Properties

Label 1152.2.l.a.287.3
Level $1152$
Weight $2$
Character 1152.287
Analytic conductor $9.199$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,2,Mod(287,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.l (of order \(4\), degree \(2\), 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: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 287.3
Root \(1.36166 - 0.381939i\) of defining polynomial
Character \(\chi\) \(=\) 1152.287
Dual form 1152.2.l.a.863.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.763878 + 0.763878i) q^{5} +1.33620 q^{7} +O(q^{10})\) \(q+(-0.763878 + 0.763878i) q^{5} +1.33620 q^{7} +(-1.95945 - 1.95945i) q^{11} +(-4.18757 + 4.18757i) q^{13} -4.03243i q^{17} +(4.26785 + 4.26785i) q^{19} +8.86408i q^{23} +3.83298i q^{25} +(-1.23934 - 1.23934i) q^{29} +2.87835i q^{31} +(-1.02070 + 1.02070i) q^{35} +(-0.434870 - 0.434870i) q^{37} -7.81179 q^{41} +(5.49678 - 5.49678i) q^{43} -3.20723 q^{47} -5.21456 q^{49} +(-4.06777 + 4.06777i) q^{53} +2.99355 q^{55} +(4.71811 + 4.71811i) q^{59} +(-3.26785 + 3.26785i) q^{61} -6.39758i q^{65} +(5.44348 + 5.44348i) q^{67} -3.76718i q^{71} +10.5357i q^{73} +(-2.61822 - 2.61822i) q^{77} +11.1995i q^{79} +(-9.73306 + 9.73306i) q^{83} +(3.08029 + 3.08029i) q^{85} -1.64130 q^{89} +(-5.59544 + 5.59544i) q^{91} -6.52023 q^{95} -5.70272 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} - 64 q^{55} + 32 q^{61} + 16 q^{67} + 32 q^{85} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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
\(5\) −0.763878 + 0.763878i −0.341617 + 0.341617i −0.856975 0.515358i \(-0.827659\pi\)
0.515358 + 0.856975i \(0.327659\pi\)
\(6\) 0 0
\(7\) 1.33620 0.505038 0.252519 0.967592i \(-0.418741\pi\)
0.252519 + 0.967592i \(0.418741\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.95945 1.95945i −0.590795 0.590795i 0.347051 0.937846i \(-0.387183\pi\)
−0.937846 + 0.347051i \(0.887183\pi\)
\(12\) 0 0
\(13\) −4.18757 + 4.18757i −1.16142 + 1.16142i −0.177257 + 0.984165i \(0.556722\pi\)
−0.984165 + 0.177257i \(0.943278\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.03243i 0.978009i −0.872281 0.489004i \(-0.837360\pi\)
0.872281 0.489004i \(-0.162640\pi\)
\(18\) 0 0
\(19\) 4.26785 + 4.26785i 0.979112 + 0.979112i 0.999786 0.0206739i \(-0.00658119\pi\)
−0.0206739 + 0.999786i \(0.506581\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.86408i 1.84829i 0.382044 + 0.924144i \(0.375220\pi\)
−0.382044 + 0.924144i \(0.624780\pi\)
\(24\) 0 0
\(25\) 3.83298i 0.766596i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.23934 1.23934i −0.230140 0.230140i 0.582611 0.812751i \(-0.302031\pi\)
−0.812751 + 0.582611i \(0.802031\pi\)
\(30\) 0 0
\(31\) 2.87835i 0.516968i 0.966016 + 0.258484i \(0.0832229\pi\)
−0.966016 + 0.258484i \(0.916777\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.02070 + 1.02070i −0.172529 + 0.172529i
\(36\) 0 0
\(37\) −0.434870 0.434870i −0.0714922 0.0714922i 0.670457 0.741949i \(-0.266098\pi\)
−0.741949 + 0.670457i \(0.766098\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.81179 −1.22000 −0.609998 0.792403i \(-0.708830\pi\)
−0.609998 + 0.792403i \(0.708830\pi\)
\(42\) 0 0
\(43\) 5.49678 5.49678i 0.838251 0.838251i −0.150378 0.988629i \(-0.548049\pi\)
0.988629 + 0.150378i \(0.0480491\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.20723 −0.467822 −0.233911 0.972258i \(-0.575152\pi\)
−0.233911 + 0.972258i \(0.575152\pi\)
\(48\) 0 0
\(49\) −5.21456 −0.744937
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.06777 + 4.06777i −0.558751 + 0.558751i −0.928952 0.370201i \(-0.879289\pi\)
0.370201 + 0.928952i \(0.379289\pi\)
\(54\) 0 0
\(55\) 2.99355 0.403651
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.71811 + 4.71811i 0.614245 + 0.614245i 0.944049 0.329804i \(-0.106983\pi\)
−0.329804 + 0.944049i \(0.606983\pi\)
\(60\) 0 0
\(61\) −3.26785 + 3.26785i −0.418406 + 0.418406i −0.884654 0.466248i \(-0.845605\pi\)
0.466248 + 0.884654i \(0.345605\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.39758i 0.793522i
\(66\) 0 0
\(67\) 5.44348 + 5.44348i 0.665027 + 0.665027i 0.956561 0.291533i \(-0.0941654\pi\)
−0.291533 + 0.956561i \(0.594165\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.76718i 0.447082i −0.974695 0.223541i \(-0.928238\pi\)
0.974695 0.223541i \(-0.0717616\pi\)
\(72\) 0 0
\(73\) 10.5357i 1.23311i 0.787312 + 0.616555i \(0.211472\pi\)
−0.787312 + 0.616555i \(0.788528\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.61822 2.61822i −0.298374 0.298374i
\(78\) 0 0
\(79\) 11.1995i 1.26004i 0.776578 + 0.630021i \(0.216954\pi\)
−0.776578 + 0.630021i \(0.783046\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.73306 + 9.73306i −1.06834 + 1.06834i −0.0708558 + 0.997487i \(0.522573\pi\)
−0.997487 + 0.0708558i \(0.977427\pi\)
\(84\) 0 0
\(85\) 3.08029 + 3.08029i 0.334104 + 0.334104i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.64130 −0.173977 −0.0869886 0.996209i \(-0.527724\pi\)
−0.0869886 + 0.996209i \(0.527724\pi\)
\(90\) 0 0
\(91\) −5.59544 + 5.59544i −0.586562 + 0.586562i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.52023 −0.668962
\(96\) 0 0
\(97\) −5.70272 −0.579024 −0.289512 0.957174i \(-0.593493\pi\)
−0.289512 + 0.957174i \(0.593493\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.68599 6.68599i 0.665281 0.665281i −0.291339 0.956620i \(-0.594101\pi\)
0.956620 + 0.291339i \(0.0941008\pi\)
\(102\) 0 0
\(103\) 10.8784 1.07188 0.535938 0.844257i \(-0.319958\pi\)
0.535938 + 0.844257i \(0.319958\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.31755 + 1.31755i 0.127372 + 0.127372i 0.767919 0.640547i \(-0.221292\pi\)
−0.640547 + 0.767919i \(0.721292\pi\)
\(108\) 0 0
\(109\) 3.51516 3.51516i 0.336691 0.336691i −0.518429 0.855120i \(-0.673483\pi\)
0.855120 + 0.518429i \(0.173483\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.3139i 1.53469i 0.641236 + 0.767344i \(0.278422\pi\)
−0.641236 + 0.767344i \(0.721578\pi\)
\(114\) 0 0
\(115\) −6.77107 6.77107i −0.631406 0.631406i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.38815i 0.493931i
\(120\) 0 0
\(121\) 3.32115i 0.301922i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.74732 6.74732i −0.603499 0.603499i
\(126\) 0 0
\(127\) 20.7416i 1.84052i −0.391303 0.920262i \(-0.627976\pi\)
0.391303 0.920262i \(-0.372024\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.43621 9.43621i 0.824446 0.824446i −0.162296 0.986742i \(-0.551890\pi\)
0.986742 + 0.162296i \(0.0518901\pi\)
\(132\) 0 0
\(133\) 5.70272 + 5.70272i 0.494489 + 0.494489i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.8211 1.09538 0.547692 0.836680i \(-0.315506\pi\)
0.547692 + 0.836680i \(0.315506\pi\)
\(138\) 0 0
\(139\) −1.44348 + 1.44348i −0.122435 + 0.122435i −0.765669 0.643235i \(-0.777592\pi\)
0.643235 + 0.765669i \(0.277592\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 16.4106 1.37232
\(144\) 0 0
\(145\) 1.89341 0.157239
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.42073 6.42073i 0.526007 0.526007i −0.393372 0.919379i \(-0.628692\pi\)
0.919379 + 0.393372i \(0.128692\pi\)
\(150\) 0 0
\(151\) 0.205945 0.0167596 0.00837978 0.999965i \(-0.497333\pi\)
0.00837978 + 0.999965i \(0.497333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.19871 2.19871i −0.176605 0.176605i
\(156\) 0 0
\(157\) 1.26785 1.26785i 0.101186 0.101186i −0.654702 0.755887i \(-0.727206\pi\)
0.755887 + 0.654702i \(0.227206\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 11.8442i 0.933456i
\(162\) 0 0
\(163\) −0.169186 0.169186i −0.0132517 0.0132517i 0.700450 0.713702i \(-0.252983\pi\)
−0.713702 + 0.700450i \(0.752983\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.4503i 0.808671i −0.914611 0.404335i \(-0.867503\pi\)
0.914611 0.404335i \(-0.132497\pi\)
\(168\) 0 0
\(169\) 22.0714i 1.69780i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.974085 0.974085i −0.0740583 0.0740583i 0.669107 0.743166i \(-0.266677\pi\)
−0.743166 + 0.669107i \(0.766677\pi\)
\(174\) 0 0
\(175\) 5.12165i 0.387160i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.59560 + 6.59560i −0.492979 + 0.492979i −0.909243 0.416265i \(-0.863339\pi\)
0.416265 + 0.909243i \(0.363339\pi\)
\(180\) 0 0
\(181\) −10.1876 10.1876i −0.757236 0.757236i 0.218583 0.975818i \(-0.429857\pi\)
−0.975818 + 0.218583i \(0.929857\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.664376 0.0488459
\(186\) 0 0
\(187\) −7.90133 + 7.90133i −0.577803 + 0.577803i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −14.2297 −1.02962 −0.514812 0.857303i \(-0.672138\pi\)
−0.514812 + 0.857303i \(0.672138\pi\)
\(192\) 0 0
\(193\) 6.53570 0.470450 0.235225 0.971941i \(-0.424417\pi\)
0.235225 + 0.971941i \(0.424417\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.07713 + 9.07713i −0.646718 + 0.646718i −0.952198 0.305480i \(-0.901183\pi\)
0.305480 + 0.952198i \(0.401183\pi\)
\(198\) 0 0
\(199\) −10.0865 −0.715011 −0.357505 0.933911i \(-0.616373\pi\)
−0.357505 + 0.933911i \(0.616373\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.65601 1.65601i −0.116229 0.116229i
\(204\) 0 0
\(205\) 5.96725 5.96725i 0.416771 0.416771i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.7252i 1.15691i
\(210\) 0 0
\(211\) −17.9792 17.9792i −1.23774 1.23774i −0.960923 0.276815i \(-0.910721\pi\)
−0.276815 0.960923i \(-0.589279\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.39773i 0.572721i
\(216\) 0 0
\(217\) 3.84607i 0.261088i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 16.8861 + 16.8861i 1.13588 + 1.13588i
\(222\) 0 0
\(223\) 4.00861i 0.268437i −0.990952 0.134218i \(-0.957148\pi\)
0.990952 0.134218i \(-0.0428523\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 13.7915 13.7915i 0.915373 0.915373i −0.0813152 0.996688i \(-0.525912\pi\)
0.996688 + 0.0813152i \(0.0259120\pi\)
\(228\) 0 0
\(229\) −3.47840 3.47840i −0.229859 0.229859i 0.582775 0.812634i \(-0.301967\pi\)
−0.812634 + 0.582775i \(0.801967\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.23973 0.212241 0.106121 0.994353i \(-0.466157\pi\)
0.106121 + 0.994353i \(0.466157\pi\)
\(234\) 0 0
\(235\) 2.44993 2.44993i 0.159816 0.159816i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.2484 1.56850 0.784249 0.620446i \(-0.213048\pi\)
0.784249 + 0.620446i \(0.213048\pi\)
\(240\) 0 0
\(241\) 16.5596 1.06670 0.533348 0.845896i \(-0.320934\pi\)
0.533348 + 0.845896i \(0.320934\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.98329 3.98329i 0.254483 0.254483i
\(246\) 0 0
\(247\) −35.7438 −2.27432
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 11.3957 + 11.3957i 0.719287 + 0.719287i 0.968459 0.249172i \(-0.0801584\pi\)
−0.249172 + 0.968459i \(0.580158\pi\)
\(252\) 0 0
\(253\) 17.3687 17.3687i 1.09196 1.09196i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.54316i 0.221016i 0.993875 + 0.110508i \(0.0352478\pi\)
−0.993875 + 0.110508i \(0.964752\pi\)
\(258\) 0 0
\(259\) −0.581076 0.581076i −0.0361063 0.0361063i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 21.0534i 1.29821i −0.760701 0.649103i \(-0.775145\pi\)
0.760701 0.649103i \(-0.224855\pi\)
\(264\) 0 0
\(265\) 6.21456i 0.381757i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 20.4077 + 20.4077i 1.24428 + 1.24428i 0.958208 + 0.286072i \(0.0923496\pi\)
0.286072 + 0.958208i \(0.407650\pi\)
\(270\) 0 0
\(271\) 5.06279i 0.307543i −0.988106 0.153771i \(-0.950858\pi\)
0.988106 0.153771i \(-0.0491419\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.51052 7.51052i 0.452901 0.452901i
\(276\) 0 0
\(277\) 5.86642 + 5.86642i 0.352479 + 0.352479i 0.861031 0.508552i \(-0.169819\pi\)
−0.508552 + 0.861031i \(0.669819\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.77316 0.284743 0.142371 0.989813i \(-0.454527\pi\)
0.142371 + 0.989813i \(0.454527\pi\)
\(282\) 0 0
\(283\) 10.0779 10.0779i 0.599066 0.599066i −0.340998 0.940064i \(-0.610765\pi\)
0.940064 + 0.340998i \(0.110765\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10.4381 −0.616144
\(288\) 0 0
\(289\) 0.739481 0.0434989
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −11.7829 + 11.7829i −0.688364 + 0.688364i −0.961870 0.273506i \(-0.911817\pi\)
0.273506 + 0.961870i \(0.411817\pi\)
\(294\) 0 0
\(295\) −7.20811 −0.419673
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −37.1189 37.1189i −2.14664 2.14664i
\(300\) 0 0
\(301\) 7.34482 7.34482i 0.423348 0.423348i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.99248i 0.285869i
\(306\) 0 0
\(307\) 4.31322 + 4.31322i 0.246169 + 0.246169i 0.819396 0.573228i \(-0.194309\pi\)
−0.573228 + 0.819396i \(0.694309\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.4434i 1.55617i 0.628156 + 0.778087i \(0.283810\pi\)
−0.628156 + 0.778087i \(0.716190\pi\)
\(312\) 0 0
\(313\) 18.8568i 1.06585i −0.846162 0.532926i \(-0.821092\pi\)
0.846162 0.532926i \(-0.178908\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.154552 + 0.154552i 0.00868053 + 0.00868053i 0.711434 0.702753i \(-0.248046\pi\)
−0.702753 + 0.711434i \(0.748046\pi\)
\(318\) 0 0
\(319\) 4.85685i 0.271931i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 17.2098 17.2098i 0.957580 0.957580i
\(324\) 0 0
\(325\) −16.0509 16.0509i −0.890342 0.890342i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.28551 −0.236268
\(330\) 0 0
\(331\) 16.3132 16.3132i 0.896656 0.896656i −0.0984829 0.995139i \(-0.531399\pi\)
0.995139 + 0.0984829i \(0.0313990\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.31631 −0.454369
\(336\) 0 0
\(337\) 1.89341 0.103141 0.0515704 0.998669i \(-0.483577\pi\)
0.0515704 + 0.998669i \(0.483577\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.63998 5.63998i 0.305422 0.305422i
\(342\) 0 0
\(343\) −16.3211 −0.881259
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.7630 + 19.7630i 1.06093 + 1.06093i 0.998019 + 0.0629131i \(0.0200391\pi\)
0.0629131 + 0.998019i \(0.479961\pi\)
\(348\) 0 0
\(349\) −18.1008 + 18.1008i −0.968915 + 0.968915i −0.999531 0.0306158i \(-0.990253\pi\)
0.0306158 + 0.999531i \(0.490253\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 19.3695i 1.03093i −0.856910 0.515466i \(-0.827619\pi\)
0.856910 0.515466i \(-0.172381\pi\)
\(354\) 0 0
\(355\) 2.87766 + 2.87766i 0.152730 + 0.152730i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.09236i 0.374321i −0.982329 0.187160i \(-0.940072\pi\)
0.982329 0.187160i \(-0.0599284\pi\)
\(360\) 0 0
\(361\) 17.4291i 0.917322i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.04799 8.04799i −0.421251 0.421251i
\(366\) 0 0
\(367\) 1.65735i 0.0865130i −0.999064 0.0432565i \(-0.986227\pi\)
0.999064 0.0432565i \(-0.0137733\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.43537 + 5.43537i −0.282190 + 0.282190i
\(372\) 0 0
\(373\) 3.48241 + 3.48241i 0.180312 + 0.180312i 0.791492 0.611180i \(-0.209305\pi\)
−0.611180 + 0.791492i \(0.709305\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.3797 0.534579
\(378\) 0 0
\(379\) −19.1548 + 19.1548i −0.983917 + 0.983917i −0.999873 0.0159558i \(-0.994921\pi\)
0.0159558 + 0.999873i \(0.494921\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.2907 0.781319 0.390660 0.920535i \(-0.372247\pi\)
0.390660 + 0.920535i \(0.372247\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 22.7144 22.7144i 1.15166 1.15166i 0.165445 0.986219i \(-0.447094\pi\)
0.986219 0.165445i \(-0.0529060\pi\)
\(390\) 0 0
\(391\) 35.7438 1.80764
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.55505 8.55505i −0.430451 0.430451i
\(396\) 0 0
\(397\) −7.45854 + 7.45854i −0.374334 + 0.374334i −0.869053 0.494719i \(-0.835271\pi\)
0.494719 + 0.869053i \(0.335271\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.5911i 0.578834i 0.957203 + 0.289417i \(0.0934615\pi\)
−0.957203 + 0.289417i \(0.906539\pi\)
\(402\) 0 0
\(403\) −12.0533 12.0533i −0.600417 0.600417i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.70421i 0.0844745i
\(408\) 0 0
\(409\) 12.4659i 0.616398i −0.951322 0.308199i \(-0.900274\pi\)
0.951322 0.308199i \(-0.0997262\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.30435 + 6.30435i 0.310217 + 0.310217i
\(414\) 0 0
\(415\) 14.8697i 0.729927i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.4096 + 14.4096i −0.703953 + 0.703953i −0.965257 0.261304i \(-0.915848\pi\)
0.261304 + 0.965257i \(0.415848\pi\)
\(420\) 0 0
\(421\) 5.83630 + 5.83630i 0.284444 + 0.284444i 0.834878 0.550434i \(-0.185538\pi\)
−0.550434 + 0.834878i \(0.685538\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 15.4562 0.749738
\(426\) 0 0
\(427\) −4.36652 + 4.36652i −0.211311 + 0.211311i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.4510 0.888755 0.444377 0.895840i \(-0.353425\pi\)
0.444377 + 0.895840i \(0.353425\pi\)
\(432\) 0 0
\(433\) 6.58166 0.316295 0.158147 0.987416i \(-0.449448\pi\)
0.158147 + 0.987416i \(0.449448\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −37.8306 + 37.8306i −1.80968 + 1.80968i
\(438\) 0 0
\(439\) 3.68747 0.175993 0.0879966 0.996121i \(-0.471954\pi\)
0.0879966 + 0.996121i \(0.471954\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.31861 + 3.31861i 0.157672 + 0.157672i 0.781534 0.623862i \(-0.214437\pi\)
−0.623862 + 0.781534i \(0.714437\pi\)
\(444\) 0 0
\(445\) 1.25375 1.25375i 0.0594335 0.0594335i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.3555i 0.960635i 0.877095 + 0.480318i \(0.159479\pi\)
−0.877095 + 0.480318i \(0.840521\pi\)
\(450\) 0 0
\(451\) 15.3068 + 15.3068i 0.720768 + 0.720768i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.54847i 0.400758i
\(456\) 0 0
\(457\) 10.0239i 0.468897i 0.972129 + 0.234448i \(0.0753284\pi\)
−0.972129 + 0.234448i \(0.924672\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16.4043 16.4043i −0.764026 0.764026i 0.213021 0.977048i \(-0.431670\pi\)
−0.977048 + 0.213021i \(0.931670\pi\)
\(462\) 0 0
\(463\) 0.997833i 0.0463732i 0.999731 + 0.0231866i \(0.00738119\pi\)
−0.999731 + 0.0231866i \(0.992619\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −9.73306 + 9.73306i −0.450392 + 0.450392i −0.895485 0.445092i \(-0.853171\pi\)
0.445092 + 0.895485i \(0.353171\pi\)
\(468\) 0 0
\(469\) 7.27361 + 7.27361i 0.335864 + 0.335864i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −21.5413 −0.990469
\(474\) 0 0
\(475\) −16.3586 + 16.3586i −0.750584 + 0.750584i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −37.0669 −1.69363 −0.846816 0.531886i \(-0.821483\pi\)
−0.846816 + 0.531886i \(0.821483\pi\)
\(480\) 0 0
\(481\) 3.64210 0.166065
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.35618 4.35618i 0.197804 0.197804i
\(486\) 0 0
\(487\) 33.1866 1.50383 0.751914 0.659261i \(-0.229131\pi\)
0.751914 + 0.659261i \(0.229131\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.2580 + 10.2580i 0.462936 + 0.462936i 0.899617 0.436681i \(-0.143846\pi\)
−0.436681 + 0.899617i \(0.643846\pi\)
\(492\) 0 0
\(493\) −4.99757 + 4.99757i −0.225079 + 0.225079i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.03372i 0.225793i
\(498\) 0 0
\(499\) −4.19733 4.19733i −0.187898 0.187898i 0.606889 0.794787i \(-0.292417\pi\)
−0.794787 + 0.606889i \(0.792417\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 21.0655i 0.939266i 0.882862 + 0.469633i \(0.155614\pi\)
−0.882862 + 0.469633i \(0.844386\pi\)
\(504\) 0 0
\(505\) 10.2146i 0.454542i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −22.6424 22.6424i −1.00361 1.00361i −0.999993 0.00361481i \(-0.998849\pi\)
−0.00361481 0.999993i \(-0.501151\pi\)
\(510\) 0 0
\(511\) 14.0779i 0.622768i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.30973 + 8.30973i −0.366171 + 0.366171i
\(516\) 0 0
\(517\) 6.28439 + 6.28439i 0.276387 + 0.276387i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −38.3351 −1.67949 −0.839746 0.542980i \(-0.817296\pi\)
−0.839746 + 0.542980i \(0.817296\pi\)
\(522\) 0 0
\(523\) 5.39811 5.39811i 0.236043 0.236043i −0.579166 0.815209i \(-0.696622\pi\)
0.815209 + 0.579166i \(0.196622\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.6068 0.505599
\(528\) 0 0
\(529\) −55.5719 −2.41617
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 32.7124 32.7124i 1.41693 1.41693i
\(534\) 0 0
\(535\) −2.01289 −0.0870249
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.2176 + 10.2176i 0.440105 + 0.440105i
\(540\) 0 0
\(541\) −5.51516 + 5.51516i −0.237115 + 0.237115i −0.815654 0.578539i \(-0.803623\pi\)
0.578539 + 0.815654i \(0.303623\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.37030i 0.230038i
\(546\) 0 0
\(547\) 23.2535 + 23.2535i 0.994247 + 0.994247i 0.999984 0.00573636i \(-0.00182595\pi\)
−0.00573636 + 0.999984i \(0.501826\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.5787i 0.450666i
\(552\) 0 0
\(553\) 14.9648i 0.636369i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −26.9066 26.9066i −1.14007 1.14007i −0.988437 0.151635i \(-0.951546\pi\)
−0.151635 0.988437i \(-0.548454\pi\)
\(558\) 0 0
\(559\) 46.0362i 1.94712i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21.4144 + 21.4144i −0.902508 + 0.902508i −0.995653 0.0931446i \(-0.970308\pi\)
0.0931446 + 0.995653i \(0.470308\pi\)
\(564\) 0 0
\(565\) −12.4619 12.4619i −0.524275 0.524275i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.54807 −0.0648986 −0.0324493 0.999473i \(-0.510331\pi\)
−0.0324493 + 0.999473i \(0.510331\pi\)
\(570\) 0 0
\(571\) 12.3384 12.3384i 0.516345 0.516345i −0.400119 0.916463i \(-0.631031\pi\)
0.916463 + 0.400119i \(0.131031\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −33.9759 −1.41689
\(576\) 0 0
\(577\) −24.9648 −1.03930 −0.519650 0.854380i \(-0.673937\pi\)
−0.519650 + 0.854380i \(0.673937\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −13.0054 + 13.0054i −0.539553 + 0.539553i
\(582\) 0 0
\(583\) 15.9411 0.660215
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.99426 + 3.99426i 0.164861 + 0.164861i 0.784716 0.619855i \(-0.212809\pi\)
−0.619855 + 0.784716i \(0.712809\pi\)
\(588\) 0 0
\(589\) −12.2844 + 12.2844i −0.506169 + 0.506169i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 21.0442i 0.864183i 0.901830 + 0.432092i \(0.142224\pi\)
−0.901830 + 0.432092i \(0.857776\pi\)
\(594\) 0 0
\(595\) 4.11589 + 4.11589i 0.168735 + 0.168735i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.7718i 0.521840i −0.965360 0.260920i \(-0.915974\pi\)
0.965360 0.260920i \(-0.0840259\pi\)
\(600\) 0 0
\(601\) 44.2967i 1.80690i 0.428691 + 0.903451i \(0.358975\pi\)
−0.428691 + 0.903451i \(0.641025\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.53695 + 2.53695i 0.103142 + 0.103142i
\(606\) 0 0
\(607\) 26.8784i 1.09096i −0.838124 0.545479i \(-0.816348\pi\)
0.838124 0.545479i \(-0.183652\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 13.4305 13.4305i 0.543339 0.543339i
\(612\) 0 0
\(613\) 27.6602 + 27.6602i 1.11719 + 1.11719i 0.992153 + 0.125033i \(0.0399036\pi\)
0.125033 + 0.992153i \(0.460096\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.6148 0.789661 0.394831 0.918754i \(-0.370803\pi\)
0.394831 + 0.918754i \(0.370803\pi\)
\(618\) 0 0
\(619\) 11.7854 11.7854i 0.473697 0.473697i −0.429412 0.903109i \(-0.641279\pi\)
0.903109 + 0.429412i \(0.141279\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.19311 −0.0878651
\(624\) 0 0
\(625\) −8.85665 −0.354266
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.75359 + 1.75359i −0.0699200 + 0.0699200i
\(630\) 0 0
\(631\) 18.8195 0.749193 0.374596 0.927188i \(-0.377781\pi\)
0.374596 + 0.927188i \(0.377781\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 15.8441 + 15.8441i 0.628753 + 0.628753i
\(636\) 0 0
\(637\) 21.8363 21.8363i 0.865186 0.865186i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.9718i 0.709845i −0.934896 0.354923i \(-0.884507\pi\)
0.934896 0.354923i \(-0.115493\pi\)
\(642\) 0 0
\(643\) 9.92589 + 9.92589i 0.391439 + 0.391439i 0.875200 0.483761i \(-0.160730\pi\)
−0.483761 + 0.875200i \(0.660730\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.6801i 0.537820i 0.963165 + 0.268910i \(0.0866635\pi\)
−0.963165 + 0.268910i \(0.913337\pi\)
\(648\) 0 0
\(649\) 18.4897i 0.725786i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 31.1940 + 31.1940i 1.22071 + 1.22071i 0.967378 + 0.253336i \(0.0815278\pi\)
0.253336 + 0.967378i \(0.418472\pi\)
\(654\) 0 0
\(655\) 14.4162i 0.563288i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −27.2397 + 27.2397i −1.06111 + 1.06111i −0.0631026 + 0.998007i \(0.520100\pi\)
−0.998007 + 0.0631026i \(0.979900\pi\)
\(660\) 0 0
\(661\) 12.0770 + 12.0770i 0.469740 + 0.469740i 0.901830 0.432091i \(-0.142224\pi\)
−0.432091 + 0.901830i \(0.642224\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.71237 −0.337851
\(666\) 0 0
\(667\) 10.9856 10.9856i 0.425365 0.425365i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.8064 0.494384
\(672\) 0 0
\(673\) 11.0108 0.424434 0.212217 0.977223i \(-0.431932\pi\)
0.212217 + 0.977223i \(0.431932\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.26708 5.26708i 0.202430 0.202430i −0.598610 0.801040i \(-0.704280\pi\)
0.801040 + 0.598610i \(0.204280\pi\)
\(678\) 0 0
\(679\) −7.62000 −0.292429
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 16.0080 + 16.0080i 0.612528 + 0.612528i 0.943604 0.331076i \(-0.107412\pi\)
−0.331076 + 0.943604i \(0.607412\pi\)
\(684\) 0 0
\(685\) −9.79379 + 9.79379i −0.374201 + 0.374201i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 34.0681i 1.29789i
\(690\) 0 0
\(691\) −7.63348 7.63348i −0.290391 0.290391i 0.546843 0.837235i \(-0.315829\pi\)
−0.837235 + 0.546843i \(0.815829\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.20529i 0.0836514i
\(696\) 0 0
\(697\) 31.5005i 1.19317i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.3494 18.3494i −0.693049 0.693049i 0.269853 0.962902i \(-0.413025\pi\)
−0.962902 + 0.269853i \(0.913025\pi\)
\(702\) 0 0
\(703\) 3.71192i 0.139998i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.93385 8.93385i 0.335992 0.335992i
\(708\) 0 0
\(709\) −19.8774 19.8774i −0.746511 0.746511i 0.227311 0.973822i \(-0.427007\pi\)
−0.973822 + 0.227311i \(0.927007\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −25.5140 −0.955505
\(714\) 0 0
\(715\) −12.5357 + 12.5357i −0.468809 + 0.468809i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.1176 0.601083 0.300542 0.953769i \(-0.402833\pi\)
0.300542 + 0.953769i \(0.402833\pi\)
\(720\) 0 0
\(721\) 14.5357 0.541338
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.75038 4.75038i 0.176425 0.176425i
\(726\) 0 0
\(727\) 16.4536 0.610229 0.305115 0.952316i \(-0.401305\pi\)
0.305115 + 0.952316i \(0.401305\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −22.1654 22.1654i −0.819816 0.819816i
\(732\) 0 0
\(733\) −24.5995 + 24.5995i −0.908602 + 0.908602i −0.996159 0.0875578i \(-0.972094\pi\)
0.0875578 + 0.996159i \(0.472094\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21.3324i 0.785790i
\(738\) 0 0
\(739\) −3.06707 3.06707i −0.112824 0.112824i 0.648441 0.761265i \(-0.275421\pi\)
−0.761265 + 0.648441i \(0.775421\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.16681i 0.0428061i 0.999771 + 0.0214031i \(0.00681333\pi\)
−0.999771 + 0.0214031i \(0.993187\pi\)
\(744\) 0 0
\(745\) 9.80931i 0.359385i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.76051 + 1.76051i 0.0643278 + 0.0643278i
\(750\) 0 0
\(751\) 2.12809i 0.0776552i 0.999246 + 0.0388276i \(0.0123623\pi\)
−0.999246 + 0.0388276i \(0.987638\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.157317 + 0.157317i −0.00572534 + 0.00572534i
\(756\) 0 0
\(757\) −19.1573 19.1573i −0.696282 0.696282i 0.267324 0.963607i \(-0.413860\pi\)
−0.963607 + 0.267324i \(0.913860\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 10.2848 0.372824 0.186412 0.982472i \(-0.440314\pi\)
0.186412 + 0.982472i \(0.440314\pi\)
\(762\) 0 0
\(763\) 4.69697 4.69697i 0.170042 0.170042i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −39.5147 −1.42679
\(768\) 0 0
\(769\) −15.2860 −0.551226 −0.275613 0.961269i \(-0.588881\pi\)
−0.275613 + 0.961269i \(0.588881\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −23.6903 + 23.6903i −0.852081 + 0.852081i −0.990389 0.138308i \(-0.955834\pi\)
0.138308 + 0.990389i \(0.455834\pi\)
\(774\) 0 0
\(775\) −11.0327 −0.396305
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −33.3396 33.3396i −1.19451 1.19451i
\(780\) 0 0
\(781\) −7.38158 + 7.38158i −0.264134 + 0.264134i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.93697i 0.0691333i
\(786\) 0 0
\(787\) 29.3220 + 29.3220i 1.04522 + 1.04522i 0.998928 + 0.0462895i \(0.0147397\pi\)
0.0462895 + 0.998928i \(0.485260\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 21.7988i 0.775075i
\(792\) 0 0
\(793\) 27.3687i 0.971890i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.41153 + 3.41153i 0.120843 + 0.120843i 0.764942 0.644099i \(-0.222768\pi\)
−0.644099 + 0.764942i \(0.722768\pi\)
\(798\) 0 0
\(799\) 12.9329i 0.457534i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 20.6441 20.6441i 0.728516 0.728516i
\(804\) 0 0
\(805\) −9.04754 9.04754i −0.318884 0.318884i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −9.15877 −0.322005 −0.161003 0.986954i \(-0.551473\pi\)
−0.161003 + 0.986954i \(0.551473\pi\)
\(810\) 0 0
\(811\) 8.91014 8.91014i 0.312877 0.312877i −0.533146 0.846023i \(-0.678990\pi\)
0.846023 + 0.533146i \(0.178990\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.258474 0.00905397
\(816\) 0 0
\(817\) 46.9189 1.64148
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −27.0618 + 27.0618i −0.944464 + 0.944464i −0.998537 0.0540734i \(-0.982779\pi\)
0.0540734 + 0.998537i \(0.482779\pi\)
\(822\) 0 0
\(823\) 11.6286 0.405348 0.202674 0.979246i \(-0.435037\pi\)
0.202674 + 0.979246i \(0.435037\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8.57283 8.57283i −0.298107 0.298107i 0.542165 0.840272i \(-0.317605\pi\)
−0.840272 + 0.542165i \(0.817605\pi\)
\(828\) 0 0
\(829\) −19.5629 + 19.5629i −0.679447 + 0.679447i −0.959875 0.280428i \(-0.909524\pi\)
0.280428 + 0.959875i \(0.409524\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 21.0274i 0.728555i
\(834\) 0 0
\(835\) 7.98277 + 7.98277i 0.276255 + 0.276255i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 14.8018i 0.511015i 0.966807 + 0.255508i \(0.0822426\pi\)
−0.966807 + 0.255508i \(0.917757\pi\)
\(840\) 0 0
\(841\) 25.9281i 0.894071i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 16.8599 + 16.8599i 0.579997 + 0.579997i
\(846\) 0 0
\(847\) 4.43773i 0.152482i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.85473 3.85473i 0.132138 0.132138i
\(852\) 0 0
\(853\) 27.2550 + 27.2550i 0.933192 + 0.933192i 0.997904 0.0647119i \(-0.0206129\pi\)
−0.0647119 + 0.997904i \(0.520613\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.7462 0.981951 0.490976 0.871173i \(-0.336640\pi\)
0.490976 + 0.871173i \(0.336640\pi\)
\(858\) 0 0
\(859\) −29.2578 + 29.2578i −0.998264 + 0.998264i −0.999998 0.00173461i \(-0.999448\pi\)
0.00173461 + 0.999998i \(0.499448\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 40.6816 1.38482 0.692408 0.721506i \(-0.256550\pi\)
0.692408 + 0.721506i \(0.256550\pi\)
\(864\) 0 0
\(865\) 1.48816 0.0505991
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 21.9448 21.9448i 0.744427 0.744427i
\(870\) 0 0
\(871\) −45.5899 −1.54475
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −9.01580 9.01580i −0.304790 0.304790i
\(876\) 0 0
\(877\) 28.9577 28.9577i 0.977831 0.977831i −0.0219281 0.999760i \(-0.506980\pi\)
0.999760 + 0.0219281i \(0.00698049\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 17.2311i 0.580531i 0.956946 + 0.290266i \(0.0937437\pi\)
−0.956946 + 0.290266i \(0.906256\pi\)
\(882\) 0 0
\(883\) −24.8129 24.8129i −0.835019 0.835019i 0.153179 0.988198i \(-0.451049\pi\)
−0.988198 + 0.153179i \(0.951049\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 39.0606i 1.31153i 0.754967 + 0.655763i \(0.227653\pi\)
−0.754967 + 0.655763i \(0.772347\pi\)
\(888\) 0 0
\(889\) 27.7151i 0.929534i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −13.6880 13.6880i −0.458050 0.458050i
\(894\) 0 0
\(895\) 10.0765i 0.336819i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.56727 3.56727i 0.118975 0.118975i
\(900\) 0 0
\(901\) 16.4030 + 16.4030i 0.546463 + 0.546463i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15.5641 0.517369
\(906\) 0 0
\(907\) −0.366518 + 0.366518i −0.0121700 + 0.0121700i −0.713166 0.700996i \(-0.752739\pi\)
0.700996 + 0.713166i \(0.252739\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 18.8950 0.626018 0.313009 0.949750i \(-0.398663\pi\)
0.313009 + 0.949750i \(0.398663\pi\)
\(912\) 0 0
\(913\) 38.1428 1.26234
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.6087 12.6087i 0.416376 0.416376i
\(918\) 0 0
\(919\) −54.5805 −1.80045 −0.900223 0.435429i \(-0.856597\pi\)
−0.900223 + 0.435429i \(0.856597\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 15.7753 + 15.7753i 0.519250 + 0.519250i
\(924\) 0 0
\(925\) 1.66685 1.66685i 0.0548057 0.0548057i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 21.2118i 0.695938i −0.937506 0.347969i \(-0.886871\pi\)
0.937506 0.347969i \(-0.113129\pi\)
\(930\) 0 0
\(931\) −22.2550 22.2550i −0.729377 0.729377i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 12.0713i 0.394774i
\(936\) 0 0
\(937\) 15.7395i 0.514186i −0.966387 0.257093i \(-0.917235\pi\)
0.966387 0.257093i \(-0.0827647\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.43393 + 1.43393i 0.0467447 + 0.0467447i 0.730093 0.683348i \(-0.239477\pi\)
−0.683348 + 0.730093i \(0.739477\pi\)
\(942\) 0 0
\(943\) 69.2443i 2.25491i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.5461 28.5461i 0.927623 0.927623i −0.0699288 0.997552i \(-0.522277\pi\)
0.997552 + 0.0699288i \(0.0222772\pi\)
\(948\) 0 0
\(949\) −44.1189 44.1189i −1.43216 1.43216i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −14.6986 −0.476136 −0.238068 0.971249i \(-0.576514\pi\)
−0.238068 + 0.971249i \(0.576514\pi\)
\(954\) 0 0
\(955\) 10.8697 10.8697i 0.351737 0.351737i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 17.1317 0.553210
\(960\) 0 0
\(961\) 22.7151 0.732745
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −4.99248 + 4.99248i −0.160714 + 0.160714i
\(966\) 0 0
\(967\) −7.34054 −0.236056 −0.118028 0.993010i \(-0.537657\pi\)
−0.118028 + 0.993010i \(0.537657\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.48251 3.48251i −0.111759 0.111759i 0.649016 0.760775i \(-0.275181\pi\)
−0.760775 + 0.649016i \(0.775181\pi\)
\(972\) 0 0
\(973\) −1.92879 + 1.92879i −0.0618341 + 0.0618341i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.95013i 0.318333i 0.987252 + 0.159166i \(0.0508806\pi\)
−0.987252 + 0.159166i \(0.949119\pi\)
\(978\) 0 0
\(979\) 3.21603 + 3.21603i 0.102785 + 0.102785i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18.7648i 0.598505i 0.954174 + 0.299253i \(0.0967373\pi\)
−0.954174 + 0.299253i \(0.903263\pi\)
\(984\) 0 0
\(985\) 13.8676i 0.441859i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 48.7239 + 48.7239i 1.54933 + 1.54933i
\(990\) 0 0
\(991\) 3.18227i 0.101088i 0.998722 + 0.0505441i \(0.0160955\pi\)
−0.998722 + 0.0505441i \(0.983904\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 7.70483 7.70483i 0.244259 0.244259i
\(996\) 0 0
\(997\) −26.9944 26.9944i −0.854923 0.854923i 0.135812 0.990735i \(-0.456636\pi\)
−0.990735 + 0.135812i \(0.956636\pi\)
\(998\) 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 1152.2.l.a.287.3 16
3.2 odd 2 inner 1152.2.l.a.287.6 16
4.3 odd 2 1152.2.l.b.287.3 16
8.3 odd 2 576.2.l.a.143.6 16
8.5 even 2 144.2.l.a.107.2 yes 16
12.11 even 2 1152.2.l.b.287.6 16
16.3 odd 4 inner 1152.2.l.a.863.6 16
16.5 even 4 576.2.l.a.431.3 16
16.11 odd 4 144.2.l.a.35.7 yes 16
16.13 even 4 1152.2.l.b.863.6 16
24.5 odd 2 144.2.l.a.107.7 yes 16
24.11 even 2 576.2.l.a.143.3 16
48.5 odd 4 576.2.l.a.431.6 16
48.11 even 4 144.2.l.a.35.2 16
48.29 odd 4 1152.2.l.b.863.3 16
48.35 even 4 inner 1152.2.l.a.863.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.l.a.35.2 16 48.11 even 4
144.2.l.a.35.7 yes 16 16.11 odd 4
144.2.l.a.107.2 yes 16 8.5 even 2
144.2.l.a.107.7 yes 16 24.5 odd 2
576.2.l.a.143.3 16 24.11 even 2
576.2.l.a.143.6 16 8.3 odd 2
576.2.l.a.431.3 16 16.5 even 4
576.2.l.a.431.6 16 48.5 odd 4
1152.2.l.a.287.3 16 1.1 even 1 trivial
1152.2.l.a.287.6 16 3.2 odd 2 inner
1152.2.l.a.863.3 16 48.35 even 4 inner
1152.2.l.a.863.6 16 16.3 odd 4 inner
1152.2.l.b.287.3 16 4.3 odd 2
1152.2.l.b.287.6 16 12.11 even 2
1152.2.l.b.863.3 16 48.29 odd 4
1152.2.l.b.863.6 16 16.13 even 4