Properties

Label 1152.3.g.f.127.1
Level $1152$
Weight $3$
Character 1152.127
Analytic conductor $31.390$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,3,Mod(127,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.g (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: \(31.3897264543\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{21} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1152.127
Dual form 1152.3.g.f.127.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.29253 q^{5} -2.75787i q^{7} +O(q^{10})\) \(q-4.29253 q^{5} -2.75787i q^{7} +13.7980i q^{11} +14.5266 q^{13} -22.8673 q^{17} -16.0399i q^{19} -17.1117i q^{23} -6.57420 q^{25} +21.8667 q^{29} +38.6944i q^{31} +11.8383i q^{35} +66.4204 q^{37} -23.2392 q^{41} -47.9230i q^{43} -14.8512i q^{47} +41.3941 q^{49} -65.5589 q^{53} -59.2281i q^{55} -65.8428i q^{59} -40.1123 q^{61} -62.3559 q^{65} -74.8105i q^{67} -122.681i q^{71} -144.904 q^{73} +38.0530 q^{77} -128.657i q^{79} +22.0417i q^{83} +98.1584 q^{85} +122.075 q^{89} -40.0626i q^{91} +68.8516i q^{95} -88.3072 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{5} + 48 q^{13} - 16 q^{17} - 8 q^{25} + 80 q^{29} - 16 q^{37} - 80 q^{41} - 88 q^{49} - 176 q^{53} - 272 q^{61} + 160 q^{65} - 16 q^{73} - 320 q^{77} + 32 q^{85} + 240 q^{89} + 400 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) −4.29253 −0.858506 −0.429253 0.903184i \(-0.641223\pi\)
−0.429253 + 0.903184i \(0.641223\pi\)
\(6\) 0 0
\(7\) − 2.75787i − 0.393982i −0.980405 0.196991i \(-0.936883\pi\)
0.980405 0.196991i \(-0.0631170\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 13.7980i 1.25436i 0.778874 + 0.627180i \(0.215791\pi\)
−0.778874 + 0.627180i \(0.784209\pi\)
\(12\) 0 0
\(13\) 14.5266 1.11743 0.558716 0.829359i \(-0.311294\pi\)
0.558716 + 0.829359i \(0.311294\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −22.8673 −1.34513 −0.672567 0.740037i \(-0.734808\pi\)
−0.672567 + 0.740037i \(0.734808\pi\)
\(18\) 0 0
\(19\) − 16.0399i − 0.844204i −0.906548 0.422102i \(-0.861292\pi\)
0.906548 0.422102i \(-0.138708\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 17.1117i − 0.743986i −0.928236 0.371993i \(-0.878675\pi\)
0.928236 0.371993i \(-0.121325\pi\)
\(24\) 0 0
\(25\) −6.57420 −0.262968
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 21.8667 0.754025 0.377013 0.926208i \(-0.376951\pi\)
0.377013 + 0.926208i \(0.376951\pi\)
\(30\) 0 0
\(31\) 38.6944i 1.24821i 0.781341 + 0.624104i \(0.214536\pi\)
−0.781341 + 0.624104i \(0.785464\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 11.8383i 0.338236i
\(36\) 0 0
\(37\) 66.4204 1.79515 0.897573 0.440866i \(-0.145329\pi\)
0.897573 + 0.440866i \(0.145329\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −23.2392 −0.566809 −0.283405 0.959000i \(-0.591464\pi\)
−0.283405 + 0.959000i \(0.591464\pi\)
\(42\) 0 0
\(43\) − 47.9230i − 1.11449i −0.830349 0.557244i \(-0.811859\pi\)
0.830349 0.557244i \(-0.188141\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 14.8512i − 0.315983i −0.987441 0.157991i \(-0.949498\pi\)
0.987441 0.157991i \(-0.0505018\pi\)
\(48\) 0 0
\(49\) 41.3941 0.844778
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −65.5589 −1.23696 −0.618480 0.785800i \(-0.712251\pi\)
−0.618480 + 0.785800i \(0.712251\pi\)
\(54\) 0 0
\(55\) − 59.2281i − 1.07688i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 65.8428i − 1.11598i −0.829848 0.557990i \(-0.811573\pi\)
0.829848 0.557990i \(-0.188427\pi\)
\(60\) 0 0
\(61\) −40.1123 −0.657579 −0.328790 0.944403i \(-0.606641\pi\)
−0.328790 + 0.944403i \(0.606641\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −62.3559 −0.959321
\(66\) 0 0
\(67\) − 74.8105i − 1.11657i −0.829648 0.558287i \(-0.811459\pi\)
0.829648 0.558287i \(-0.188541\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 122.681i − 1.72790i −0.503578 0.863950i \(-0.667983\pi\)
0.503578 0.863950i \(-0.332017\pi\)
\(72\) 0 0
\(73\) −144.904 −1.98498 −0.992492 0.122312i \(-0.960969\pi\)
−0.992492 + 0.122312i \(0.960969\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 38.0530 0.494195
\(78\) 0 0
\(79\) − 128.657i − 1.62857i −0.580467 0.814284i \(-0.697130\pi\)
0.580467 0.814284i \(-0.302870\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 22.0417i 0.265563i 0.991145 + 0.132781i \(0.0423908\pi\)
−0.991145 + 0.132781i \(0.957609\pi\)
\(84\) 0 0
\(85\) 98.1584 1.15480
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 122.075 1.37163 0.685813 0.727778i \(-0.259447\pi\)
0.685813 + 0.727778i \(0.259447\pi\)
\(90\) 0 0
\(91\) − 40.0626i − 0.440248i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 68.8516i 0.724754i
\(96\) 0 0
\(97\) −88.3072 −0.910384 −0.455192 0.890393i \(-0.650429\pi\)
−0.455192 + 0.890393i \(0.650429\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 104.345 1.03312 0.516560 0.856251i \(-0.327212\pi\)
0.516560 + 0.856251i \(0.327212\pi\)
\(102\) 0 0
\(103\) − 69.0609i − 0.670494i −0.942130 0.335247i \(-0.891180\pi\)
0.942130 0.335247i \(-0.108820\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 149.429i − 1.39653i −0.715839 0.698265i \(-0.753956\pi\)
0.715839 0.698265i \(-0.246044\pi\)
\(108\) 0 0
\(109\) 54.3341 0.498478 0.249239 0.968442i \(-0.419820\pi\)
0.249239 + 0.968442i \(0.419820\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.32846 −0.0560041 −0.0280021 0.999608i \(-0.508914\pi\)
−0.0280021 + 0.999608i \(0.508914\pi\)
\(114\) 0 0
\(115\) 73.4523i 0.638716i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 63.0651i 0.529958i
\(120\) 0 0
\(121\) −69.3837 −0.573419
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 135.533 1.08427
\(126\) 0 0
\(127\) − 16.5228i − 0.130101i −0.997882 0.0650504i \(-0.979279\pi\)
0.997882 0.0650504i \(-0.0207208\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 50.9799i − 0.389159i −0.980887 0.194580i \(-0.937666\pi\)
0.980887 0.194580i \(-0.0623343\pi\)
\(132\) 0 0
\(133\) −44.2360 −0.332601
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.0157 −0.0731070 −0.0365535 0.999332i \(-0.511638\pi\)
−0.0365535 + 0.999332i \(0.511638\pi\)
\(138\) 0 0
\(139\) − 65.7088i − 0.472725i −0.971665 0.236363i \(-0.924045\pi\)
0.971665 0.236363i \(-0.0759553\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 200.438i 1.40166i
\(144\) 0 0
\(145\) −93.8635 −0.647335
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 94.7716 0.636051 0.318026 0.948082i \(-0.396980\pi\)
0.318026 + 0.948082i \(0.396980\pi\)
\(150\) 0 0
\(151\) − 269.769i − 1.78655i −0.449508 0.893276i \(-0.648401\pi\)
0.449508 0.893276i \(-0.351599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 166.097i − 1.07159i
\(156\) 0 0
\(157\) 31.5058 0.200674 0.100337 0.994954i \(-0.468008\pi\)
0.100337 + 0.994954i \(0.468008\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −47.1918 −0.293117
\(162\) 0 0
\(163\) 201.159i 1.23410i 0.786923 + 0.617051i \(0.211673\pi\)
−0.786923 + 0.617051i \(0.788327\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 266.393i − 1.59517i −0.603208 0.797584i \(-0.706111\pi\)
0.603208 0.797584i \(-0.293889\pi\)
\(168\) 0 0
\(169\) 42.0224 0.248653
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 43.8456 0.253443 0.126721 0.991938i \(-0.459555\pi\)
0.126721 + 0.991938i \(0.459555\pi\)
\(174\) 0 0
\(175\) 18.1308i 0.103605i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 275.778i 1.54066i 0.637646 + 0.770330i \(0.279908\pi\)
−0.637646 + 0.770330i \(0.720092\pi\)
\(180\) 0 0
\(181\) −79.0033 −0.436482 −0.218241 0.975895i \(-0.570032\pi\)
−0.218241 + 0.975895i \(0.570032\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −285.111 −1.54114
\(186\) 0 0
\(187\) − 315.522i − 1.68728i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 142.409i − 0.745598i −0.927912 0.372799i \(-0.878398\pi\)
0.927912 0.372799i \(-0.121602\pi\)
\(192\) 0 0
\(193\) −277.818 −1.43947 −0.719736 0.694248i \(-0.755737\pi\)
−0.719736 + 0.694248i \(0.755737\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.40848 −0.0173019 −0.00865095 0.999963i \(-0.502754\pi\)
−0.00865095 + 0.999963i \(0.502754\pi\)
\(198\) 0 0
\(199\) 314.323i 1.57951i 0.613421 + 0.789756i \(0.289793\pi\)
−0.613421 + 0.789756i \(0.710207\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 60.3057i − 0.297072i
\(204\) 0 0
\(205\) 99.7548 0.486609
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 221.317 1.05894
\(210\) 0 0
\(211\) 1.54933i 0.00734281i 0.999993 + 0.00367140i \(0.00116865\pi\)
−0.999993 + 0.00367140i \(0.998831\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 205.711i 0.956794i
\(216\) 0 0
\(217\) 106.714 0.491772
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −332.184 −1.50309
\(222\) 0 0
\(223\) 148.964i 0.668001i 0.942573 + 0.334000i \(0.108399\pi\)
−0.942573 + 0.334000i \(0.891601\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 237.803i 1.04759i 0.851844 + 0.523796i \(0.175485\pi\)
−0.851844 + 0.523796i \(0.824515\pi\)
\(228\) 0 0
\(229\) 70.0590 0.305935 0.152967 0.988231i \(-0.451117\pi\)
0.152967 + 0.988231i \(0.451117\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.18107 −0.0308200 −0.0154100 0.999881i \(-0.504905\pi\)
−0.0154100 + 0.999881i \(0.504905\pi\)
\(234\) 0 0
\(235\) 63.7491i 0.271273i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 216.620i 0.906359i 0.891419 + 0.453180i \(0.149710\pi\)
−0.891419 + 0.453180i \(0.850290\pi\)
\(240\) 0 0
\(241\) 385.001 1.59751 0.798757 0.601653i \(-0.205491\pi\)
0.798757 + 0.601653i \(0.205491\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −177.685 −0.725247
\(246\) 0 0
\(247\) − 233.005i − 0.943340i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 85.4882i 0.340590i 0.985393 + 0.170295i \(0.0544721\pi\)
−0.985393 + 0.170295i \(0.945528\pi\)
\(252\) 0 0
\(253\) 236.106 0.933226
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 54.2981 0.211277 0.105638 0.994405i \(-0.466311\pi\)
0.105638 + 0.994405i \(0.466311\pi\)
\(258\) 0 0
\(259\) − 183.179i − 0.707255i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 501.827i − 1.90809i −0.299670 0.954043i \(-0.596877\pi\)
0.299670 0.954043i \(-0.403123\pi\)
\(264\) 0 0
\(265\) 281.413 1.06194
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −284.911 −1.05915 −0.529574 0.848264i \(-0.677648\pi\)
−0.529574 + 0.848264i \(0.677648\pi\)
\(270\) 0 0
\(271\) 241.190i 0.889998i 0.895531 + 0.444999i \(0.146796\pi\)
−0.895531 + 0.444999i \(0.853204\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 90.7105i − 0.329856i
\(276\) 0 0
\(277\) −242.877 −0.876813 −0.438407 0.898777i \(-0.644457\pi\)
−0.438407 + 0.898777i \(0.644457\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −105.341 −0.374880 −0.187440 0.982276i \(-0.560019\pi\)
−0.187440 + 0.982276i \(0.560019\pi\)
\(282\) 0 0
\(283\) 223.867i 0.791049i 0.918455 + 0.395525i \(0.129437\pi\)
−0.918455 + 0.395525i \(0.870563\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 64.0907i 0.223313i
\(288\) 0 0
\(289\) 233.912 0.809384
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 421.057 1.43705 0.718527 0.695499i \(-0.244817\pi\)
0.718527 + 0.695499i \(0.244817\pi\)
\(294\) 0 0
\(295\) 282.632i 0.958075i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 248.575i − 0.831353i
\(300\) 0 0
\(301\) −132.166 −0.439088
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 172.183 0.564536
\(306\) 0 0
\(307\) − 122.865i − 0.400211i −0.979774 0.200106i \(-0.935871\pi\)
0.979774 0.200106i \(-0.0641285\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 437.535i − 1.40686i −0.710762 0.703432i \(-0.751650\pi\)
0.710762 0.703432i \(-0.248350\pi\)
\(312\) 0 0
\(313\) 375.413 1.19940 0.599702 0.800223i \(-0.295286\pi\)
0.599702 + 0.800223i \(0.295286\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −388.867 −1.22671 −0.613355 0.789808i \(-0.710180\pi\)
−0.613355 + 0.789808i \(0.710180\pi\)
\(318\) 0 0
\(319\) 301.716i 0.945819i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 366.788i 1.13557i
\(324\) 0 0
\(325\) −95.5008 −0.293849
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −40.9577 −0.124491
\(330\) 0 0
\(331\) − 15.4690i − 0.0467341i −0.999727 0.0233670i \(-0.992561\pi\)
0.999727 0.0233670i \(-0.00743864\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 321.126i 0.958585i
\(336\) 0 0
\(337\) −88.0105 −0.261159 −0.130579 0.991438i \(-0.541684\pi\)
−0.130579 + 0.991438i \(0.541684\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −533.904 −1.56570
\(342\) 0 0
\(343\) − 249.296i − 0.726810i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 183.589i 0.529076i 0.964375 + 0.264538i \(0.0852194\pi\)
−0.964375 + 0.264538i \(0.914781\pi\)
\(348\) 0 0
\(349\) −242.556 −0.695003 −0.347501 0.937679i \(-0.612970\pi\)
−0.347501 + 0.937679i \(0.612970\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.2850 −0.0291360 −0.0145680 0.999894i \(-0.504637\pi\)
−0.0145680 + 0.999894i \(0.504637\pi\)
\(354\) 0 0
\(355\) 526.611i 1.48341i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 137.976i 0.384334i 0.981362 + 0.192167i \(0.0615515\pi\)
−0.981362 + 0.192167i \(0.938448\pi\)
\(360\) 0 0
\(361\) 103.723 0.287320
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 622.004 1.70412
\(366\) 0 0
\(367\) 400.180i 1.09041i 0.838303 + 0.545205i \(0.183548\pi\)
−0.838303 + 0.545205i \(0.816452\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 180.803i 0.487340i
\(372\) 0 0
\(373\) −644.881 −1.72890 −0.864451 0.502717i \(-0.832334\pi\)
−0.864451 + 0.502717i \(0.832334\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 317.649 0.842571
\(378\) 0 0
\(379\) 485.900i 1.28206i 0.767517 + 0.641029i \(0.221492\pi\)
−0.767517 + 0.641029i \(0.778508\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 168.828i 0.440804i 0.975409 + 0.220402i \(0.0707370\pi\)
−0.975409 + 0.220402i \(0.929263\pi\)
\(384\) 0 0
\(385\) −163.344 −0.424270
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 192.592 0.495095 0.247548 0.968876i \(-0.420375\pi\)
0.247548 + 0.968876i \(0.420375\pi\)
\(390\) 0 0
\(391\) 391.297i 1.00076i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 552.263i 1.39813i
\(396\) 0 0
\(397\) −62.0483 −0.156293 −0.0781465 0.996942i \(-0.524900\pi\)
−0.0781465 + 0.996942i \(0.524900\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −353.066 −0.880464 −0.440232 0.897884i \(-0.645104\pi\)
−0.440232 + 0.897884i \(0.645104\pi\)
\(402\) 0 0
\(403\) 562.099i 1.39479i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 916.466i 2.25176i
\(408\) 0 0
\(409\) −62.9725 −0.153967 −0.0769835 0.997032i \(-0.524529\pi\)
−0.0769835 + 0.997032i \(0.524529\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −181.586 −0.439676
\(414\) 0 0
\(415\) − 94.6146i − 0.227987i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 398.741i − 0.951650i −0.879540 0.475825i \(-0.842150\pi\)
0.879540 0.475825i \(-0.157850\pi\)
\(420\) 0 0
\(421\) 533.059 1.26617 0.633086 0.774081i \(-0.281788\pi\)
0.633086 + 0.774081i \(0.281788\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 150.334 0.353727
\(426\) 0 0
\(427\) 110.625i 0.259075i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 284.282i 0.659587i 0.944053 + 0.329793i \(0.106979\pi\)
−0.944053 + 0.329793i \(0.893021\pi\)
\(432\) 0 0
\(433\) −304.599 −0.703462 −0.351731 0.936101i \(-0.614407\pi\)
−0.351731 + 0.936101i \(0.614407\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −274.469 −0.628075
\(438\) 0 0
\(439\) − 204.615i − 0.466094i −0.972465 0.233047i \(-0.925130\pi\)
0.972465 0.233047i \(-0.0748696\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 625.828i − 1.41270i −0.707861 0.706352i \(-0.750340\pi\)
0.707861 0.706352i \(-0.249660\pi\)
\(444\) 0 0
\(445\) −524.009 −1.17755
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 557.532 1.24172 0.620860 0.783921i \(-0.286783\pi\)
0.620860 + 0.783921i \(0.286783\pi\)
\(450\) 0 0
\(451\) − 320.653i − 0.710983i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 171.970i 0.377955i
\(456\) 0 0
\(457\) −312.142 −0.683024 −0.341512 0.939877i \(-0.610939\pi\)
−0.341512 + 0.939877i \(0.610939\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 603.187 1.30843 0.654216 0.756308i \(-0.272999\pi\)
0.654216 + 0.756308i \(0.272999\pi\)
\(462\) 0 0
\(463\) − 707.866i − 1.52887i −0.644702 0.764434i \(-0.723019\pi\)
0.644702 0.764434i \(-0.276981\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 195.220i − 0.418031i −0.977912 0.209015i \(-0.932974\pi\)
0.977912 0.209015i \(-0.0670259\pi\)
\(468\) 0 0
\(469\) −206.318 −0.439910
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 661.239 1.39797
\(474\) 0 0
\(475\) 105.449i 0.221998i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 233.729i − 0.487952i −0.969781 0.243976i \(-0.921548\pi\)
0.969781 0.243976i \(-0.0784517\pi\)
\(480\) 0 0
\(481\) 964.863 2.00595
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 379.061 0.781569
\(486\) 0 0
\(487\) 224.058i 0.460078i 0.973181 + 0.230039i \(0.0738854\pi\)
−0.973181 + 0.230039i \(0.926115\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 194.772i − 0.396684i −0.980133 0.198342i \(-0.936444\pi\)
0.980133 0.198342i \(-0.0635556\pi\)
\(492\) 0 0
\(493\) −500.032 −1.01426
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −338.339 −0.680762
\(498\) 0 0
\(499\) 99.7462i 0.199892i 0.994993 + 0.0999461i \(0.0318670\pi\)
−0.994993 + 0.0999461i \(0.968133\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 772.172i 1.53513i 0.640969 + 0.767567i \(0.278533\pi\)
−0.640969 + 0.767567i \(0.721467\pi\)
\(504\) 0 0
\(505\) −447.904 −0.886939
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 292.306 0.574276 0.287138 0.957889i \(-0.407296\pi\)
0.287138 + 0.957889i \(0.407296\pi\)
\(510\) 0 0
\(511\) 399.627i 0.782048i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 296.446i 0.575623i
\(516\) 0 0
\(517\) 204.916 0.396356
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 412.035 0.790853 0.395427 0.918498i \(-0.370597\pi\)
0.395427 + 0.918498i \(0.370597\pi\)
\(522\) 0 0
\(523\) − 124.693i − 0.238420i −0.992869 0.119210i \(-0.961964\pi\)
0.992869 0.119210i \(-0.0380361\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 884.836i − 1.67901i
\(528\) 0 0
\(529\) 236.191 0.446486
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −337.586 −0.633370
\(534\) 0 0
\(535\) 641.427i 1.19893i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 571.154i 1.05966i
\(540\) 0 0
\(541\) −572.988 −1.05913 −0.529564 0.848270i \(-0.677644\pi\)
−0.529564 + 0.848270i \(0.677644\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −233.231 −0.427946
\(546\) 0 0
\(547\) 682.433i 1.24759i 0.781587 + 0.623796i \(0.214410\pi\)
−0.781587 + 0.623796i \(0.785590\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 350.739i − 0.636551i
\(552\) 0 0
\(553\) −354.820 −0.641627
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −856.125 −1.53703 −0.768515 0.639832i \(-0.779004\pi\)
−0.768515 + 0.639832i \(0.779004\pi\)
\(558\) 0 0
\(559\) − 696.158i − 1.24536i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 680.745i − 1.20914i −0.796552 0.604570i \(-0.793345\pi\)
0.796552 0.604570i \(-0.206655\pi\)
\(564\) 0 0
\(565\) 27.1651 0.0480798
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −343.401 −0.603516 −0.301758 0.953384i \(-0.597574\pi\)
−0.301758 + 0.953384i \(0.597574\pi\)
\(570\) 0 0
\(571\) − 329.137i − 0.576422i −0.957567 0.288211i \(-0.906940\pi\)
0.957567 0.288211i \(-0.0930604\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 112.495i 0.195644i
\(576\) 0 0
\(577\) 734.738 1.27338 0.636688 0.771122i \(-0.280304\pi\)
0.636688 + 0.771122i \(0.280304\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 60.7883 0.104627
\(582\) 0 0
\(583\) − 904.579i − 1.55159i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 433.815i 0.739037i 0.929223 + 0.369519i \(0.120477\pi\)
−0.929223 + 0.369519i \(0.879523\pi\)
\(588\) 0 0
\(589\) 620.654 1.05374
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 425.075 0.716822 0.358411 0.933564i \(-0.383319\pi\)
0.358411 + 0.933564i \(0.383319\pi\)
\(594\) 0 0
\(595\) − 270.709i − 0.454972i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 1048.73i − 1.75081i −0.483394 0.875403i \(-0.660596\pi\)
0.483394 0.875403i \(-0.339404\pi\)
\(600\) 0 0
\(601\) −478.816 −0.796699 −0.398349 0.917234i \(-0.630417\pi\)
−0.398349 + 0.917234i \(0.630417\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 297.831 0.492283
\(606\) 0 0
\(607\) − 18.7009i − 0.0308088i −0.999881 0.0154044i \(-0.995096\pi\)
0.999881 0.0154044i \(-0.00490356\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 215.737i − 0.353089i
\(612\) 0 0
\(613\) 314.964 0.513807 0.256904 0.966437i \(-0.417298\pi\)
0.256904 + 0.966437i \(0.417298\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −786.590 −1.27486 −0.637431 0.770507i \(-0.720003\pi\)
−0.637431 + 0.770507i \(0.720003\pi\)
\(618\) 0 0
\(619\) 505.431i 0.816528i 0.912864 + 0.408264i \(0.133866\pi\)
−0.912864 + 0.408264i \(0.866134\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 336.667i − 0.540396i
\(624\) 0 0
\(625\) −417.425 −0.667880
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1518.85 −2.41471
\(630\) 0 0
\(631\) 682.834i 1.08215i 0.840976 + 0.541073i \(0.181982\pi\)
−0.840976 + 0.541073i \(0.818018\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 70.9246i 0.111692i
\(636\) 0 0
\(637\) 601.316 0.943982
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −816.719 −1.27413 −0.637066 0.770809i \(-0.719852\pi\)
−0.637066 + 0.770809i \(0.719852\pi\)
\(642\) 0 0
\(643\) − 1164.05i − 1.81034i −0.425045 0.905172i \(-0.639742\pi\)
0.425045 0.905172i \(-0.360258\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 471.519i 0.728778i 0.931247 + 0.364389i \(0.118722\pi\)
−0.931247 + 0.364389i \(0.881278\pi\)
\(648\) 0 0
\(649\) 908.496 1.39984
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1187.25 1.81814 0.909071 0.416642i \(-0.136793\pi\)
0.909071 + 0.416642i \(0.136793\pi\)
\(654\) 0 0
\(655\) 218.833i 0.334096i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 245.640i − 0.372747i −0.982479 0.186373i \(-0.940327\pi\)
0.982479 0.186373i \(-0.0596734\pi\)
\(660\) 0 0
\(661\) 1244.70 1.88305 0.941527 0.336936i \(-0.109391\pi\)
0.941527 + 0.336936i \(0.109391\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 189.884 0.285540
\(666\) 0 0
\(667\) − 374.176i − 0.560984i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 553.468i − 0.824841i
\(672\) 0 0
\(673\) −852.278 −1.26639 −0.633193 0.773994i \(-0.718256\pi\)
−0.633193 + 0.773994i \(0.718256\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 555.765 0.820923 0.410461 0.911878i \(-0.365368\pi\)
0.410461 + 0.911878i \(0.365368\pi\)
\(678\) 0 0
\(679\) 243.540i 0.358675i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 335.241i − 0.490836i −0.969417 0.245418i \(-0.921075\pi\)
0.969417 0.245418i \(-0.0789253\pi\)
\(684\) 0 0
\(685\) 42.9925 0.0627628
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −952.349 −1.38222
\(690\) 0 0
\(691\) − 1274.20i − 1.84400i −0.387192 0.921999i \(-0.626555\pi\)
0.387192 0.921999i \(-0.373445\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 282.057i 0.405837i
\(696\) 0 0
\(697\) 531.416 0.762434
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −676.056 −0.964416 −0.482208 0.876057i \(-0.660165\pi\)
−0.482208 + 0.876057i \(0.660165\pi\)
\(702\) 0 0
\(703\) − 1065.37i − 1.51547i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 287.771i − 0.407031i
\(708\) 0 0
\(709\) −62.3418 −0.0879292 −0.0439646 0.999033i \(-0.513999\pi\)
−0.0439646 + 0.999033i \(0.513999\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 662.127 0.928649
\(714\) 0 0
\(715\) − 860.384i − 1.20333i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 322.341i − 0.448318i −0.974553 0.224159i \(-0.928036\pi\)
0.974553 0.224159i \(-0.0719636\pi\)
\(720\) 0 0
\(721\) −190.461 −0.264163
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −143.756 −0.198284
\(726\) 0 0
\(727\) 368.070i 0.506287i 0.967429 + 0.253143i \(0.0814644\pi\)
−0.967429 + 0.253143i \(0.918536\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1095.87i 1.49913i
\(732\) 0 0
\(733\) 430.587 0.587431 0.293715 0.955893i \(-0.405108\pi\)
0.293715 + 0.955893i \(0.405108\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1032.23 1.40059
\(738\) 0 0
\(739\) − 818.050i − 1.10697i −0.832859 0.553485i \(-0.813298\pi\)
0.832859 0.553485i \(-0.186702\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 685.438i 0.922528i 0.887263 + 0.461264i \(0.152604\pi\)
−0.887263 + 0.461264i \(0.847396\pi\)
\(744\) 0 0
\(745\) −406.810 −0.546053
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −412.106 −0.550208
\(750\) 0 0
\(751\) − 104.336i − 0.138929i −0.997584 0.0694647i \(-0.977871\pi\)
0.997584 0.0694647i \(-0.0221291\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1157.99i 1.53377i
\(756\) 0 0
\(757\) −539.949 −0.713274 −0.356637 0.934243i \(-0.616077\pi\)
−0.356637 + 0.934243i \(0.616077\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1192.21 −1.56663 −0.783317 0.621623i \(-0.786474\pi\)
−0.783317 + 0.621623i \(0.786474\pi\)
\(762\) 0 0
\(763\) − 149.847i − 0.196391i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 956.473i − 1.24703i
\(768\) 0 0
\(769\) −321.506 −0.418083 −0.209042 0.977907i \(-0.567034\pi\)
−0.209042 + 0.977907i \(0.567034\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1453.29 1.88006 0.940029 0.341094i \(-0.110798\pi\)
0.940029 + 0.341094i \(0.110798\pi\)
\(774\) 0 0
\(775\) − 254.385i − 0.328239i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 372.753i 0.478502i
\(780\) 0 0
\(781\) 1692.75 2.16741
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −135.240 −0.172280
\(786\) 0 0
\(787\) − 231.223i − 0.293802i −0.989151 0.146901i \(-0.953070\pi\)
0.989151 0.146901i \(-0.0469299\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 17.4531i 0.0220646i
\(792\) 0 0
\(793\) −582.696 −0.734800
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −51.5068 −0.0646259 −0.0323129 0.999478i \(-0.510287\pi\)
−0.0323129 + 0.999478i \(0.510287\pi\)
\(798\) 0 0
\(799\) 339.606i 0.425039i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 1999.38i − 2.48988i
\(804\) 0 0
\(805\) 202.572 0.251643
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −580.914 −0.718065 −0.359032 0.933325i \(-0.616893\pi\)
−0.359032 + 0.933325i \(0.616893\pi\)
\(810\) 0 0
\(811\) 1351.98i 1.66706i 0.552475 + 0.833529i \(0.313683\pi\)
−0.552475 + 0.833529i \(0.686317\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 863.480i − 1.05948i
\(816\) 0 0
\(817\) −768.678 −0.940855
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −179.511 −0.218650 −0.109325 0.994006i \(-0.534869\pi\)
−0.109325 + 0.994006i \(0.534869\pi\)
\(822\) 0 0
\(823\) − 665.446i − 0.808561i −0.914635 0.404281i \(-0.867522\pi\)
0.914635 0.404281i \(-0.132478\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 901.166i 1.08968i 0.838540 + 0.544840i \(0.183410\pi\)
−0.838540 + 0.544840i \(0.816590\pi\)
\(828\) 0 0
\(829\) −183.151 −0.220930 −0.110465 0.993880i \(-0.535234\pi\)
−0.110465 + 0.993880i \(0.535234\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −946.571 −1.13634
\(834\) 0 0
\(835\) 1143.50i 1.36946i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 846.082i 1.00844i 0.863575 + 0.504220i \(0.168220\pi\)
−0.863575 + 0.504220i \(0.831780\pi\)
\(840\) 0 0
\(841\) −362.846 −0.431446
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −180.382 −0.213470
\(846\) 0 0
\(847\) 191.351i 0.225917i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 1136.56i − 1.33556i
\(852\) 0 0
\(853\) −540.635 −0.633804 −0.316902 0.948458i \(-0.602643\pi\)
−0.316902 + 0.948458i \(0.602643\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1333.41 −1.55590 −0.777952 0.628323i \(-0.783742\pi\)
−0.777952 + 0.628323i \(0.783742\pi\)
\(858\) 0 0
\(859\) 439.034i 0.511099i 0.966796 + 0.255549i \(0.0822563\pi\)
−0.966796 + 0.255549i \(0.917744\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1164.41i 1.34926i 0.738155 + 0.674631i \(0.235697\pi\)
−0.738155 + 0.674631i \(0.764303\pi\)
\(864\) 0 0
\(865\) −188.208 −0.217582
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1775.20 2.04281
\(870\) 0 0
\(871\) − 1086.74i − 1.24769i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 373.783i − 0.427181i
\(876\) 0 0
\(877\) 404.456 0.461181 0.230591 0.973051i \(-0.425934\pi\)
0.230591 + 0.973051i \(0.425934\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 227.084 0.257757 0.128879 0.991660i \(-0.458862\pi\)
0.128879 + 0.991660i \(0.458862\pi\)
\(882\) 0 0
\(883\) − 673.730i − 0.763001i −0.924369 0.381501i \(-0.875407\pi\)
0.924369 0.381501i \(-0.124593\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1370.95i − 1.54560i −0.634648 0.772801i \(-0.718855\pi\)
0.634648 0.772801i \(-0.281145\pi\)
\(888\) 0 0
\(889\) −45.5678 −0.0512574
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −238.211 −0.266754
\(894\) 0 0
\(895\) − 1183.79i − 1.32267i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 846.121i 0.941180i
\(900\) 0 0
\(901\) 1499.15 1.66388
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 339.124 0.374723
\(906\) 0 0
\(907\) − 1469.46i − 1.62013i −0.586342 0.810063i \(-0.699433\pi\)
0.586342 0.810063i \(-0.300567\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 871.203i − 0.956315i −0.878274 0.478157i \(-0.841305\pi\)
0.878274 0.478157i \(-0.158695\pi\)
\(912\) 0 0
\(913\) −304.131 −0.333111
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −140.596 −0.153322
\(918\) 0 0
\(919\) 256.361i 0.278957i 0.990225 + 0.139478i \(0.0445426\pi\)
−0.990225 + 0.139478i \(0.955457\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 1782.14i − 1.93081i
\(924\) 0 0
\(925\) −436.661 −0.472066
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1399.93 1.50692 0.753462 0.657491i \(-0.228382\pi\)
0.753462 + 0.657491i \(0.228382\pi\)
\(930\) 0 0
\(931\) − 663.956i − 0.713165i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1354.39i 1.44854i
\(936\) 0 0
\(937\) 365.610 0.390192 0.195096 0.980784i \(-0.437498\pi\)
0.195096 + 0.980784i \(0.437498\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 407.498 0.433048 0.216524 0.976277i \(-0.430528\pi\)
0.216524 + 0.976277i \(0.430528\pi\)
\(942\) 0 0
\(943\) 397.661i 0.421698i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1150.64i − 1.21503i −0.794307 0.607516i \(-0.792166\pi\)
0.794307 0.607516i \(-0.207834\pi\)
\(948\) 0 0
\(949\) −2104.96 −2.21808
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1341.53 1.40769 0.703844 0.710355i \(-0.251466\pi\)
0.703844 + 0.710355i \(0.251466\pi\)
\(954\) 0 0
\(955\) 611.295i 0.640100i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 27.6219i 0.0288029i
\(960\) 0 0
\(961\) −536.260 −0.558023
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1192.54 1.23579
\(966\) 0 0
\(967\) − 1227.25i − 1.26914i −0.772867 0.634568i \(-0.781178\pi\)
0.772867 0.634568i \(-0.218822\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1864.31i 1.91999i 0.280020 + 0.959994i \(0.409659\pi\)
−0.280020 + 0.959994i \(0.590341\pi\)
\(972\) 0 0
\(973\) −181.217 −0.186245
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1209.56 1.23803 0.619017 0.785378i \(-0.287531\pi\)
0.619017 + 0.785378i \(0.287531\pi\)
\(978\) 0 0
\(979\) 1684.38i 1.72051i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1018.26i − 1.03587i −0.855420 0.517935i \(-0.826701\pi\)
0.855420 0.517935i \(-0.173299\pi\)
\(984\) 0 0
\(985\) 14.6310 0.0148538
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −820.042 −0.829163
\(990\) 0 0
\(991\) − 1627.52i − 1.64230i −0.570712 0.821150i \(-0.693333\pi\)
0.570712 0.821150i \(-0.306667\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1349.24i − 1.35602i
\(996\) 0 0
\(997\) 540.192 0.541817 0.270909 0.962605i \(-0.412676\pi\)
0.270909 + 0.962605i \(0.412676\pi\)
\(998\) 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 1152.3.g.f.127.1 8
3.2 odd 2 384.3.g.a.127.8 yes 8
4.3 odd 2 inner 1152.3.g.f.127.2 8
8.3 odd 2 1152.3.g.c.127.8 8
8.5 even 2 1152.3.g.c.127.7 8
12.11 even 2 384.3.g.a.127.4 8
16.3 odd 4 2304.3.b.q.127.7 8
16.5 even 4 2304.3.b.q.127.2 8
16.11 odd 4 2304.3.b.t.127.2 8
16.13 even 4 2304.3.b.t.127.7 8
24.5 odd 2 384.3.g.b.127.1 yes 8
24.11 even 2 384.3.g.b.127.5 yes 8
48.5 odd 4 768.3.b.f.127.8 8
48.11 even 4 768.3.b.e.127.4 8
48.29 odd 4 768.3.b.e.127.1 8
48.35 even 4 768.3.b.f.127.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.g.a.127.4 8 12.11 even 2
384.3.g.a.127.8 yes 8 3.2 odd 2
384.3.g.b.127.1 yes 8 24.5 odd 2
384.3.g.b.127.5 yes 8 24.11 even 2
768.3.b.e.127.1 8 48.29 odd 4
768.3.b.e.127.4 8 48.11 even 4
768.3.b.f.127.5 8 48.35 even 4
768.3.b.f.127.8 8 48.5 odd 4
1152.3.g.c.127.7 8 8.5 even 2
1152.3.g.c.127.8 8 8.3 odd 2
1152.3.g.f.127.1 8 1.1 even 1 trivial
1152.3.g.f.127.2 8 4.3 odd 2 inner
2304.3.b.q.127.2 8 16.5 even 4
2304.3.b.q.127.7 8 16.3 odd 4
2304.3.b.t.127.2 8 16.11 odd 4
2304.3.b.t.127.7 8 16.13 even 4