Properties

Label 768.2.a.h.1.1
Level $768$
Weight $2$
Character 768.1
Self dual yes
Analytic conductor $6.133$
Analytic rank $0$
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] = [768,2,Mod(1,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("768.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 768.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +4.00000 q^{13} +2.00000 q^{15} -2.00000 q^{17} -4.00000 q^{19} +2.00000 q^{21} -4.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +2.00000 q^{31} +4.00000 q^{35} +8.00000 q^{37} +4.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} +2.00000 q^{45} -12.0000 q^{47} -3.00000 q^{49} -2.00000 q^{51} +6.00000 q^{53} -4.00000 q^{57} +4.00000 q^{59} +2.00000 q^{63} +8.00000 q^{65} +12.0000 q^{67} -4.00000 q^{69} -12.0000 q^{71} +6.00000 q^{73} -1.00000 q^{75} +10.0000 q^{79} +1.00000 q^{81} -16.0000 q^{83} -4.00000 q^{85} +6.00000 q^{87} +10.0000 q^{89} +8.00000 q^{91} +2.00000 q^{93} -8.00000 q^{95} -2.00000 q^{97} +O(q^{100})\)

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
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 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
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 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
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) 2.00000 0.172133
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 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
\(165\) 0 0
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 16.0000 1.17634
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 0 0
\(195\) 8.00000 0.572892
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 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
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 0 0
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −18.0000 −1.09342 −0.546711 0.837321i \(-0.684120\pi\)
−0.546711 + 0.837321i \(0.684120\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) −10.0000 −0.574485
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 0 0
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 8.00000 0.438397
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) 0 0
\(347\) 8.00000 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) 0 0
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −24.0000 −1.27379
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 20.0000 1.00887
\(394\) 0 0
\(395\) 20.0000 1.00631
\(396\) 0 0
\(397\) 32.0000 1.60603 0.803017 0.595956i \(-0.203227\pi\)
0.803017 + 0.595956i \(0.203227\pi\)
\(398\) 0 0
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 0 0
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −32.0000 −1.57082
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 0 0
\(423\) −12.0000 −0.583460
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 0 0
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) 30.0000 1.43182 0.715911 0.698192i \(-0.246012\pi\)
0.715911 + 0.698192i \(0.246012\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 20.0000 0.948091
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 18.0000 0.845714
\(454\) 0 0
\(455\) 16.0000 0.750092
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) −8.00000 −0.368621
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 32.0000 1.45907
\(482\) 0 0
\(483\) −8.00000 −0.364013
\(484\) 0 0
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 0 0
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) −20.0000 −0.889988
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 12.0000 0.528783
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 0 0
\(525\) −2.00000 −0.0872872
\(526\) 0 0
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) 0 0
\(537\) −4.00000 −0.172613
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −20.0000 −0.859867 −0.429934 0.902861i \(-0.641463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) 0 0
\(543\) −20.0000 −0.858282
\(544\) 0 0
\(545\) −8.00000 −0.342682
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) 0 0
\(555\) 16.0000 0.679162
\(556\) 0 0
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) 0 0
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −32.0000 −1.32758
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 8.00000 0.330759
\(586\) 0 0
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −8.00000 −0.327968
\(596\) 0 0
\(597\) −10.0000 −0.409273
\(598\) 0 0
\(599\) 20.0000 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) −22.0000 −0.894427
\(606\) 0 0
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) −4.00000 −0.161296
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) −36.0000 −1.44696 −0.723481 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 20.0000 0.801283
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) −4.00000 −0.158735
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 22.0000 0.868948 0.434474 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 40.0000 1.56293
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) 0 0
\(663\) −8.00000 −0.310694
\(664\) 0 0
\(665\) −16.0000 −0.620453
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) 16.0000 0.612223 0.306111 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) 0 0
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) 4.00000 0.152610
\(688\) 0 0
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) −24.0000 −0.903892
\(706\) 0 0
\(707\) −20.0000 −0.752177
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 2.00000 0.0743808
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 42.0000 1.55769 0.778847 0.627214i \(-0.215805\pi\)
0.778847 + 0.627214i \(0.215805\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 0 0
\(747\) −16.0000 −0.585409
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) 2.00000 0.0729810 0.0364905 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 36.0000 1.31017
\(756\) 0 0
\(757\) −12.0000 −0.436147 −0.218074 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 38.0000 1.37750 0.688749 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) 16.0000 0.577727
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) −34.0000 −1.22290 −0.611448 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) −16.0000 −0.571064
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 0 0
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 12.0000 0.425596
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) −18.0000 −0.631288
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) −14.0000 −0.488009 −0.244005 0.969774i \(-0.578461\pi\)
−0.244005 + 0.969774i \(0.578461\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) 28.0000 0.971309
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 0 0
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 18.0000 0.619953
\(844\) 0 0
\(845\) 6.00000 0.206406
\(846\) 0 0
\(847\) −22.0000 −0.755929
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) −24.0000 −0.821744 −0.410872 0.911693i \(-0.634776\pi\)
−0.410872 + 0.911693i \(0.634776\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −12.0000 −0.408012
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 48.0000 1.60626
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) −40.0000 −1.32964
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 40.0000 1.32092
\(918\) 0 0
\(919\) −30.0000 −0.989609 −0.494804 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 0 0
\(927\) 6.00000 0.197066
\(928\) 0 0
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) 8.00000 0.260516
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 0 0
\(985\) −4.00000 −0.127451
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) 48.0000 1.52018 0.760088 0.649821i \(-0.225156\pi\)
0.760088 + 0.649821i \(0.225156\pi\)
\(998\) 0 0
\(999\) 8.00000 0.253109
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 768.2.a.h.1.1 1
3.2 odd 2 2304.2.a.e.1.1 1
4.3 odd 2 768.2.a.d.1.1 1
8.3 odd 2 768.2.a.e.1.1 1
8.5 even 2 768.2.a.a.1.1 1
12.11 even 2 2304.2.a.b.1.1 1
16.3 odd 4 96.2.d.a.49.1 2
16.5 even 4 24.2.d.a.13.1 2
16.11 odd 4 96.2.d.a.49.2 2
16.13 even 4 24.2.d.a.13.2 yes 2
24.5 odd 2 2304.2.a.o.1.1 1
24.11 even 2 2304.2.a.l.1.1 1
48.5 odd 4 72.2.d.b.37.2 2
48.11 even 4 288.2.d.b.145.1 2
48.29 odd 4 72.2.d.b.37.1 2
48.35 even 4 288.2.d.b.145.2 2
80.3 even 4 2400.2.d.c.49.1 2
80.13 odd 4 600.2.d.c.349.2 2
80.19 odd 4 2400.2.k.a.1201.2 2
80.27 even 4 2400.2.d.c.49.2 2
80.29 even 4 600.2.k.b.301.1 2
80.37 odd 4 600.2.d.c.349.1 2
80.43 even 4 2400.2.d.b.49.1 2
80.53 odd 4 600.2.d.b.349.2 2
80.59 odd 4 2400.2.k.a.1201.1 2
80.67 even 4 2400.2.d.b.49.2 2
80.69 even 4 600.2.k.b.301.2 2
80.77 odd 4 600.2.d.b.349.1 2
112.13 odd 4 1176.2.c.a.589.2 2
112.27 even 4 4704.2.c.a.2353.1 2
112.69 odd 4 1176.2.c.a.589.1 2
112.83 even 4 4704.2.c.a.2353.2 2
144.5 odd 12 648.2.n.c.541.1 4
144.11 even 12 2592.2.r.g.433.2 4
144.13 even 12 648.2.n.k.541.1 4
144.29 odd 12 648.2.n.c.109.1 4
144.43 odd 12 2592.2.r.f.433.1 4
144.59 even 12 2592.2.r.g.2161.1 4
144.61 even 12 648.2.n.k.109.2 4
144.67 odd 12 2592.2.r.f.2161.1 4
144.77 odd 12 648.2.n.c.541.2 4
144.83 even 12 2592.2.r.g.433.1 4
144.85 even 12 648.2.n.k.541.2 4
144.101 odd 12 648.2.n.c.109.2 4
144.115 odd 12 2592.2.r.f.433.2 4
144.131 even 12 2592.2.r.g.2161.2 4
144.133 even 12 648.2.n.k.109.1 4
144.139 odd 12 2592.2.r.f.2161.2 4
240.29 odd 4 1800.2.k.a.901.2 2
240.53 even 4 1800.2.d.i.1549.1 2
240.59 even 4 7200.2.k.d.3601.2 2
240.77 even 4 1800.2.d.i.1549.2 2
240.83 odd 4 7200.2.d.d.2449.1 2
240.107 odd 4 7200.2.d.d.2449.2 2
240.149 odd 4 1800.2.k.a.901.1 2
240.173 even 4 1800.2.d.b.1549.1 2
240.179 even 4 7200.2.k.d.3601.1 2
240.197 even 4 1800.2.d.b.1549.2 2
240.203 odd 4 7200.2.d.g.2449.1 2
240.227 odd 4 7200.2.d.g.2449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.d.a.13.1 2 16.5 even 4
24.2.d.a.13.2 yes 2 16.13 even 4
72.2.d.b.37.1 2 48.29 odd 4
72.2.d.b.37.2 2 48.5 odd 4
96.2.d.a.49.1 2 16.3 odd 4
96.2.d.a.49.2 2 16.11 odd 4
288.2.d.b.145.1 2 48.11 even 4
288.2.d.b.145.2 2 48.35 even 4
600.2.d.b.349.1 2 80.77 odd 4
600.2.d.b.349.2 2 80.53 odd 4
600.2.d.c.349.1 2 80.37 odd 4
600.2.d.c.349.2 2 80.13 odd 4
600.2.k.b.301.1 2 80.29 even 4
600.2.k.b.301.2 2 80.69 even 4
648.2.n.c.109.1 4 144.29 odd 12
648.2.n.c.109.2 4 144.101 odd 12
648.2.n.c.541.1 4 144.5 odd 12
648.2.n.c.541.2 4 144.77 odd 12
648.2.n.k.109.1 4 144.133 even 12
648.2.n.k.109.2 4 144.61 even 12
648.2.n.k.541.1 4 144.13 even 12
648.2.n.k.541.2 4 144.85 even 12
768.2.a.a.1.1 1 8.5 even 2
768.2.a.d.1.1 1 4.3 odd 2
768.2.a.e.1.1 1 8.3 odd 2
768.2.a.h.1.1 1 1.1 even 1 trivial
1176.2.c.a.589.1 2 112.69 odd 4
1176.2.c.a.589.2 2 112.13 odd 4
1800.2.d.b.1549.1 2 240.173 even 4
1800.2.d.b.1549.2 2 240.197 even 4
1800.2.d.i.1549.1 2 240.53 even 4
1800.2.d.i.1549.2 2 240.77 even 4
1800.2.k.a.901.1 2 240.149 odd 4
1800.2.k.a.901.2 2 240.29 odd 4
2304.2.a.b.1.1 1 12.11 even 2
2304.2.a.e.1.1 1 3.2 odd 2
2304.2.a.l.1.1 1 24.11 even 2
2304.2.a.o.1.1 1 24.5 odd 2
2400.2.d.b.49.1 2 80.43 even 4
2400.2.d.b.49.2 2 80.67 even 4
2400.2.d.c.49.1 2 80.3 even 4
2400.2.d.c.49.2 2 80.27 even 4
2400.2.k.a.1201.1 2 80.59 odd 4
2400.2.k.a.1201.2 2 80.19 odd 4
2592.2.r.f.433.1 4 144.43 odd 12
2592.2.r.f.433.2 4 144.115 odd 12
2592.2.r.f.2161.1 4 144.67 odd 12
2592.2.r.f.2161.2 4 144.139 odd 12
2592.2.r.g.433.1 4 144.83 even 12
2592.2.r.g.433.2 4 144.11 even 12
2592.2.r.g.2161.1 4 144.59 even 12
2592.2.r.g.2161.2 4 144.131 even 12
4704.2.c.a.2353.1 2 112.27 even 4
4704.2.c.a.2353.2 2 112.83 even 4
7200.2.d.d.2449.1 2 240.83 odd 4
7200.2.d.d.2449.2 2 240.107 odd 4
7200.2.d.g.2449.1 2 240.203 odd 4
7200.2.d.g.2449.2 2 240.227 odd 4
7200.2.k.d.3601.1 2 240.179 even 4
7200.2.k.d.3601.2 2 240.59 even 4