Properties

Label 1824.2.a.i.1.1
Level $1824$
Weight $2$
Character 1824.1
Self dual yes
Analytic conductor $14.565$
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] = [1824,2,Mod(1,1824)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1824, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1824.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1824 = 2^{5} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1824.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: \(14.5647133287\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1824.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -1.00000 q^{15} -7.00000 q^{17} -1.00000 q^{19} +1.00000 q^{21} -8.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} +2.00000 q^{31} -3.00000 q^{33} -1.00000 q^{35} +4.00000 q^{37} -4.00000 q^{41} +1.00000 q^{43} -1.00000 q^{45} +3.00000 q^{47} -6.00000 q^{49} -7.00000 q^{51} +6.00000 q^{53} +3.00000 q^{55} -1.00000 q^{57} +6.00000 q^{59} -5.00000 q^{61} +1.00000 q^{63} +2.00000 q^{67} -8.00000 q^{69} +2.00000 q^{71} -11.0000 q^{73} -4.00000 q^{75} -3.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -16.0000 q^{83} +7.00000 q^{85} -14.0000 q^{89} +2.00000 q^{93} +1.00000 q^{95} -8.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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 \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −7.00000 −0.980196
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 7.00000 0.759257
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 14.0000 1.35343 0.676716 0.736245i \(-0.263403\pi\)
0.676716 + 0.736245i \(0.263403\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −4.00000 −0.360668
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 1.00000 0.0854358 0.0427179 0.999087i \(-0.486398\pi\)
0.0427179 + 0.999087i \(0.486398\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) 1.00000 0.0819232 0.0409616 0.999161i \(-0.486958\pi\)
0.0409616 + 0.999161i \(0.486958\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 0 0
\(153\) −7.00000 −0.565916
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 26.0000 1.97674 0.988372 0.152057i \(-0.0485898\pi\)
0.988372 + 0.152057i \(0.0485898\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −5.00000 −0.369611
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 21.0000 1.53567
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −7.00000 −0.506502 −0.253251 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 17.0000 1.20510 0.602549 0.798082i \(-0.294152\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 0 0
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 0 0
\(219\) −11.0000 −0.743311
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) −23.0000 −1.51988 −0.759941 0.649992i \(-0.774772\pi\)
−0.759941 + 0.649992i \(0.774772\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −17.0000 −1.11371 −0.556854 0.830611i \(-0.687992\pi\)
−0.556854 + 0.830611i \(0.687992\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 0 0
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) 1.00000 0.0646846 0.0323423 0.999477i \(-0.489703\pi\)
0.0323423 + 0.999477i \(0.489703\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) 11.0000 0.694314 0.347157 0.937807i \(-0.387147\pi\)
0.347157 + 0.937807i \(0.387147\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 0 0
\(255\) 7.00000 0.438357
\(256\) 0 0
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) −11.0000 −0.660926 −0.330463 0.943819i \(-0.607205\pi\)
−0.330463 + 0.943819i \(0.607205\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −19.0000 −1.12943 −0.564716 0.825285i \(-0.691014\pi\)
−0.564716 + 0.825285i \(0.691014\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) −3.00000 −0.174078
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) −18.0000 −1.03407
\(304\) 0 0
\(305\) 5.00000 0.286299
\(306\) 0 0
\(307\) −18.0000 −1.02731 −0.513657 0.857996i \(-0.671710\pi\)
−0.513657 + 0.857996i \(0.671710\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 14.0000 0.781404
\(322\) 0 0
\(323\) 7.00000 0.389490
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −10.0000 −0.553001
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) −6.00000 −0.329790 −0.164895 0.986311i \(-0.552728\pi\)
−0.164895 + 0.986311i \(0.552728\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) −2.00000 −0.109272
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 0 0
\(339\) 4.00000 0.217250
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 8.00000 0.430706
\(346\) 0 0
\(347\) 23.0000 1.23470 0.617352 0.786687i \(-0.288205\pi\)
0.617352 + 0.786687i \(0.288205\pi\)
\(348\) 0 0
\(349\) 29.0000 1.55233 0.776167 0.630527i \(-0.217161\pi\)
0.776167 + 0.630527i \(0.217161\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) −2.00000 −0.106149
\(356\) 0 0
\(357\) −7.00000 −0.370479
\(358\) 0 0
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 11.0000 0.575766
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 0 0
\(375\) 9.00000 0.464758
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) 0 0
\(387\) 1.00000 0.0508329
\(388\) 0 0
\(389\) −13.0000 −0.659126 −0.329563 0.944134i \(-0.606901\pi\)
−0.329563 + 0.944134i \(0.606901\pi\)
\(390\) 0 0
\(391\) 56.0000 2.83204
\(392\) 0 0
\(393\) 3.00000 0.151330
\(394\) 0 0
\(395\) −10.0000 −0.503155
\(396\) 0 0
\(397\) 5.00000 0.250943 0.125471 0.992097i \(-0.459956\pi\)
0.125471 + 0.992097i \(0.459956\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 1.00000 0.0493264
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) 0 0
\(417\) 9.00000 0.440732
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 3.00000 0.145865
\(424\) 0 0
\(425\) 28.0000 1.35820
\(426\) 0 0
\(427\) −5.00000 −0.241967
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −28.0000 −1.34559 −0.672797 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.00000 0.382692
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −33.0000 −1.56788 −0.783939 0.620838i \(-0.786792\pi\)
−0.783939 + 0.620838i \(0.786792\pi\)
\(444\) 0 0
\(445\) 14.0000 0.663664
\(446\) 0 0
\(447\) 1.00000 0.0472984
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 6.00000 0.281905
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 15.0000 0.701670 0.350835 0.936437i \(-0.385898\pi\)
0.350835 + 0.936437i \(0.385898\pi\)
\(458\) 0 0
\(459\) −7.00000 −0.326732
\(460\) 0 0
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 0 0
\(463\) 39.0000 1.81248 0.906242 0.422760i \(-0.138939\pi\)
0.906242 + 0.422760i \(0.138939\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 0 0
\(467\) 7.00000 0.323921 0.161961 0.986797i \(-0.448218\pi\)
0.161961 + 0.986797i \(0.448218\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) −3.00000 −0.137940
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −8.00000 −0.364013
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) 0 0
\(497\) 2.00000 0.0897123
\(498\) 0 0
\(499\) 7.00000 0.313363 0.156682 0.987649i \(-0.449920\pi\)
0.156682 + 0.987649i \(0.449920\pi\)
\(500\) 0 0
\(501\) −20.0000 −0.893534
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) 0 0
\(509\) 40.0000 1.77297 0.886484 0.462758i \(-0.153140\pi\)
0.886484 + 0.462758i \(0.153140\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −14.0000 −0.616914
\(516\) 0 0
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) 26.0000 1.14127
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) −14.0000 −0.609850
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −14.0000 −0.605273
\(536\) 0 0
\(537\) −16.0000 −0.690451
\(538\) 0 0
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 0 0
\(543\) 14.0000 0.600798
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −40.0000 −1.71028 −0.855138 0.518400i \(-0.826528\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 0 0
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) 9.00000 0.381342 0.190671 0.981654i \(-0.438934\pi\)
0.190671 + 0.981654i \(0.438934\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 21.0000 0.886621
\(562\) 0 0
\(563\) 32.0000 1.34864 0.674320 0.738440i \(-0.264437\pi\)
0.674320 + 0.738440i \(0.264437\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 36.0000 1.50920 0.754599 0.656186i \(-0.227831\pi\)
0.754599 + 0.656186i \(0.227831\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) −7.00000 −0.292429
\(574\) 0 0
\(575\) 32.0000 1.33449
\(576\) 0 0
\(577\) −9.00000 −0.374675 −0.187337 0.982296i \(-0.559986\pi\)
−0.187337 + 0.982296i \(0.559986\pi\)
\(578\) 0 0
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −21.0000 −0.866763 −0.433381 0.901211i \(-0.642680\pi\)
−0.433381 + 0.901211i \(0.642680\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 7.00000 0.286972
\(596\) 0 0
\(597\) 17.0000 0.695764
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 12.0000 0.487065 0.243532 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −27.0000 −1.09052 −0.545260 0.838267i \(-0.683569\pi\)
−0.545260 + 0.838267i \(0.683569\pi\)
\(614\) 0 0
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) −39.0000 −1.57008 −0.785040 0.619445i \(-0.787358\pi\)
−0.785040 + 0.619445i \(0.787358\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) −14.0000 −0.560898
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 3.00000 0.119808
\(628\) 0 0
\(629\) −28.0000 −1.11643
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) 0 0
\(633\) 10.0000 0.397464
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −31.0000 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) 0 0
\(645\) −1.00000 −0.0393750
\(646\) 0 0
\(647\) −15.0000 −0.589711 −0.294855 0.955542i \(-0.595271\pi\)
−0.294855 + 0.955542i \(0.595271\pi\)
\(648\) 0 0
\(649\) −18.0000 −0.706562
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 0 0
\(653\) −31.0000 −1.21312 −0.606562 0.795036i \(-0.707452\pi\)
−0.606562 + 0.795036i \(0.707452\pi\)
\(654\) 0 0
\(655\) −3.00000 −0.117220
\(656\) 0 0
\(657\) −11.0000 −0.429151
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −24.0000 −0.933492 −0.466746 0.884391i \(-0.654574\pi\)
−0.466746 + 0.884391i \(0.654574\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.00000 0.0387783
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 15.0000 0.579069
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −28.0000 −1.07613 −0.538064 0.842904i \(-0.680844\pi\)
−0.538064 + 0.842904i \(0.680844\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) −1.00000 −0.0382080
\(686\) 0 0
\(687\) −23.0000 −0.877505
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −41.0000 −1.55971 −0.779857 0.625958i \(-0.784708\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 0 0
\(693\) −3.00000 −0.113961
\(694\) 0 0
\(695\) −9.00000 −0.341389
\(696\) 0 0
\(697\) 28.0000 1.06058
\(698\) 0 0
\(699\) −17.0000 −0.642999
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) −3.00000 −0.112987
\(706\) 0 0
\(707\) −18.0000 −0.676960
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.00000 0.0373457
\(718\) 0 0
\(719\) 45.0000 1.67822 0.839108 0.543964i \(-0.183077\pi\)
0.839108 + 0.543964i \(0.183077\pi\)
\(720\) 0 0
\(721\) 14.0000 0.521387
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 7.00000 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −7.00000 −0.258904
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) −6.00000 −0.221013
\(738\) 0 0
\(739\) −3.00000 −0.110357 −0.0551784 0.998477i \(-0.517573\pi\)
−0.0551784 + 0.998477i \(0.517573\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.0000 0.366864 0.183432 0.983032i \(-0.441279\pi\)
0.183432 + 0.983032i \(0.441279\pi\)
\(744\) 0 0
\(745\) −1.00000 −0.0366372
\(746\) 0 0
\(747\) −16.0000 −0.585409
\(748\) 0 0
\(749\) 14.0000 0.511549
\(750\) 0 0
\(751\) −44.0000 −1.60558 −0.802791 0.596260i \(-0.796653\pi\)
−0.802791 + 0.596260i \(0.796653\pi\)
\(752\) 0 0
\(753\) 11.0000 0.400862
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) −29.0000 −1.05402 −0.527011 0.849858i \(-0.676688\pi\)
−0.527011 + 0.849858i \(0.676688\pi\)
\(758\) 0 0
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) 0 0
\(765\) 7.00000 0.253086
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 27.0000 0.973645 0.486822 0.873501i \(-0.338156\pi\)
0.486822 + 0.873501i \(0.338156\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 0 0
\(773\) 14.0000 0.503545 0.251773 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 4.00000 0.143499
\(778\) 0 0
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 0 0
\(789\) −9.00000 −0.320408
\(790\) 0 0
\(791\) 4.00000 0.142224
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) −21.0000 −0.742927
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) 33.0000 1.16454
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) 10.0000 0.352017
\(808\) 0 0
\(809\) −1.00000 −0.0351581 −0.0175791 0.999845i \(-0.505596\pi\)
−0.0175791 + 0.999845i \(0.505596\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −23.0000 −0.802706 −0.401353 0.915924i \(-0.631460\pi\)
−0.401353 + 0.915924i \(0.631460\pi\)
\(822\) 0 0
\(823\) 49.0000 1.70803 0.854016 0.520246i \(-0.174160\pi\)
0.854016 + 0.520246i \(0.174160\pi\)
\(824\) 0 0
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) 54.0000 1.87776 0.938882 0.344239i \(-0.111863\pi\)
0.938882 + 0.344239i \(0.111863\pi\)
\(828\) 0 0
\(829\) 28.0000 0.972480 0.486240 0.873825i \(-0.338368\pi\)
0.486240 + 0.873825i \(0.338368\pi\)
\(830\) 0 0
\(831\) −11.0000 −0.381586
\(832\) 0 0
\(833\) 42.0000 1.45521
\(834\) 0 0
\(835\) 20.0000 0.692129
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 0 0
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −10.0000 −0.344418
\(844\) 0 0
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) −19.0000 −0.652078
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) −55.0000 −1.87658 −0.938288 0.345855i \(-0.887589\pi\)
−0.938288 + 0.345855i \(0.887589\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) −26.0000 −0.884027
\(866\) 0 0
\(867\) 32.0000 1.08678
\(868\) 0 0
\(869\) −30.0000 −1.01768
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −8.00000 −0.270759
\(874\) 0 0
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −35.0000 −1.17918 −0.589590 0.807703i \(-0.700711\pi\)
−0.589590 + 0.807703i \(0.700711\pi\)
\(882\) 0 0
\(883\) −3.00000 −0.100958 −0.0504790 0.998725i \(-0.516075\pi\)
−0.0504790 + 0.998725i \(0.516075\pi\)
\(884\) 0 0
\(885\) −6.00000 −0.201688
\(886\) 0 0
\(887\) −38.0000 −1.27592 −0.637958 0.770072i \(-0.720220\pi\)
−0.637958 + 0.770072i \(0.720220\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) −3.00000 −0.100391
\(894\) 0 0
\(895\) 16.0000 0.534821
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −42.0000 −1.39922
\(902\) 0 0
\(903\) 1.00000 0.0332779
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) −38.0000 −1.26177 −0.630885 0.775877i \(-0.717308\pi\)
−0.630885 + 0.775877i \(0.717308\pi\)
\(908\) 0 0
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) −58.0000 −1.92163 −0.960813 0.277198i \(-0.910594\pi\)
−0.960813 + 0.277198i \(0.910594\pi\)
\(912\) 0 0
\(913\) 48.0000 1.58857
\(914\) 0 0
\(915\) 5.00000 0.165295
\(916\) 0 0
\(917\) 3.00000 0.0990687
\(918\) 0 0
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) −18.0000 −0.593120
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −16.0000 −0.526077
\(926\) 0 0
\(927\) 14.0000 0.459820
\(928\) 0 0
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) −21.0000 −0.687509
\(934\) 0 0
\(935\) −21.0000 −0.686773
\(936\) 0 0
\(937\) −35.0000 −1.14340 −0.571700 0.820463i \(-0.693716\pi\)
−0.571700 + 0.820463i \(0.693716\pi\)
\(938\) 0 0
\(939\) 22.0000 0.717943
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) 7.00000 0.226515
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.00000 0.0322917
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 14.0000 0.451144
\(964\) 0 0
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 7.00000 0.224872
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 9.00000 0.288527
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.0000 0.319928 0.159964 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) 0 0
\(979\) 42.0000 1.34233
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 3.00000 0.0954911
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −44.0000 −1.39771 −0.698853 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) 0 0
\(993\) −6.00000 −0.190404
\(994\) 0 0
\(995\) −17.0000 −0.538936
\(996\) 0 0
\(997\) −7.00000 −0.221692 −0.110846 0.993838i \(-0.535356\pi\)
−0.110846 + 0.993838i \(0.535356\pi\)
\(998\) 0 0
\(999\) 4.00000 0.126554
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 1824.2.a.i.1.1 yes 1
3.2 odd 2 5472.2.a.r.1.1 1
4.3 odd 2 1824.2.a.b.1.1 1
8.3 odd 2 3648.2.a.bd.1.1 1
8.5 even 2 3648.2.a.m.1.1 1
12.11 even 2 5472.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1824.2.a.b.1.1 1 4.3 odd 2
1824.2.a.i.1.1 yes 1 1.1 even 1 trivial
3648.2.a.m.1.1 1 8.5 even 2
3648.2.a.bd.1.1 1 8.3 odd 2
5472.2.a.n.1.1 1 12.11 even 2
5472.2.a.r.1.1 1 3.2 odd 2