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Properties

Label 7728.2.a.s.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7728,2,Mod(1,7728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7728.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

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: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7728.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{13} +1.00000 q^{15} +4.00000 q^{17} -4.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} -1.00000 q^{29} -10.0000 q^{31} -1.00000 q^{35} -5.00000 q^{37} -1.00000 q^{39} +7.00000 q^{41} +3.00000 q^{43} +1.00000 q^{45} -3.00000 q^{47} +1.00000 q^{49} +4.00000 q^{51} -4.00000 q^{57} -6.00000 q^{59} -6.00000 q^{61} -1.00000 q^{63} -1.00000 q^{65} -4.00000 q^{67} -1.00000 q^{69} +2.00000 q^{71} -4.00000 q^{73} -4.00000 q^{75} -12.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +4.00000 q^{85} -1.00000 q^{87} -6.00000 q^{89} +1.00000 q^{91} -10.0000 q^{93} -4.00000 q^{95} -1.00000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 − 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 − 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) 3.00000 0.457496 0.228748 − 0.973486i \(-0.426537\pi\)
0.228748 + 0.973486i \(0.426537\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 − 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 − 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 17.0000 1.67506 0.837530 − 0.546392i \(-0.183999\pi\)
0.837530 + 0.546392i \(0.183999\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −17.0000 −1.62830 −0.814152 − 0.580651i \(-0.802798\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) 0 0
\(111\) −5.00000 −0.474579
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 − 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 7.00000 0.631169
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 13.0000 1.15356 0.576782 − 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) 0 0
\(129\) 3.00000 0.264135
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 − 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 − 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 1.00000 0.0848189 0.0424094 − 0.999100i \(-0.486497\pi\)
0.0424094 + 0.999100i \(0.486497\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −4.00000 −0.327693 −0.163846 − 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 − 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) 12.0000 0.957704 0.478852 − 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 0 0
\(179\) −3.00000 −0.224231 −0.112115 − 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 − 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) −5.00000 −0.367607
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 − 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) −17.0000 −1.22369 −0.611843 − 0.790979i \(-0.709572\pi\)
−0.611843 + 0.790979i \(0.709572\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) 5.00000 0.356235 0.178118 − 0.984009i \(-0.442999\pi\)
0.178118 + 0.984009i \(0.442999\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 − 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 1.00000 0.0701862
\(204\) 0 0
\(205\) 7.00000 0.488901
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 0 0
\(215\) 3.00000 0.204598
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 − 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) −21.0000 −1.39382 −0.696909 − 0.717159i \(-0.745442\pi\)
−0.696909 + 0.717159i \(0.745442\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 − 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 0 0
\(237\) −12.0000 −0.779484
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −3.00000 −0.193247 −0.0966235 − 0.995321i \(-0.530804\pi\)
−0.0966235 + 0.995321i \(0.530804\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 − 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 0 0
\(257\) −22.0000 −1.37232 −0.686161 − 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) 5.00000 0.310685
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) −25.0000 −1.54157 −0.770783 − 0.637098i \(-0.780135\pi\)
−0.770783 + 0.637098i \(0.780135\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 − 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −32.0000 −1.92269 −0.961347 − 0.275340i \(-0.911209\pi\)
−0.961347 + 0.275340i \(0.911209\pi\)
\(278\) 0 0
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 23.0000 1.37206 0.686032 − 0.727571i \(-0.259351\pi\)
0.686032 + 0.727571i \(0.259351\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 − 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −7.00000 −0.413197
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) 0 0
\(293\) −14.0000 −0.817889 −0.408944 − 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) 23.0000 1.31268 0.656340 − 0.754466i \(-0.272104\pi\)
0.656340 + 0.754466i \(0.272104\pi\)
\(308\) 0 0
\(309\) 17.0000 0.967096
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 − 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) 17.0000 0.954815 0.477408 − 0.878682i \(-0.341577\pi\)
0.477408 + 0.878682i \(0.341577\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) −17.0000 −0.940102
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 − 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) −5.00000 −0.273998
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 24.0000 1.30736 0.653682 − 0.756770i \(-0.273224\pi\)
0.653682 + 0.756770i \(0.273224\pi\)
\(338\) 0 0
\(339\) 1.00000 0.0543125
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) 33.0000 1.77153 0.885766 − 0.464131i \(-0.153633\pi\)
0.885766 + 0.464131i \(0.153633\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 − 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 13.0000 0.691920 0.345960 − 0.938249i \(-0.387553\pi\)
0.345960 + 0.938249i \(0.387553\pi\)
\(354\) 0 0
\(355\) 2.00000 0.106149
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) −3.00000 −0.158334 −0.0791670 − 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −13.0000 −0.678594 −0.339297 − 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) 0 0
\(369\) 7.00000 0.364405
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) −23.0000 −1.18143 −0.590715 − 0.806880i \(-0.701154\pi\)
−0.590715 + 0.806880i \(0.701154\pi\)
\(380\) 0 0
\(381\) 13.0000 0.666010
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 − 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.00000 0.152499
\(388\) 0 0
\(389\) 2.00000 0.101404 0.0507020 − 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 14.0000 0.706207
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −2.00000 −0.0998752 −0.0499376 − 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) 10.0000 0.498135
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 28.0000 1.38451 0.692255 − 0.721653i \(-0.256617\pi\)
0.692255 + 0.721653i \(0.256617\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 1.00000 0.0489702
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −9.00000 −0.438633 −0.219317 − 0.975654i \(-0.570383\pi\)
−0.219317 + 0.975654i \(0.570383\pi\)
\(422\) 0 0
\(423\) −3.00000 −0.145865
\(424\) 0 0
\(425\) −16.0000 −0.776114
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 − 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) 0 0
\(433\) −7.00000 −0.336399 −0.168199 − 0.985753i \(-0.553795\pi\)
−0.168199 + 0.985753i \(0.553795\pi\)
\(434\) 0 0
\(435\) −1.00000 −0.0479463
\(436\) 0 0
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 34.0000 1.62273 0.811366 − 0.584539i \(-0.198725\pi\)
0.811366 + 0.584539i \(0.198725\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −1.00000 −0.0475114 −0.0237557 − 0.999718i \(-0.507562\pi\)
−0.0237557 + 0.999718i \(0.507562\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) −4.00000 −0.189194
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 − 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 5.00000 0.234920
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 − 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) −25.0000 −1.16185 −0.580924 − 0.813958i \(-0.697309\pi\)
−0.580924 + 0.813958i \(0.697309\pi\)
\(464\) 0 0
\(465\) −10.0000 −0.463739
\(466\) 0 0
\(467\) 29.0000 1.34196 0.670980 − 0.741475i \(-0.265874\pi\)
0.670980 + 0.741475i \(0.265874\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 12.0000 0.552931
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 16.0000 0.734130
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −10.0000 −0.456912 −0.228456 − 0.973554i \(-0.573368\pi\)
−0.228456 + 0.973554i \(0.573368\pi\)
\(480\) 0 0
\(481\) 5.00000 0.227980
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −1.00000 −0.0454077
\(486\) 0 0
\(487\) 15.0000 0.679715 0.339857 − 0.940477i \(-0.389621\pi\)
0.339857 + 0.940477i \(0.389621\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.00000 −0.0897123
\(498\) 0 0
\(499\) −18.0000 −0.805791 −0.402895 − 0.915246i \(-0.631996\pi\)
−0.402895 + 0.915246i \(0.631996\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) −2.00000 −0.0891756 −0.0445878 − 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) −16.0000 −0.709188 −0.354594 − 0.935020i \(-0.615381\pi\)
−0.354594 + 0.935020i \(0.615381\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 17.0000 0.749110
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −6.00000 −0.262362 −0.131181 − 0.991358i \(-0.541877\pi\)
−0.131181 + 0.991358i \(0.541877\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) −40.0000 −1.74243
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) −7.00000 −0.303204
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.00000 −0.129460
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 4.00000 0.171973 0.0859867 − 0.996296i \(-0.472596\pi\)
0.0859867 + 0.996296i \(0.472596\pi\)
\(542\) 0 0
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) −17.0000 −0.728200
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 − 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 0 0
\(555\) −5.00000 −0.212238
\(556\) 0 0
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −3.00000 −0.126886
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 39.0000 1.64365 0.821827 − 0.569737i \(-0.192955\pi\)
0.821827 + 0.569737i \(0.192955\pi\)
\(564\) 0 0
\(565\) 1.00000 0.0420703
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 27.0000 1.13190 0.565949 − 0.824440i \(-0.308510\pi\)
0.565949 + 0.824440i \(0.308510\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 20.0000 0.835512
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 6.00000 0.249783 0.124892 − 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) 0 0
\(579\) −17.0000 −0.706496
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 40.0000 1.64817
\(590\) 0 0
\(591\) 5.00000 0.205673
\(592\) 0 0
\(593\) −21.0000 −0.862367 −0.431183 − 0.902264i \(-0.641904\pi\)
−0.431183 + 0.902264i \(0.641904\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 0 0
\(597\) 7.00000 0.286491
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 − 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −4.00000 −0.163163 −0.0815817 − 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) −11.0000 −0.447214
\(606\) 0 0
\(607\) −18.0000 −0.730597 −0.365299 − 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 0 0
\(609\) 1.00000 0.0405220
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) 0 0
\(613\) −27.0000 −1.09052 −0.545260 − 0.838267i \(-0.683569\pi\)
−0.545260 + 0.838267i \(0.683569\pi\)
\(614\) 0 0
\(615\) 7.00000 0.282267
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 − 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 − 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 13.0000 0.515889
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 − 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 − 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) 3.00000 0.118125
\(646\) 0 0
\(647\) 16.0000 0.629025 0.314512 − 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 0 0
\(653\) 11.0000 0.430463 0.215232 − 0.976563i \(-0.430949\pi\)
0.215232 + 0.976563i \(0.430949\pi\)
\(654\) 0 0
\(655\) 14.0000 0.547025
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 − 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) 2.00000 0.0773245
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 43.0000 1.65753 0.828764 − 0.559598i \(-0.189045\pi\)
0.828764 + 0.559598i \(0.189045\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −14.0000 −0.538064 −0.269032 − 0.963131i \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) 0 0
\(679\) 1.00000 0.0383765
\(680\) 0 0
\(681\) −21.0000 −0.804722
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 9.00000 0.343872
\(686\) 0 0
\(687\) 4.00000 0.152610
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −29.0000 −1.10321 −0.551606 − 0.834105i \(-0.685985\pi\)
−0.551606 + 0.834105i \(0.685985\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.00000 0.0379322
\(696\) 0 0
\(697\) 28.0000 1.06058
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −52.0000 −1.96401 −0.982006 − 0.188847i \(-0.939525\pi\)
−0.982006 + 0.188847i \(0.939525\pi\)
\(702\) 0 0
\(703\) 20.0000 0.754314
\(704\) 0 0
\(705\) −3.00000 −0.112987
\(706\) 0 0
\(707\) −2.00000 −0.0752177
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 − 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 0 0
\(713\) 10.0000 0.374503
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) −51.0000 −1.90198 −0.950990 − 0.309223i \(-0.899931\pi\)
−0.950990 + 0.309223i \(0.899931\pi\)
\(720\) 0 0
\(721\) −17.0000 −0.633113
\(722\) 0 0
\(723\) −3.00000 −0.111571
\(724\) 0 0
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 − 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 0 0
\(733\) 20.0000 0.738717 0.369358 − 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.00000 0.0735712 0.0367856 − 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) −4.00000 −0.146549
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 − 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) −21.0000 −0.765283
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 − 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −54.0000 −1.95750 −0.978749 − 0.205061i \(-0.934261\pi\)
−0.978749 + 0.205061i \(0.934261\pi\)
\(762\) 0 0
\(763\) 17.0000 0.615441
\(764\) 0 0
\(765\) 4.00000 0.144620
\(766\) 0 0
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) 51.0000 1.83911 0.919554 − 0.392965i \(-0.128551\pi\)
0.919554 + 0.392965i \(0.128551\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 0 0
\(773\) −23.0000 −0.827253 −0.413626 − 0.910447i \(-0.635738\pi\)
−0.413626 + 0.910447i \(0.635738\pi\)
\(774\) 0 0
\(775\) 40.0000 1.43684
\(776\) 0 0
\(777\) 5.00000 0.179374
\(778\) 0 0
\(779\) −28.0000 −1.00320
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 − 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 0 0
\(789\) −25.0000 −0.890024
\(790\) 0 0
\(791\) −1.00000 −0.0355559
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −27.0000 −0.956389 −0.478195 − 0.878254i \(-0.658709\pi\)
−0.478195 + 0.878254i \(0.658709\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 1.00000 0.0352454
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 0 0
\(809\) 20.0000 0.703163 0.351581 − 0.936157i \(-0.385644\pi\)
0.351581 + 0.936157i \(0.385644\pi\)
\(810\) 0 0
\(811\) 17.0000 0.596951 0.298475 − 0.954417i \(-0.403522\pi\)
0.298475 + 0.954417i \(0.403522\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −46.0000 −1.60541 −0.802706 − 0.596376i \(-0.796607\pi\)
−0.802706 + 0.596376i \(0.796607\pi\)
\(822\) 0 0
\(823\) −19.0000 −0.662298 −0.331149 − 0.943578i \(-0.607436\pi\)
−0.331149 + 0.943578i \(0.607436\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22.0000 0.765015 0.382507 − 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 − 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) −32.0000 −1.11007
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) 0 0
\(839\) 2.00000 0.0690477 0.0345238 − 0.999404i \(-0.489009\pi\)
0.0345238 + 0.999404i \(0.489009\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 23.0000 0.792162
\(844\) 0 0
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 0 0
\(849\) 6.00000 0.205919
\(850\) 0 0
\(851\) 5.00000 0.171398
\(852\) 0 0
\(853\) −27.0000 −0.924462 −0.462231 − 0.886759i \(-0.652951\pi\)
−0.462231 + 0.886759i \(0.652951\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) 7.00000 0.239115 0.119558 − 0.992827i \(-0.461852\pi\)
0.119558 + 0.992827i \(0.461852\pi\)
\(858\) 0 0
\(859\) −43.0000 −1.46714 −0.733571 − 0.679613i \(-0.762148\pi\)
−0.733571 + 0.679613i \(0.762148\pi\)
\(860\) 0 0
\(861\) −7.00000 −0.238559
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 − 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) −1.00000 −0.0338449
\(874\) 0 0
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) −52.0000 −1.75592 −0.877958 − 0.478738i \(-0.841094\pi\)
−0.877958 + 0.478738i \(0.841094\pi\)
\(878\) 0 0
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 46.0000 1.54802 0.774012 − 0.633171i \(-0.218247\pi\)
0.774012 + 0.633171i \(0.218247\pi\)
\(884\) 0 0
\(885\) −6.00000 −0.201688
\(886\) 0 0
\(887\) 48.0000 1.61168 0.805841 − 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) −13.0000 −0.436006
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) −3.00000 −0.100279
\(896\) 0 0
\(897\) 1.00000 0.0333890
\(898\) 0 0
\(899\) 10.0000 0.333519
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −3.00000 −0.0998337
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) −19.0000 −0.630885 −0.315442 − 0.948945i \(-0.602153\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(908\) 0 0
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) 5.00000 0.165657 0.0828287 − 0.996564i \(-0.473605\pi\)
0.0828287 + 0.996564i \(0.473605\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −6.00000 −0.198354
\(916\) 0 0
\(917\) −14.0000 −0.462321
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 23.0000 0.757876
\(922\) 0 0
\(923\) −2.00000 −0.0658308
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) 0 0
\(927\) 17.0000 0.558353
\(928\) 0 0
\(929\) −25.0000 −0.820223 −0.410112 − 0.912035i \(-0.634510\pi\)
−0.410112 + 0.912035i \(0.634510\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 0 0
\(933\) −4.00000 −0.130954
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.00000 −0.0326686 −0.0163343 − 0.999867i \(-0.505200\pi\)
−0.0163343 + 0.999867i \(0.505200\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) 57.0000 1.85815 0.929073 − 0.369895i \(-0.120606\pi\)
0.929073 + 0.369895i \(0.120606\pi\)
\(942\) 0 0
\(943\) −7.00000 −0.227951
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) −33.0000 −1.07236 −0.536178 − 0.844105i \(-0.680132\pi\)
−0.536178 + 0.844105i \(0.680132\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 17.0000 0.551263
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 − 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 0 0
\(955\) 20.0000 0.647185
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −17.0000 −0.547249
\(966\) 0 0
\(967\) 52.0000 1.67221 0.836104 − 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 0 0
\(969\) −16.0000 −0.513994
\(970\) 0 0
\(971\) 56.0000 1.79713 0.898563 − 0.438845i \(-0.144612\pi\)
0.898563 + 0.438845i \(0.144612\pi\)
\(972\) 0 0
\(973\) −1.00000 −0.0320585
\(974\) 0 0
\(975\) 4.00000 0.128103
\(976\) 0 0
\(977\) −27.0000 −0.863807 −0.431903 − 0.901920i \(-0.642158\pi\)
−0.431903 + 0.901920i \(0.642158\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −17.0000 −0.542768
\(982\) 0 0
\(983\) −58.0000 −1.84991 −0.924956 − 0.380073i \(-0.875899\pi\)
−0.924956 + 0.380073i \(0.875899\pi\)
\(984\) 0 0
\(985\) 5.00000 0.159313
\(986\) 0 0
\(987\) 3.00000 0.0954911
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) 12.0000 0.381193 0.190596 − 0.981669i \(-0.438958\pi\)
0.190596 + 0.981669i \(0.438958\pi\)
\(992\) 0 0
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) 7.00000 0.221915
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 − 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 0 0
\(999\) −5.00000 −0.158193
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 7728.2.a.s.1.1 1
4.3 odd 2 966.2.a.h.1.1 ✓ 1
12.11 even 2 2898.2.a.d.1.1 1
28.27 even 2 6762.2.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.h.1.1 ✓ 1 4.3 odd 2
2898.2.a.d.1.1 1 12.11 even 2
6762.2.a.bi.1.1 1 28.27 even 2
7728.2.a.s.1.1 1 1.1 even 1 trivial