Properties

Label 1764.3.d.h.685.7
Level $1764$
Weight $3$
Character 1764.685
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
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] = [1764,3,Mod(685,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, 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("1764.685");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 685.7
Root \(1.60021 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 1764.685
Dual form 1764.3.d.h.685.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.37964i q^{5} +O(q^{10})\) \(q+5.37964i q^{5} +8.59159 q^{11} +21.0158i q^{13} -5.48622i q^{17} +7.24098i q^{19} -28.0556 q^{23} -3.94054 q^{25} -40.3447 q^{29} +40.5101i q^{31} +66.6370 q^{37} +33.6357i q^{41} +0.932907 q^{43} -85.6544i q^{47} +44.5954 q^{53} +46.2197i q^{55} -63.6950i q^{59} -32.0084i q^{61} -113.058 q^{65} -47.7341 q^{67} -14.9676 q^{71} +140.298i q^{73} -122.307 q^{79} +33.1852i q^{83} +29.5139 q^{85} +36.1246i q^{89} -38.9539 q^{95} -16.2175i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{23} + 72 q^{25} - 80 q^{29} + 128 q^{37} - 112 q^{43} + 144 q^{53} - 240 q^{65} - 64 q^{67} - 224 q^{71} - 432 q^{79} - 96 q^{85} + 272 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 5.37964i 1.07593i 0.842968 + 0.537964i \(0.180806\pi\)
−0.842968 + 0.537964i \(0.819194\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 8.59159 0.781054 0.390527 0.920592i \(-0.372293\pi\)
0.390527 + 0.920592i \(0.372293\pi\)
\(12\) 0 0
\(13\) 21.0158i 1.61660i 0.588769 + 0.808301i \(0.299613\pi\)
−0.588769 + 0.808301i \(0.700387\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.48622i − 0.322719i −0.986896 0.161359i \(-0.948412\pi\)
0.986896 0.161359i \(-0.0515878\pi\)
\(18\) 0 0
\(19\) 7.24098i 0.381104i 0.981677 + 0.190552i \(0.0610278\pi\)
−0.981677 + 0.190552i \(0.938972\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −28.0556 −1.21981 −0.609905 0.792474i \(-0.708792\pi\)
−0.609905 + 0.792474i \(0.708792\pi\)
\(24\) 0 0
\(25\) −3.94054 −0.157621
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −40.3447 −1.39120 −0.695599 0.718431i \(-0.744861\pi\)
−0.695599 + 0.718431i \(0.744861\pi\)
\(30\) 0 0
\(31\) 40.5101i 1.30678i 0.757023 + 0.653388i \(0.226653\pi\)
−0.757023 + 0.653388i \(0.773347\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 66.6370 1.80100 0.900500 0.434856i \(-0.143201\pi\)
0.900500 + 0.434856i \(0.143201\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 33.6357i 0.820383i 0.911999 + 0.410191i \(0.134538\pi\)
−0.911999 + 0.410191i \(0.865462\pi\)
\(42\) 0 0
\(43\) 0.932907 0.0216955 0.0108478 0.999941i \(-0.496547\pi\)
0.0108478 + 0.999941i \(0.496547\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 85.6544i − 1.82243i −0.411927 0.911217i \(-0.635144\pi\)
0.411927 0.911217i \(-0.364856\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 44.5954 0.841422 0.420711 0.907195i \(-0.361781\pi\)
0.420711 + 0.907195i \(0.361781\pi\)
\(54\) 0 0
\(55\) 46.2197i 0.840358i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 63.6950i − 1.07958i −0.841801 0.539788i \(-0.818504\pi\)
0.841801 0.539788i \(-0.181496\pi\)
\(60\) 0 0
\(61\) − 32.0084i − 0.524727i −0.964969 0.262364i \(-0.915498\pi\)
0.964969 0.262364i \(-0.0845020\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −113.058 −1.73935
\(66\) 0 0
\(67\) −47.7341 −0.712450 −0.356225 0.934400i \(-0.615936\pi\)
−0.356225 + 0.934400i \(0.615936\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.9676 −0.210811 −0.105405 0.994429i \(-0.533614\pi\)
−0.105405 + 0.994429i \(0.533614\pi\)
\(72\) 0 0
\(73\) 140.298i 1.92188i 0.276750 + 0.960942i \(0.410742\pi\)
−0.276750 + 0.960942i \(0.589258\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −122.307 −1.54820 −0.774098 0.633066i \(-0.781796\pi\)
−0.774098 + 0.633066i \(0.781796\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 33.1852i 0.399822i 0.979814 + 0.199911i \(0.0640652\pi\)
−0.979814 + 0.199911i \(0.935935\pi\)
\(84\) 0 0
\(85\) 29.5139 0.347222
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 36.1246i 0.405894i 0.979190 + 0.202947i \(0.0650519\pi\)
−0.979190 + 0.202947i \(0.934948\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −38.9539 −0.410041
\(96\) 0 0
\(97\) − 16.2175i − 0.167191i −0.996500 0.0835956i \(-0.973360\pi\)
0.996500 0.0835956i \(-0.0266404\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 119.464i 1.18281i 0.806373 + 0.591407i \(0.201427\pi\)
−0.806373 + 0.591407i \(0.798573\pi\)
\(102\) 0 0
\(103\) 8.93188i 0.0867173i 0.999060 + 0.0433586i \(0.0138058\pi\)
−0.999060 + 0.0433586i \(0.986194\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −173.856 −1.62482 −0.812410 0.583086i \(-0.801845\pi\)
−0.812410 + 0.583086i \(0.801845\pi\)
\(108\) 0 0
\(109\) −160.315 −1.47078 −0.735388 0.677646i \(-0.763000\pi\)
−0.735388 + 0.677646i \(0.763000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 81.4420 0.720725 0.360363 0.932812i \(-0.382653\pi\)
0.360363 + 0.932812i \(0.382653\pi\)
\(114\) 0 0
\(115\) − 150.929i − 1.31243i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −47.1846 −0.389955
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 113.292i 0.906339i
\(126\) 0 0
\(127\) 117.172 0.922613 0.461307 0.887241i \(-0.347381\pi\)
0.461307 + 0.887241i \(0.347381\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 244.150i − 1.86374i −0.362795 0.931869i \(-0.618177\pi\)
0.362795 0.931869i \(-0.381823\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −245.412 −1.79133 −0.895663 0.444733i \(-0.853299\pi\)
−0.895663 + 0.444733i \(0.853299\pi\)
\(138\) 0 0
\(139\) 17.1371i 0.123288i 0.998098 + 0.0616441i \(0.0196344\pi\)
−0.998098 + 0.0616441i \(0.980366\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 180.559i 1.26265i
\(144\) 0 0
\(145\) − 217.040i − 1.49683i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −134.242 −0.900952 −0.450476 0.892789i \(-0.648746\pi\)
−0.450476 + 0.892789i \(0.648746\pi\)
\(150\) 0 0
\(151\) 199.009 1.31794 0.658971 0.752168i \(-0.270992\pi\)
0.658971 + 0.752168i \(0.270992\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −217.930 −1.40600
\(156\) 0 0
\(157\) − 77.0189i − 0.490566i −0.969451 0.245283i \(-0.921119\pi\)
0.969451 0.245283i \(-0.0788810\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −230.379 −1.41337 −0.706684 0.707529i \(-0.749810\pi\)
−0.706684 + 0.707529i \(0.749810\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 229.231i 1.37264i 0.727298 + 0.686321i \(0.240776\pi\)
−0.727298 + 0.686321i \(0.759224\pi\)
\(168\) 0 0
\(169\) −272.665 −1.61340
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 112.473i − 0.650133i −0.945691 0.325066i \(-0.894613\pi\)
0.945691 0.325066i \(-0.105387\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −105.494 −0.589352 −0.294676 0.955597i \(-0.595212\pi\)
−0.294676 + 0.955597i \(0.595212\pi\)
\(180\) 0 0
\(181\) − 15.2683i − 0.0843553i −0.999110 0.0421776i \(-0.986570\pi\)
0.999110 0.0421776i \(-0.0134295\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 358.483i 1.93775i
\(186\) 0 0
\(187\) − 47.1353i − 0.252061i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 306.186 1.60307 0.801534 0.597950i \(-0.204018\pi\)
0.801534 + 0.597950i \(0.204018\pi\)
\(192\) 0 0
\(193\) −1.84100 −0.00953885 −0.00476943 0.999989i \(-0.501518\pi\)
−0.00476943 + 0.999989i \(0.501518\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 255.334 1.29611 0.648056 0.761593i \(-0.275582\pi\)
0.648056 + 0.761593i \(0.275582\pi\)
\(198\) 0 0
\(199\) 36.6483i 0.184162i 0.995752 + 0.0920812i \(0.0293519\pi\)
−0.995752 + 0.0920812i \(0.970648\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −180.948 −0.882673
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 62.2116i 0.297663i
\(210\) 0 0
\(211\) −126.571 −0.599862 −0.299931 0.953961i \(-0.596964\pi\)
−0.299931 + 0.953961i \(0.596964\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.01871i 0.0233428i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 115.297 0.521708
\(222\) 0 0
\(223\) 212.193i 0.951536i 0.879571 + 0.475768i \(0.157830\pi\)
−0.879571 + 0.475768i \(0.842170\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 105.489i − 0.464708i −0.972631 0.232354i \(-0.925357\pi\)
0.972631 0.232354i \(-0.0746428\pi\)
\(228\) 0 0
\(229\) − 6.99086i − 0.0305278i −0.999884 0.0152639i \(-0.995141\pi\)
0.999884 0.0152639i \(-0.00485883\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −336.046 −1.44226 −0.721128 0.692801i \(-0.756376\pi\)
−0.721128 + 0.692801i \(0.756376\pi\)
\(234\) 0 0
\(235\) 460.790 1.96081
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 232.382 0.972309 0.486155 0.873873i \(-0.338399\pi\)
0.486155 + 0.873873i \(0.338399\pi\)
\(240\) 0 0
\(241\) 63.7346i 0.264459i 0.991219 + 0.132229i \(0.0422136\pi\)
−0.991219 + 0.132229i \(0.957786\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −152.175 −0.616094
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 415.450i 1.65518i 0.561334 + 0.827589i \(0.310288\pi\)
−0.561334 + 0.827589i \(0.689712\pi\)
\(252\) 0 0
\(253\) −241.043 −0.952737
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 265.598i 1.03345i 0.856150 + 0.516727i \(0.172850\pi\)
−0.856150 + 0.516727i \(0.827150\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −156.373 −0.594575 −0.297288 0.954788i \(-0.596082\pi\)
−0.297288 + 0.954788i \(0.596082\pi\)
\(264\) 0 0
\(265\) 239.907i 0.905310i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 332.861i 1.23740i 0.785626 + 0.618701i \(0.212341\pi\)
−0.785626 + 0.618701i \(0.787659\pi\)
\(270\) 0 0
\(271\) 209.089i 0.771546i 0.922594 + 0.385773i \(0.126065\pi\)
−0.922594 + 0.385773i \(0.873935\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −33.8555 −0.123111
\(276\) 0 0
\(277\) −263.231 −0.950293 −0.475146 0.879907i \(-0.657605\pi\)
−0.475146 + 0.879907i \(0.657605\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −391.519 −1.39331 −0.696653 0.717408i \(-0.745328\pi\)
−0.696653 + 0.717408i \(0.745328\pi\)
\(282\) 0 0
\(283\) − 109.394i − 0.386550i −0.981145 0.193275i \(-0.938089\pi\)
0.981145 0.193275i \(-0.0619109\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 258.901 0.895853
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 35.3685i 0.120712i 0.998177 + 0.0603558i \(0.0192235\pi\)
−0.998177 + 0.0603558i \(0.980776\pi\)
\(294\) 0 0
\(295\) 342.656 1.16155
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 589.613i − 1.97195i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 172.194 0.564569
\(306\) 0 0
\(307\) − 125.621i − 0.409189i −0.978847 0.204594i \(-0.934412\pi\)
0.978847 0.204594i \(-0.0655875\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 315.735i − 1.01523i −0.861585 0.507613i \(-0.830528\pi\)
0.861585 0.507613i \(-0.169472\pi\)
\(312\) 0 0
\(313\) 51.6046i 0.164871i 0.996596 + 0.0824355i \(0.0262698\pi\)
−0.996596 + 0.0824355i \(0.973730\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.99391 0.0157537 0.00787683 0.999969i \(-0.497493\pi\)
0.00787683 + 0.999969i \(0.497493\pi\)
\(318\) 0 0
\(319\) −346.625 −1.08660
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 39.7256 0.122990
\(324\) 0 0
\(325\) − 82.8137i − 0.254811i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 454.780 1.37396 0.686980 0.726677i \(-0.258936\pi\)
0.686980 + 0.726677i \(0.258936\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 256.792i − 0.766545i
\(336\) 0 0
\(337\) −183.824 −0.545471 −0.272736 0.962089i \(-0.587928\pi\)
−0.272736 + 0.962089i \(0.587928\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 348.046i 1.02066i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −144.776 −0.417223 −0.208611 0.977999i \(-0.566894\pi\)
−0.208611 + 0.977999i \(0.566894\pi\)
\(348\) 0 0
\(349\) − 187.069i − 0.536015i −0.963417 0.268007i \(-0.913635\pi\)
0.963417 0.268007i \(-0.0863651\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 222.601i − 0.630597i −0.948993 0.315299i \(-0.897895\pi\)
0.948993 0.315299i \(-0.102105\pi\)
\(354\) 0 0
\(355\) − 80.5201i − 0.226817i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −62.9856 −0.175447 −0.0877237 0.996145i \(-0.527959\pi\)
−0.0877237 + 0.996145i \(0.527959\pi\)
\(360\) 0 0
\(361\) 308.568 0.854759
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −754.750 −2.06781
\(366\) 0 0
\(367\) 547.100i 1.49074i 0.666653 + 0.745368i \(0.267726\pi\)
−0.666653 + 0.745368i \(0.732274\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −32.4417 −0.0869750 −0.0434875 0.999054i \(-0.513847\pi\)
−0.0434875 + 0.999054i \(0.513847\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 847.878i − 2.24901i
\(378\) 0 0
\(379\) 508.859 1.34263 0.671317 0.741170i \(-0.265729\pi\)
0.671317 + 0.741170i \(0.265729\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 26.2839i 0.0686263i 0.999411 + 0.0343131i \(0.0109244\pi\)
−0.999411 + 0.0343131i \(0.989076\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 360.128 0.925779 0.462890 0.886416i \(-0.346813\pi\)
0.462890 + 0.886416i \(0.346813\pi\)
\(390\) 0 0
\(391\) 153.919i 0.393656i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 657.970i − 1.66575i
\(396\) 0 0
\(397\) − 570.992i − 1.43827i −0.694872 0.719134i \(-0.744539\pi\)
0.694872 0.719134i \(-0.255461\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −68.4014 −0.170577 −0.0852885 0.996356i \(-0.527181\pi\)
−0.0852885 + 0.996356i \(0.527181\pi\)
\(402\) 0 0
\(403\) −851.353 −2.11254
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 572.518 1.40668
\(408\) 0 0
\(409\) 15.3649i 0.0375671i 0.999824 + 0.0187835i \(0.00597934\pi\)
−0.999824 + 0.0187835i \(0.994021\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −178.524 −0.430179
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 366.079i − 0.873696i −0.899535 0.436848i \(-0.856095\pi\)
0.899535 0.436848i \(-0.143905\pi\)
\(420\) 0 0
\(421\) 607.135 1.44213 0.721063 0.692870i \(-0.243654\pi\)
0.721063 + 0.692870i \(0.243654\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 21.6186i 0.0508674i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 586.353 1.36045 0.680224 0.733004i \(-0.261882\pi\)
0.680224 + 0.733004i \(0.261882\pi\)
\(432\) 0 0
\(433\) − 518.769i − 1.19808i −0.800719 0.599040i \(-0.795549\pi\)
0.800719 0.599040i \(-0.204451\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 203.150i − 0.464875i
\(438\) 0 0
\(439\) 221.524i 0.504610i 0.967648 + 0.252305i \(0.0811886\pi\)
−0.967648 + 0.252305i \(0.918811\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 226.415 0.511095 0.255548 0.966796i \(-0.417744\pi\)
0.255548 + 0.966796i \(0.417744\pi\)
\(444\) 0 0
\(445\) −194.337 −0.436713
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 378.422 0.842810 0.421405 0.906873i \(-0.361537\pi\)
0.421405 + 0.906873i \(0.361537\pi\)
\(450\) 0 0
\(451\) 288.984i 0.640763i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 438.153 0.958760 0.479380 0.877607i \(-0.340862\pi\)
0.479380 + 0.877607i \(0.340862\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 46.3981i 0.100647i 0.998733 + 0.0503234i \(0.0160252\pi\)
−0.998733 + 0.0503234i \(0.983975\pi\)
\(462\) 0 0
\(463\) 367.455 0.793639 0.396820 0.917897i \(-0.370114\pi\)
0.396820 + 0.917897i \(0.370114\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 529.465i 1.13376i 0.823801 + 0.566879i \(0.191849\pi\)
−0.823801 + 0.566879i \(0.808151\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.01516 0.0169454
\(474\) 0 0
\(475\) − 28.5334i − 0.0600702i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 289.761i − 0.604928i −0.953161 0.302464i \(-0.902191\pi\)
0.953161 0.302464i \(-0.0978093\pi\)
\(480\) 0 0
\(481\) 1400.43i 2.91150i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 87.2445 0.179886
\(486\) 0 0
\(487\) 439.636 0.902744 0.451372 0.892336i \(-0.350935\pi\)
0.451372 + 0.892336i \(0.350935\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 320.561 0.652874 0.326437 0.945219i \(-0.394152\pi\)
0.326437 + 0.945219i \(0.394152\pi\)
\(492\) 0 0
\(493\) 221.340i 0.448965i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 771.973 1.54704 0.773520 0.633772i \(-0.218494\pi\)
0.773520 + 0.633772i \(0.218494\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 101.632i 0.202052i 0.994884 + 0.101026i \(0.0322126\pi\)
−0.994884 + 0.101026i \(0.967787\pi\)
\(504\) 0 0
\(505\) −642.675 −1.27262
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 605.407i 1.18941i 0.803946 + 0.594703i \(0.202730\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −48.0503 −0.0933016
\(516\) 0 0
\(517\) − 735.907i − 1.42342i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 383.293i − 0.735688i −0.929888 0.367844i \(-0.880096\pi\)
0.929888 0.367844i \(-0.119904\pi\)
\(522\) 0 0
\(523\) − 448.095i − 0.856777i −0.903595 0.428389i \(-0.859081\pi\)
0.903595 0.428389i \(-0.140919\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 222.247 0.421721
\(528\) 0 0
\(529\) 258.119 0.487937
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −706.882 −1.32623
\(534\) 0 0
\(535\) − 935.282i − 1.74819i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −271.258 −0.501400 −0.250700 0.968065i \(-0.580661\pi\)
−0.250700 + 0.968065i \(0.580661\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 862.435i − 1.58245i
\(546\) 0 0
\(547\) −590.544 −1.07961 −0.539803 0.841791i \(-0.681501\pi\)
−0.539803 + 0.841791i \(0.681501\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 292.135i − 0.530191i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −188.295 −0.338051 −0.169026 0.985612i \(-0.554062\pi\)
−0.169026 + 0.985612i \(0.554062\pi\)
\(558\) 0 0
\(559\) 19.6058i 0.0350730i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 204.157i 0.362623i 0.983426 + 0.181312i \(0.0580342\pi\)
−0.983426 + 0.181312i \(0.941966\pi\)
\(564\) 0 0
\(565\) 438.129i 0.775449i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 660.913 1.16153 0.580767 0.814070i \(-0.302753\pi\)
0.580767 + 0.814070i \(0.302753\pi\)
\(570\) 0 0
\(571\) −533.978 −0.935162 −0.467581 0.883950i \(-0.654874\pi\)
−0.467581 + 0.883950i \(0.654874\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 110.554 0.192268
\(576\) 0 0
\(577\) 53.4337i 0.0926062i 0.998927 + 0.0463031i \(0.0147440\pi\)
−0.998927 + 0.0463031i \(0.985256\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 383.145 0.657196
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 413.063i 0.703684i 0.936059 + 0.351842i \(0.114445\pi\)
−0.936059 + 0.351842i \(0.885555\pi\)
\(588\) 0 0
\(589\) −293.333 −0.498018
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 122.452i − 0.206496i −0.994656 0.103248i \(-0.967076\pi\)
0.994656 0.103248i \(-0.0329235\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 645.136 1.07702 0.538511 0.842619i \(-0.318987\pi\)
0.538511 + 0.842619i \(0.318987\pi\)
\(600\) 0 0
\(601\) 683.488i 1.13725i 0.822596 + 0.568626i \(0.192525\pi\)
−0.822596 + 0.568626i \(0.807475\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 253.836i − 0.419564i
\(606\) 0 0
\(607\) − 257.253i − 0.423811i −0.977290 0.211905i \(-0.932033\pi\)
0.977290 0.211905i \(-0.0679669\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1800.10 2.94615
\(612\) 0 0
\(613\) 885.713 1.44488 0.722441 0.691433i \(-0.243020\pi\)
0.722441 + 0.691433i \(0.243020\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −46.0724 −0.0746717 −0.0373358 0.999303i \(-0.511887\pi\)
−0.0373358 + 0.999303i \(0.511887\pi\)
\(618\) 0 0
\(619\) − 1096.33i − 1.77112i −0.464522 0.885562i \(-0.653774\pi\)
0.464522 0.885562i \(-0.346226\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −707.986 −1.13278
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 365.585i − 0.581217i
\(630\) 0 0
\(631\) 606.319 0.960886 0.480443 0.877026i \(-0.340476\pi\)
0.480443 + 0.877026i \(0.340476\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 630.343i 0.992666i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −939.184 −1.46519 −0.732593 0.680667i \(-0.761690\pi\)
−0.732593 + 0.680667i \(0.761690\pi\)
\(642\) 0 0
\(643\) 992.960i 1.54426i 0.635464 + 0.772131i \(0.280809\pi\)
−0.635464 + 0.772131i \(0.719191\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 265.184i 0.409867i 0.978776 + 0.204933i \(0.0656978\pi\)
−0.978776 + 0.204933i \(0.934302\pi\)
\(648\) 0 0
\(649\) − 547.241i − 0.843206i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1134.14 1.73682 0.868409 0.495849i \(-0.165143\pi\)
0.868409 + 0.495849i \(0.165143\pi\)
\(654\) 0 0
\(655\) 1313.44 2.00525
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 924.147 1.40235 0.701174 0.712990i \(-0.252660\pi\)
0.701174 + 0.712990i \(0.252660\pi\)
\(660\) 0 0
\(661\) − 234.367i − 0.354564i −0.984160 0.177282i \(-0.943269\pi\)
0.984160 0.177282i \(-0.0567305\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1131.90 1.69700
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 275.003i − 0.409840i
\(672\) 0 0
\(673\) −465.127 −0.691125 −0.345563 0.938396i \(-0.612312\pi\)
−0.345563 + 0.938396i \(0.612312\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1295.93i 1.91422i 0.289724 + 0.957110i \(0.406437\pi\)
−0.289724 + 0.957110i \(0.593563\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −16.2612 −0.0238085 −0.0119042 0.999929i \(-0.503789\pi\)
−0.0119042 + 0.999929i \(0.503789\pi\)
\(684\) 0 0
\(685\) − 1320.23i − 1.92734i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 937.209i 1.36025i
\(690\) 0 0
\(691\) 68.5364i 0.0991843i 0.998770 + 0.0495922i \(0.0157922\pi\)
−0.998770 + 0.0495922i \(0.984208\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −92.1912 −0.132649
\(696\) 0 0
\(697\) 184.533 0.264753
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −923.360 −1.31720 −0.658602 0.752491i \(-0.728852\pi\)
−0.658602 + 0.752491i \(0.728852\pi\)
\(702\) 0 0
\(703\) 482.518i 0.686369i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −912.863 −1.28754 −0.643768 0.765221i \(-0.722630\pi\)
−0.643768 + 0.765221i \(0.722630\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 1136.54i − 1.59402i
\(714\) 0 0
\(715\) −971.345 −1.35852
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 545.020i 0.758025i 0.925391 + 0.379013i \(0.123736\pi\)
−0.925391 + 0.379013i \(0.876264\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 158.980 0.219283
\(726\) 0 0
\(727\) − 750.292i − 1.03204i −0.856577 0.516019i \(-0.827413\pi\)
0.856577 0.516019i \(-0.172587\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 5.11813i − 0.00700155i
\(732\) 0 0
\(733\) 926.599i 1.26412i 0.774920 + 0.632059i \(0.217790\pi\)
−0.774920 + 0.632059i \(0.782210\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −410.112 −0.556461
\(738\) 0 0
\(739\) 488.147 0.660551 0.330275 0.943885i \(-0.392858\pi\)
0.330275 + 0.943885i \(0.392858\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1091.72 1.46935 0.734674 0.678421i \(-0.237335\pi\)
0.734674 + 0.678421i \(0.237335\pi\)
\(744\) 0 0
\(745\) − 722.173i − 0.969360i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −480.443 −0.639738 −0.319869 0.947462i \(-0.603639\pi\)
−0.319869 + 0.947462i \(0.603639\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1070.60i 1.41801i
\(756\) 0 0
\(757\) 941.400 1.24359 0.621796 0.783179i \(-0.286403\pi\)
0.621796 + 0.783179i \(0.286403\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 1192.34i − 1.56681i −0.621509 0.783407i \(-0.713480\pi\)
0.621509 0.783407i \(-0.286520\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1338.60 1.74524
\(768\) 0 0
\(769\) 908.294i 1.18114i 0.806988 + 0.590568i \(0.201096\pi\)
−0.806988 + 0.590568i \(0.798904\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1189.38i 1.53866i 0.638854 + 0.769328i \(0.279409\pi\)
−0.638854 + 0.769328i \(0.720591\pi\)
\(774\) 0 0
\(775\) − 159.632i − 0.205976i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −243.556 −0.312652
\(780\) 0 0
\(781\) −128.595 −0.164654
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 414.334 0.527814
\(786\) 0 0
\(787\) − 1057.12i − 1.34323i −0.740902 0.671613i \(-0.765602\pi\)
0.740902 0.671613i \(-0.234398\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 672.682 0.848275
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1037.94i 1.30231i 0.758945 + 0.651154i \(0.225715\pi\)
−0.758945 + 0.651154i \(0.774285\pi\)
\(798\) 0 0
\(799\) −469.919 −0.588133
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1205.38i 1.50109i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1087.36 −1.34408 −0.672041 0.740514i \(-0.734582\pi\)
−0.672041 + 0.740514i \(0.734582\pi\)
\(810\) 0 0
\(811\) − 926.995i − 1.14303i −0.820593 0.571514i \(-0.806356\pi\)
0.820593 0.571514i \(-0.193644\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1239.36i − 1.52068i
\(816\) 0 0
\(817\) 6.75517i 0.00826826i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 430.300 0.524117 0.262058 0.965052i \(-0.415599\pi\)
0.262058 + 0.965052i \(0.415599\pi\)
\(822\) 0 0
\(823\) −1044.74 −1.26942 −0.634712 0.772749i \(-0.718881\pi\)
−0.634712 + 0.772749i \(0.718881\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 610.313 0.737984 0.368992 0.929433i \(-0.379703\pi\)
0.368992 + 0.929433i \(0.379703\pi\)
\(828\) 0 0
\(829\) 1413.90i 1.70555i 0.522279 + 0.852775i \(0.325082\pi\)
−0.522279 + 0.852775i \(0.674918\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1233.18 −1.47687
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 46.3303i 0.0552209i 0.999619 + 0.0276105i \(0.00878980\pi\)
−0.999619 + 0.0276105i \(0.991210\pi\)
\(840\) 0 0
\(841\) 786.696 0.935430
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 1466.84i − 1.73591i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1869.54 −2.19688
\(852\) 0 0
\(853\) − 773.111i − 0.906343i −0.891423 0.453172i \(-0.850292\pi\)
0.891423 0.453172i \(-0.149708\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 438.159i 0.511271i 0.966773 + 0.255636i \(0.0822847\pi\)
−0.966773 + 0.255636i \(0.917715\pi\)
\(858\) 0 0
\(859\) − 72.3419i − 0.0842164i −0.999113 0.0421082i \(-0.986593\pi\)
0.999113 0.0421082i \(-0.0134074\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −426.495 −0.494200 −0.247100 0.968990i \(-0.579478\pi\)
−0.247100 + 0.968990i \(0.579478\pi\)
\(864\) 0 0
\(865\) 605.064 0.699496
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1050.82 −1.20922
\(870\) 0 0
\(871\) − 1003.17i − 1.15175i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 416.485 0.474898 0.237449 0.971400i \(-0.423689\pi\)
0.237449 + 0.971400i \(0.423689\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 1249.01i − 1.41771i −0.705353 0.708857i \(-0.749211\pi\)
0.705353 0.708857i \(-0.250789\pi\)
\(882\) 0 0
\(883\) 81.5906 0.0924016 0.0462008 0.998932i \(-0.485289\pi\)
0.0462008 + 0.998932i \(0.485289\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 917.730i − 1.03464i −0.855791 0.517322i \(-0.826929\pi\)
0.855791 0.517322i \(-0.173071\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 620.222 0.694537
\(894\) 0 0
\(895\) − 567.520i − 0.634100i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 1634.37i − 1.81798i
\(900\) 0 0
\(901\) − 244.660i − 0.271543i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 82.1380 0.0907602
\(906\) 0 0
\(907\) −1451.40 −1.60022 −0.800110 0.599853i \(-0.795226\pi\)
−0.800110 + 0.599853i \(0.795226\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 562.064 0.616975 0.308487 0.951228i \(-0.400177\pi\)
0.308487 + 0.951228i \(0.400177\pi\)
\(912\) 0 0
\(913\) 285.114i 0.312282i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 538.158 0.585591 0.292795 0.956175i \(-0.405414\pi\)
0.292795 + 0.956175i \(0.405414\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 314.556i − 0.340797i
\(924\) 0 0
\(925\) −262.586 −0.283876
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 827.080i 0.890291i 0.895458 + 0.445145i \(0.146848\pi\)
−0.895458 + 0.445145i \(0.853152\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 253.571 0.271199
\(936\) 0 0
\(937\) 1501.10i 1.60203i 0.598644 + 0.801016i \(0.295707\pi\)
−0.598644 + 0.801016i \(0.704293\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 766.672i − 0.814742i −0.913263 0.407371i \(-0.866446\pi\)
0.913263 0.407371i \(-0.133554\pi\)
\(942\) 0 0
\(943\) − 943.671i − 1.00071i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1362.23 −1.43847 −0.719237 0.694765i \(-0.755508\pi\)
−0.719237 + 0.694765i \(0.755508\pi\)
\(948\) 0 0
\(949\) −2948.47 −3.10692
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1431.37 −1.50196 −0.750980 0.660325i \(-0.770419\pi\)
−0.750980 + 0.660325i \(0.770419\pi\)
\(954\) 0 0
\(955\) 1647.17i 1.72479i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −680.067 −0.707666
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 9.90391i − 0.0102631i
\(966\) 0 0
\(967\) −426.276 −0.440823 −0.220411 0.975407i \(-0.570740\pi\)
−0.220411 + 0.975407i \(0.570740\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1728.41i − 1.78003i −0.455934 0.890013i \(-0.650695\pi\)
0.455934 0.890013i \(-0.349305\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 712.320 0.729089 0.364545 0.931186i \(-0.381225\pi\)
0.364545 + 0.931186i \(0.381225\pi\)
\(978\) 0 0
\(979\) 310.368i 0.317025i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1042.56i 1.06059i 0.847812 + 0.530297i \(0.177920\pi\)
−0.847812 + 0.530297i \(0.822080\pi\)
\(984\) 0 0
\(985\) 1373.61i 1.39452i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −26.1733 −0.0264644
\(990\) 0 0
\(991\) 1397.41 1.41010 0.705052 0.709156i \(-0.250924\pi\)
0.705052 + 0.709156i \(0.250924\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −197.155 −0.198146
\(996\) 0 0
\(997\) 717.962i 0.720122i 0.932929 + 0.360061i \(0.117244\pi\)
−0.932929 + 0.360061i \(0.882756\pi\)
\(998\) 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 1764.3.d.h.685.7 8
3.2 odd 2 588.3.d.c.97.5 yes 8
7.2 even 3 1764.3.z.m.325.4 8
7.3 odd 6 1764.3.z.m.901.4 8
7.4 even 3 1764.3.z.l.901.1 8
7.5 odd 6 1764.3.z.l.325.1 8
7.6 odd 2 inner 1764.3.d.h.685.2 8
12.11 even 2 2352.3.f.j.97.1 8
21.2 odd 6 588.3.m.e.325.1 8
21.5 even 6 588.3.m.f.325.4 8
21.11 odd 6 588.3.m.f.313.4 8
21.17 even 6 588.3.m.e.313.1 8
21.20 even 2 588.3.d.c.97.4 8
84.83 odd 2 2352.3.f.j.97.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.3.d.c.97.4 8 21.20 even 2
588.3.d.c.97.5 yes 8 3.2 odd 2
588.3.m.e.313.1 8 21.17 even 6
588.3.m.e.325.1 8 21.2 odd 6
588.3.m.f.313.4 8 21.11 odd 6
588.3.m.f.325.4 8 21.5 even 6
1764.3.d.h.685.2 8 7.6 odd 2 inner
1764.3.d.h.685.7 8 1.1 even 1 trivial
1764.3.z.l.325.1 8 7.5 odd 6
1764.3.z.l.901.1 8 7.4 even 3
1764.3.z.m.325.4 8 7.2 even 3
1764.3.z.m.901.4 8 7.3 odd 6
2352.3.f.j.97.1 8 12.11 even 2
2352.3.f.j.97.8 8 84.83 odd 2