Properties

Label 2304.3.e.i.1025.1
Level $2304$
Weight $3$
Character 2304.1025
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
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] = [2304,3,Mod(1025,2304)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1025");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.e (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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.1
Root \(-0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1025
Dual form 2304.3.e.i.1025.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.44949i q^{5} -3.46410 q^{7} +O(q^{10})\) \(q-2.44949i q^{5} -3.46410 q^{7} -8.48528i q^{11} +10.3923 q^{13} -12.7279i q^{17} +4.00000 q^{19} -34.2929i q^{23} +19.0000 q^{25} +46.5403i q^{29} +38.1051 q^{31} +8.48528i q^{35} -27.7128 q^{37} +29.6985i q^{41} -56.0000 q^{43} +63.6867i q^{47} -37.0000 q^{49} -71.0352i q^{53} -20.7846 q^{55} -101.823i q^{59} +69.2820 q^{61} -25.4558i q^{65} -28.0000 q^{67} -53.8888i q^{71} -20.0000 q^{73} +29.3939i q^{77} +72.7461 q^{79} -76.3675i q^{83} -31.1769 q^{85} +55.1543i q^{89} -36.0000 q^{91} -9.79796i q^{95} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{19} + 76 q^{25} - 224 q^{43} - 148 q^{49} - 112 q^{67} - 80 q^{73} - 144 q^{91} - 32 q^{97}+O(q^{100}) \) 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
\(5\) − 2.44949i − 0.489898i −0.969536 0.244949i \(-0.921229\pi\)
0.969536 0.244949i \(-0.0787712\pi\)
\(6\) 0 0
\(7\) −3.46410 −0.494872 −0.247436 0.968904i \(-0.579588\pi\)
−0.247436 + 0.968904i \(0.579588\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 8.48528i − 0.771389i −0.922627 0.385695i \(-0.873962\pi\)
0.922627 0.385695i \(-0.126038\pi\)
\(12\) 0 0
\(13\) 10.3923 0.799408 0.399704 0.916644i \(-0.369113\pi\)
0.399704 + 0.916644i \(0.369113\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 12.7279i − 0.748701i −0.927287 0.374351i \(-0.877866\pi\)
0.927287 0.374351i \(-0.122134\pi\)
\(18\) 0 0
\(19\) 4.00000 0.210526 0.105263 0.994444i \(-0.466432\pi\)
0.105263 + 0.994444i \(0.466432\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 34.2929i − 1.49099i −0.666509 0.745497i \(-0.732212\pi\)
0.666509 0.745497i \(-0.267788\pi\)
\(24\) 0 0
\(25\) 19.0000 0.760000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 46.5403i 1.60484i 0.596761 + 0.802419i \(0.296454\pi\)
−0.596761 + 0.802419i \(0.703546\pi\)
\(30\) 0 0
\(31\) 38.1051 1.22920 0.614599 0.788840i \(-0.289318\pi\)
0.614599 + 0.788840i \(0.289318\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.48528i 0.242437i
\(36\) 0 0
\(37\) −27.7128 −0.748995 −0.374497 0.927228i \(-0.622185\pi\)
−0.374497 + 0.927228i \(0.622185\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 29.6985i 0.724353i 0.932109 + 0.362177i \(0.117966\pi\)
−0.932109 + 0.362177i \(0.882034\pi\)
\(42\) 0 0
\(43\) −56.0000 −1.30233 −0.651163 0.758938i \(-0.725718\pi\)
−0.651163 + 0.758938i \(0.725718\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 63.6867i 1.35504i 0.735506 + 0.677518i \(0.236945\pi\)
−0.735506 + 0.677518i \(0.763055\pi\)
\(48\) 0 0
\(49\) −37.0000 −0.755102
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 71.0352i − 1.34029i −0.742232 0.670143i \(-0.766233\pi\)
0.742232 0.670143i \(-0.233767\pi\)
\(54\) 0 0
\(55\) −20.7846 −0.377902
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 101.823i − 1.72582i −0.505358 0.862910i \(-0.668639\pi\)
0.505358 0.862910i \(-0.331361\pi\)
\(60\) 0 0
\(61\) 69.2820 1.13577 0.567886 0.823108i \(-0.307762\pi\)
0.567886 + 0.823108i \(0.307762\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 25.4558i − 0.391628i
\(66\) 0 0
\(67\) −28.0000 −0.417910 −0.208955 0.977925i \(-0.567006\pi\)
−0.208955 + 0.977925i \(0.567006\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 53.8888i − 0.758997i −0.925192 0.379498i \(-0.876097\pi\)
0.925192 0.379498i \(-0.123903\pi\)
\(72\) 0 0
\(73\) −20.0000 −0.273973 −0.136986 0.990573i \(-0.543742\pi\)
−0.136986 + 0.990573i \(0.543742\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 29.3939i 0.381739i
\(78\) 0 0
\(79\) 72.7461 0.920837 0.460419 0.887702i \(-0.347699\pi\)
0.460419 + 0.887702i \(0.347699\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 76.3675i − 0.920091i −0.887895 0.460045i \(-0.847833\pi\)
0.887895 0.460045i \(-0.152167\pi\)
\(84\) 0 0
\(85\) −31.1769 −0.366787
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 55.1543i 0.619712i 0.950784 + 0.309856i \(0.100281\pi\)
−0.950784 + 0.309856i \(0.899719\pi\)
\(90\) 0 0
\(91\) −36.0000 −0.395604
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 9.79796i − 0.103136i
\(96\) 0 0
\(97\) −8.00000 −0.0824742 −0.0412371 0.999149i \(-0.513130\pi\)
−0.0412371 + 0.999149i \(0.513130\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.44949i 0.0242524i 0.999926 + 0.0121262i \(0.00385998\pi\)
−0.999926 + 0.0121262i \(0.996140\pi\)
\(102\) 0 0
\(103\) −24.2487 −0.235424 −0.117712 0.993048i \(-0.537556\pi\)
−0.117712 + 0.993048i \(0.537556\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 101.823i − 0.951620i −0.879548 0.475810i \(-0.842155\pi\)
0.879548 0.475810i \(-0.157845\pi\)
\(108\) 0 0
\(109\) −107.387 −0.985203 −0.492602 0.870255i \(-0.663954\pi\)
−0.492602 + 0.870255i \(0.663954\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.24264i 0.0375455i 0.999824 + 0.0187727i \(0.00597590\pi\)
−0.999824 + 0.0187727i \(0.994024\pi\)
\(114\) 0 0
\(115\) −84.0000 −0.730435
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 44.0908i 0.370511i
\(120\) 0 0
\(121\) 49.0000 0.404959
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 107.778i − 0.862220i
\(126\) 0 0
\(127\) −225.167 −1.77297 −0.886483 0.462762i \(-0.846859\pi\)
−0.886483 + 0.462762i \(0.846859\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 220.617i − 1.68410i −0.539398 0.842051i \(-0.681348\pi\)
0.539398 0.842051i \(-0.318652\pi\)
\(132\) 0 0
\(133\) −13.8564 −0.104184
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 199.404i − 1.45550i −0.685840 0.727752i \(-0.740565\pi\)
0.685840 0.727752i \(-0.259435\pi\)
\(138\) 0 0
\(139\) −148.000 −1.06475 −0.532374 0.846509i \(-0.678700\pi\)
−0.532374 + 0.846509i \(0.678700\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 88.1816i − 0.616655i
\(144\) 0 0
\(145\) 114.000 0.786207
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 80.8332i 0.542504i 0.962508 + 0.271252i \(0.0874377\pi\)
−0.962508 + 0.271252i \(0.912562\pi\)
\(150\) 0 0
\(151\) −3.46410 −0.0229411 −0.0114705 0.999934i \(-0.503651\pi\)
−0.0114705 + 0.999934i \(0.503651\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 93.3381i − 0.602181i
\(156\) 0 0
\(157\) −69.2820 −0.441287 −0.220643 0.975355i \(-0.570816\pi\)
−0.220643 + 0.975355i \(0.570816\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 118.794i 0.737851i
\(162\) 0 0
\(163\) −272.000 −1.66871 −0.834356 0.551226i \(-0.814160\pi\)
−0.834356 + 0.551226i \(0.814160\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 254.747i 1.52543i 0.646734 + 0.762715i \(0.276134\pi\)
−0.646734 + 0.762715i \(0.723866\pi\)
\(168\) 0 0
\(169\) −61.0000 −0.360947
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 129.823i − 0.750422i −0.926939 0.375211i \(-0.877570\pi\)
0.926939 0.375211i \(-0.122430\pi\)
\(174\) 0 0
\(175\) −65.8179 −0.376102
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 288.500i − 1.61173i −0.592100 0.805865i \(-0.701701\pi\)
0.592100 0.805865i \(-0.298299\pi\)
\(180\) 0 0
\(181\) 349.874 1.93301 0.966503 0.256653i \(-0.0826199\pi\)
0.966503 + 0.256653i \(0.0826199\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 67.8823i 0.366931i
\(186\) 0 0
\(187\) −108.000 −0.577540
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 137.171i − 0.718175i −0.933304 0.359088i \(-0.883088\pi\)
0.933304 0.359088i \(-0.116912\pi\)
\(192\) 0 0
\(193\) 86.0000 0.445596 0.222798 0.974865i \(-0.428481\pi\)
0.222798 + 0.974865i \(0.428481\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 286.590i 1.45477i 0.686228 + 0.727387i \(0.259265\pi\)
−0.686228 + 0.727387i \(0.740735\pi\)
\(198\) 0 0
\(199\) 239.023 1.20112 0.600560 0.799579i \(-0.294944\pi\)
0.600560 + 0.799579i \(0.294944\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 161.220i − 0.794189i
\(204\) 0 0
\(205\) 72.7461 0.354859
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 33.9411i − 0.162398i
\(210\) 0 0
\(211\) −244.000 −1.15640 −0.578199 0.815896i \(-0.696244\pi\)
−0.578199 + 0.815896i \(0.696244\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 137.171i 0.638007i
\(216\) 0 0
\(217\) −132.000 −0.608295
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 132.272i − 0.598518i
\(222\) 0 0
\(223\) 107.387 0.481557 0.240778 0.970580i \(-0.422597\pi\)
0.240778 + 0.970580i \(0.422597\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 296.985i − 1.30830i −0.756363 0.654152i \(-0.773026\pi\)
0.756363 0.654152i \(-0.226974\pi\)
\(228\) 0 0
\(229\) −128.172 −0.559702 −0.279851 0.960043i \(-0.590285\pi\)
−0.279851 + 0.960043i \(0.590285\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 173.948i − 0.746559i −0.927719 0.373280i \(-0.878233\pi\)
0.927719 0.373280i \(-0.121767\pi\)
\(234\) 0 0
\(235\) 156.000 0.663830
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.79796i 0.0409956i 0.999790 + 0.0204978i \(0.00652512\pi\)
−0.999790 + 0.0204978i \(0.993475\pi\)
\(240\) 0 0
\(241\) −196.000 −0.813278 −0.406639 0.913589i \(-0.633299\pi\)
−0.406639 + 0.913589i \(0.633299\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 90.6311i 0.369923i
\(246\) 0 0
\(247\) 41.5692 0.168296
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 161.220i − 0.642312i −0.947026 0.321156i \(-0.895929\pi\)
0.947026 0.321156i \(-0.104071\pi\)
\(252\) 0 0
\(253\) −290.985 −1.15014
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 250.316i 0.973991i 0.873404 + 0.486996i \(0.161907\pi\)
−0.873404 + 0.486996i \(0.838093\pi\)
\(258\) 0 0
\(259\) 96.0000 0.370656
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 29.3939i − 0.111764i −0.998437 0.0558819i \(-0.982203\pi\)
0.998437 0.0558819i \(-0.0177970\pi\)
\(264\) 0 0
\(265\) −174.000 −0.656604
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 257.196i 0.956121i 0.878327 + 0.478060i \(0.158660\pi\)
−0.878327 + 0.478060i \(0.841340\pi\)
\(270\) 0 0
\(271\) −266.736 −0.984265 −0.492133 0.870520i \(-0.663782\pi\)
−0.492133 + 0.870520i \(0.663782\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 161.220i − 0.586256i
\(276\) 0 0
\(277\) −65.8179 −0.237610 −0.118805 0.992918i \(-0.537906\pi\)
−0.118805 + 0.992918i \(0.537906\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 275.772i 0.981394i 0.871330 + 0.490697i \(0.163258\pi\)
−0.871330 + 0.490697i \(0.836742\pi\)
\(282\) 0 0
\(283\) −496.000 −1.75265 −0.876325 0.481720i \(-0.840012\pi\)
−0.876325 + 0.481720i \(0.840012\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 102.879i − 0.358462i
\(288\) 0 0
\(289\) 127.000 0.439446
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 188.611i − 0.643723i −0.946787 0.321861i \(-0.895692\pi\)
0.946787 0.321861i \(-0.104308\pi\)
\(294\) 0 0
\(295\) −249.415 −0.845476
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 356.382i − 1.19191i
\(300\) 0 0
\(301\) 193.990 0.644484
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 169.706i − 0.556412i
\(306\) 0 0
\(307\) −220.000 −0.716612 −0.358306 0.933604i \(-0.616646\pi\)
−0.358306 + 0.933604i \(0.616646\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 362.524i − 1.16567i −0.812589 0.582837i \(-0.801943\pi\)
0.812589 0.582837i \(-0.198057\pi\)
\(312\) 0 0
\(313\) 554.000 1.76997 0.884984 0.465621i \(-0.154169\pi\)
0.884984 + 0.465621i \(0.154169\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.0454i 0.0695439i 0.999395 + 0.0347719i \(0.0110705\pi\)
−0.999395 + 0.0347719i \(0.988930\pi\)
\(318\) 0 0
\(319\) 394.908 1.23795
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 50.9117i − 0.157621i
\(324\) 0 0
\(325\) 197.454 0.607550
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 220.617i − 0.670569i
\(330\) 0 0
\(331\) −244.000 −0.737160 −0.368580 0.929596i \(-0.620156\pi\)
−0.368580 + 0.929596i \(0.620156\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 68.5857i 0.204733i
\(336\) 0 0
\(337\) −188.000 −0.557864 −0.278932 0.960311i \(-0.589980\pi\)
−0.278932 + 0.960311i \(0.589980\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 323.333i − 0.948190i
\(342\) 0 0
\(343\) 297.913 0.868550
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 25.4558i − 0.0733598i −0.999327 0.0366799i \(-0.988322\pi\)
0.999327 0.0366799i \(-0.0116782\pi\)
\(348\) 0 0
\(349\) 609.682 1.74694 0.873470 0.486878i \(-0.161865\pi\)
0.873470 + 0.486878i \(0.161865\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 267.286i − 0.757185i −0.925563 0.378593i \(-0.876408\pi\)
0.925563 0.378593i \(-0.123592\pi\)
\(354\) 0 0
\(355\) −132.000 −0.371831
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 612.372i − 1.70577i −0.522097 0.852886i \(-0.674850\pi\)
0.522097 0.852886i \(-0.325150\pi\)
\(360\) 0 0
\(361\) −345.000 −0.955679
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 48.9898i 0.134219i
\(366\) 0 0
\(367\) 481.510 1.31202 0.656008 0.754754i \(-0.272244\pi\)
0.656008 + 0.754754i \(0.272244\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 246.073i 0.663270i
\(372\) 0 0
\(373\) −540.400 −1.44879 −0.724397 0.689383i \(-0.757882\pi\)
−0.724397 + 0.689383i \(0.757882\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 483.661i 1.28292i
\(378\) 0 0
\(379\) 268.000 0.707124 0.353562 0.935411i \(-0.384970\pi\)
0.353562 + 0.935411i \(0.384970\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 480.100i 1.25352i 0.779210 + 0.626762i \(0.215620\pi\)
−0.779210 + 0.626762i \(0.784380\pi\)
\(384\) 0 0
\(385\) 72.0000 0.187013
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 438.459i − 1.12714i −0.826067 0.563572i \(-0.809427\pi\)
0.826067 0.563572i \(-0.190573\pi\)
\(390\) 0 0
\(391\) −436.477 −1.11631
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 178.191i − 0.451116i
\(396\) 0 0
\(397\) −526.543 −1.32631 −0.663153 0.748484i \(-0.730782\pi\)
−0.663153 + 0.748484i \(0.730782\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 420.021i − 1.04743i −0.851892 0.523717i \(-0.824545\pi\)
0.851892 0.523717i \(-0.175455\pi\)
\(402\) 0 0
\(403\) 396.000 0.982630
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 235.151i 0.577767i
\(408\) 0 0
\(409\) 640.000 1.56479 0.782396 0.622781i \(-0.213997\pi\)
0.782396 + 0.622781i \(0.213997\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 352.727i 0.854059i
\(414\) 0 0
\(415\) −187.061 −0.450751
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 364.867i 0.870805i 0.900236 + 0.435402i \(0.143394\pi\)
−0.900236 + 0.435402i \(0.856606\pi\)
\(420\) 0 0
\(421\) −495.367 −1.17664 −0.588321 0.808627i \(-0.700211\pi\)
−0.588321 + 0.808627i \(0.700211\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 241.831i − 0.569013i
\(426\) 0 0
\(427\) −240.000 −0.562061
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 83.2827i 0.193231i 0.995322 + 0.0966156i \(0.0308017\pi\)
−0.995322 + 0.0966156i \(0.969198\pi\)
\(432\) 0 0
\(433\) 106.000 0.244804 0.122402 0.992481i \(-0.460940\pi\)
0.122402 + 0.992481i \(0.460940\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 137.171i − 0.313893i
\(438\) 0 0
\(439\) −613.146 −1.39669 −0.698344 0.715762i \(-0.746079\pi\)
−0.698344 + 0.715762i \(0.746079\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 212.132i − 0.478853i −0.970914 0.239427i \(-0.923041\pi\)
0.970914 0.239427i \(-0.0769595\pi\)
\(444\) 0 0
\(445\) 135.100 0.303595
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 182.434i 0.406311i 0.979147 + 0.203155i \(0.0651197\pi\)
−0.979147 + 0.203155i \(0.934880\pi\)
\(450\) 0 0
\(451\) 252.000 0.558758
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 88.1816i 0.193806i
\(456\) 0 0
\(457\) −344.000 −0.752735 −0.376368 0.926470i \(-0.622827\pi\)
−0.376368 + 0.926470i \(0.622827\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 110.227i 0.239104i 0.992828 + 0.119552i \(0.0381459\pi\)
−0.992828 + 0.119552i \(0.961854\pi\)
\(462\) 0 0
\(463\) 114.315 0.246901 0.123451 0.992351i \(-0.460604\pi\)
0.123451 + 0.992351i \(0.460604\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.48528i 0.0181698i 0.999959 + 0.00908488i \(0.00289185\pi\)
−0.999959 + 0.00908488i \(0.997108\pi\)
\(468\) 0 0
\(469\) 96.9948 0.206812
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 475.176i 1.00460i
\(474\) 0 0
\(475\) 76.0000 0.160000
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 778.938i 1.62617i 0.582142 + 0.813087i \(0.302215\pi\)
−0.582142 + 0.813087i \(0.697785\pi\)
\(480\) 0 0
\(481\) −288.000 −0.598753
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 19.5959i 0.0404040i
\(486\) 0 0
\(487\) 820.992 1.68582 0.842908 0.538058i \(-0.180842\pi\)
0.842908 + 0.538058i \(0.180842\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 390.323i 0.794955i 0.917612 + 0.397478i \(0.130114\pi\)
−0.917612 + 0.397478i \(0.869886\pi\)
\(492\) 0 0
\(493\) 592.361 1.20154
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 186.676i 0.375606i
\(498\) 0 0
\(499\) 128.000 0.256513 0.128257 0.991741i \(-0.459062\pi\)
0.128257 + 0.991741i \(0.459062\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 181.262i − 0.360362i −0.983633 0.180181i \(-0.942332\pi\)
0.983633 0.180181i \(-0.0576684\pi\)
\(504\) 0 0
\(505\) 6.00000 0.0118812
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 438.459i 0.861412i 0.902492 + 0.430706i \(0.141735\pi\)
−0.902492 + 0.430706i \(0.858265\pi\)
\(510\) 0 0
\(511\) 69.2820 0.135581
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 59.3970i 0.115334i
\(516\) 0 0
\(517\) 540.400 1.04526
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 691.550i − 1.32735i −0.748020 0.663676i \(-0.768995\pi\)
0.748020 0.663676i \(-0.231005\pi\)
\(522\) 0 0
\(523\) 484.000 0.925430 0.462715 0.886507i \(-0.346875\pi\)
0.462715 + 0.886507i \(0.346875\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 484.999i − 0.920302i
\(528\) 0 0
\(529\) −647.000 −1.22306
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 308.636i 0.579054i
\(534\) 0 0
\(535\) −249.415 −0.466197
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 313.955i 0.582478i
\(540\) 0 0
\(541\) −523.079 −0.966875 −0.483437 0.875379i \(-0.660612\pi\)
−0.483437 + 0.875379i \(0.660612\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 263.044i 0.482649i
\(546\) 0 0
\(547\) 184.000 0.336380 0.168190 0.985755i \(-0.446208\pi\)
0.168190 + 0.985755i \(0.446208\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 186.161i 0.337861i
\(552\) 0 0
\(553\) −252.000 −0.455696
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 918.559i − 1.64912i −0.565776 0.824559i \(-0.691423\pi\)
0.565776 0.824559i \(-0.308577\pi\)
\(558\) 0 0
\(559\) −581.969 −1.04109
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 330.926i 0.587790i 0.955838 + 0.293895i \(0.0949517\pi\)
−0.955838 + 0.293895i \(0.905048\pi\)
\(564\) 0 0
\(565\) 10.3923 0.0183935
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1056.42i 1.85662i 0.371805 + 0.928311i \(0.378739\pi\)
−0.371805 + 0.928311i \(0.621261\pi\)
\(570\) 0 0
\(571\) 152.000 0.266200 0.133100 0.991103i \(-0.457507\pi\)
0.133100 + 0.991103i \(0.457507\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 651.564i − 1.13316i
\(576\) 0 0
\(577\) 790.000 1.36915 0.684575 0.728942i \(-0.259988\pi\)
0.684575 + 0.728942i \(0.259988\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 264.545i 0.455327i
\(582\) 0 0
\(583\) −602.754 −1.03388
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1154.00i 1.96593i 0.183805 + 0.982963i \(0.441159\pi\)
−0.183805 + 0.982963i \(0.558841\pi\)
\(588\) 0 0
\(589\) 152.420 0.258778
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 377.595i − 0.636754i −0.947964 0.318377i \(-0.896862\pi\)
0.947964 0.318377i \(-0.103138\pi\)
\(594\) 0 0
\(595\) 108.000 0.181513
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 318.434i 0.531609i 0.964027 + 0.265804i \(0.0856375\pi\)
−0.964027 + 0.265804i \(0.914362\pi\)
\(600\) 0 0
\(601\) 202.000 0.336106 0.168053 0.985778i \(-0.446252\pi\)
0.168053 + 0.985778i \(0.446252\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 120.025i − 0.198388i
\(606\) 0 0
\(607\) −391.443 −0.644882 −0.322441 0.946590i \(-0.604503\pi\)
−0.322441 + 0.946590i \(0.604503\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 661.852i 1.08323i
\(612\) 0 0
\(613\) 803.672 1.31105 0.655523 0.755175i \(-0.272448\pi\)
0.655523 + 0.755175i \(0.272448\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 1115.81i − 1.80845i −0.427055 0.904226i \(-0.640449\pi\)
0.427055 0.904226i \(-0.359551\pi\)
\(618\) 0 0
\(619\) 200.000 0.323102 0.161551 0.986864i \(-0.448350\pi\)
0.161551 + 0.986864i \(0.448350\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 191.060i − 0.306678i
\(624\) 0 0
\(625\) 211.000 0.337600
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 352.727i 0.560773i
\(630\) 0 0
\(631\) 668.572 1.05954 0.529771 0.848140i \(-0.322278\pi\)
0.529771 + 0.848140i \(0.322278\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 551.543i 0.868572i
\(636\) 0 0
\(637\) −384.515 −0.603635
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1090.36i 1.70103i 0.525953 + 0.850514i \(0.323709\pi\)
−0.525953 + 0.850514i \(0.676291\pi\)
\(642\) 0 0
\(643\) −416.000 −0.646967 −0.323484 0.946234i \(-0.604854\pi\)
−0.323484 + 0.946234i \(0.604854\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 680.958i 1.05249i 0.850334 + 0.526243i \(0.176400\pi\)
−0.850334 + 0.526243i \(0.823600\pi\)
\(648\) 0 0
\(649\) −864.000 −1.33128
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 110.227i − 0.168801i −0.996432 0.0844005i \(-0.973102\pi\)
0.996432 0.0844005i \(-0.0268975\pi\)
\(654\) 0 0
\(655\) −540.400 −0.825038
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 441.235i 0.669552i 0.942298 + 0.334776i \(0.108661\pi\)
−0.942298 + 0.334776i \(0.891339\pi\)
\(660\) 0 0
\(661\) −484.974 −0.733698 −0.366849 0.930281i \(-0.619563\pi\)
−0.366849 + 0.930281i \(0.619563\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 33.9411i 0.0510393i
\(666\) 0 0
\(667\) 1596.00 2.39280
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 587.878i − 0.876122i
\(672\) 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
\(677\) 879.367i 1.29892i 0.760397 + 0.649459i \(0.225004\pi\)
−0.760397 + 0.649459i \(0.774996\pi\)
\(678\) 0 0
\(679\) 27.7128 0.0408142
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 568.514i 0.832378i 0.909278 + 0.416189i \(0.136634\pi\)
−0.909278 + 0.416189i \(0.863366\pi\)
\(684\) 0 0
\(685\) −488.438 −0.713049
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 738.219i − 1.07144i
\(690\) 0 0
\(691\) 832.000 1.20405 0.602026 0.798476i \(-0.294360\pi\)
0.602026 + 0.798476i \(0.294360\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 362.524i 0.521618i
\(696\) 0 0
\(697\) 378.000 0.542324
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 433.560i − 0.618487i −0.950983 0.309244i \(-0.899924\pi\)
0.950983 0.309244i \(-0.100076\pi\)
\(702\) 0 0
\(703\) −110.851 −0.157683
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 8.48528i − 0.0120018i
\(708\) 0 0
\(709\) −197.454 −0.278496 −0.139248 0.990258i \(-0.544469\pi\)
−0.139248 + 0.990258i \(0.544469\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 1306.73i − 1.83273i
\(714\) 0 0
\(715\) −216.000 −0.302098
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 220.454i 0.306612i 0.988179 + 0.153306i \(0.0489920\pi\)
−0.988179 + 0.153306i \(0.951008\pi\)
\(720\) 0 0
\(721\) 84.0000 0.116505
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 884.266i 1.21968i
\(726\) 0 0
\(727\) 862.561 1.18647 0.593233 0.805031i \(-0.297851\pi\)
0.593233 + 0.805031i \(0.297851\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 712.764i 0.975053i
\(732\) 0 0
\(733\) −169.741 −0.231570 −0.115785 0.993274i \(-0.536938\pi\)
−0.115785 + 0.993274i \(0.536938\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 237.588i 0.322372i
\(738\) 0 0
\(739\) 872.000 1.17997 0.589986 0.807413i \(-0.299133\pi\)
0.589986 + 0.807413i \(0.299133\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 195.959i 0.263740i 0.991267 + 0.131870i \(0.0420982\pi\)
−0.991267 + 0.131870i \(0.957902\pi\)
\(744\) 0 0
\(745\) 198.000 0.265772
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 352.727i 0.470930i
\(750\) 0 0
\(751\) 481.510 0.641159 0.320579 0.947222i \(-0.396122\pi\)
0.320579 + 0.947222i \(0.396122\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.48528i 0.0112388i
\(756\) 0 0
\(757\) 1202.04 1.58790 0.793952 0.607981i \(-0.208020\pi\)
0.793952 + 0.607981i \(0.208020\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 929.138i 1.22094i 0.792038 + 0.610472i \(0.209020\pi\)
−0.792038 + 0.610472i \(0.790980\pi\)
\(762\) 0 0
\(763\) 372.000 0.487549
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1058.18i − 1.37963i
\(768\) 0 0
\(769\) 218.000 0.283485 0.141743 0.989904i \(-0.454729\pi\)
0.141743 + 0.989904i \(0.454729\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 1041.03i − 1.34674i −0.739304 0.673372i \(-0.764845\pi\)
0.739304 0.673372i \(-0.235155\pi\)
\(774\) 0 0
\(775\) 723.997 0.934190
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 118.794i 0.152495i
\(780\) 0 0
\(781\) −457.261 −0.585482
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 169.706i 0.216186i
\(786\) 0 0
\(787\) −404.000 −0.513342 −0.256671 0.966499i \(-0.582626\pi\)
−0.256671 + 0.966499i \(0.582626\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 14.6969i − 0.0185802i
\(792\) 0 0
\(793\) 720.000 0.907945
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1511.34i − 1.89628i −0.317853 0.948140i \(-0.602962\pi\)
0.317853 0.948140i \(-0.397038\pi\)
\(798\) 0 0
\(799\) 810.600 1.01452
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 169.706i 0.211340i
\(804\) 0 0
\(805\) 290.985 0.361471
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 106.066i − 0.131108i −0.997849 0.0655538i \(-0.979119\pi\)
0.997849 0.0655538i \(-0.0208814\pi\)
\(810\) 0 0
\(811\) 1132.00 1.39581 0.697904 0.716191i \(-0.254116\pi\)
0.697904 + 0.716191i \(0.254116\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 666.261i 0.817498i
\(816\) 0 0
\(817\) −224.000 −0.274174
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 722.599i − 0.880146i −0.897962 0.440073i \(-0.854953\pi\)
0.897962 0.440073i \(-0.145047\pi\)
\(822\) 0 0
\(823\) 10.3923 0.0126273 0.00631367 0.999980i \(-0.497990\pi\)
0.00631367 + 0.999980i \(0.497990\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1306.73i 1.58009i 0.613050 + 0.790044i \(0.289943\pi\)
−0.613050 + 0.790044i \(0.710057\pi\)
\(828\) 0 0
\(829\) 530.008 0.639334 0.319667 0.947530i \(-0.396429\pi\)
0.319667 + 0.947530i \(0.396429\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 470.933i 0.565346i
\(834\) 0 0
\(835\) 624.000 0.747305
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 827.928i − 0.986803i −0.869802 0.493401i \(-0.835753\pi\)
0.869802 0.493401i \(-0.164247\pi\)
\(840\) 0 0
\(841\) −1325.00 −1.57551
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 149.419i 0.176827i
\(846\) 0 0
\(847\) −169.741 −0.200403
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 950.352i 1.11675i
\(852\) 0 0
\(853\) −1011.52 −1.18584 −0.592918 0.805263i \(-0.702024\pi\)
−0.592918 + 0.805263i \(0.702024\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 538.815i 0.628723i 0.949303 + 0.314361i \(0.101790\pi\)
−0.949303 + 0.314361i \(0.898210\pi\)
\(858\) 0 0
\(859\) −1160.00 −1.35041 −0.675204 0.737631i \(-0.735944\pi\)
−0.675204 + 0.737631i \(0.735944\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 391.918i 0.454135i 0.973879 + 0.227067i \(0.0729138\pi\)
−0.973879 + 0.227067i \(0.927086\pi\)
\(864\) 0 0
\(865\) −318.000 −0.367630
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 617.271i − 0.710324i
\(870\) 0 0
\(871\) −290.985 −0.334081
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 373.352i 0.426688i
\(876\) 0 0
\(877\) 526.543 0.600392 0.300196 0.953878i \(-0.402948\pi\)
0.300196 + 0.953878i \(0.402948\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 258.801i − 0.293758i −0.989154 0.146879i \(-0.953077\pi\)
0.989154 0.146879i \(-0.0469228\pi\)
\(882\) 0 0
\(883\) 520.000 0.588901 0.294451 0.955667i \(-0.404863\pi\)
0.294451 + 0.955667i \(0.404863\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 29.3939i 0.0331385i 0.999863 + 0.0165693i \(0.00527440\pi\)
−0.999863 + 0.0165693i \(0.994726\pi\)
\(888\) 0 0
\(889\) 780.000 0.877390
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 254.747i 0.285271i
\(894\) 0 0
\(895\) −706.677 −0.789583
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1773.42i 1.97266i
\(900\) 0 0
\(901\) −904.131 −1.00347
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 857.013i − 0.946976i
\(906\) 0 0
\(907\) 1288.00 1.42007 0.710033 0.704168i \(-0.248680\pi\)
0.710033 + 0.704168i \(0.248680\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 39.1918i − 0.0430207i −0.999769 0.0215103i \(-0.993153\pi\)
0.999769 0.0215103i \(-0.00684748\pi\)
\(912\) 0 0
\(913\) −648.000 −0.709748
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 764.241i 0.833414i
\(918\) 0 0
\(919\) −1569.24 −1.70755 −0.853775 0.520643i \(-0.825692\pi\)
−0.853775 + 0.520643i \(0.825692\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 560.029i − 0.606748i
\(924\) 0 0
\(925\) −526.543 −0.569236
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 997.021i − 1.07322i −0.843831 0.536610i \(-0.819705\pi\)
0.843831 0.536610i \(-0.180295\pi\)
\(930\) 0 0
\(931\) −148.000 −0.158969
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 264.545i 0.282936i
\(936\) 0 0
\(937\) −874.000 −0.932764 −0.466382 0.884583i \(-0.654443\pi\)
−0.466382 + 0.884583i \(0.654443\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1383.96i 1.47074i 0.677668 + 0.735368i \(0.262990\pi\)
−0.677668 + 0.735368i \(0.737010\pi\)
\(942\) 0 0
\(943\) 1018.45 1.08001
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 678.823i 0.716814i 0.933566 + 0.358407i \(0.116680\pi\)
−0.933566 + 0.358407i \(0.883320\pi\)
\(948\) 0 0
\(949\) −207.846 −0.219016
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1650.39i 1.73178i 0.500234 + 0.865890i \(0.333247\pi\)
−0.500234 + 0.865890i \(0.666753\pi\)
\(954\) 0 0
\(955\) −336.000 −0.351832
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 690.756i 0.720288i
\(960\) 0 0
\(961\) 491.000 0.510926
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 210.656i − 0.218296i
\(966\) 0 0
\(967\) 543.864 0.562424 0.281212 0.959646i \(-0.409264\pi\)
0.281212 + 0.959646i \(0.409264\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 449.720i − 0.463151i −0.972817 0.231576i \(-0.925612\pi\)
0.972817 0.231576i \(-0.0743881\pi\)
\(972\) 0 0
\(973\) 512.687 0.526914
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 80.6102i 0.0825079i 0.999149 + 0.0412539i \(0.0131353\pi\)
−0.999149 + 0.0412539i \(0.986865\pi\)
\(978\) 0 0
\(979\) 468.000 0.478039
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1734.24i − 1.76423i −0.471033 0.882115i \(-0.656119\pi\)
0.471033 0.882115i \(-0.343881\pi\)
\(984\) 0 0
\(985\) 702.000 0.712690
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1920.40i 1.94176i
\(990\) 0 0
\(991\) 232.095 0.234203 0.117101 0.993120i \(-0.462640\pi\)
0.117101 + 0.993120i \(0.462640\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 585.484i − 0.588427i
\(996\) 0 0
\(997\) −290.985 −0.291860 −0.145930 0.989295i \(-0.546617\pi\)
−0.145930 + 0.989295i \(0.546617\pi\)
\(998\) 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.e.i.1025.1 4
3.2 odd 2 inner 2304.3.e.i.1025.3 4
4.3 odd 2 2304.3.e.h.1025.2 4
8.3 odd 2 inner 2304.3.e.i.1025.4 4
8.5 even 2 2304.3.e.h.1025.3 4
12.11 even 2 2304.3.e.h.1025.4 4
16.3 odd 4 576.3.h.a.161.2 yes 8
16.5 even 4 576.3.h.a.161.7 yes 8
16.11 odd 4 576.3.h.a.161.5 yes 8
16.13 even 4 576.3.h.a.161.4 yes 8
24.5 odd 2 2304.3.e.h.1025.1 4
24.11 even 2 inner 2304.3.e.i.1025.2 4
48.5 odd 4 576.3.h.a.161.3 yes 8
48.11 even 4 576.3.h.a.161.1 8
48.29 odd 4 576.3.h.a.161.8 yes 8
48.35 even 4 576.3.h.a.161.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.3.h.a.161.1 8 48.11 even 4
576.3.h.a.161.2 yes 8 16.3 odd 4
576.3.h.a.161.3 yes 8 48.5 odd 4
576.3.h.a.161.4 yes 8 16.13 even 4
576.3.h.a.161.5 yes 8 16.11 odd 4
576.3.h.a.161.6 yes 8 48.35 even 4
576.3.h.a.161.7 yes 8 16.5 even 4
576.3.h.a.161.8 yes 8 48.29 odd 4
2304.3.e.h.1025.1 4 24.5 odd 2
2304.3.e.h.1025.2 4 4.3 odd 2
2304.3.e.h.1025.3 4 8.5 even 2
2304.3.e.h.1025.4 4 12.11 even 2
2304.3.e.i.1025.1 4 1.1 even 1 trivial
2304.3.e.i.1025.2 4 24.11 even 2 inner
2304.3.e.i.1025.3 4 3.2 odd 2 inner
2304.3.e.i.1025.4 4 8.3 odd 2 inner