Properties

Label 2304.3.b.b.127.2
Level $2304$
Weight $3$
Character 2304.127
Analytic conductor $62.779$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(127,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2304.127
Dual form 2304.3.b.b.127.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000i q^{5} -4.00000i q^{7} +O(q^{10})\) \(q+4.00000i q^{5} -4.00000i q^{7} -16.0000 q^{11} +2.00000i q^{13} +24.0000 q^{17} -24.0000 q^{19} -32.0000i q^{23} +9.00000 q^{25} -44.0000i q^{29} +52.0000i q^{31} +16.0000 q^{35} +18.0000i q^{37} -8.00000 q^{41} +56.0000 q^{43} +32.0000i q^{47} +33.0000 q^{49} -36.0000i q^{53} -64.0000i q^{55} +32.0000 q^{59} +62.0000i q^{61} -8.00000 q^{65} -80.0000 q^{67} +128.000i q^{71} +66.0000 q^{73} +64.0000i q^{77} +20.0000i q^{79} -16.0000 q^{83} +96.0000i q^{85} +144.000 q^{89} +8.00000 q^{91} -96.0000i q^{95} +94.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{11} + 48 q^{17} - 48 q^{19} + 18 q^{25} + 32 q^{35} - 16 q^{41} + 112 q^{43} + 66 q^{49} + 64 q^{59} - 16 q^{65} - 160 q^{67} + 132 q^{73} - 32 q^{83} + 288 q^{89} + 16 q^{91} + 188 q^{97}+O(q^{100}) \) 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
\(5\) 4.00000i 0.800000i 0.916515 + 0.400000i \(0.130990\pi\)
−0.916515 + 0.400000i \(0.869010\pi\)
\(6\) 0 0
\(7\) − 4.00000i − 0.571429i −0.958315 0.285714i \(-0.907769\pi\)
0.958315 0.285714i \(-0.0922308\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −16.0000 −1.45455 −0.727273 0.686349i \(-0.759212\pi\)
−0.727273 + 0.686349i \(0.759212\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.153846i 0.997037 + 0.0769231i \(0.0245096\pi\)
−0.997037 + 0.0769231i \(0.975490\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 24.0000 1.41176 0.705882 0.708329i \(-0.250551\pi\)
0.705882 + 0.708329i \(0.250551\pi\)
\(18\) 0 0
\(19\) −24.0000 −1.26316 −0.631579 0.775312i \(-0.717593\pi\)
−0.631579 + 0.775312i \(0.717593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 32.0000i − 1.39130i −0.718379 0.695652i \(-0.755116\pi\)
0.718379 0.695652i \(-0.244884\pi\)
\(24\) 0 0
\(25\) 9.00000 0.360000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 44.0000i − 1.51724i −0.651533 0.758621i \(-0.725874\pi\)
0.651533 0.758621i \(-0.274126\pi\)
\(30\) 0 0
\(31\) 52.0000i 1.67742i 0.544579 + 0.838710i \(0.316690\pi\)
−0.544579 + 0.838710i \(0.683310\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 16.0000 0.457143
\(36\) 0 0
\(37\) 18.0000i 0.486486i 0.969965 + 0.243243i \(0.0782113\pi\)
−0.969965 + 0.243243i \(0.921789\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −0.195122 −0.0975610 0.995230i \(-0.531104\pi\)
−0.0975610 + 0.995230i \(0.531104\pi\)
\(42\) 0 0
\(43\) 56.0000 1.30233 0.651163 0.758938i \(-0.274282\pi\)
0.651163 + 0.758938i \(0.274282\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 32.0000i 0.680851i 0.940271 + 0.340426i \(0.110571\pi\)
−0.940271 + 0.340426i \(0.889429\pi\)
\(48\) 0 0
\(49\) 33.0000 0.673469
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 36.0000i − 0.679245i −0.940562 0.339623i \(-0.889701\pi\)
0.940562 0.339623i \(-0.110299\pi\)
\(54\) 0 0
\(55\) − 64.0000i − 1.16364i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 32.0000 0.542373 0.271186 0.962527i \(-0.412584\pi\)
0.271186 + 0.962527i \(0.412584\pi\)
\(60\) 0 0
\(61\) 62.0000i 1.01639i 0.861241 + 0.508197i \(0.169688\pi\)
−0.861241 + 0.508197i \(0.830312\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.00000 −0.123077
\(66\) 0 0
\(67\) −80.0000 −1.19403 −0.597015 0.802230i \(-0.703647\pi\)
−0.597015 + 0.802230i \(0.703647\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 128.000i 1.80282i 0.432970 + 0.901408i \(0.357466\pi\)
−0.432970 + 0.901408i \(0.642534\pi\)
\(72\) 0 0
\(73\) 66.0000 0.904110 0.452055 0.891990i \(-0.350691\pi\)
0.452055 + 0.891990i \(0.350691\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 64.0000i 0.831169i
\(78\) 0 0
\(79\) 20.0000i 0.253165i 0.991956 + 0.126582i \(0.0404008\pi\)
−0.991956 + 0.126582i \(0.959599\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −16.0000 −0.192771 −0.0963855 0.995344i \(-0.530728\pi\)
−0.0963855 + 0.995344i \(0.530728\pi\)
\(84\) 0 0
\(85\) 96.0000i 1.12941i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 144.000 1.61798 0.808989 0.587824i \(-0.200015\pi\)
0.808989 + 0.587824i \(0.200015\pi\)
\(90\) 0 0
\(91\) 8.00000 0.0879121
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 96.0000i − 1.01053i
\(96\) 0 0
\(97\) 94.0000 0.969072 0.484536 0.874771i \(-0.338988\pi\)
0.484536 + 0.874771i \(0.338988\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 20.0000i − 0.198020i −0.995086 0.0990099i \(-0.968432\pi\)
0.995086 0.0990099i \(-0.0315676\pi\)
\(102\) 0 0
\(103\) − 44.0000i − 0.427184i −0.976923 0.213592i \(-0.931484\pi\)
0.976923 0.213592i \(-0.0685164\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 96.0000 0.897196 0.448598 0.893734i \(-0.351923\pi\)
0.448598 + 0.893734i \(0.351923\pi\)
\(108\) 0 0
\(109\) 66.0000i 0.605505i 0.953069 + 0.302752i \(0.0979055\pi\)
−0.953069 + 0.302752i \(0.902095\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −80.0000 −0.707965 −0.353982 0.935252i \(-0.615173\pi\)
−0.353982 + 0.935252i \(0.615173\pi\)
\(114\) 0 0
\(115\) 128.000 1.11304
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 96.0000i − 0.806723i
\(120\) 0 0
\(121\) 135.000 1.11570
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 136.000i 1.08800i
\(126\) 0 0
\(127\) − 132.000i − 1.03937i −0.854358 0.519685i \(-0.826049\pi\)
0.854358 0.519685i \(-0.173951\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −128.000 −0.977099 −0.488550 0.872536i \(-0.662474\pi\)
−0.488550 + 0.872536i \(0.662474\pi\)
\(132\) 0 0
\(133\) 96.0000i 0.721805i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −248.000 −1.81022 −0.905109 0.425179i \(-0.860211\pi\)
−0.905109 + 0.425179i \(0.860211\pi\)
\(138\) 0 0
\(139\) 16.0000 0.115108 0.0575540 0.998342i \(-0.481670\pi\)
0.0575540 + 0.998342i \(0.481670\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 32.0000i − 0.223776i
\(144\) 0 0
\(145\) 176.000 1.21379
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 116.000i 0.778523i 0.921127 + 0.389262i \(0.127270\pi\)
−0.921127 + 0.389262i \(0.872730\pi\)
\(150\) 0 0
\(151\) 220.000i 1.45695i 0.685070 + 0.728477i \(0.259771\pi\)
−0.685070 + 0.728477i \(0.740229\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −208.000 −1.34194
\(156\) 0 0
\(157\) 286.000i 1.82166i 0.412786 + 0.910828i \(0.364556\pi\)
−0.412786 + 0.910828i \(0.635444\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −128.000 −0.795031
\(162\) 0 0
\(163\) 168.000 1.03067 0.515337 0.856987i \(-0.327667\pi\)
0.515337 + 0.856987i \(0.327667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 160.000i − 0.958084i −0.877792 0.479042i \(-0.840984\pi\)
0.877792 0.479042i \(-0.159016\pi\)
\(168\) 0 0
\(169\) 165.000 0.976331
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 244.000i − 1.41040i −0.709006 0.705202i \(-0.750856\pi\)
0.709006 0.705202i \(-0.249144\pi\)
\(174\) 0 0
\(175\) − 36.0000i − 0.205714i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 96.0000 0.536313 0.268156 0.963375i \(-0.413586\pi\)
0.268156 + 0.963375i \(0.413586\pi\)
\(180\) 0 0
\(181\) 270.000i 1.49171i 0.666107 + 0.745856i \(0.267959\pi\)
−0.666107 + 0.745856i \(0.732041\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −72.0000 −0.389189
\(186\) 0 0
\(187\) −384.000 −2.05348
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 224.000i 1.17277i 0.810031 + 0.586387i \(0.199450\pi\)
−0.810031 + 0.586387i \(0.800550\pi\)
\(192\) 0 0
\(193\) 178.000 0.922280 0.461140 0.887327i \(-0.347441\pi\)
0.461140 + 0.887327i \(0.347441\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 92.0000i 0.467005i 0.972356 + 0.233503i \(0.0750187\pi\)
−0.972356 + 0.233503i \(0.924981\pi\)
\(198\) 0 0
\(199\) 164.000i 0.824121i 0.911157 + 0.412060i \(0.135191\pi\)
−0.911157 + 0.412060i \(0.864809\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −176.000 −0.866995
\(204\) 0 0
\(205\) − 32.0000i − 0.156098i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 384.000 1.83732
\(210\) 0 0
\(211\) −80.0000 −0.379147 −0.189573 0.981867i \(-0.560711\pi\)
−0.189573 + 0.981867i \(0.560711\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 224.000i 1.04186i
\(216\) 0 0
\(217\) 208.000 0.958525
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 48.0000i 0.217195i
\(222\) 0 0
\(223\) − 212.000i − 0.950673i −0.879804 0.475336i \(-0.842326\pi\)
0.879804 0.475336i \(-0.157674\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 400.000 1.76211 0.881057 0.473010i \(-0.156832\pi\)
0.881057 + 0.473010i \(0.156832\pi\)
\(228\) 0 0
\(229\) 110.000i 0.480349i 0.970730 + 0.240175i \(0.0772047\pi\)
−0.970730 + 0.240175i \(0.922795\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −16.0000 −0.0686695 −0.0343348 0.999410i \(-0.510931\pi\)
−0.0343348 + 0.999410i \(0.510931\pi\)
\(234\) 0 0
\(235\) −128.000 −0.544681
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 32.0000i 0.133891i 0.997757 + 0.0669456i \(0.0213254\pi\)
−0.997757 + 0.0669456i \(0.978675\pi\)
\(240\) 0 0
\(241\) 110.000 0.456432 0.228216 0.973611i \(-0.426711\pi\)
0.228216 + 0.973611i \(0.426711\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 132.000i 0.538776i
\(246\) 0 0
\(247\) − 48.0000i − 0.194332i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 432.000 1.72112 0.860558 0.509353i \(-0.170115\pi\)
0.860558 + 0.509353i \(0.170115\pi\)
\(252\) 0 0
\(253\) 512.000i 2.02372i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −32.0000 −0.124514 −0.0622568 0.998060i \(-0.519830\pi\)
−0.0622568 + 0.998060i \(0.519830\pi\)
\(258\) 0 0
\(259\) 72.0000 0.277992
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 256.000i − 0.973384i −0.873574 0.486692i \(-0.838203\pi\)
0.873574 0.486692i \(-0.161797\pi\)
\(264\) 0 0
\(265\) 144.000 0.543396
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 132.000i 0.490706i 0.969434 + 0.245353i \(0.0789039\pi\)
−0.969434 + 0.245353i \(0.921096\pi\)
\(270\) 0 0
\(271\) 12.0000i 0.0442804i 0.999755 + 0.0221402i \(0.00704803\pi\)
−0.999755 + 0.0221402i \(0.992952\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −144.000 −0.523636
\(276\) 0 0
\(277\) 206.000i 0.743682i 0.928296 + 0.371841i \(0.121273\pi\)
−0.928296 + 0.371841i \(0.878727\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 352.000 1.25267 0.626335 0.779554i \(-0.284554\pi\)
0.626335 + 0.779554i \(0.284554\pi\)
\(282\) 0 0
\(283\) 352.000 1.24382 0.621908 0.783090i \(-0.286358\pi\)
0.621908 + 0.783090i \(0.286358\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 32.0000i 0.111498i
\(288\) 0 0
\(289\) 287.000 0.993080
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 260.000i − 0.887372i −0.896182 0.443686i \(-0.853671\pi\)
0.896182 0.443686i \(-0.146329\pi\)
\(294\) 0 0
\(295\) 128.000i 0.433898i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 64.0000 0.214047
\(300\) 0 0
\(301\) − 224.000i − 0.744186i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −248.000 −0.813115
\(306\) 0 0
\(307\) 48.0000 0.156352 0.0781759 0.996940i \(-0.475090\pi\)
0.0781759 + 0.996940i \(0.475090\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 448.000i 1.44051i 0.693707 + 0.720257i \(0.255976\pi\)
−0.693707 + 0.720257i \(0.744024\pi\)
\(312\) 0 0
\(313\) 94.0000 0.300319 0.150160 0.988662i \(-0.452021\pi\)
0.150160 + 0.988662i \(0.452021\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 332.000i − 1.04732i −0.851928 0.523659i \(-0.824566\pi\)
0.851928 0.523659i \(-0.175434\pi\)
\(318\) 0 0
\(319\) 704.000i 2.20690i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −576.000 −1.78328
\(324\) 0 0
\(325\) 18.0000i 0.0553846i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 128.000 0.389058
\(330\) 0 0
\(331\) −32.0000 −0.0966767 −0.0483384 0.998831i \(-0.515393\pi\)
−0.0483384 + 0.998831i \(0.515393\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 320.000i − 0.955224i
\(336\) 0 0
\(337\) 110.000 0.326409 0.163205 0.986592i \(-0.447817\pi\)
0.163205 + 0.986592i \(0.447817\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 832.000i − 2.43988i
\(342\) 0 0
\(343\) − 328.000i − 0.956268i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −176.000 −0.507205 −0.253602 0.967309i \(-0.581615\pi\)
−0.253602 + 0.967309i \(0.581615\pi\)
\(348\) 0 0
\(349\) 382.000i 1.09456i 0.836951 + 0.547278i \(0.184336\pi\)
−0.836951 + 0.547278i \(0.815664\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 176.000 0.498584 0.249292 0.968428i \(-0.419802\pi\)
0.249292 + 0.968428i \(0.419802\pi\)
\(354\) 0 0
\(355\) −512.000 −1.44225
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 215.000 0.595568
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 264.000i 0.723288i
\(366\) 0 0
\(367\) − 364.000i − 0.991826i −0.868372 0.495913i \(-0.834834\pi\)
0.868372 0.495913i \(-0.165166\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −144.000 −0.388140
\(372\) 0 0
\(373\) 274.000i 0.734584i 0.930106 + 0.367292i \(0.119715\pi\)
−0.930106 + 0.367292i \(0.880285\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 88.0000 0.233422
\(378\) 0 0
\(379\) 632.000 1.66755 0.833773 0.552107i \(-0.186176\pi\)
0.833773 + 0.552107i \(0.186176\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 416.000i − 1.08616i −0.839680 0.543081i \(-0.817258\pi\)
0.839680 0.543081i \(-0.182742\pi\)
\(384\) 0 0
\(385\) −256.000 −0.664935
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 148.000i 0.380463i 0.981739 + 0.190231i \(0.0609238\pi\)
−0.981739 + 0.190231i \(0.939076\pi\)
\(390\) 0 0
\(391\) − 768.000i − 1.96419i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −80.0000 −0.202532
\(396\) 0 0
\(397\) − 450.000i − 1.13350i −0.823889 0.566751i \(-0.808200\pi\)
0.823889 0.566751i \(-0.191800\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −376.000 −0.937656 −0.468828 0.883290i \(-0.655324\pi\)
−0.468828 + 0.883290i \(0.655324\pi\)
\(402\) 0 0
\(403\) −104.000 −0.258065
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 288.000i − 0.707617i
\(408\) 0 0
\(409\) −110.000 −0.268949 −0.134474 0.990917i \(-0.542935\pi\)
−0.134474 + 0.990917i \(0.542935\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 128.000i − 0.309927i
\(414\) 0 0
\(415\) − 64.0000i − 0.154217i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 272.000 0.649165 0.324582 0.945857i \(-0.394776\pi\)
0.324582 + 0.945857i \(0.394776\pi\)
\(420\) 0 0
\(421\) − 530.000i − 1.25891i −0.777038 0.629454i \(-0.783279\pi\)
0.777038 0.629454i \(-0.216721\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 216.000 0.508235
\(426\) 0 0
\(427\) 248.000 0.580796
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 96.0000i − 0.222738i −0.993779 0.111369i \(-0.964476\pi\)
0.993779 0.111369i \(-0.0355235\pi\)
\(432\) 0 0
\(433\) 66.0000 0.152425 0.0762125 0.997092i \(-0.475717\pi\)
0.0762125 + 0.997092i \(0.475717\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 768.000i 1.75744i
\(438\) 0 0
\(439\) − 188.000i − 0.428246i −0.976807 0.214123i \(-0.931311\pi\)
0.976807 0.214123i \(-0.0686893\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −176.000 −0.397291 −0.198646 0.980071i \(-0.563654\pi\)
−0.198646 + 0.980071i \(0.563654\pi\)
\(444\) 0 0
\(445\) 576.000i 1.29438i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 312.000 0.694878 0.347439 0.937703i \(-0.387051\pi\)
0.347439 + 0.937703i \(0.387051\pi\)
\(450\) 0 0
\(451\) 128.000 0.283814
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 32.0000i 0.0703297i
\(456\) 0 0
\(457\) −526.000 −1.15098 −0.575492 0.817807i \(-0.695190\pi\)
−0.575492 + 0.817807i \(0.695190\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 316.000i 0.685466i 0.939433 + 0.342733i \(0.111353\pi\)
−0.939433 + 0.342733i \(0.888647\pi\)
\(462\) 0 0
\(463\) 796.000i 1.71922i 0.510949 + 0.859611i \(0.329294\pi\)
−0.510949 + 0.859611i \(0.670706\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 336.000 0.719486 0.359743 0.933051i \(-0.382864\pi\)
0.359743 + 0.933051i \(0.382864\pi\)
\(468\) 0 0
\(469\) 320.000i 0.682303i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −896.000 −1.89429
\(474\) 0 0
\(475\) −216.000 −0.454737
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 640.000i 1.33612i 0.744109 + 0.668058i \(0.232874\pi\)
−0.744109 + 0.668058i \(0.767126\pi\)
\(480\) 0 0
\(481\) −36.0000 −0.0748441
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 376.000i 0.775258i
\(486\) 0 0
\(487\) 300.000i 0.616016i 0.951384 + 0.308008i \(0.0996624\pi\)
−0.951384 + 0.308008i \(0.900338\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 512.000 1.04277 0.521385 0.853322i \(-0.325416\pi\)
0.521385 + 0.853322i \(0.325416\pi\)
\(492\) 0 0
\(493\) − 1056.00i − 2.14199i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 512.000 1.03018
\(498\) 0 0
\(499\) −736.000 −1.47495 −0.737475 0.675374i \(-0.763982\pi\)
−0.737475 + 0.675374i \(0.763982\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 704.000i − 1.39960i −0.714338 0.699801i \(-0.753272\pi\)
0.714338 0.699801i \(-0.246728\pi\)
\(504\) 0 0
\(505\) 80.0000 0.158416
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 100.000i 0.196464i 0.995164 + 0.0982318i \(0.0313187\pi\)
−0.995164 + 0.0982318i \(0.968681\pi\)
\(510\) 0 0
\(511\) − 264.000i − 0.516634i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 176.000 0.341748
\(516\) 0 0
\(517\) − 512.000i − 0.990329i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 88.0000 0.168906 0.0844530 0.996427i \(-0.473086\pi\)
0.0844530 + 0.996427i \(0.473086\pi\)
\(522\) 0 0
\(523\) −200.000 −0.382409 −0.191205 0.981550i \(-0.561239\pi\)
−0.191205 + 0.981550i \(0.561239\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1248.00i 2.36812i
\(528\) 0 0
\(529\) −495.000 −0.935728
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 16.0000i − 0.0300188i
\(534\) 0 0
\(535\) 384.000i 0.717757i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −528.000 −0.979592
\(540\) 0 0
\(541\) 578.000i 1.06839i 0.845361 + 0.534196i \(0.179386\pi\)
−0.845361 + 0.534196i \(0.820614\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −264.000 −0.484404
\(546\) 0 0
\(547\) 824.000 1.50640 0.753199 0.657792i \(-0.228510\pi\)
0.753199 + 0.657792i \(0.228510\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1056.00i 1.91652i
\(552\) 0 0
\(553\) 80.0000 0.144665
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 1044.00i − 1.87433i −0.348891 0.937163i \(-0.613442\pi\)
0.348891 0.937163i \(-0.386558\pi\)
\(558\) 0 0
\(559\) 112.000i 0.200358i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 464.000 0.824156 0.412078 0.911149i \(-0.364803\pi\)
0.412078 + 0.911149i \(0.364803\pi\)
\(564\) 0 0
\(565\) − 320.000i − 0.566372i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −616.000 −1.08260 −0.541301 0.840829i \(-0.682068\pi\)
−0.541301 + 0.840829i \(0.682068\pi\)
\(570\) 0 0
\(571\) 272.000 0.476357 0.238179 0.971221i \(-0.423450\pi\)
0.238179 + 0.971221i \(0.423450\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 288.000i − 0.500870i
\(576\) 0 0
\(577\) −462.000 −0.800693 −0.400347 0.916364i \(-0.631110\pi\)
−0.400347 + 0.916364i \(0.631110\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 64.0000i 0.110155i
\(582\) 0 0
\(583\) 576.000i 0.987993i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −128.000 −0.218058 −0.109029 0.994039i \(-0.534774\pi\)
−0.109029 + 0.994039i \(0.534774\pi\)
\(588\) 0 0
\(589\) − 1248.00i − 2.11885i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −432.000 −0.728499 −0.364250 0.931301i \(-0.618674\pi\)
−0.364250 + 0.931301i \(0.618674\pi\)
\(594\) 0 0
\(595\) 384.000 0.645378
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 352.000i − 0.587646i −0.955860 0.293823i \(-0.905072\pi\)
0.955860 0.293823i \(-0.0949276\pi\)
\(600\) 0 0
\(601\) −994.000 −1.65391 −0.826955 0.562268i \(-0.809929\pi\)
−0.826955 + 0.562268i \(0.809929\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 540.000i 0.892562i
\(606\) 0 0
\(607\) − 580.000i − 0.955519i −0.878491 0.477759i \(-0.841449\pi\)
0.878491 0.477759i \(-0.158551\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −64.0000 −0.104746
\(612\) 0 0
\(613\) − 462.000i − 0.753670i −0.926280 0.376835i \(-0.877012\pi\)
0.926280 0.376835i \(-0.122988\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −32.0000 −0.0518639 −0.0259319 0.999664i \(-0.508255\pi\)
−0.0259319 + 0.999664i \(0.508255\pi\)
\(618\) 0 0
\(619\) −800.000 −1.29241 −0.646204 0.763165i \(-0.723644\pi\)
−0.646204 + 0.763165i \(0.723644\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 576.000i − 0.924559i
\(624\) 0 0
\(625\) −319.000 −0.510400
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 432.000i 0.686804i
\(630\) 0 0
\(631\) 204.000i 0.323296i 0.986848 + 0.161648i \(0.0516810\pi\)
−0.986848 + 0.161648i \(0.948319\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 528.000 0.831496
\(636\) 0 0
\(637\) 66.0000i 0.103611i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −680.000 −1.06084 −0.530421 0.847734i \(-0.677966\pi\)
−0.530421 + 0.847734i \(0.677966\pi\)
\(642\) 0 0
\(643\) −200.000 −0.311042 −0.155521 0.987833i \(-0.549706\pi\)
−0.155521 + 0.987833i \(0.549706\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 96.0000i − 0.148377i −0.997244 0.0741886i \(-0.976363\pi\)
0.997244 0.0741886i \(-0.0236367\pi\)
\(648\) 0 0
\(649\) −512.000 −0.788906
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 380.000i 0.581930i 0.956734 + 0.290965i \(0.0939762\pi\)
−0.956734 + 0.290965i \(0.906024\pi\)
\(654\) 0 0
\(655\) − 512.000i − 0.781679i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −704.000 −1.06829 −0.534143 0.845394i \(-0.679365\pi\)
−0.534143 + 0.845394i \(0.679365\pi\)
\(660\) 0 0
\(661\) − 238.000i − 0.360061i −0.983661 0.180030i \(-0.942380\pi\)
0.983661 0.180030i \(-0.0576196\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −384.000 −0.577444
\(666\) 0 0
\(667\) −1408.00 −2.11094
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 992.000i − 1.47839i
\(672\) 0 0
\(673\) −350.000 −0.520059 −0.260030 0.965601i \(-0.583732\pi\)
−0.260030 + 0.965601i \(0.583732\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1100.00i − 1.62482i −0.583090 0.812408i \(-0.698156\pi\)
0.583090 0.812408i \(-0.301844\pi\)
\(678\) 0 0
\(679\) − 376.000i − 0.553756i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −912.000 −1.33529 −0.667643 0.744482i \(-0.732697\pi\)
−0.667643 + 0.744482i \(0.732697\pi\)
\(684\) 0 0
\(685\) − 992.000i − 1.44818i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 72.0000 0.104499
\(690\) 0 0
\(691\) 616.000 0.891462 0.445731 0.895167i \(-0.352944\pi\)
0.445731 + 0.895167i \(0.352944\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 64.0000i 0.0920863i
\(696\) 0 0
\(697\) −192.000 −0.275466
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 900.000i 1.28388i 0.766755 + 0.641940i \(0.221870\pi\)
−0.766755 + 0.641940i \(0.778130\pi\)
\(702\) 0 0
\(703\) − 432.000i − 0.614509i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −80.0000 −0.113154
\(708\) 0 0
\(709\) − 1234.00i − 1.74048i −0.492628 0.870240i \(-0.663964\pi\)
0.492628 0.870240i \(-0.336036\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1664.00 2.33380
\(714\) 0 0
\(715\) 128.000 0.179021
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 576.000i − 0.801113i −0.916272 0.400556i \(-0.868817\pi\)
0.916272 0.400556i \(-0.131183\pi\)
\(720\) 0 0
\(721\) −176.000 −0.244105
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 396.000i − 0.546207i
\(726\) 0 0
\(727\) 1228.00i 1.68913i 0.535450 + 0.844567i \(0.320142\pi\)
−0.535450 + 0.844567i \(0.679858\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1344.00 1.83858
\(732\) 0 0
\(733\) 770.000i 1.05048i 0.850955 + 0.525239i \(0.176024\pi\)
−0.850955 + 0.525239i \(0.823976\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1280.00 1.73677
\(738\) 0 0
\(739\) −592.000 −0.801083 −0.400541 0.916279i \(-0.631178\pi\)
−0.400541 + 0.916279i \(0.631178\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 160.000i − 0.215343i −0.994187 0.107672i \(-0.965660\pi\)
0.994187 0.107672i \(-0.0343395\pi\)
\(744\) 0 0
\(745\) −464.000 −0.622819
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 384.000i − 0.512684i
\(750\) 0 0
\(751\) − 508.000i − 0.676431i −0.941069 0.338216i \(-0.890177\pi\)
0.941069 0.338216i \(-0.109823\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −880.000 −1.16556
\(756\) 0 0
\(757\) 398.000i 0.525760i 0.964829 + 0.262880i \(0.0846723\pi\)
−0.964829 + 0.262880i \(0.915328\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1384.00 1.81866 0.909330 0.416076i \(-0.136595\pi\)
0.909330 + 0.416076i \(0.136595\pi\)
\(762\) 0 0
\(763\) 264.000 0.346003
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 64.0000i 0.0834420i
\(768\) 0 0
\(769\) −910.000 −1.18336 −0.591678 0.806175i \(-0.701534\pi\)
−0.591678 + 0.806175i \(0.701534\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 492.000i 0.636481i 0.948010 + 0.318241i \(0.103092\pi\)
−0.948010 + 0.318241i \(0.896908\pi\)
\(774\) 0 0
\(775\) 468.000i 0.603871i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 192.000 0.246470
\(780\) 0 0
\(781\) − 2048.00i − 2.62228i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1144.00 −1.45732
\(786\) 0 0
\(787\) 1496.00 1.90089 0.950445 0.310894i \(-0.100628\pi\)
0.950445 + 0.310894i \(0.100628\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 320.000i 0.404551i
\(792\) 0 0
\(793\) −124.000 −0.156368
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 836.000i − 1.04893i −0.851431 0.524467i \(-0.824265\pi\)
0.851431 0.524467i \(-0.175735\pi\)
\(798\) 0 0
\(799\) 768.000i 0.961202i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1056.00 −1.31507
\(804\) 0 0
\(805\) − 512.000i − 0.636025i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −312.000 −0.385661 −0.192831 0.981232i \(-0.561767\pi\)
−0.192831 + 0.981232i \(0.561767\pi\)
\(810\) 0 0
\(811\) 648.000 0.799014 0.399507 0.916730i \(-0.369181\pi\)
0.399507 + 0.916730i \(0.369181\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 672.000i 0.824540i
\(816\) 0 0
\(817\) −1344.00 −1.64504
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 964.000i − 1.17418i −0.809522 0.587089i \(-0.800274\pi\)
0.809522 0.587089i \(-0.199726\pi\)
\(822\) 0 0
\(823\) − 1300.00i − 1.57959i −0.613373 0.789793i \(-0.710188\pi\)
0.613373 0.789793i \(-0.289812\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1056.00 −1.27690 −0.638452 0.769661i \(-0.720425\pi\)
−0.638452 + 0.769661i \(0.720425\pi\)
\(828\) 0 0
\(829\) 930.000i 1.12183i 0.827872 + 0.560917i \(0.189551\pi\)
−0.827872 + 0.560917i \(0.810449\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 792.000 0.950780
\(834\) 0 0
\(835\) 640.000 0.766467
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 256.000i 0.305125i 0.988294 + 0.152563i \(0.0487526\pi\)
−0.988294 + 0.152563i \(0.951247\pi\)
\(840\) 0 0
\(841\) −1095.00 −1.30202
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 660.000i 0.781065i
\(846\) 0 0
\(847\) − 540.000i − 0.637544i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 576.000 0.676851
\(852\) 0 0
\(853\) − 302.000i − 0.354045i −0.984207 0.177022i \(-0.943354\pi\)
0.984207 0.177022i \(-0.0566465\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −584.000 −0.681447 −0.340723 0.940164i \(-0.610672\pi\)
−0.340723 + 0.940164i \(0.610672\pi\)
\(858\) 0 0
\(859\) 584.000 0.679860 0.339930 0.940451i \(-0.389597\pi\)
0.339930 + 0.940451i \(0.389597\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1056.00i 1.22364i 0.790998 + 0.611819i \(0.209562\pi\)
−0.790998 + 0.611819i \(0.790438\pi\)
\(864\) 0 0
\(865\) 976.000 1.12832
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 320.000i − 0.368239i
\(870\) 0 0
\(871\) − 160.000i − 0.183697i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 544.000 0.621714
\(876\) 0 0
\(877\) − 1378.00i − 1.57127i −0.618693 0.785633i \(-0.712338\pi\)
0.618693 0.785633i \(-0.287662\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 480.000 0.544835 0.272418 0.962179i \(-0.412177\pi\)
0.272418 + 0.962179i \(0.412177\pi\)
\(882\) 0 0
\(883\) −792.000 −0.896942 −0.448471 0.893797i \(-0.648031\pi\)
−0.448471 + 0.893797i \(0.648031\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 832.000i 0.937993i 0.883200 + 0.468997i \(0.155384\pi\)
−0.883200 + 0.468997i \(0.844616\pi\)
\(888\) 0 0
\(889\) −528.000 −0.593926
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 768.000i − 0.860022i
\(894\) 0 0
\(895\) 384.000i 0.429050i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2288.00 2.54505
\(900\) 0 0
\(901\) − 864.000i − 0.958935i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1080.00 −1.19337
\(906\) 0 0
\(907\) 488.000 0.538037 0.269019 0.963135i \(-0.413301\pi\)
0.269019 + 0.963135i \(0.413301\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 992.000i − 1.08891i −0.838789 0.544457i \(-0.816736\pi\)
0.838789 0.544457i \(-0.183264\pi\)
\(912\) 0 0
\(913\) 256.000 0.280394
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 512.000i 0.558342i
\(918\) 0 0
\(919\) − 852.000i − 0.927095i −0.886072 0.463547i \(-0.846576\pi\)
0.886072 0.463547i \(-0.153424\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −256.000 −0.277356
\(924\) 0 0
\(925\) 162.000i 0.175135i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −952.000 −1.02476 −0.512379 0.858759i \(-0.671236\pi\)
−0.512379 + 0.858759i \(0.671236\pi\)
\(930\) 0 0
\(931\) −792.000 −0.850698
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 1536.00i − 1.64278i
\(936\) 0 0
\(937\) 750.000 0.800427 0.400213 0.916422i \(-0.368936\pi\)
0.400213 + 0.916422i \(0.368936\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 772.000i − 0.820404i −0.911995 0.410202i \(-0.865458\pi\)
0.911995 0.410202i \(-0.134542\pi\)
\(942\) 0 0
\(943\) 256.000i 0.271474i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1312.00 1.38543 0.692714 0.721213i \(-0.256415\pi\)
0.692714 + 0.721213i \(0.256415\pi\)
\(948\) 0 0
\(949\) 132.000i 0.139094i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −264.000 −0.277020 −0.138510 0.990361i \(-0.544231\pi\)
−0.138510 + 0.990361i \(0.544231\pi\)
\(954\) 0 0
\(955\) −896.000 −0.938220
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 992.000i 1.03441i
\(960\) 0 0
\(961\) −1743.00 −1.81374
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 712.000i 0.737824i
\(966\) 0 0
\(967\) 212.000i 0.219235i 0.993974 + 0.109617i \(0.0349626\pi\)
−0.993974 + 0.109617i \(0.965037\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 880.000 0.906282 0.453141 0.891439i \(-0.350303\pi\)
0.453141 + 0.891439i \(0.350303\pi\)
\(972\) 0 0
\(973\) − 64.0000i − 0.0657760i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1384.00 1.41658 0.708291 0.705921i \(-0.249467\pi\)
0.708291 + 0.705921i \(0.249467\pi\)
\(978\) 0 0
\(979\) −2304.00 −2.35342
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 704.000i 0.716175i 0.933688 + 0.358087i \(0.116571\pi\)
−0.933688 + 0.358087i \(0.883429\pi\)
\(984\) 0 0
\(985\) −368.000 −0.373604
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 1792.00i − 1.81193i
\(990\) 0 0
\(991\) 1092.00i 1.10192i 0.834533 + 0.550959i \(0.185738\pi\)
−0.834533 + 0.550959i \(0.814262\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −656.000 −0.659296
\(996\) 0 0
\(997\) − 1198.00i − 1.20160i −0.799398 0.600802i \(-0.794848\pi\)
0.799398 0.600802i \(-0.205152\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2304.3.b.b.127.2 2
3.2 odd 2 2304.3.b.h.127.1 2
4.3 odd 2 2304.3.b.i.127.2 2
8.3 odd 2 inner 2304.3.b.b.127.1 2
8.5 even 2 2304.3.b.i.127.1 2
12.11 even 2 2304.3.b.a.127.1 2
16.3 odd 4 288.3.g.c.127.1 yes 2
16.5 even 4 576.3.g.d.127.2 2
16.11 odd 4 576.3.g.d.127.1 2
16.13 even 4 288.3.g.c.127.2 yes 2
24.5 odd 2 2304.3.b.a.127.2 2
24.11 even 2 2304.3.b.h.127.2 2
48.5 odd 4 576.3.g.h.127.2 2
48.11 even 4 576.3.g.h.127.1 2
48.29 odd 4 288.3.g.a.127.2 yes 2
48.35 even 4 288.3.g.a.127.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.3.g.a.127.1 2 48.35 even 4
288.3.g.a.127.2 yes 2 48.29 odd 4
288.3.g.c.127.1 yes 2 16.3 odd 4
288.3.g.c.127.2 yes 2 16.13 even 4
576.3.g.d.127.1 2 16.11 odd 4
576.3.g.d.127.2 2 16.5 even 4
576.3.g.h.127.1 2 48.11 even 4
576.3.g.h.127.2 2 48.5 odd 4
2304.3.b.a.127.1 2 12.11 even 2
2304.3.b.a.127.2 2 24.5 odd 2
2304.3.b.b.127.1 2 8.3 odd 2 inner
2304.3.b.b.127.2 2 1.1 even 1 trivial
2304.3.b.h.127.1 2 3.2 odd 2
2304.3.b.h.127.2 2 24.11 even 2
2304.3.b.i.127.1 2 8.5 even 2
2304.3.b.i.127.2 2 4.3 odd 2