.0245970i \(-0.00783027\pi\)
\(948\) 0 0
\(949\) − 43.2047i − 1.40249i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 28.4605i − 0.921926i −0.887419 0.460963i \(-0.847504\pi\)
0.887419 0.460963i \(-0.152496\pi\)
\(954\) 0 0
\(955\) −23.2379 −0.751961
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.24264 −0.137002
\(960\) 0 0
\(961\) 27.0000 0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −93.9641 −3.02481
\(966\) 0 0
\(967\) 55.7082i 1.79146i 0.444603 + 0.895728i \(0.353345\pi\)
−0.444603 + 0.895728i \(0.646655\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 30.5504i 0.980408i 0.871608 + 0.490204i \(0.163078\pi\)
−0.871608 + 0.490204i \(0.836922\pi\)
\(972\) 0 0
\(973\) − 11.2101i − 0.359378i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26.4574i 0.846447i 0.906025 + 0.423224i \(0.139102\pi\)
−0.906025 + 0.423224i \(0.860898\pi\)
\(978\) 0 0
\(979\) −44.0225 −1.40697
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 42.4264 1.35319 0.676596 0.736354i \(-0.263454\pi\)
0.676596 + 0.736354i \(0.263454\pi\)
\(984\) 0 0
\(985\) 5.12461 0.163284
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 20.3107 0.645842
\(990\) 0 0
\(991\) 42.8328i 1.36063i 0.732920 + 0.680315i \(0.238157\pi\)
−0.732920 + 0.680315i \(0.761843\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 45.2473i − 1.43444i
\(996\) 0 0
\(997\) − 2.64634i − 0.0838104i −0.999122 0.0419052i \(-0.986657\pi\)
0.999122 0.0419052i \(-0.0133427\pi\)
\(998\) 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 4032.2.j.f.2591.3 yes 16
3.2 odd 2 inner 4032.2.j.f.2591.15 yes 16
4.3 odd 2 inner 4032.2.j.f.2591.1 16
8.3 odd 2 inner 4032.2.j.f.2591.14 yes 16
8.5 even 2 inner 4032.2.j.f.2591.16 yes 16
12.11 even 2 inner 4032.2.j.f.2591.13 yes 16
24.5 odd 2 inner 4032.2.j.f.2591.4 yes 16
24.11 even 2 inner 4032.2.j.f.2591.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4032.2.j.f.2591.1 16 4.3 odd 2 inner
4032.2.j.f.2591.2 yes 16 24.11 even 2 inner
4032.2.j.f.2591.3 yes 16 1.1 even 1 trivial
4032.2.j.f.2591.4 yes 16 24.5 odd 2 inner
4032.2.j.f.2591.13 yes 16 12.11 even 2 inner
4032.2.j.f.2591.14 yes 16 8.3 odd 2 inner
4032.2.j.f.2591.15 yes 16 3.2 odd 2 inner
4032.2.j.f.2591.16 yes 16 8.5 even 2 inner