Properties

Label 2304.3.b.r.127.1
Level $2304$
Weight $3$
Character 2304.127
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(127,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (of order \(2\), degree \(1\), not 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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(1.28897 - 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 2304.127
Dual form 2304.3.b.r.127.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.11993i q^{5} -9.48331i q^{7} +O(q^{10})\) \(q-8.11993i q^{5} -9.48331i q^{7} -10.5830 q^{11} -20.9666i q^{13} +4.92615 q^{17} -26.9666 q^{19} -43.7934i q^{23} -40.9333 q^{25} +14.5075i q^{29} +25.4833i q^{31} -77.0038 q^{35} -12.9666i q^{37} +50.1810 q^{41} -5.03337 q^{43} -40.8706i q^{47} -40.9333 q^{49} -27.2826i q^{53} +85.9333i q^{55} +24.0888 q^{59} +28.9666i q^{61} -170.248 q^{65} +65.7998 q^{67} -90.5097i q^{71} +127.800 q^{73} +100.362i q^{77} +35.5501i q^{79} -101.093 q^{83} -40.0000i q^{85} +15.6979 q^{89} -198.833 q^{91} +218.967i q^{95} +73.8665 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 96 q^{19} - 88 q^{25} - 160 q^{43} - 88 q^{49} - 192 q^{67} + 304 q^{73} - 992 q^{91} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(5\) − 8.11993i − 1.62399i −0.583667 0.811993i \(-0.698383\pi\)
0.583667 0.811993i \(-0.301617\pi\)
\(6\) 0 0
\(7\) − 9.48331i − 1.35476i −0.735634 0.677380i \(-0.763115\pi\)
0.735634 0.677380i \(-0.236885\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −10.5830 −0.962091 −0.481046 0.876696i \(-0.659743\pi\)
−0.481046 + 0.876696i \(0.659743\pi\)
\(12\) 0 0
\(13\) − 20.9666i − 1.61282i −0.591358 0.806409i \(-0.701408\pi\)
0.591358 0.806409i \(-0.298592\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.92615 0.289774 0.144887 0.989448i \(-0.453718\pi\)
0.144887 + 0.989448i \(0.453718\pi\)
\(18\) 0 0
\(19\) −26.9666 −1.41930 −0.709648 0.704556i \(-0.751146\pi\)
−0.709648 + 0.704556i \(0.751146\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 43.7934i − 1.90406i −0.306001 0.952031i \(-0.598991\pi\)
0.306001 0.952031i \(-0.401009\pi\)
\(24\) 0 0
\(25\) −40.9333 −1.63733
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 14.5075i 0.500258i 0.968212 + 0.250129i \(0.0804731\pi\)
−0.968212 + 0.250129i \(0.919527\pi\)
\(30\) 0 0
\(31\) 25.4833i 0.822042i 0.911626 + 0.411021i \(0.134828\pi\)
−0.911626 + 0.411021i \(0.865172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −77.0038 −2.20011
\(36\) 0 0
\(37\) − 12.9666i − 0.350449i −0.984528 0.175225i \(-0.943935\pi\)
0.984528 0.175225i \(-0.0560652\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 50.1810 1.22393 0.611963 0.790886i \(-0.290380\pi\)
0.611963 + 0.790886i \(0.290380\pi\)
\(42\) 0 0
\(43\) −5.03337 −0.117055 −0.0585276 0.998286i \(-0.518641\pi\)
−0.0585276 + 0.998286i \(0.518641\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 40.8706i − 0.869588i −0.900530 0.434794i \(-0.856821\pi\)
0.900530 0.434794i \(-0.143179\pi\)
\(48\) 0 0
\(49\) −40.9333 −0.835373
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 27.2826i − 0.514766i −0.966309 0.257383i \(-0.917140\pi\)
0.966309 0.257383i \(-0.0828602\pi\)
\(54\) 0 0
\(55\) 85.9333i 1.56242i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 24.0888 0.408285 0.204143 0.978941i \(-0.434559\pi\)
0.204143 + 0.978941i \(0.434559\pi\)
\(60\) 0 0
\(61\) 28.9666i 0.474863i 0.971404 + 0.237431i \(0.0763055\pi\)
−0.971404 + 0.237431i \(0.923695\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −170.248 −2.61919
\(66\) 0 0
\(67\) 65.7998 0.982086 0.491043 0.871135i \(-0.336616\pi\)
0.491043 + 0.871135i \(0.336616\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 90.5097i − 1.27478i −0.770540 0.637392i \(-0.780013\pi\)
0.770540 0.637392i \(-0.219987\pi\)
\(72\) 0 0
\(73\) 127.800 1.75068 0.875341 0.483506i \(-0.160637\pi\)
0.875341 + 0.483506i \(0.160637\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 100.362i 1.30340i
\(78\) 0 0
\(79\) 35.5501i 0.450001i 0.974359 + 0.225000i \(0.0722383\pi\)
−0.974359 + 0.225000i \(0.927762\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −101.093 −1.21798 −0.608992 0.793176i \(-0.708426\pi\)
−0.608992 + 0.793176i \(0.708426\pi\)
\(84\) 0 0
\(85\) − 40.0000i − 0.470588i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.6979 0.176381 0.0881906 0.996104i \(-0.471892\pi\)
0.0881906 + 0.996104i \(0.471892\pi\)
\(90\) 0 0
\(91\) −198.833 −2.18498
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 218.967i 2.30492i
\(96\) 0 0
\(97\) 73.8665 0.761510 0.380755 0.924676i \(-0.375664\pi\)
0.380755 + 0.924676i \(0.375664\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 114.869i − 1.13732i −0.822572 0.568661i \(-0.807462\pi\)
0.822572 0.568661i \(-0.192538\pi\)
\(102\) 0 0
\(103\) 121.483i 1.17945i 0.807604 + 0.589725i \(0.200764\pi\)
−0.807604 + 0.589725i \(0.799236\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 156.931 1.46664 0.733320 0.679883i \(-0.237970\pi\)
0.733320 + 0.679883i \(0.237970\pi\)
\(108\) 0 0
\(109\) 114.766i 1.05290i 0.850205 + 0.526451i \(0.176478\pi\)
−0.850205 + 0.526451i \(0.823522\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 179.180 1.58567 0.792834 0.609438i \(-0.208605\pi\)
0.792834 + 0.609438i \(0.208605\pi\)
\(114\) 0 0
\(115\) −355.600 −3.09217
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 46.7162i − 0.392573i
\(120\) 0 0
\(121\) −9.00000 −0.0743802
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 129.377i 1.03502i
\(126\) 0 0
\(127\) 96.5834i 0.760499i 0.924884 + 0.380250i \(0.124162\pi\)
−0.924884 + 0.380250i \(0.875838\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 169.328 1.29258 0.646290 0.763092i \(-0.276319\pi\)
0.646290 + 0.763092i \(0.276319\pi\)
\(132\) 0 0
\(133\) 255.733i 1.92280i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 235.207 1.71684 0.858420 0.512948i \(-0.171447\pi\)
0.858420 + 0.512948i \(0.171447\pi\)
\(138\) 0 0
\(139\) −30.2002 −0.217268 −0.108634 0.994082i \(-0.534648\pi\)
−0.108634 + 0.994082i \(0.534648\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 221.890i 1.55168i
\(144\) 0 0
\(145\) 117.800 0.812412
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 170.519i − 1.14442i −0.820107 0.572210i \(-0.806086\pi\)
0.820107 0.572210i \(-0.193914\pi\)
\(150\) 0 0
\(151\) − 87.1496i − 0.577150i −0.957457 0.288575i \(-0.906819\pi\)
0.957457 0.288575i \(-0.0931814\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 206.923 1.33499
\(156\) 0 0
\(157\) − 98.7664i − 0.629085i −0.949243 0.314543i \(-0.898149\pi\)
0.949243 0.314543i \(-0.101851\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −415.307 −2.57955
\(162\) 0 0
\(163\) −67.2336 −0.412476 −0.206238 0.978502i \(-0.566122\pi\)
−0.206238 + 0.978502i \(0.566122\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 216.044i 1.29368i 0.762626 + 0.646839i \(0.223910\pi\)
−0.762626 + 0.646839i \(0.776090\pi\)
\(168\) 0 0
\(169\) −270.600 −1.60118
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 118.876i − 0.687145i −0.939126 0.343573i \(-0.888363\pi\)
0.939126 0.343573i \(-0.111637\pi\)
\(174\) 0 0
\(175\) 388.183i 2.21819i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −24.0888 −0.134574 −0.0672872 0.997734i \(-0.521434\pi\)
−0.0672872 + 0.997734i \(0.521434\pi\)
\(180\) 0 0
\(181\) − 131.033i − 0.723941i −0.932189 0.361971i \(-0.882104\pi\)
0.932189 0.361971i \(-0.117896\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −105.288 −0.569125
\(186\) 0 0
\(187\) −52.1335 −0.278789
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 128.457i − 0.672552i −0.941763 0.336276i \(-0.890832\pi\)
0.941763 0.336276i \(-0.109168\pi\)
\(192\) 0 0
\(193\) −167.666 −0.868737 −0.434369 0.900735i \(-0.643028\pi\)
−0.434369 + 0.900735i \(0.643028\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 122.341i 0.621020i 0.950570 + 0.310510i \(0.100500\pi\)
−0.950570 + 0.310510i \(0.899500\pi\)
\(198\) 0 0
\(199\) 363.283i 1.82554i 0.408470 + 0.912772i \(0.366062\pi\)
−0.408470 + 0.912772i \(0.633938\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 137.579 0.677729
\(204\) 0 0
\(205\) − 407.466i − 1.98764i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 285.388 1.36549
\(210\) 0 0
\(211\) −138.067 −0.654345 −0.327172 0.944965i \(-0.606096\pi\)
−0.327172 + 0.944965i \(0.606096\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 40.8706i 0.190096i
\(216\) 0 0
\(217\) 241.666 1.11367
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 103.285i − 0.467352i
\(222\) 0 0
\(223\) 264.316i 1.18528i 0.805469 + 0.592638i \(0.201913\pi\)
−0.805469 + 0.592638i \(0.798087\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −243.409 −1.07229 −0.536143 0.844127i \(-0.680119\pi\)
−0.536143 + 0.844127i \(0.680119\pi\)
\(228\) 0 0
\(229\) − 234.232i − 1.02285i −0.859328 0.511425i \(-0.829118\pi\)
0.859328 0.511425i \(-0.170882\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 35.4025 0.151942 0.0759711 0.997110i \(-0.475794\pi\)
0.0759711 + 0.997110i \(0.475794\pi\)
\(234\) 0 0
\(235\) −331.867 −1.41220
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 32.1022i 0.134319i 0.997742 + 0.0671594i \(0.0213936\pi\)
−0.997742 + 0.0671594i \(0.978606\pi\)
\(240\) 0 0
\(241\) −103.800 −0.430704 −0.215352 0.976536i \(-0.569090\pi\)
−0.215352 + 0.976536i \(0.569090\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 332.375i 1.35663i
\(246\) 0 0
\(247\) 565.399i 2.28907i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −312.753 −1.24603 −0.623014 0.782211i \(-0.714092\pi\)
−0.623014 + 0.782211i \(0.714092\pi\)
\(252\) 0 0
\(253\) 463.466i 1.83188i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −33.5636 −0.130598 −0.0652988 0.997866i \(-0.520800\pi\)
−0.0652988 + 0.997866i \(0.520800\pi\)
\(258\) 0 0
\(259\) −122.967 −0.474775
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 338.656i − 1.28767i −0.765166 0.643833i \(-0.777343\pi\)
0.765166 0.643833i \(-0.222657\pi\)
\(264\) 0 0
\(265\) −221.533 −0.835973
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 310.832i 1.15551i 0.816211 + 0.577754i \(0.196071\pi\)
−0.816211 + 0.577754i \(0.803929\pi\)
\(270\) 0 0
\(271\) − 266.749i − 0.984314i −0.870506 0.492157i \(-0.836209\pi\)
0.870506 0.492157i \(-0.163791\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 433.197 1.57526
\(276\) 0 0
\(277\) − 458.766i − 1.65620i −0.560583 0.828098i \(-0.689423\pi\)
0.560583 0.828098i \(-0.310577\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −395.602 −1.40784 −0.703919 0.710281i \(-0.748568\pi\)
−0.703919 + 0.710281i \(0.748568\pi\)
\(282\) 0 0
\(283\) −375.733 −1.32768 −0.663839 0.747875i \(-0.731074\pi\)
−0.663839 + 0.747875i \(0.731074\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 475.882i − 1.65813i
\(288\) 0 0
\(289\) −264.733 −0.916031
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 9.74572i − 0.0332619i −0.999862 0.0166309i \(-0.994706\pi\)
0.999862 0.0166309i \(-0.00529403\pi\)
\(294\) 0 0
\(295\) − 195.600i − 0.663049i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −918.201 −3.07090
\(300\) 0 0
\(301\) 47.7330i 0.158582i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 235.207 0.771170
\(306\) 0 0
\(307\) 89.5328 0.291638 0.145819 0.989311i \(-0.453418\pi\)
0.145819 + 0.989311i \(0.453418\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 274.452i 0.882482i 0.897389 + 0.441241i \(0.145462\pi\)
−0.897389 + 0.441241i \(0.854538\pi\)
\(312\) 0 0
\(313\) 151.666 0.484557 0.242278 0.970207i \(-0.422105\pi\)
0.242278 + 0.970207i \(0.422105\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 201.914i − 0.636954i −0.947931 0.318477i \(-0.896829\pi\)
0.947931 0.318477i \(-0.103171\pi\)
\(318\) 0 0
\(319\) − 153.533i − 0.481294i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −132.842 −0.411275
\(324\) 0 0
\(325\) 858.232i 2.64072i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −387.589 −1.17808
\(330\) 0 0
\(331\) −195.600 −0.590935 −0.295468 0.955353i \(-0.595475\pi\)
−0.295468 + 0.955353i \(0.595475\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 534.290i − 1.59489i
\(336\) 0 0
\(337\) 23.9333 0.0710186 0.0355093 0.999369i \(-0.488695\pi\)
0.0355093 + 0.999369i \(0.488695\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 269.690i − 0.790880i
\(342\) 0 0
\(343\) − 76.4994i − 0.223030i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −139.796 −0.402869 −0.201435 0.979502i \(-0.564560\pi\)
−0.201435 + 0.979502i \(0.564560\pi\)
\(348\) 0 0
\(349\) 21.2336i 0.0608412i 0.999537 + 0.0304206i \(0.00968468\pi\)
−0.999537 + 0.0304206i \(0.990315\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 515.669 1.46082 0.730409 0.683010i \(-0.239329\pi\)
0.730409 + 0.683010i \(0.239329\pi\)
\(354\) 0 0
\(355\) −734.932 −2.07023
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 253.992i − 0.707499i −0.935340 0.353749i \(-0.884907\pi\)
0.935340 0.353749i \(-0.115093\pi\)
\(360\) 0 0
\(361\) 366.199 1.01440
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 1037.73i − 2.84308i
\(366\) 0 0
\(367\) 384.682i 1.04818i 0.851663 + 0.524090i \(0.175595\pi\)
−0.851663 + 0.524090i \(0.824405\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −258.730 −0.697384
\(372\) 0 0
\(373\) − 197.501i − 0.529492i −0.964318 0.264746i \(-0.914712\pi\)
0.964318 0.264746i \(-0.0852881\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 304.173 0.806825
\(378\) 0 0
\(379\) 234.433 0.618556 0.309278 0.950972i \(-0.399913\pi\)
0.309278 + 0.950972i \(0.399913\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 315.322i − 0.823296i −0.911343 0.411648i \(-0.864953\pi\)
0.911343 0.411648i \(-0.135047\pi\)
\(384\) 0 0
\(385\) 814.932 2.11671
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 486.760i 1.25131i 0.780099 + 0.625656i \(0.215169\pi\)
−0.780099 + 0.625656i \(0.784831\pi\)
\(390\) 0 0
\(391\) − 215.733i − 0.551747i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 288.664 0.730795
\(396\) 0 0
\(397\) − 115.300i − 0.290429i −0.989400 0.145215i \(-0.953613\pi\)
0.989400 0.145215i \(-0.0463872\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −52.3487 −0.130545 −0.0652727 0.997867i \(-0.520792\pi\)
−0.0652727 + 0.997867i \(0.520792\pi\)
\(402\) 0 0
\(403\) 534.299 1.32580
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 137.226i 0.337164i
\(408\) 0 0
\(409\) −345.333 −0.844334 −0.422167 0.906518i \(-0.638730\pi\)
−0.422167 + 0.906518i \(0.638730\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 228.442i − 0.553128i
\(414\) 0 0
\(415\) 820.865i 1.97799i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 644.151 1.53735 0.768676 0.639638i \(-0.220916\pi\)
0.768676 + 0.639638i \(0.220916\pi\)
\(420\) 0 0
\(421\) 756.700i 1.79739i 0.438578 + 0.898693i \(0.355482\pi\)
−0.438578 + 0.898693i \(0.644518\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −201.643 −0.474455
\(426\) 0 0
\(427\) 274.700 0.643325
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 49.6391i 0.115172i 0.998341 + 0.0575859i \(0.0183403\pi\)
−0.998341 + 0.0575859i \(0.981660\pi\)
\(432\) 0 0
\(433\) 581.066 1.34195 0.670976 0.741479i \(-0.265875\pi\)
0.670976 + 0.741479i \(0.265875\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1180.96i 2.70243i
\(438\) 0 0
\(439\) 274.482i 0.625244i 0.949878 + 0.312622i \(0.101207\pi\)
−0.949878 + 0.312622i \(0.898793\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −466.760 −1.05364 −0.526818 0.849978i \(-0.676615\pi\)
−0.526818 + 0.849978i \(0.676615\pi\)
\(444\) 0 0
\(445\) − 127.466i − 0.286441i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 479.347 1.06759 0.533794 0.845615i \(-0.320766\pi\)
0.533794 + 0.845615i \(0.320766\pi\)
\(450\) 0 0
\(451\) −531.066 −1.17753
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1614.51i 3.54838i
\(456\) 0 0
\(457\) 405.600 0.887526 0.443763 0.896144i \(-0.353643\pi\)
0.443763 + 0.896144i \(0.353643\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 636.171i 1.37998i 0.723819 + 0.689990i \(0.242385\pi\)
−0.723819 + 0.689990i \(0.757615\pi\)
\(462\) 0 0
\(463\) − 323.016i − 0.697659i −0.937186 0.348830i \(-0.886579\pi\)
0.937186 0.348830i \(-0.113421\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −128.104 −0.274313 −0.137157 0.990549i \(-0.543796\pi\)
−0.137157 + 0.990549i \(0.543796\pi\)
\(468\) 0 0
\(469\) − 624.000i − 1.33049i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 53.2682 0.112618
\(474\) 0 0
\(475\) 1103.83 2.32386
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 67.1272i 0.140140i 0.997542 + 0.0700701i \(0.0223223\pi\)
−0.997542 + 0.0700701i \(0.977678\pi\)
\(480\) 0 0
\(481\) −271.867 −0.565211
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 599.791i − 1.23668i
\(486\) 0 0
\(487\) 180.183i 0.369986i 0.982740 + 0.184993i \(0.0592262\pi\)
−0.982740 + 0.184993i \(0.940774\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 423.320 0.862159 0.431080 0.902314i \(-0.358133\pi\)
0.431080 + 0.902314i \(0.358133\pi\)
\(492\) 0 0
\(493\) 71.4661i 0.144962i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −858.332 −1.72703
\(498\) 0 0
\(499\) 830.932 1.66519 0.832597 0.553879i \(-0.186853\pi\)
0.832597 + 0.553879i \(0.186853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 359.116i − 0.713948i −0.934114 0.356974i \(-0.883808\pi\)
0.934114 0.356974i \(-0.116192\pi\)
\(504\) 0 0
\(505\) −932.732 −1.84699
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 43.3093i 0.0850870i 0.999095 + 0.0425435i \(0.0135461\pi\)
−0.999095 + 0.0425435i \(0.986454\pi\)
\(510\) 0 0
\(511\) − 1211.97i − 2.37175i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 986.436 1.91541
\(516\) 0 0
\(517\) 432.534i 0.836623i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −426.407 −0.818440 −0.409220 0.912436i \(-0.634199\pi\)
−0.409220 + 0.912436i \(0.634199\pi\)
\(522\) 0 0
\(523\) −384.366 −0.734925 −0.367463 0.930038i \(-0.619773\pi\)
−0.367463 + 0.930038i \(0.619773\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 125.535i 0.238206i
\(528\) 0 0
\(529\) −1388.86 −2.62545
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 1052.13i − 1.97397i
\(534\) 0 0
\(535\) − 1274.26i − 2.38180i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 433.197 0.803705
\(540\) 0 0
\(541\) 403.033i 0.744979i 0.928037 + 0.372489i \(0.121496\pi\)
−0.928037 + 0.372489i \(0.878504\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 931.895 1.70990
\(546\) 0 0
\(547\) 1066.43 1.94960 0.974801 0.223075i \(-0.0716094\pi\)
0.974801 + 0.223075i \(0.0716094\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 391.218i − 0.710015i
\(552\) 0 0
\(553\) 337.132 0.609643
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 221.406i − 0.397497i −0.980051 0.198749i \(-0.936312\pi\)
0.980051 0.198749i \(-0.0636877\pi\)
\(558\) 0 0
\(559\) 105.533i 0.188789i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 747.764 1.32818 0.664089 0.747654i \(-0.268820\pi\)
0.664089 + 0.747654i \(0.268820\pi\)
\(564\) 0 0
\(565\) − 1454.93i − 2.57510i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −489.199 −0.859752 −0.429876 0.902888i \(-0.641443\pi\)
−0.429876 + 0.902888i \(0.641443\pi\)
\(570\) 0 0
\(571\) −784.198 −1.37338 −0.686688 0.726952i \(-0.740936\pi\)
−0.686688 + 0.726952i \(0.740936\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1792.61i 3.11758i
\(576\) 0 0
\(577\) −239.132 −0.414441 −0.207220 0.978294i \(-0.566442\pi\)
−0.207220 + 0.978294i \(0.566442\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 958.694i 1.65008i
\(582\) 0 0
\(583\) 288.732i 0.495252i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −84.6640 −0.144232 −0.0721159 0.997396i \(-0.522975\pi\)
−0.0721159 + 0.997396i \(0.522975\pi\)
\(588\) 0 0
\(589\) − 687.199i − 1.16672i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1135.38 −1.91463 −0.957317 0.289041i \(-0.906664\pi\)
−0.957317 + 0.289041i \(0.906664\pi\)
\(594\) 0 0
\(595\) −379.333 −0.637534
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 411.678i 0.687275i 0.939102 + 0.343638i \(0.111659\pi\)
−0.939102 + 0.343638i \(0.888341\pi\)
\(600\) 0 0
\(601\) 463.666 0.771491 0.385746 0.922605i \(-0.373944\pi\)
0.385746 + 0.922605i \(0.373944\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 73.0794i 0.120792i
\(606\) 0 0
\(607\) − 803.016i − 1.32293i −0.749978 0.661463i \(-0.769936\pi\)
0.749978 0.661463i \(-0.230064\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −856.919 −1.40249
\(612\) 0 0
\(613\) 459.567i 0.749702i 0.927085 + 0.374851i \(0.122306\pi\)
−0.927085 + 0.374851i \(0.877694\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −370.052 −0.599760 −0.299880 0.953977i \(-0.596947\pi\)
−0.299880 + 0.953977i \(0.596947\pi\)
\(618\) 0 0
\(619\) −10.7987 −0.0174453 −0.00872267 0.999962i \(-0.502777\pi\)
−0.00872267 + 0.999962i \(0.502777\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 148.868i − 0.238954i
\(624\) 0 0
\(625\) 27.2002 0.0435204
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 63.8756i − 0.101551i
\(630\) 0 0
\(631\) − 63.9505i − 0.101348i −0.998715 0.0506739i \(-0.983863\pi\)
0.998715 0.0506739i \(-0.0161369\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 784.251 1.23504
\(636\) 0 0
\(637\) 858.232i 1.34730i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 772.748 1.20554 0.602768 0.797917i \(-0.294065\pi\)
0.602768 + 0.797917i \(0.294065\pi\)
\(642\) 0 0
\(643\) 23.3671 0.0363407 0.0181704 0.999835i \(-0.494216\pi\)
0.0181704 + 0.999835i \(0.494216\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 686.130i − 1.06048i −0.847848 0.530239i \(-0.822102\pi\)
0.847848 0.530239i \(-0.177898\pi\)
\(648\) 0 0
\(649\) −254.932 −0.392808
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 899.686i 1.37777i 0.724869 + 0.688887i \(0.241900\pi\)
−0.724869 + 0.688887i \(0.758100\pi\)
\(654\) 0 0
\(655\) − 1374.93i − 2.09913i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −217.506 −0.330054 −0.165027 0.986289i \(-0.552771\pi\)
−0.165027 + 0.986289i \(0.552771\pi\)
\(660\) 0 0
\(661\) − 829.234i − 1.25451i −0.778812 0.627257i \(-0.784178\pi\)
0.778812 0.627257i \(-0.215822\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2076.53 3.12261
\(666\) 0 0
\(667\) 635.333 0.952523
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 306.554i − 0.456861i
\(672\) 0 0
\(673\) −234.400 −0.348292 −0.174146 0.984720i \(-0.555716\pi\)
−0.174146 + 0.984720i \(0.555716\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 772.690i − 1.14134i −0.821178 0.570672i \(-0.806683\pi\)
0.821178 0.570672i \(-0.193317\pi\)
\(678\) 0 0
\(679\) − 700.499i − 1.03166i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 640.522 0.937806 0.468903 0.883250i \(-0.344649\pi\)
0.468903 + 0.883250i \(0.344649\pi\)
\(684\) 0 0
\(685\) − 1909.86i − 2.78812i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −572.024 −0.830224
\(690\) 0 0
\(691\) 973.300 1.40854 0.704269 0.709933i \(-0.251275\pi\)
0.704269 + 0.709933i \(0.251275\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 245.224i 0.352840i
\(696\) 0 0
\(697\) 247.199 0.354662
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 759.045i 1.08280i 0.840764 + 0.541401i \(0.182106\pi\)
−0.840764 + 0.541401i \(0.817894\pi\)
\(702\) 0 0
\(703\) 349.666i 0.497392i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1089.34 −1.54080
\(708\) 0 0
\(709\) 334.034i 0.471135i 0.971858 + 0.235567i \(0.0756948\pi\)
−0.971858 + 0.235567i \(0.924305\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1116.00 1.56522
\(714\) 0 0
\(715\) 1801.73 2.51990
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 318.196i − 0.442554i −0.975211 0.221277i \(-0.928977\pi\)
0.975211 0.221277i \(-0.0710225\pi\)
\(720\) 0 0
\(721\) 1152.06 1.59787
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 593.839i − 0.819088i
\(726\) 0 0
\(727\) − 877.617i − 1.20718i −0.797296 0.603588i \(-0.793737\pi\)
0.797296 0.603588i \(-0.206263\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −24.7951 −0.0339195
\(732\) 0 0
\(733\) 1001.70i 1.36657i 0.730150 + 0.683287i \(0.239450\pi\)
−0.730150 + 0.683287i \(0.760550\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −696.359 −0.944857
\(738\) 0 0
\(739\) 648.999 0.878212 0.439106 0.898435i \(-0.355295\pi\)
0.439106 + 0.898435i \(0.355295\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 411.678i − 0.554075i −0.960859 0.277038i \(-0.910647\pi\)
0.960859 0.277038i \(-0.0893526\pi\)
\(744\) 0 0
\(745\) −1384.60 −1.85852
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1488.22i − 1.98694i
\(750\) 0 0
\(751\) 471.882i 0.628338i 0.949367 + 0.314169i \(0.101726\pi\)
−0.949367 + 0.314169i \(0.898274\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −707.649 −0.937283
\(756\) 0 0
\(757\) − 164.101i − 0.216778i −0.994109 0.108389i \(-0.965431\pi\)
0.994109 0.108389i \(-0.0345693\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 331.562 0.435693 0.217846 0.975983i \(-0.430097\pi\)
0.217846 + 0.975983i \(0.430097\pi\)
\(762\) 0 0
\(763\) 1088.37 1.42643
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 505.061i − 0.658489i
\(768\) 0 0
\(769\) 847.533 1.10212 0.551062 0.834465i \(-0.314223\pi\)
0.551062 + 0.834465i \(0.314223\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 980.557i − 1.26851i −0.773125 0.634254i \(-0.781307\pi\)
0.773125 0.634254i \(-0.218693\pi\)
\(774\) 0 0
\(775\) − 1043.12i − 1.34596i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1353.21 −1.73711
\(780\) 0 0
\(781\) 957.864i 1.22646i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −801.976 −1.02163
\(786\) 0 0
\(787\) −335.832 −0.426724 −0.213362 0.976973i \(-0.568441\pi\)
−0.213362 + 0.976973i \(0.568441\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1699.22i − 2.14820i
\(792\) 0 0
\(793\) 607.333 0.765867
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 283.114i − 0.355224i −0.984101 0.177612i \(-0.943163\pi\)
0.984101 0.177612i \(-0.0568372\pi\)
\(798\) 0 0
\(799\) − 201.335i − 0.251984i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1352.51 −1.68432
\(804\) 0 0
\(805\) 3372.26i 4.18915i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −117.308 −0.145004 −0.0725019 0.997368i \(-0.523098\pi\)
−0.0725019 + 0.997368i \(0.523098\pi\)
\(810\) 0 0
\(811\) 1235.43 1.52334 0.761672 0.647963i \(-0.224379\pi\)
0.761672 + 0.647963i \(0.224379\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 545.932i 0.669855i
\(816\) 0 0
\(817\) 135.733 0.166136
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 823.364i 1.00288i 0.865192 + 0.501440i \(0.167196\pi\)
−0.865192 + 0.501440i \(0.832804\pi\)
\(822\) 0 0
\(823\) 743.249i 0.903097i 0.892247 + 0.451548i \(0.149128\pi\)
−0.892247 + 0.451548i \(0.850872\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 542.352 0.655806 0.327903 0.944711i \(-0.393658\pi\)
0.327903 + 0.944711i \(0.393658\pi\)
\(828\) 0 0
\(829\) − 906.564i − 1.09356i −0.837275 0.546782i \(-0.815853\pi\)
0.837275 0.546782i \(-0.184147\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −201.643 −0.242069
\(834\) 0 0
\(835\) 1754.26 2.10092
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 1015.97i − 1.21093i −0.795873 0.605464i \(-0.792988\pi\)
0.795873 0.605464i \(-0.207012\pi\)
\(840\) 0 0
\(841\) 630.533 0.749742
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2197.25i 2.60030i
\(846\) 0 0
\(847\) 85.3498i 0.100767i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −567.853 −0.667277
\(852\) 0 0
\(853\) − 102.568i − 0.120244i −0.998191 0.0601222i \(-0.980851\pi\)
0.998191 0.0601222i \(-0.0191490\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1081.85 −1.26237 −0.631183 0.775634i \(-0.717430\pi\)
−0.631183 + 0.775634i \(0.717430\pi\)
\(858\) 0 0
\(859\) −597.231 −0.695264 −0.347632 0.937631i \(-0.613014\pi\)
−0.347632 + 0.937631i \(0.613014\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 537.261i 0.622551i 0.950320 + 0.311275i \(0.100756\pi\)
−0.950320 + 0.311275i \(0.899244\pi\)
\(864\) 0 0
\(865\) −965.266 −1.11591
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 376.226i − 0.432942i
\(870\) 0 0
\(871\) − 1379.60i − 1.58393i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1226.92 1.40220
\(876\) 0 0
\(877\) 395.899i 0.451424i 0.974194 + 0.225712i \(0.0724708\pi\)
−0.974194 + 0.225712i \(0.927529\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −289.066 −0.328111 −0.164056 0.986451i \(-0.552458\pi\)
−0.164056 + 0.986451i \(0.552458\pi\)
\(882\) 0 0
\(883\) −1548.96 −1.75421 −0.877103 0.480302i \(-0.840527\pi\)
−0.877103 + 0.480302i \(0.840527\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1544.41i − 1.74116i −0.492023 0.870582i \(-0.663743\pi\)
0.492023 0.870582i \(-0.336257\pi\)
\(888\) 0 0
\(889\) 915.931 1.03029
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1102.14i 1.23420i
\(894\) 0 0
\(895\) 195.600i 0.218547i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −369.699 −0.411233
\(900\) 0 0
\(901\) − 134.398i − 0.149166i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1063.98 −1.17567
\(906\) 0 0
\(907\) −472.831 −0.521313 −0.260657 0.965432i \(-0.583939\pi\)
−0.260657 + 0.965432i \(0.583939\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1094.84i 1.20180i 0.799326 + 0.600898i \(0.205190\pi\)
−0.799326 + 0.600898i \(0.794810\pi\)
\(912\) 0 0
\(913\) 1069.86 1.17181
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1605.79i − 1.75114i
\(918\) 0 0
\(919\) 152.850i 0.166323i 0.996536 + 0.0831613i \(0.0265017\pi\)
−0.996536 + 0.0831613i \(0.973498\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1897.68 −2.05599
\(924\) 0 0
\(925\) 530.766i 0.573802i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 63.7112 0.0685804 0.0342902 0.999412i \(-0.489083\pi\)
0.0342902 + 0.999412i \(0.489083\pi\)
\(930\) 0 0
\(931\) 1103.83 1.18564
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 423.320i 0.452749i
\(936\) 0 0
\(937\) −1320.67 −1.40946 −0.704731 0.709475i \(-0.748932\pi\)
−0.704731 + 0.709475i \(0.748932\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 5.41027i − 0.00574949i −0.999996 0.00287475i \(-0.999085\pi\)
0.999996 0.00287475i \(-0.000915061\pi\)
\(942\) 0 0
\(943\) − 2197.60i − 2.33043i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −83.9577 −0.0886565 −0.0443283 0.999017i \(-0.514115\pi\)
−0.0443283 + 0.999017i \(0.514115\pi\)
\(948\) 0 0
\(949\) − 2679.53i − 2.82353i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1577.15 1.65494 0.827468 0.561513i \(-0.189780\pi\)
0.827468 + 0.561513i \(0.189780\pi\)
\(954\) 0 0
\(955\) −1043.07 −1.09222
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 2230.54i − 2.32590i
\(960\) 0 0
\(961\) 311.601 0.324246
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1361.44i 1.41082i
\(966\) 0 0
\(967\) 278.616i 0.288124i 0.989569 + 0.144062i \(0.0460164\pi\)
−0.989569 + 0.144062i \(0.953984\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 380.684 0.392053 0.196027 0.980599i \(-0.437196\pi\)
0.196027 + 0.980599i \(0.437196\pi\)
\(972\) 0 0
\(973\) 286.398i 0.294346i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1630.75 1.66914 0.834571 0.550901i \(-0.185716\pi\)
0.834571 + 0.550901i \(0.185716\pi\)
\(978\) 0 0
\(979\) −166.131 −0.169695
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 928.382i 0.944437i 0.881482 + 0.472219i \(0.156547\pi\)
−0.881482 + 0.472219i \(0.843453\pi\)
\(984\) 0 0
\(985\) 993.399 1.00853
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 220.429i 0.222880i
\(990\) 0 0
\(991\) − 1399.98i − 1.41269i −0.707865 0.706347i \(-0.750342\pi\)
0.707865 0.706347i \(-0.249658\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2949.83 2.96466
\(996\) 0 0
\(997\) 595.567i 0.597359i 0.954353 + 0.298680i \(0.0965462\pi\)
−0.954353 + 0.298680i \(0.903454\pi\)
\(998\) 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 2304.3.b.r.127.1 8
3.2 odd 2 inner 2304.3.b.r.127.7 8
4.3 odd 2 2304.3.b.s.127.2 8
8.3 odd 2 inner 2304.3.b.r.127.8 8
8.5 even 2 2304.3.b.s.127.7 8
12.11 even 2 2304.3.b.s.127.8 8
16.3 odd 4 1152.3.g.e.127.1 yes 8
16.5 even 4 1152.3.g.d.127.8 yes 8
16.11 odd 4 1152.3.g.d.127.7 yes 8
16.13 even 4 1152.3.g.e.127.2 yes 8
24.5 odd 2 2304.3.b.s.127.1 8
24.11 even 2 inner 2304.3.b.r.127.2 8
48.5 odd 4 1152.3.g.d.127.2 yes 8
48.11 even 4 1152.3.g.d.127.1 8
48.29 odd 4 1152.3.g.e.127.8 yes 8
48.35 even 4 1152.3.g.e.127.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.g.d.127.1 8 48.11 even 4
1152.3.g.d.127.2 yes 8 48.5 odd 4
1152.3.g.d.127.7 yes 8 16.11 odd 4
1152.3.g.d.127.8 yes 8 16.5 even 4
1152.3.g.e.127.1 yes 8 16.3 odd 4
1152.3.g.e.127.2 yes 8 16.13 even 4
1152.3.g.e.127.7 yes 8 48.35 even 4
1152.3.g.e.127.8 yes 8 48.29 odd 4
2304.3.b.r.127.1 8 1.1 even 1 trivial
2304.3.b.r.127.2 8 24.11 even 2 inner
2304.3.b.r.127.7 8 3.2 odd 2 inner
2304.3.b.r.127.8 8 8.3 odd 2 inner
2304.3.b.s.127.1 8 24.5 odd 2
2304.3.b.s.127.2 8 4.3 odd 2
2304.3.b.s.127.7 8 8.5 even 2
2304.3.b.s.127.8 8 12.11 even 2