Properties

Label 1152.3.b.i.703.5
Level $1152$
Weight $3$
Character 1152.703
Analytic conductor $31.390$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,3,Mod(703,1152)]
 
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, 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("1152.703");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.b (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 703.5
Root \(0.581861 + 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 1152.703
Dual form 1152.3.b.i.703.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.29150i q^{5} -7.48331i q^{7} +O(q^{10})\) \(q+5.29150i q^{5} -7.48331i q^{7} +5.65685 q^{11} -4.00000i q^{13} +21.1660 q^{17} -29.9333 q^{19} -22.6274i q^{23} -3.00000 q^{25} +5.29150i q^{29} -22.4499i q^{31} +39.5980 q^{35} +28.0000i q^{37} -63.4980 q^{41} +29.9333 q^{43} -67.8823i q^{47} -7.00000 q^{49} -47.6235i q^{53} +29.9333i q^{55} +101.823 q^{59} -76.0000i q^{61} +21.1660 q^{65} +59.8665 q^{67} -90.5097i q^{71} +26.0000 q^{73} -42.3320i q^{77} +127.216i q^{79} +118.794 q^{83} +112.000i q^{85} +42.3320 q^{89} -29.9333 q^{91} -158.392i q^{95} +18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{25} - 56 q^{49} + 208 q^{73} + 144 q^{97}+O(q^{100}) \) 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
\(5\) 5.29150i 1.05830i 0.848528 + 0.529150i \(0.177489\pi\)
−0.848528 + 0.529150i \(0.822511\pi\)
\(6\) 0 0
\(7\) − 7.48331i − 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685 0.514259 0.257130 0.966377i \(-0.417223\pi\)
0.257130 + 0.966377i \(0.417223\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 0.307692i −0.988095 0.153846i \(-0.950834\pi\)
0.988095 0.153846i \(-0.0491660\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 21.1660 1.24506 0.622530 0.782596i \(-0.286105\pi\)
0.622530 + 0.782596i \(0.286105\pi\)
\(18\) 0 0
\(19\) −29.9333 −1.57543 −0.787717 0.616037i \(-0.788737\pi\)
−0.787717 + 0.616037i \(0.788737\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 22.6274i − 0.983801i −0.870651 0.491900i \(-0.836302\pi\)
0.870651 0.491900i \(-0.163698\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.120000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.29150i 0.182466i 0.995830 + 0.0912328i \(0.0290807\pi\)
−0.995830 + 0.0912328i \(0.970919\pi\)
\(30\) 0 0
\(31\) − 22.4499i − 0.724192i −0.932141 0.362096i \(-0.882061\pi\)
0.932141 0.362096i \(-0.117939\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 39.5980 1.13137
\(36\) 0 0
\(37\) 28.0000i 0.756757i 0.925651 + 0.378378i \(0.123518\pi\)
−0.925651 + 0.378378i \(0.876482\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −63.4980 −1.54873 −0.774366 0.632738i \(-0.781931\pi\)
−0.774366 + 0.632738i \(0.781931\pi\)
\(42\) 0 0
\(43\) 29.9333 0.696122 0.348061 0.937472i \(-0.386840\pi\)
0.348061 + 0.937472i \(0.386840\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 67.8823i − 1.44430i −0.691735 0.722152i \(-0.743153\pi\)
0.691735 0.722152i \(-0.256847\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 47.6235i − 0.898557i −0.893392 0.449279i \(-0.851681\pi\)
0.893392 0.449279i \(-0.148319\pi\)
\(54\) 0 0
\(55\) 29.9333i 0.544241i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 101.823 1.72582 0.862910 0.505358i \(-0.168639\pi\)
0.862910 + 0.505358i \(0.168639\pi\)
\(60\) 0 0
\(61\) − 76.0000i − 1.24590i −0.782261 0.622951i \(-0.785934\pi\)
0.782261 0.622951i \(-0.214066\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 21.1660 0.325631
\(66\) 0 0
\(67\) 59.8665 0.893530 0.446765 0.894651i \(-0.352576\pi\)
0.446765 + 0.894651i \(0.352576\pi\)
\(68\) 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
\(73\) 26.0000 0.356164 0.178082 0.984016i \(-0.443011\pi\)
0.178082 + 0.984016i \(0.443011\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 42.3320i − 0.549767i
\(78\) 0 0
\(79\) 127.216i 1.61033i 0.593048 + 0.805167i \(0.297924\pi\)
−0.593048 + 0.805167i \(0.702076\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 118.794 1.43125 0.715626 0.698484i \(-0.246141\pi\)
0.715626 + 0.698484i \(0.246141\pi\)
\(84\) 0 0
\(85\) 112.000i 1.31765i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 42.3320 0.475641 0.237820 0.971309i \(-0.423567\pi\)
0.237820 + 0.971309i \(0.423567\pi\)
\(90\) 0 0
\(91\) −29.9333 −0.328937
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 158.392i − 1.66728i
\(96\) 0 0
\(97\) 18.0000 0.185567 0.0927835 0.995686i \(-0.470424\pi\)
0.0927835 + 0.995686i \(0.470424\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 164.037i 1.62412i 0.583571 + 0.812062i \(0.301655\pi\)
−0.583571 + 0.812062i \(0.698345\pi\)
\(102\) 0 0
\(103\) − 22.4499i − 0.217961i −0.994044 0.108980i \(-0.965241\pi\)
0.994044 0.108980i \(-0.0347585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 101.823 0.951620 0.475810 0.879548i \(-0.342155\pi\)
0.475810 + 0.879548i \(0.342155\pi\)
\(108\) 0 0
\(109\) − 148.000i − 1.35780i −0.734232 0.678899i \(-0.762457\pi\)
0.734232 0.678899i \(-0.237543\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −42.3320 −0.374620 −0.187310 0.982301i \(-0.559977\pi\)
−0.187310 + 0.982301i \(0.559977\pi\)
\(114\) 0 0
\(115\) 119.733 1.04116
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 158.392i − 1.33102i
\(120\) 0 0
\(121\) −89.0000 −0.735537
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 116.413i 0.931304i
\(126\) 0 0
\(127\) − 217.016i − 1.70879i −0.519625 0.854394i \(-0.673928\pi\)
0.519625 0.854394i \(-0.326072\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 158.392 1.20910 0.604549 0.796568i \(-0.293353\pi\)
0.604549 + 0.796568i \(0.293353\pi\)
\(132\) 0 0
\(133\) 224.000i 1.68421i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 232.826 1.69946 0.849730 0.527218i \(-0.176765\pi\)
0.849730 + 0.527218i \(0.176765\pi\)
\(138\) 0 0
\(139\) −179.600 −1.29208 −0.646042 0.763302i \(-0.723577\pi\)
−0.646042 + 0.763302i \(0.723577\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 22.6274i − 0.158234i
\(144\) 0 0
\(145\) −28.0000 −0.193103
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 248.701i − 1.66913i −0.550908 0.834566i \(-0.685719\pi\)
0.550908 0.834566i \(-0.314281\pi\)
\(150\) 0 0
\(151\) 22.4499i 0.148675i 0.997233 + 0.0743376i \(0.0236842\pi\)
−0.997233 + 0.0743376i \(0.976316\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 118.794 0.766413
\(156\) 0 0
\(157\) − 236.000i − 1.50318i −0.659628 0.751592i \(-0.729286\pi\)
0.659628 0.751592i \(-0.270714\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −169.328 −1.05173
\(162\) 0 0
\(163\) −269.399 −1.65276 −0.826378 0.563115i \(-0.809603\pi\)
−0.826378 + 0.563115i \(0.809603\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 113.137i − 0.677468i −0.940882 0.338734i \(-0.890001\pi\)
0.940882 0.338734i \(-0.109999\pi\)
\(168\) 0 0
\(169\) 153.000 0.905325
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 89.9555i − 0.519974i −0.965612 0.259987i \(-0.916282\pi\)
0.965612 0.259987i \(-0.0837183\pi\)
\(174\) 0 0
\(175\) 22.4499i 0.128285i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −33.9411 −0.189615 −0.0948076 0.995496i \(-0.530224\pi\)
−0.0948076 + 0.995496i \(0.530224\pi\)
\(180\) 0 0
\(181\) 228.000i 1.25967i 0.776730 + 0.629834i \(0.216877\pi\)
−0.776730 + 0.629834i \(0.783123\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −148.162 −0.800876
\(186\) 0 0
\(187\) 119.733 0.640284
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 22.6274i − 0.118468i −0.998244 0.0592341i \(-0.981134\pi\)
0.998244 0.0592341i \(-0.0188658\pi\)
\(192\) 0 0
\(193\) 70.0000 0.362694 0.181347 0.983419i \(-0.441954\pi\)
0.181347 + 0.983419i \(0.441954\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 121.705i 0.617790i 0.951096 + 0.308895i \(0.0999591\pi\)
−0.951096 + 0.308895i \(0.900041\pi\)
\(198\) 0 0
\(199\) − 202.049i − 1.01532i −0.861556 0.507662i \(-0.830510\pi\)
0.861556 0.507662i \(-0.169490\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 39.5980 0.195064
\(204\) 0 0
\(205\) − 336.000i − 1.63902i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −169.328 −0.810182
\(210\) 0 0
\(211\) −59.8665 −0.283728 −0.141864 0.989886i \(-0.545309\pi\)
−0.141864 + 0.989886i \(0.545309\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 158.392i 0.736707i
\(216\) 0 0
\(217\) −168.000 −0.774194
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 84.6640i − 0.383095i
\(222\) 0 0
\(223\) 172.116i 0.771822i 0.922536 + 0.385911i \(0.126113\pi\)
−0.922536 + 0.385911i \(0.873887\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −96.1665 −0.423641 −0.211821 0.977309i \(-0.567939\pi\)
−0.211821 + 0.977309i \(0.567939\pi\)
\(228\) 0 0
\(229\) 244.000i 1.06550i 0.846272 + 0.532751i \(0.178842\pi\)
−0.846272 + 0.532751i \(0.821158\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −42.3320 −0.181682 −0.0908412 0.995865i \(-0.528956\pi\)
−0.0908412 + 0.995865i \(0.528956\pi\)
\(234\) 0 0
\(235\) 359.199 1.52851
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 67.8823i − 0.284026i −0.989865 0.142013i \(-0.954642\pi\)
0.989865 0.142013i \(-0.0453575\pi\)
\(240\) 0 0
\(241\) −166.000 −0.688797 −0.344398 0.938824i \(-0.611917\pi\)
−0.344398 + 0.938824i \(0.611917\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 37.0405i − 0.151186i
\(246\) 0 0
\(247\) 119.733i 0.484749i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −401.637 −1.60015 −0.800073 0.599903i \(-0.795206\pi\)
−0.800073 + 0.599903i \(0.795206\pi\)
\(252\) 0 0
\(253\) − 128.000i − 0.505929i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 423.320 1.64716 0.823580 0.567200i \(-0.191973\pi\)
0.823580 + 0.567200i \(0.191973\pi\)
\(258\) 0 0
\(259\) 209.533 0.809007
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 271.529i 1.03243i 0.856459 + 0.516215i \(0.172659\pi\)
−0.856459 + 0.516215i \(0.827341\pi\)
\(264\) 0 0
\(265\) 252.000 0.950943
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 37.0405i − 0.137697i −0.997627 0.0688485i \(-0.978067\pi\)
0.997627 0.0688485i \(-0.0219325\pi\)
\(270\) 0 0
\(271\) 52.3832i 0.193296i 0.995319 + 0.0966480i \(0.0308121\pi\)
−0.995319 + 0.0966480i \(0.969188\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −16.9706 −0.0617111
\(276\) 0 0
\(277\) 372.000i 1.34296i 0.741023 + 0.671480i \(0.234341\pi\)
−0.741023 + 0.671480i \(0.765659\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 84.6640 0.301296 0.150648 0.988588i \(-0.451864\pi\)
0.150648 + 0.988588i \(0.451864\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 475.176i 1.65566i
\(288\) 0 0
\(289\) 159.000 0.550173
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 291.033i 0.993285i 0.867955 + 0.496643i \(0.165434\pi\)
−0.867955 + 0.496643i \(0.834566\pi\)
\(294\) 0 0
\(295\) 538.799i 1.82644i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −90.5097 −0.302708
\(300\) 0 0
\(301\) − 224.000i − 0.744186i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 402.154 1.31854
\(306\) 0 0
\(307\) −179.600 −0.585015 −0.292507 0.956263i \(-0.594490\pi\)
−0.292507 + 0.956263i \(0.594490\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 135.765i 0.436542i 0.975888 + 0.218271i \(0.0700416\pi\)
−0.975888 + 0.218271i \(0.929958\pi\)
\(312\) 0 0
\(313\) −470.000 −1.50160 −0.750799 0.660531i \(-0.770331\pi\)
−0.750799 + 0.660531i \(0.770331\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 386.280i 1.21855i 0.792960 + 0.609274i \(0.208539\pi\)
−0.792960 + 0.609274i \(0.791461\pi\)
\(318\) 0 0
\(319\) 29.9333i 0.0938347i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −633.568 −1.96151
\(324\) 0 0
\(325\) 12.0000i 0.0369231i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −507.984 −1.54403
\(330\) 0 0
\(331\) −119.733 −0.361731 −0.180866 0.983508i \(-0.557890\pi\)
−0.180866 + 0.983508i \(0.557890\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 316.784i 0.945623i
\(336\) 0 0
\(337\) −42.0000 −0.124629 −0.0623145 0.998057i \(-0.519848\pi\)
−0.0623145 + 0.998057i \(0.519848\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 126.996i − 0.372422i
\(342\) 0 0
\(343\) − 314.299i − 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −164.049 −0.472763 −0.236382 0.971660i \(-0.575962\pi\)
−0.236382 + 0.971660i \(0.575962\pi\)
\(348\) 0 0
\(349\) − 284.000i − 0.813754i −0.913483 0.406877i \(-0.866618\pi\)
0.913483 0.406877i \(-0.133382\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −211.660 −0.599604 −0.299802 0.954001i \(-0.596921\pi\)
−0.299802 + 0.954001i \(0.596921\pi\)
\(354\) 0 0
\(355\) 478.932 1.34910
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 452.548i 1.26058i 0.776360 + 0.630290i \(0.217064\pi\)
−0.776360 + 0.630290i \(0.782936\pi\)
\(360\) 0 0
\(361\) 535.000 1.48199
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 137.579i 0.376929i
\(366\) 0 0
\(367\) − 561.249i − 1.52929i −0.644453 0.764644i \(-0.722915\pi\)
0.644453 0.764644i \(-0.277085\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −356.382 −0.960598
\(372\) 0 0
\(373\) − 84.0000i − 0.225201i −0.993640 0.112601i \(-0.964082\pi\)
0.993640 0.112601i \(-0.0359180\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 21.1660 0.0561433
\(378\) 0 0
\(379\) −448.999 −1.18469 −0.592347 0.805683i \(-0.701798\pi\)
−0.592347 + 0.805683i \(0.701798\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 158.392i 0.413556i 0.978388 + 0.206778i \(0.0662978\pi\)
−0.978388 + 0.206778i \(0.933702\pi\)
\(384\) 0 0
\(385\) 224.000 0.581818
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 714.353i − 1.83638i −0.396137 0.918191i \(-0.629650\pi\)
0.396137 0.918191i \(-0.370350\pi\)
\(390\) 0 0
\(391\) − 478.932i − 1.22489i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −673.166 −1.70422
\(396\) 0 0
\(397\) 244.000i 0.614610i 0.951611 + 0.307305i \(0.0994271\pi\)
−0.951611 + 0.307305i \(0.900573\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 232.826 0.580614 0.290307 0.956934i \(-0.406243\pi\)
0.290307 + 0.956934i \(0.406243\pi\)
\(402\) 0 0
\(403\) −89.7998 −0.222828
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 158.392i 0.389169i
\(408\) 0 0
\(409\) −418.000 −1.02200 −0.511002 0.859579i \(-0.670726\pi\)
−0.511002 + 0.859579i \(0.670726\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 761.976i − 1.84498i
\(414\) 0 0
\(415\) 628.598i 1.51470i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 152.735 0.364523 0.182261 0.983250i \(-0.441658\pi\)
0.182261 + 0.983250i \(0.441658\pi\)
\(420\) 0 0
\(421\) 212.000i 0.503563i 0.967784 + 0.251781i \(0.0810164\pi\)
−0.967784 + 0.251781i \(0.918984\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −63.4980 −0.149407
\(426\) 0 0
\(427\) −568.732 −1.33192
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 746.705i 1.73249i 0.499616 + 0.866247i \(0.333474\pi\)
−0.499616 + 0.866247i \(0.666526\pi\)
\(432\) 0 0
\(433\) −466.000 −1.07621 −0.538106 0.842877i \(-0.680860\pi\)
−0.538106 + 0.842877i \(0.680860\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 677.312i 1.54991i
\(438\) 0 0
\(439\) − 381.649i − 0.869360i −0.900585 0.434680i \(-0.856861\pi\)
0.900585 0.434680i \(-0.143139\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −254.558 −0.574624 −0.287312 0.957837i \(-0.592762\pi\)
−0.287312 + 0.957837i \(0.592762\pi\)
\(444\) 0 0
\(445\) 224.000i 0.503371i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −571.482 −1.27279 −0.636395 0.771364i \(-0.719575\pi\)
−0.636395 + 0.771364i \(0.719575\pi\)
\(450\) 0 0
\(451\) −359.199 −0.796450
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 158.392i − 0.348114i
\(456\) 0 0
\(457\) 542.000 1.18600 0.592998 0.805204i \(-0.297944\pi\)
0.592998 + 0.805204i \(0.297944\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 174.620i − 0.378784i −0.981902 0.189392i \(-0.939348\pi\)
0.981902 0.189392i \(-0.0606517\pi\)
\(462\) 0 0
\(463\) − 336.749i − 0.727320i −0.931532 0.363660i \(-0.881527\pi\)
0.931532 0.363660i \(-0.118473\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −526.087 −1.12653 −0.563263 0.826278i \(-0.690454\pi\)
−0.563263 + 0.826278i \(0.690454\pi\)
\(468\) 0 0
\(469\) − 448.000i − 0.955224i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 169.328 0.357987
\(474\) 0 0
\(475\) 89.7998 0.189052
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 181.019i − 0.377911i −0.981986 0.188955i \(-0.939490\pi\)
0.981986 0.188955i \(-0.0605102\pi\)
\(480\) 0 0
\(481\) 112.000 0.232848
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 95.2470i 0.196386i
\(486\) 0 0
\(487\) 651.048i 1.33686i 0.743777 + 0.668428i \(0.233032\pi\)
−0.743777 + 0.668428i \(0.766968\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 248.902 0.506928 0.253464 0.967345i \(-0.418430\pi\)
0.253464 + 0.967345i \(0.418430\pi\)
\(492\) 0 0
\(493\) 112.000i 0.227181i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −677.312 −1.36280
\(498\) 0 0
\(499\) 239.466 0.479892 0.239946 0.970786i \(-0.422870\pi\)
0.239946 + 0.970786i \(0.422870\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 950.352i 1.88937i 0.327984 + 0.944683i \(0.393631\pi\)
−0.327984 + 0.944683i \(0.606369\pi\)
\(504\) 0 0
\(505\) −868.000 −1.71881
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 428.612i 0.842066i 0.907045 + 0.421033i \(0.138332\pi\)
−0.907045 + 0.421033i \(0.861668\pi\)
\(510\) 0 0
\(511\) − 194.566i − 0.380756i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 118.794 0.230668
\(516\) 0 0
\(517\) − 384.000i − 0.742747i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 698.478 1.34065 0.670325 0.742068i \(-0.266155\pi\)
0.670325 + 0.742068i \(0.266155\pi\)
\(522\) 0 0
\(523\) 868.065 1.65978 0.829890 0.557928i \(-0.188403\pi\)
0.829890 + 0.557928i \(0.188403\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 475.176i − 0.901662i
\(528\) 0 0
\(529\) 17.0000 0.0321361
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 253.992i 0.476533i
\(534\) 0 0
\(535\) 538.799i 1.00710i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −39.5980 −0.0734656
\(540\) 0 0
\(541\) 556.000i 1.02773i 0.857872 + 0.513863i \(0.171786\pi\)
−0.857872 + 0.513863i \(0.828214\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 783.142 1.43696
\(546\) 0 0
\(547\) −448.999 −0.820839 −0.410419 0.911897i \(-0.634618\pi\)
−0.410419 + 0.911897i \(0.634618\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 158.392i − 0.287463i
\(552\) 0 0
\(553\) 952.000 1.72152
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 799.017i 1.43450i 0.696816 + 0.717250i \(0.254600\pi\)
−0.696816 + 0.717250i \(0.745400\pi\)
\(558\) 0 0
\(559\) − 119.733i − 0.214191i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 220.617 0.391860 0.195930 0.980618i \(-0.437227\pi\)
0.195930 + 0.980618i \(0.437227\pi\)
\(564\) 0 0
\(565\) − 224.000i − 0.396460i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 529.150 0.929965 0.464983 0.885320i \(-0.346061\pi\)
0.464983 + 0.885320i \(0.346061\pi\)
\(570\) 0 0
\(571\) 897.998 1.57268 0.786338 0.617797i \(-0.211975\pi\)
0.786338 + 0.617797i \(0.211975\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 67.8823i 0.118056i
\(576\) 0 0
\(577\) 586.000 1.01560 0.507799 0.861476i \(-0.330459\pi\)
0.507799 + 0.861476i \(0.330459\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 888.972i − 1.53007i
\(582\) 0 0
\(583\) − 269.399i − 0.462091i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.2548 0.0770951 0.0385476 0.999257i \(-0.487727\pi\)
0.0385476 + 0.999257i \(0.487727\pi\)
\(588\) 0 0
\(589\) 672.000i 1.14092i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −296.324 −0.499703 −0.249852 0.968284i \(-0.580382\pi\)
−0.249852 + 0.968284i \(0.580382\pi\)
\(594\) 0 0
\(595\) 838.131 1.40862
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 1154.00i − 1.92654i −0.268532 0.963271i \(-0.586539\pi\)
0.268532 0.963271i \(-0.413461\pi\)
\(600\) 0 0
\(601\) −634.000 −1.05491 −0.527454 0.849583i \(-0.676853\pi\)
−0.527454 + 0.849583i \(0.676853\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 470.944i − 0.778419i
\(606\) 0 0
\(607\) 800.715i 1.31913i 0.751646 + 0.659567i \(0.229260\pi\)
−0.751646 + 0.659567i \(0.770740\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −271.529 −0.444401
\(612\) 0 0
\(613\) − 1140.00i − 1.85971i −0.367931 0.929853i \(-0.619934\pi\)
0.367931 0.929853i \(-0.380066\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 592.648 0.960532 0.480266 0.877123i \(-0.340540\pi\)
0.480266 + 0.877123i \(0.340540\pi\)
\(618\) 0 0
\(619\) −359.199 −0.580289 −0.290145 0.956983i \(-0.593703\pi\)
−0.290145 + 0.956983i \(0.593703\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 316.784i − 0.508481i
\(624\) 0 0
\(625\) −691.000 −1.10560
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 592.648i 0.942207i
\(630\) 0 0
\(631\) 291.849i 0.462519i 0.972892 + 0.231259i \(0.0742846\pi\)
−0.972892 + 0.231259i \(0.925715\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1148.34 1.80841
\(636\) 0 0
\(637\) 28.0000i 0.0439560i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −486.818 −0.759467 −0.379733 0.925096i \(-0.623984\pi\)
−0.379733 + 0.925096i \(0.623984\pi\)
\(642\) 0 0
\(643\) −808.198 −1.25692 −0.628459 0.777843i \(-0.716314\pi\)
−0.628459 + 0.777843i \(0.716314\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 294.156i 0.454647i 0.973819 + 0.227323i \(0.0729974\pi\)
−0.973819 + 0.227323i \(0.927003\pi\)
\(648\) 0 0
\(649\) 576.000 0.887519
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 79.3725i 0.121551i 0.998151 + 0.0607753i \(0.0193573\pi\)
−0.998151 + 0.0607753i \(0.980643\pi\)
\(654\) 0 0
\(655\) 838.131i 1.27959i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1040.86 1.57946 0.789728 0.613457i \(-0.210222\pi\)
0.789728 + 0.613457i \(0.210222\pi\)
\(660\) 0 0
\(661\) − 548.000i − 0.829047i −0.910039 0.414523i \(-0.863948\pi\)
0.910039 0.414523i \(-0.136052\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1185.30 −1.78240
\(666\) 0 0
\(667\) 119.733 0.179510
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 429.921i − 0.640717i
\(672\) 0 0
\(673\) 334.000 0.496285 0.248143 0.968724i \(-0.420180\pi\)
0.248143 + 0.968724i \(0.420180\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 79.3725i − 0.117242i −0.998280 0.0586208i \(-0.981330\pi\)
0.998280 0.0586208i \(-0.0186703\pi\)
\(678\) 0 0
\(679\) − 134.700i − 0.198379i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −627.911 −0.919342 −0.459671 0.888089i \(-0.652033\pi\)
−0.459671 + 0.888089i \(0.652033\pi\)
\(684\) 0 0
\(685\) 1232.00i 1.79854i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −190.494 −0.276479
\(690\) 0 0
\(691\) −748.331 −1.08297 −0.541484 0.840711i \(-0.682137\pi\)
−0.541484 + 0.840711i \(0.682137\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 950.352i − 1.36741i
\(696\) 0 0
\(697\) −1344.00 −1.92826
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 121.705i − 0.173616i −0.996225 0.0868078i \(-0.972333\pi\)
0.996225 0.0868078i \(-0.0276666\pi\)
\(702\) 0 0
\(703\) − 838.131i − 1.19222i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1227.54 1.73626
\(708\) 0 0
\(709\) 4.00000i 0.00564175i 0.999996 + 0.00282087i \(0.000897913\pi\)
−0.999996 + 0.00282087i \(0.999102\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −507.984 −0.712460
\(714\) 0 0
\(715\) 119.733 0.167459
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 316.784i − 0.440589i −0.975433 0.220295i \(-0.929298\pi\)
0.975433 0.220295i \(-0.0707019\pi\)
\(720\) 0 0
\(721\) −168.000 −0.233010
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 15.8745i − 0.0218959i
\(726\) 0 0
\(727\) − 1174.88i − 1.61607i −0.589137 0.808033i \(-0.700532\pi\)
0.589137 0.808033i \(-0.299468\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 633.568 0.866714
\(732\) 0 0
\(733\) 1212.00i 1.65348i 0.562585 + 0.826739i \(0.309807\pi\)
−0.562585 + 0.826739i \(0.690193\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 338.656 0.459506
\(738\) 0 0
\(739\) 1257.20 1.70121 0.850607 0.525802i \(-0.176235\pi\)
0.850607 + 0.525802i \(0.176235\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 1018.23i − 1.37044i −0.728338 0.685218i \(-0.759707\pi\)
0.728338 0.685218i \(-0.240293\pi\)
\(744\) 0 0
\(745\) 1316.00 1.76644
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 761.976i − 1.01732i
\(750\) 0 0
\(751\) − 411.582i − 0.548046i −0.961723 0.274023i \(-0.911646\pi\)
0.961723 0.274023i \(-0.0883544\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −118.794 −0.157343
\(756\) 0 0
\(757\) − 1020.00i − 1.34742i −0.738994 0.673712i \(-0.764699\pi\)
0.738994 0.673712i \(-0.235301\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −529.150 −0.695335 −0.347668 0.937618i \(-0.613026\pi\)
−0.347668 + 0.937618i \(0.613026\pi\)
\(762\) 0 0
\(763\) −1107.53 −1.45155
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 407.294i − 0.531022i
\(768\) 0 0
\(769\) 262.000 0.340702 0.170351 0.985383i \(-0.445510\pi\)
0.170351 + 0.985383i \(0.445510\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 851.932i − 1.10211i −0.834469 0.551056i \(-0.814225\pi\)
0.834469 0.551056i \(-0.185775\pi\)
\(774\) 0 0
\(775\) 67.3498i 0.0869030i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1900.70 2.43993
\(780\) 0 0
\(781\) − 512.000i − 0.655570i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1248.79 1.59082
\(786\) 0 0
\(787\) −568.732 −0.722658 −0.361329 0.932438i \(-0.617677\pi\)
−0.361329 + 0.932438i \(0.617677\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 316.784i 0.400485i
\(792\) 0 0
\(793\) −304.000 −0.383354
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 894.264i − 1.12204i −0.827803 0.561019i \(-0.810410\pi\)
0.827803 0.561019i \(-0.189590\pi\)
\(798\) 0 0
\(799\) − 1436.80i − 1.79824i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 147.078 0.183161
\(804\) 0 0
\(805\) − 896.000i − 1.11304i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −105.830 −0.130816 −0.0654079 0.997859i \(-0.520835\pi\)
−0.0654079 + 0.997859i \(0.520835\pi\)
\(810\) 0 0
\(811\) −987.798 −1.21800 −0.609000 0.793170i \(-0.708429\pi\)
−0.609000 + 0.793170i \(0.708429\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1425.53i − 1.74911i
\(816\) 0 0
\(817\) −896.000 −1.09670
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1010.68i 1.23103i 0.788125 + 0.615516i \(0.211052\pi\)
−0.788125 + 0.615516i \(0.788948\pi\)
\(822\) 0 0
\(823\) 830.648i 1.00929i 0.863326 + 0.504646i \(0.168377\pi\)
−0.863326 + 0.504646i \(0.831623\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −328.098 −0.396732 −0.198366 0.980128i \(-0.563563\pi\)
−0.198366 + 0.980128i \(0.563563\pi\)
\(828\) 0 0
\(829\) 1164.00i 1.40410i 0.712127 + 0.702051i \(0.247732\pi\)
−0.712127 + 0.702051i \(0.752268\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −148.162 −0.177866
\(834\) 0 0
\(835\) 598.665 0.716964
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 995.606i − 1.18666i −0.804960 0.593329i \(-0.797813\pi\)
0.804960 0.593329i \(-0.202187\pi\)
\(840\) 0 0
\(841\) 813.000 0.966706
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 809.600i 0.958106i
\(846\) 0 0
\(847\) 666.015i 0.786322i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 633.568 0.744498
\(852\) 0 0
\(853\) − 276.000i − 0.323564i −0.986827 0.161782i \(-0.948276\pi\)
0.986827 0.161782i \(-0.0517241\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 275.158 0.321071 0.160536 0.987030i \(-0.448678\pi\)
0.160536 + 0.987030i \(0.448678\pi\)
\(858\) 0 0
\(859\) −628.598 −0.731779 −0.365890 0.930658i \(-0.619235\pi\)
−0.365890 + 0.930658i \(0.619235\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 294.156i 0.340853i 0.985370 + 0.170427i \(0.0545146\pi\)
−0.985370 + 0.170427i \(0.945485\pi\)
\(864\) 0 0
\(865\) 476.000 0.550289
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 719.644i 0.828129i
\(870\) 0 0
\(871\) − 239.466i − 0.274932i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 871.156 0.995606
\(876\) 0 0
\(877\) 340.000i 0.387685i 0.981033 + 0.193843i \(0.0620951\pi\)
−0.981033 + 0.193843i \(0.937905\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 761.976 0.864899 0.432450 0.901658i \(-0.357649\pi\)
0.432450 + 0.901658i \(0.357649\pi\)
\(882\) 0 0
\(883\) 89.7998 0.101699 0.0508493 0.998706i \(-0.483807\pi\)
0.0508493 + 0.998706i \(0.483807\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 859.842i − 0.969382i −0.874685 0.484691i \(-0.838932\pi\)
0.874685 0.484691i \(-0.161068\pi\)
\(888\) 0 0
\(889\) −1624.00 −1.82677
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2031.94i 2.27541i
\(894\) 0 0
\(895\) − 179.600i − 0.200670i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 118.794 0.132140
\(900\) 0 0
\(901\) − 1008.00i − 1.11876i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1206.46 −1.33311
\(906\) 0 0
\(907\) 448.999 0.495037 0.247519 0.968883i \(-0.420385\pi\)
0.247519 + 0.968883i \(0.420385\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 248.902i − 0.273218i −0.990625 0.136609i \(-0.956380\pi\)
0.990625 0.136609i \(-0.0436204\pi\)
\(912\) 0 0
\(913\) 672.000 0.736035
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1185.30i − 1.29258i
\(918\) 0 0
\(919\) 710.915i 0.773574i 0.922169 + 0.386787i \(0.126415\pi\)
−0.922169 + 0.386787i \(0.873585\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −362.039 −0.392241
\(924\) 0 0
\(925\) − 84.0000i − 0.0908108i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 910.138 0.979697 0.489848 0.871808i \(-0.337052\pi\)
0.489848 + 0.871808i \(0.337052\pi\)
\(930\) 0 0
\(931\) 209.533 0.225062
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 633.568i 0.677612i
\(936\) 0 0
\(937\) 1166.00 1.24440 0.622199 0.782860i \(-0.286240\pi\)
0.622199 + 0.782860i \(0.286240\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 418.029i 0.444239i 0.975019 + 0.222119i \(0.0712975\pi\)
−0.975019 + 0.222119i \(0.928703\pi\)
\(942\) 0 0
\(943\) 1436.80i 1.52364i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 893.783 0.943805 0.471902 0.881651i \(-0.343567\pi\)
0.471902 + 0.881651i \(0.343567\pi\)
\(948\) 0 0
\(949\) − 104.000i − 0.109589i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1291.13 1.35480 0.677401 0.735614i \(-0.263106\pi\)
0.677401 + 0.735614i \(0.263106\pi\)
\(954\) 0 0
\(955\) 119.733 0.125375
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 1742.31i − 1.81680i
\(960\) 0 0
\(961\) 457.000 0.475546
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 370.405i 0.383840i
\(966\) 0 0
\(967\) − 1070.11i − 1.10663i −0.832971 0.553316i \(-0.813362\pi\)
0.832971 0.553316i \(-0.186638\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −514.774 −0.530148 −0.265074 0.964228i \(-0.585396\pi\)
−0.265074 + 0.964228i \(0.585396\pi\)
\(972\) 0 0
\(973\) 1344.00i 1.38129i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1206.46 −1.23486 −0.617432 0.786624i \(-0.711827\pi\)
−0.617432 + 0.786624i \(0.711827\pi\)
\(978\) 0 0
\(979\) 239.466 0.244603
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 950.352i 0.966787i 0.875403 + 0.483393i \(0.160596\pi\)
−0.875403 + 0.483393i \(0.839404\pi\)
\(984\) 0 0
\(985\) −644.000 −0.653807
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 677.312i − 0.684846i
\(990\) 0 0
\(991\) 486.415i 0.490833i 0.969418 + 0.245416i \(0.0789247\pi\)
−0.969418 + 0.245416i \(0.921075\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1069.15 1.07452
\(996\) 0 0
\(997\) 1244.00i 1.24774i 0.781527 + 0.623872i \(0.214441\pi\)
−0.781527 + 0.623872i \(0.785559\pi\)
\(998\) 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.b.i.703.5 yes 8
3.2 odd 2 inner 1152.3.b.i.703.1 8
4.3 odd 2 inner 1152.3.b.i.703.7 yes 8
8.3 odd 2 inner 1152.3.b.i.703.4 yes 8
8.5 even 2 inner 1152.3.b.i.703.2 yes 8
12.11 even 2 inner 1152.3.b.i.703.3 yes 8
16.3 odd 4 2304.3.g.w.1279.3 4
16.5 even 4 2304.3.g.p.1279.2 4
16.11 odd 4 2304.3.g.p.1279.1 4
16.13 even 4 2304.3.g.w.1279.4 4
24.5 odd 2 inner 1152.3.b.i.703.6 yes 8
24.11 even 2 inner 1152.3.b.i.703.8 yes 8
48.5 odd 4 2304.3.g.p.1279.4 4
48.11 even 4 2304.3.g.p.1279.3 4
48.29 odd 4 2304.3.g.w.1279.2 4
48.35 even 4 2304.3.g.w.1279.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.b.i.703.1 8 3.2 odd 2 inner
1152.3.b.i.703.2 yes 8 8.5 even 2 inner
1152.3.b.i.703.3 yes 8 12.11 even 2 inner
1152.3.b.i.703.4 yes 8 8.3 odd 2 inner
1152.3.b.i.703.5 yes 8 1.1 even 1 trivial
1152.3.b.i.703.6 yes 8 24.5 odd 2 inner
1152.3.b.i.703.7 yes 8 4.3 odd 2 inner
1152.3.b.i.703.8 yes 8 24.11 even 2 inner
2304.3.g.p.1279.1 4 16.11 odd 4
2304.3.g.p.1279.2 4 16.5 even 4
2304.3.g.p.1279.3 4 48.11 even 4
2304.3.g.p.1279.4 4 48.5 odd 4
2304.3.g.w.1279.1 4 48.35 even 4
2304.3.g.w.1279.2 4 48.29 odd 4
2304.3.g.w.1279.3 4 16.3 odd 4
2304.3.g.w.1279.4 4 16.13 even 4