Properties

Label 1152.3.g.d.127.8
Level $1152$
Weight $3$
Character 1152.127
Analytic conductor $31.390$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: 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: \( 2^{22} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.8
Root \(-0.581861 - 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 1152.127
Dual form 1152.3.g.d.127.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.11993 q^{5} +9.48331i q^{7} +O(q^{10})\) \(q+8.11993 q^{5} +9.48331i q^{7} -10.5830i q^{11} -20.9666 q^{13} +4.92615 q^{17} +26.9666i q^{19} +43.7934i q^{23} +40.9333 q^{25} +14.5075 q^{29} +25.4833i q^{31} +77.0038i q^{35} +12.9666 q^{37} -50.1810 q^{41} -5.03337i q^{43} -40.8706i q^{47} -40.9333 q^{49} +27.2826 q^{53} -85.9333i q^{55} +24.0888i q^{59} +28.9666 q^{61} -170.248 q^{65} -65.7998i q^{67} +90.5097i q^{71} -127.800 q^{73} +100.362 q^{77} +35.5501i q^{79} +101.093i q^{83} +40.0000 q^{85} -15.6979 q^{89} -198.833i q^{91} +218.967i q^{95} +73.8665 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 48 q^{13} + 88 q^{25} - 16 q^{37} - 88 q^{49} + 112 q^{61} - 304 q^{73} + 320 q^{85} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 8.11993 1.62399 0.811993 0.583667i \(-0.198383\pi\)
0.811993 + 0.583667i \(0.198383\pi\)
\(6\) 0 0
\(7\) 9.48331i 1.35476i 0.735634 + 0.677380i \(0.236885\pi\)
−0.735634 + 0.677380i \(0.763115\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 10.5830i − 0.962091i −0.876696 0.481046i \(-0.840257\pi\)
0.876696 0.481046i \(-0.159743\pi\)
\(12\) 0 0
\(13\) −20.9666 −1.61282 −0.806409 0.591358i \(-0.798592\pi\)
−0.806409 + 0.591358i \(0.798592\pi\)
\(14\) 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
\(19\) 26.9666i 1.41930i 0.704556 + 0.709648i \(0.251146\pi\)
−0.704556 + 0.709648i \(0.748854\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 43.7934i 1.90406i 0.306001 + 0.952031i \(0.401009\pi\)
−0.306001 + 0.952031i \(0.598991\pi\)
\(24\) 0 0
\(25\) 40.9333 1.63733
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 14.5075 0.500258 0.250129 0.968212i \(-0.419527\pi\)
0.250129 + 0.968212i \(0.419527\pi\)
\(30\) 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
\(35\) 77.0038i 2.20011i
\(36\) 0 0
\(37\) 12.9666 0.350449 0.175225 0.984528i \(-0.443935\pi\)
0.175225 + 0.984528i \(0.443935\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −50.1810 −1.22393 −0.611963 0.790886i \(-0.709620\pi\)
−0.611963 + 0.790886i \(0.709620\pi\)
\(42\) 0 0
\(43\) − 5.03337i − 0.117055i −0.998286 0.0585276i \(-0.981359\pi\)
0.998286 0.0585276i \(-0.0186406\pi\)
\(44\) 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
\(53\) 27.2826 0.514766 0.257383 0.966309i \(-0.417140\pi\)
0.257383 + 0.966309i \(0.417140\pi\)
\(54\) 0 0
\(55\) − 85.9333i − 1.56242i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 24.0888i 0.408285i 0.978941 + 0.204143i \(0.0654406\pi\)
−0.978941 + 0.204143i \(0.934559\pi\)
\(60\) 0 0
\(61\) 28.9666 0.474863 0.237431 0.971404i \(-0.423695\pi\)
0.237431 + 0.971404i \(0.423695\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −170.248 −2.61919
\(66\) 0 0
\(67\) − 65.7998i − 0.982086i −0.871135 0.491043i \(-0.836616\pi\)
0.871135 0.491043i \(-0.163384\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 90.5097i 1.27478i 0.770540 + 0.637392i \(0.219987\pi\)
−0.770540 + 0.637392i \(0.780013\pi\)
\(72\) 0 0
\(73\) −127.800 −1.75068 −0.875341 0.483506i \(-0.839363\pi\)
−0.875341 + 0.483506i \(0.839363\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 100.362 1.30340
\(78\) 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
\(83\) 101.093i 1.21798i 0.793176 + 0.608992i \(0.208426\pi\)
−0.793176 + 0.608992i \(0.791574\pi\)
\(84\) 0 0
\(85\) 40.0000 0.470588
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −15.6979 −0.176381 −0.0881906 0.996104i \(-0.528108\pi\)
−0.0881906 + 0.996104i \(0.528108\pi\)
\(90\) 0 0
\(91\) − 198.833i − 2.18498i
\(92\) 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
\(101\) 114.869 1.13732 0.568661 0.822572i \(-0.307462\pi\)
0.568661 + 0.822572i \(0.307462\pi\)
\(102\) 0 0
\(103\) − 121.483i − 1.17945i −0.807604 0.589725i \(-0.799236\pi\)
0.807604 0.589725i \(-0.200764\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 156.931i 1.46664i 0.679883 + 0.733320i \(0.262030\pi\)
−0.679883 + 0.733320i \(0.737970\pi\)
\(108\) 0 0
\(109\) 114.766 1.05290 0.526451 0.850205i \(-0.323522\pi\)
0.526451 + 0.850205i \(0.323522\pi\)
\(110\) 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
\(115\) 355.600i 3.09217i
\(116\) 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
\(125\) 129.377 1.03502
\(126\) 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
\(131\) − 169.328i − 1.29258i −0.763092 0.646290i \(-0.776319\pi\)
0.763092 0.646290i \(-0.223681\pi\)
\(132\) 0 0
\(133\) −255.733 −1.92280
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −235.207 −1.71684 −0.858420 0.512948i \(-0.828553\pi\)
−0.858420 + 0.512948i \(0.828553\pi\)
\(138\) 0 0
\(139\) − 30.2002i − 0.217268i −0.994082 0.108634i \(-0.965352\pi\)
0.994082 0.108634i \(-0.0346476\pi\)
\(140\) 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
\(149\) 170.519 1.14442 0.572210 0.820107i \(-0.306086\pi\)
0.572210 + 0.820107i \(0.306086\pi\)
\(150\) 0 0
\(151\) 87.1496i 0.577150i 0.957457 + 0.288575i \(0.0931814\pi\)
−0.957457 + 0.288575i \(0.906819\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 206.923i 1.33499i
\(156\) 0 0
\(157\) −98.7664 −0.629085 −0.314543 0.949243i \(-0.601851\pi\)
−0.314543 + 0.949243i \(0.601851\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −415.307 −2.57955
\(162\) 0 0
\(163\) 67.2336i 0.412476i 0.978502 + 0.206238i \(0.0661221\pi\)
−0.978502 + 0.206238i \(0.933878\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 216.044i − 1.29368i −0.762626 0.646839i \(-0.776090\pi\)
0.762626 0.646839i \(-0.223910\pi\)
\(168\) 0 0
\(169\) 270.600 1.60118
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −118.876 −0.687145 −0.343573 0.939126i \(-0.611637\pi\)
−0.343573 + 0.939126i \(0.611637\pi\)
\(174\) 0 0
\(175\) 388.183i 2.21819i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.0888i 0.134574i 0.997734 + 0.0672872i \(0.0214344\pi\)
−0.997734 + 0.0672872i \(0.978566\pi\)
\(180\) 0 0
\(181\) 131.033 0.723941 0.361971 0.932189i \(-0.382104\pi\)
0.361971 + 0.932189i \(0.382104\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 105.288 0.569125
\(186\) 0 0
\(187\) − 52.1335i − 0.278789i
\(188\) 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
\(197\) −122.341 −0.621020 −0.310510 0.950570i \(-0.600500\pi\)
−0.310510 + 0.950570i \(0.600500\pi\)
\(198\) 0 0
\(199\) − 363.283i − 1.82554i −0.408470 0.912772i \(-0.633938\pi\)
0.408470 0.912772i \(-0.366062\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 137.579i 0.677729i
\(204\) 0 0
\(205\) −407.466 −1.98764
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 285.388 1.36549
\(210\) 0 0
\(211\) 138.067i 0.654345i 0.944965 + 0.327172i \(0.106096\pi\)
−0.944965 + 0.327172i \(0.893904\pi\)
\(212\) 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
\(221\) −103.285 −0.467352
\(222\) 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
\(227\) 243.409i 1.07229i 0.844127 + 0.536143i \(0.180119\pi\)
−0.844127 + 0.536143i \(0.819881\pi\)
\(228\) 0 0
\(229\) 234.232 1.02285 0.511425 0.859328i \(-0.329118\pi\)
0.511425 + 0.859328i \(0.329118\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −35.4025 −0.151942 −0.0759711 0.997110i \(-0.524206\pi\)
−0.0759711 + 0.997110i \(0.524206\pi\)
\(234\) 0 0
\(235\) − 331.867i − 1.41220i
\(236\) 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
\(245\) −332.375 −1.35663
\(246\) 0 0
\(247\) − 565.399i − 2.28907i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 312.753i − 1.24603i −0.782211 0.623014i \(-0.785908\pi\)
0.782211 0.623014i \(-0.214092\pi\)
\(252\) 0 0
\(253\) 463.466 1.83188
\(254\) 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
\(259\) 122.967i 0.474775i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 338.656i 1.28767i 0.765166 + 0.643833i \(0.222657\pi\)
−0.765166 + 0.643833i \(0.777343\pi\)
\(264\) 0 0
\(265\) 221.533 0.835973
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 310.832 1.15551 0.577754 0.816211i \(-0.303929\pi\)
0.577754 + 0.816211i \(0.303929\pi\)
\(270\) 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
\(275\) − 433.197i − 1.57526i
\(276\) 0 0
\(277\) 458.766 1.65620 0.828098 0.560583i \(-0.189423\pi\)
0.828098 + 0.560583i \(0.189423\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 395.602 1.40784 0.703919 0.710281i \(-0.251432\pi\)
0.703919 + 0.710281i \(0.251432\pi\)
\(282\) 0 0
\(283\) − 375.733i − 1.32768i −0.747875 0.663839i \(-0.768926\pi\)
0.747875 0.663839i \(-0.231074\pi\)
\(284\) 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
\(293\) 9.74572 0.0332619 0.0166309 0.999862i \(-0.494706\pi\)
0.0166309 + 0.999862i \(0.494706\pi\)
\(294\) 0 0
\(295\) 195.600i 0.663049i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 918.201i − 3.07090i
\(300\) 0 0
\(301\) 47.7330 0.158582
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 235.207 0.771170
\(306\) 0 0
\(307\) − 89.5328i − 0.291638i −0.989311 0.145819i \(-0.953418\pi\)
0.989311 0.145819i \(-0.0465817\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 274.452i − 0.882482i −0.897389 0.441241i \(-0.854538\pi\)
0.897389 0.441241i \(-0.145462\pi\)
\(312\) 0 0
\(313\) −151.666 −0.484557 −0.242278 0.970207i \(-0.577895\pi\)
−0.242278 + 0.970207i \(0.577895\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −201.914 −0.636954 −0.318477 0.947931i \(-0.603171\pi\)
−0.318477 + 0.947931i \(0.603171\pi\)
\(318\) 0 0
\(319\) − 153.533i − 0.481294i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 132.842i 0.411275i
\(324\) 0 0
\(325\) −858.232 −2.64072
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 387.589 1.17808
\(330\) 0 0
\(331\) − 195.600i − 0.590935i −0.955353 0.295468i \(-0.904525\pi\)
0.955353 0.295468i \(-0.0954754\pi\)
\(332\) 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
\(341\) 269.690 0.790880
\(342\) 0 0
\(343\) 76.4994i 0.223030i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 139.796i − 0.402869i −0.979502 0.201435i \(-0.935440\pi\)
0.979502 0.201435i \(-0.0645603\pi\)
\(348\) 0 0
\(349\) 21.2336 0.0608412 0.0304206 0.999537i \(-0.490315\pi\)
0.0304206 + 0.999537i \(0.490315\pi\)
\(350\) 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
\(355\) 734.932i 2.07023i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 253.992i 0.707499i 0.935340 + 0.353749i \(0.115093\pi\)
−0.935340 + 0.353749i \(0.884907\pi\)
\(360\) 0 0
\(361\) −366.199 −1.01440
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1037.73 −2.84308
\(366\) 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
\(371\) 258.730i 0.697384i
\(372\) 0 0
\(373\) 197.501 0.529492 0.264746 0.964318i \(-0.414712\pi\)
0.264746 + 0.964318i \(0.414712\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −304.173 −0.806825
\(378\) 0 0
\(379\) 234.433i 0.618556i 0.950972 + 0.309278i \(0.100087\pi\)
−0.950972 + 0.309278i \(0.899913\pi\)
\(380\) 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
\(389\) −486.760 −1.25131 −0.625656 0.780099i \(-0.715169\pi\)
−0.625656 + 0.780099i \(0.715169\pi\)
\(390\) 0 0
\(391\) 215.733i 0.551747i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 288.664i 0.730795i
\(396\) 0 0
\(397\) −115.300 −0.290429 −0.145215 0.989400i \(-0.546387\pi\)
−0.145215 + 0.989400i \(0.546387\pi\)
\(398\) 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
\(403\) − 534.299i − 1.32580i
\(404\) 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 \(-0.361270\pi\)
0.422167 + 0.906518i \(0.361270\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −228.442 −0.553128
\(414\) 0 0
\(415\) 820.865i 1.97799i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 644.151i − 1.53735i −0.639638 0.768676i \(-0.720916\pi\)
0.639638 0.768676i \(-0.279084\pi\)
\(420\) 0 0
\(421\) −756.700 −1.79739 −0.898693 0.438578i \(-0.855482\pi\)
−0.898693 + 0.438578i \(0.855482\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 201.643 0.474455
\(426\) 0 0
\(427\) 274.700i 0.643325i
\(428\) 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
\(437\) −1180.96 −2.70243
\(438\) 0 0
\(439\) − 274.482i − 0.625244i −0.949878 0.312622i \(-0.898793\pi\)
0.949878 0.312622i \(-0.101207\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 466.760i − 1.05364i −0.849978 0.526818i \(-0.823385\pi\)
0.849978 0.526818i \(-0.176615\pi\)
\(444\) 0 0
\(445\) −127.466 −0.286441
\(446\) 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
\(451\) 531.066i 1.17753i
\(452\) 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 \(-0.646357\pi\)
−0.443763 + 0.896144i \(0.646357\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 636.171 1.37998 0.689990 0.723819i \(-0.257615\pi\)
0.689990 + 0.723819i \(0.257615\pi\)
\(462\) 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
\(467\) 128.104i 0.274313i 0.990549 + 0.137157i \(0.0437964\pi\)
−0.990549 + 0.137157i \(0.956204\pi\)
\(468\) 0 0
\(469\) 624.000 1.33049
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −53.2682 −0.112618
\(474\) 0 0
\(475\) 1103.83i 2.32386i
\(476\) 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
\(485\) 599.791 1.23668
\(486\) 0 0
\(487\) − 180.183i − 0.369986i −0.982740 0.184993i \(-0.940774\pi\)
0.982740 0.184993i \(-0.0592262\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 423.320i 0.862159i 0.902314 + 0.431080i \(0.141867\pi\)
−0.902314 + 0.431080i \(0.858133\pi\)
\(492\) 0 0
\(493\) 71.4661 0.144962
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −858.332 −1.72703
\(498\) 0 0
\(499\) − 830.932i − 1.66519i −0.553879 0.832597i \(-0.686853\pi\)
0.553879 0.832597i \(-0.313147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 359.116i 0.713948i 0.934114 + 0.356974i \(0.116192\pi\)
−0.934114 + 0.356974i \(0.883808\pi\)
\(504\) 0 0
\(505\) 932.732 1.84699
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 43.3093 0.0850870 0.0425435 0.999095i \(-0.486454\pi\)
0.0425435 + 0.999095i \(0.486454\pi\)
\(510\) 0 0
\(511\) − 1211.97i − 2.37175i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 986.436i − 1.91541i
\(516\) 0 0
\(517\) −432.534 −0.836623
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 426.407 0.818440 0.409220 0.912436i \(-0.365801\pi\)
0.409220 + 0.912436i \(0.365801\pi\)
\(522\) 0 0
\(523\) − 384.366i − 0.734925i −0.930038 0.367463i \(-0.880227\pi\)
0.930038 0.367463i \(-0.119773\pi\)
\(524\) 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
\(533\) 1052.13 1.97397
\(534\) 0 0
\(535\) 1274.26i 2.38180i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 433.197i 0.803705i
\(540\) 0 0
\(541\) 403.033 0.744979 0.372489 0.928037i \(-0.378504\pi\)
0.372489 + 0.928037i \(0.378504\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 931.895 1.70990
\(546\) 0 0
\(547\) − 1066.43i − 1.94960i −0.223075 0.974801i \(-0.571609\pi\)
0.223075 0.974801i \(-0.428391\pi\)
\(548\) 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
\(557\) −221.406 −0.397497 −0.198749 0.980051i \(-0.563688\pi\)
−0.198749 + 0.980051i \(0.563688\pi\)
\(558\) 0 0
\(559\) 105.533i 0.188789i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 747.764i − 1.32818i −0.747654 0.664089i \(-0.768820\pi\)
0.747654 0.664089i \(-0.231180\pi\)
\(564\) 0 0
\(565\) 1454.93 2.57510
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 489.199 0.859752 0.429876 0.902888i \(-0.358557\pi\)
0.429876 + 0.902888i \(0.358557\pi\)
\(570\) 0 0
\(571\) − 784.198i − 1.37338i −0.726952 0.686688i \(-0.759064\pi\)
0.726952 0.686688i \(-0.240936\pi\)
\(572\) 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
\(581\) −958.694 −1.65008
\(582\) 0 0
\(583\) − 288.732i − 0.495252i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 84.6640i − 0.144232i −0.997396 0.0721159i \(-0.977025\pi\)
0.997396 0.0721159i \(-0.0229751\pi\)
\(588\) 0 0
\(589\) −687.199 −1.16672
\(590\) 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
\(595\) 379.333i 0.637534i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 411.678i − 0.687275i −0.939102 0.343638i \(-0.888341\pi\)
0.939102 0.343638i \(-0.111659\pi\)
\(600\) 0 0
\(601\) −463.666 −0.771491 −0.385746 0.922605i \(-0.626056\pi\)
−0.385746 + 0.922605i \(0.626056\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 73.0794 0.120792
\(606\) 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
\(611\) 856.919i 1.40249i
\(612\) 0 0
\(613\) −459.567 −0.749702 −0.374851 0.927085i \(-0.622306\pi\)
−0.374851 + 0.927085i \(0.622306\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 370.052 0.599760 0.299880 0.953977i \(-0.403053\pi\)
0.299880 + 0.953977i \(0.403053\pi\)
\(618\) 0 0
\(619\) − 10.7987i − 0.0174453i −0.999962 0.00872267i \(-0.997223\pi\)
0.999962 0.00872267i \(-0.00277655\pi\)
\(620\) 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
\(629\) 63.8756 0.101551
\(630\) 0 0
\(631\) 63.9505i 0.101348i 0.998715 + 0.0506739i \(0.0161369\pi\)
−0.998715 + 0.0506739i \(0.983863\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 784.251i 1.23504i
\(636\) 0 0
\(637\) 858.232 1.34730
\(638\) 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
\(643\) − 23.3671i − 0.0363407i −0.999835 0.0181704i \(-0.994216\pi\)
0.999835 0.0181704i \(-0.00578412\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 686.130i 1.06048i 0.847848 + 0.530239i \(0.177898\pi\)
−0.847848 + 0.530239i \(0.822102\pi\)
\(648\) 0 0
\(649\) 254.932 0.392808
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 899.686 1.37777 0.688887 0.724869i \(-0.258100\pi\)
0.688887 + 0.724869i \(0.258100\pi\)
\(654\) 0 0
\(655\) − 1374.93i − 2.09913i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 217.506i 0.330054i 0.986289 + 0.165027i \(0.0527712\pi\)
−0.986289 + 0.165027i \(0.947229\pi\)
\(660\) 0 0
\(661\) 829.234 1.25451 0.627257 0.778812i \(-0.284178\pi\)
0.627257 + 0.778812i \(0.284178\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2076.53 −3.12261
\(666\) 0 0
\(667\) 635.333i 0.952523i
\(668\) 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
\(677\) 772.690 1.14134 0.570672 0.821178i \(-0.306683\pi\)
0.570672 + 0.821178i \(0.306683\pi\)
\(678\) 0 0
\(679\) 700.499i 1.03166i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 640.522i 0.937806i 0.883250 + 0.468903i \(0.155351\pi\)
−0.883250 + 0.468903i \(0.844649\pi\)
\(684\) 0 0
\(685\) −1909.86 −2.78812
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −572.024 −0.830224
\(690\) 0 0
\(691\) − 973.300i − 1.40854i −0.709933 0.704269i \(-0.751275\pi\)
0.709933 0.704269i \(-0.248725\pi\)
\(692\) 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
\(701\) 759.045 1.08280 0.541401 0.840764i \(-0.317894\pi\)
0.541401 + 0.840764i \(0.317894\pi\)
\(702\) 0 0
\(703\) 349.666i 0.497392i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1089.34i 1.54080i
\(708\) 0 0
\(709\) −334.034 −0.471135 −0.235567 0.971858i \(-0.575695\pi\)
−0.235567 + 0.971858i \(0.575695\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1116.00 −1.56522
\(714\) 0 0
\(715\) 1801.73i 2.51990i
\(716\) 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
\(725\) 593.839 0.819088
\(726\) 0 0
\(727\) 877.617i 1.20718i 0.797296 + 0.603588i \(0.206263\pi\)
−0.797296 + 0.603588i \(0.793737\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 24.7951i − 0.0339195i
\(732\) 0 0
\(733\) 1001.70 1.36657 0.683287 0.730150i \(-0.260550\pi\)
0.683287 + 0.730150i \(0.260550\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −696.359 −0.944857
\(738\) 0 0
\(739\) − 648.999i − 0.878212i −0.898435 0.439106i \(-0.855295\pi\)
0.898435 0.439106i \(-0.144705\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 411.678i 0.554075i 0.960859 + 0.277038i \(0.0893526\pi\)
−0.960859 + 0.277038i \(0.910647\pi\)
\(744\) 0 0
\(745\) 1384.60 1.85852
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1488.22 −1.98694
\(750\) 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
\(755\) 707.649i 0.937283i
\(756\) 0 0
\(757\) 164.101 0.216778 0.108389 0.994109i \(-0.465431\pi\)
0.108389 + 0.994109i \(0.465431\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −331.562 −0.435693 −0.217846 0.975983i \(-0.569903\pi\)
−0.217846 + 0.975983i \(0.569903\pi\)
\(762\) 0 0
\(763\) 1088.37i 1.42643i
\(764\) 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
\(773\) 980.557 1.26851 0.634254 0.773125i \(-0.281307\pi\)
0.634254 + 0.773125i \(0.281307\pi\)
\(774\) 0 0
\(775\) 1043.12i 1.34596i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 1353.21i − 1.73711i
\(780\) 0 0
\(781\) 957.864 1.22646
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −801.976 −1.02163
\(786\) 0 0
\(787\) 335.832i 0.426724i 0.976973 + 0.213362i \(0.0684414\pi\)
−0.976973 + 0.213362i \(0.931559\pi\)
\(788\) 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
\(797\) −283.114 −0.355224 −0.177612 0.984101i \(-0.556837\pi\)
−0.177612 + 0.984101i \(0.556837\pi\)
\(798\) 0 0
\(799\) − 201.335i − 0.251984i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1352.51i 1.68432i
\(804\) 0 0
\(805\) −3372.26 −4.18915
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 117.308 0.145004 0.0725019 0.997368i \(-0.476902\pi\)
0.0725019 + 0.997368i \(0.476902\pi\)
\(810\) 0 0
\(811\) 1235.43i 1.52334i 0.647963 + 0.761672i \(0.275621\pi\)
−0.647963 + 0.761672i \(0.724379\pi\)
\(812\) 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
\(821\) −823.364 −1.00288 −0.501440 0.865192i \(-0.667196\pi\)
−0.501440 + 0.865192i \(0.667196\pi\)
\(822\) 0 0
\(823\) − 743.249i − 0.903097i −0.892247 0.451548i \(-0.850872\pi\)
0.892247 0.451548i \(-0.149128\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 542.352i 0.655806i 0.944711 + 0.327903i \(0.106342\pi\)
−0.944711 + 0.327903i \(0.893658\pi\)
\(828\) 0 0
\(829\) −906.564 −1.09356 −0.546782 0.837275i \(-0.684147\pi\)
−0.546782 + 0.837275i \(0.684147\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −201.643 −0.242069
\(834\) 0 0
\(835\) − 1754.26i − 2.10092i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1015.97i 1.21093i 0.795873 + 0.605464i \(0.207012\pi\)
−0.795873 + 0.605464i \(0.792988\pi\)
\(840\) 0 0
\(841\) −630.533 −0.749742
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2197.25 2.60030
\(846\) 0 0
\(847\) 85.3498i 0.100767i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 567.853i 0.667277i
\(852\) 0 0
\(853\) 102.568 0.120244 0.0601222 0.998191i \(-0.480851\pi\)
0.0601222 + 0.998191i \(0.480851\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1081.85 1.26237 0.631183 0.775634i \(-0.282570\pi\)
0.631183 + 0.775634i \(0.282570\pi\)
\(858\) 0 0
\(859\) − 597.231i − 0.695264i −0.937631 0.347632i \(-0.886986\pi\)
0.937631 0.347632i \(-0.113014\pi\)
\(860\) 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
\(869\) 376.226 0.432942
\(870\) 0 0
\(871\) 1379.60i 1.58393i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1226.92i 1.40220i
\(876\) 0 0
\(877\) 395.899 0.451424 0.225712 0.974194i \(-0.427529\pi\)
0.225712 + 0.974194i \(0.427529\pi\)
\(878\) 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
\(883\) 1548.96i 1.75421i 0.480302 + 0.877103i \(0.340527\pi\)
−0.480302 + 0.877103i \(0.659473\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1544.41i 1.74116i 0.492023 + 0.870582i \(0.336257\pi\)
−0.492023 + 0.870582i \(0.663743\pi\)
\(888\) 0 0
\(889\) −915.931 −1.03029
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1102.14 1.23420
\(894\) 0 0
\(895\) 195.600i 0.218547i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 369.699i 0.411233i
\(900\) 0 0
\(901\) 134.398 0.149166
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1063.98 1.17567
\(906\) 0 0
\(907\) − 472.831i − 0.521313i −0.965432 0.260657i \(-0.916061\pi\)
0.965432 0.260657i \(-0.0839390\pi\)
\(908\) 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
\(917\) 1605.79 1.75114
\(918\) 0 0
\(919\) − 152.850i − 0.166323i −0.996536 0.0831613i \(-0.973498\pi\)
0.996536 0.0831613i \(-0.0265017\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 1897.68i − 2.05599i
\(924\) 0 0
\(925\) 530.766 0.573802
\(926\) 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
\(931\) − 1103.83i − 1.18564i
\(932\) 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 \(-0.251068\pi\)
0.704731 + 0.709475i \(0.251068\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −5.41027 −0.00574949 −0.00287475 0.999996i \(-0.500915\pi\)
−0.00287475 + 0.999996i \(0.500915\pi\)
\(942\) 0 0
\(943\) − 2197.60i − 2.33043i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 83.9577i 0.0886565i 0.999017 + 0.0443283i \(0.0141147\pi\)
−0.999017 + 0.0443283i \(0.985885\pi\)
\(948\) 0 0
\(949\) 2679.53 2.82353
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1577.15 −1.65494 −0.827468 0.561513i \(-0.810220\pi\)
−0.827468 + 0.561513i \(0.810220\pi\)
\(954\) 0 0
\(955\) − 1043.07i − 1.09222i
\(956\) 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
\(965\) −1361.44 −1.41082
\(966\) 0 0
\(967\) − 278.616i − 0.288124i −0.989569 0.144062i \(-0.953984\pi\)
0.989569 0.144062i \(-0.0460164\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 380.684i 0.392053i 0.980599 + 0.196027i \(0.0628039\pi\)
−0.980599 + 0.196027i \(0.937196\pi\)
\(972\) 0 0
\(973\) 286.398 0.294346
\(974\) 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
\(979\) 166.131i 0.169695i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 928.382i − 0.944437i −0.881482 0.472219i \(-0.843453\pi\)
0.881482 0.472219i \(-0.156547\pi\)
\(984\) 0 0
\(985\) −993.399 −1.00853
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 220.429 0.222880
\(990\) 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
\(995\) − 2949.83i − 2.96466i
\(996\) 0 0
\(997\) −595.567 −0.597359 −0.298680 0.954353i \(-0.596546\pi\)
−0.298680 + 0.954353i \(0.596546\pi\)
\(998\) 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.d.127.8 yes 8
3.2 odd 2 inner 1152.3.g.d.127.2 yes 8
4.3 odd 2 inner 1152.3.g.d.127.7 yes 8
8.3 odd 2 1152.3.g.e.127.1 yes 8
8.5 even 2 1152.3.g.e.127.2 yes 8
12.11 even 2 inner 1152.3.g.d.127.1 8
16.3 odd 4 2304.3.b.s.127.2 8
16.5 even 4 2304.3.b.s.127.7 8
16.11 odd 4 2304.3.b.r.127.8 8
16.13 even 4 2304.3.b.r.127.1 8
24.5 odd 2 1152.3.g.e.127.8 yes 8
24.11 even 2 1152.3.g.e.127.7 yes 8
48.5 odd 4 2304.3.b.s.127.1 8
48.11 even 4 2304.3.b.r.127.2 8
48.29 odd 4 2304.3.b.r.127.7 8
48.35 even 4 2304.3.b.s.127.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.g.d.127.1 8 12.11 even 2 inner
1152.3.g.d.127.2 yes 8 3.2 odd 2 inner
1152.3.g.d.127.7 yes 8 4.3 odd 2 inner
1152.3.g.d.127.8 yes 8 1.1 even 1 trivial
1152.3.g.e.127.1 yes 8 8.3 odd 2
1152.3.g.e.127.2 yes 8 8.5 even 2
1152.3.g.e.127.7 yes 8 24.11 even 2
1152.3.g.e.127.8 yes 8 24.5 odd 2
2304.3.b.r.127.1 8 16.13 even 4
2304.3.b.r.127.2 8 48.11 even 4
2304.3.b.r.127.7 8 48.29 odd 4
2304.3.b.r.127.8 8 16.11 odd 4
2304.3.b.s.127.1 8 48.5 odd 4
2304.3.b.s.127.2 8 16.3 odd 4
2304.3.b.s.127.7 8 16.5 even 4
2304.3.b.s.127.8 8 48.35 even 4