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Properties

Label 4851.2.a.l.1.1
Level $4851$
Weight $2$
Character 4851.1
Self dual yes
Analytic conductor $38.735$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4851,2,Mod(1,4851)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4851, 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("4851.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4851.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: \(38.7354300205\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4851.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} +4.00000 q^{5} +O(q^{10})\) \(q-2.00000 q^{4} +4.00000 q^{5} -1.00000 q^{11} -5.00000 q^{13} +4.00000 q^{16} +4.00000 q^{17} +3.00000 q^{19} -8.00000 q^{20} -8.00000 q^{23} +11.0000 q^{25} +4.00000 q^{29} -1.00000 q^{31} +7.00000 q^{37} -4.00000 q^{41} +1.00000 q^{43} +2.00000 q^{44} -8.00000 q^{47} +10.0000 q^{52} +12.0000 q^{53} -4.00000 q^{55} +2.00000 q^{61} -8.00000 q^{64} -20.0000 q^{65} +3.00000 q^{67} -8.00000 q^{68} +4.00000 q^{71} +11.0000 q^{73} -6.00000 q^{76} +15.0000 q^{79} +16.0000 q^{80} +12.0000 q^{83} +16.0000 q^{85} +12.0000 q^{89} +16.0000 q^{92} +12.0000 q^{95} +2.00000 q^{97} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −8.00000 −1.78885
\(21\) 0 0
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 − 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 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 \(0.475706\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 10.0000 1.38675
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −20.0000 −2.48069
\(66\) 0 0
\(67\) 3.00000 0.366508 0.183254 − 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) −8.00000 −0.970143
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 − 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 15.0000 1.68763 0.843816 − 0.536633i \(-0.180304\pi\)
0.843816 + 0.536633i \(0.180304\pi\)
\(80\) 16.0000 1.78885
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 16.0000 1.73544
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 16.0000 1.66812
\(93\) 0 0
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −22.0000 −2.20000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −9.00000 −0.886796 −0.443398 − 0.896325i \(-0.646227\pi\)
−0.443398 + 0.896325i \(0.646227\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 − 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 − 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 − 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) −32.0000 −2.98402
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 − 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 − 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) 19.0000 1.61156 0.805779 − 0.592216i \(-0.201747\pi\)
0.805779 + 0.592216i \(0.201747\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.00000 0.418121
\(144\) 0 0
\(145\) 16.0000 1.32873
\(146\) 0 0
\(147\) 0 0
\(148\) −14.0000 −1.15079
\(149\) −4.00000 −0.327693 −0.163846 − 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 − 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 − 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 8.00000 0.624695
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) 20.0000 1.52057 0.760286 − 0.649589i \(-0.225059\pi\)
0.760286 + 0.649589i \(0.225059\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 − 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 − 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 28.0000 2.05860
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 16.0000 1.16692
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 − 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −24.0000 −1.70993 −0.854965 − 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −16.0000 −1.11749
\(206\) 0 0
\(207\) 0 0
\(208\) −20.0000 −1.38675
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 − 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −24.0000 −1.64833
\(213\) 0 0
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 8.00000 0.539360
\(221\) −20.0000 −1.34535
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 − 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −16.0000 −1.06196 −0.530979 − 0.847385i \(-0.678176\pi\)
−0.530979 + 0.847385i \(0.678176\pi\)
\(228\) 0 0
\(229\) −17.0000 −1.12339 −0.561696 − 0.827344i \(-0.689851\pi\)
−0.561696 + 0.827344i \(0.689851\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 −0.786146 −0.393073 − 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) −32.0000 −2.08745
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 28.0000 1.81117 0.905585 − 0.424165i \(-0.139432\pi\)
0.905585 + 0.424165i \(0.139432\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 − 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) −15.0000 −0.954427
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 40.0000 2.48069
\(261\) 0 0
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 48.0000 2.94862
\(266\) 0 0
\(267\) 0 0
\(268\) −6.00000 −0.366508
\(269\) 24.0000 1.46331 0.731653 − 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 16.0000 0.970143
\(273\) 0 0
\(274\) 0 0
\(275\) −11.0000 −0.663325
\(276\) 0 0
\(277\) 21.0000 1.26177 0.630884 − 0.775877i \(-0.282692\pi\)
0.630884 + 0.775877i \(0.282692\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 − 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) −21.0000 −1.24832 −0.624160 − 0.781296i \(-0.714559\pi\)
−0.624160 + 0.781296i \(0.714559\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) −22.0000 −1.28745
\(293\) 24.0000 1.40209 0.701047 − 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 40.0000 2.31326
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 12.0000 0.688247
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) −23.0000 −1.31268 −0.656340 − 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 − 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 0 0
\(313\) 5.00000 0.282617 0.141308 − 0.989966i \(-0.454869\pi\)
0.141308 + 0.989966i \(0.454869\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −30.0000 −1.68763
\(317\) −12.0000 −0.673987 −0.336994 − 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) −32.0000 −1.78885
\(321\) 0 0
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) −55.0000 −3.05085
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 − 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) −24.0000 −1.31717
\(333\) 0 0
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 9.00000 0.490261 0.245131 − 0.969490i \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −32.0000 −1.73544
\(341\) 1.00000 0.0541530
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.0000 1.07366 0.536828 − 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.0000 1.27739 0.638696 − 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) −24.0000 −1.27200
\(357\) 0 0
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 − 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 44.0000 2.30307
\(366\) 0 0
\(367\) 29.0000 1.51379 0.756894 − 0.653538i \(-0.226716\pi\)
0.756894 + 0.653538i \(0.226716\pi\)
\(368\) −32.0000 −1.66812
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 23.0000 1.19089 0.595447 − 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 − 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) −24.0000 −1.23117
\(381\) 0 0
\(382\) 0 0
\(383\) −20.0000 −1.02195 −0.510976 − 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −4.00000 −0.203069
\(389\) 24.0000 1.21685 0.608424 − 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) −32.0000 −1.61831
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 60.0000 3.01893
\(396\) 0 0
\(397\) −3.00000 −0.150566 −0.0752828 − 0.997162i \(-0.523986\pi\)
−0.0752828 + 0.997162i \(0.523986\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 44.0000 2.20000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 5.00000 0.249068
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.00000 −0.346977
\(408\) 0 0
\(409\) 35.0000 1.73064 0.865319 − 0.501221i \(-0.167116\pi\)
0.865319 + 0.501221i \(0.167116\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 18.0000 0.886796
\(413\) 0 0
\(414\) 0 0
\(415\) 48.0000 2.35623
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 − 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −11.0000 −0.536107 −0.268054 − 0.963404i \(-0.586380\pi\)
−0.268054 + 0.963404i \(0.586380\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 44.0000 2.13431
\(426\) 0 0
\(427\) 0 0
\(428\) −16.0000 −0.773389
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) −35.0000 −1.68199 −0.840996 − 0.541041i \(-0.818030\pi\)
−0.840996 + 0.541041i \(0.818030\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 22.0000 1.05361
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 − 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) 48.0000 2.27542
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −36.0000 −1.69895 −0.849473 − 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 32.0000 1.50515
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 29.0000 1.35656 0.678281 − 0.734802i \(-0.262725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 64.0000 2.98402
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 7.00000 0.325318 0.162659 − 0.986682i \(-0.447993\pi\)
0.162659 + 0.986682i \(0.447993\pi\)
\(464\) 16.0000 0.742781
\(465\) 0 0
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.00000 −0.0459800
\(474\) 0 0
\(475\) 33.0000 1.51414
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.00000 0.182765 0.0913823 − 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) −35.0000 −1.59586
\(482\) 0 0
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −1.00000 −0.0453143 −0.0226572 − 0.999743i \(-0.507213\pi\)
−0.0226572 + 0.999743i \(0.507213\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 16.0000 0.720604
\(494\) 0 0
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −19.0000 −0.850557 −0.425278 − 0.905063i \(-0.639824\pi\)
−0.425278 + 0.905063i \(0.639824\pi\)
\(500\) −48.0000 −2.14663
\(501\) 0 0
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 − 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 14.0000 0.621150
\(509\) −8.00000 −0.354594 −0.177297 − 0.984157i \(-0.556735\pi\)
−0.177297 + 0.984157i \(0.556735\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −36.0000 −1.58635
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 − 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) 1.00000 0.0437269 0.0218635 − 0.999761i \(-0.493040\pi\)
0.0218635 + 0.999761i \(0.493040\pi\)
\(524\) 24.0000 1.04844
\(525\) 0 0
\(526\) 0 0
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) 0 0
\(535\) 32.0000 1.38348
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.00000 −0.300954 −0.150477 − 0.988614i \(-0.548081\pi\)
−0.150477 + 0.988614i \(0.548081\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −44.0000 −1.88475
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) −24.0000 −1.02523
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −38.0000 −1.61156
\(557\) −28.0000 −1.18640 −0.593199 − 0.805056i \(-0.702135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) −5.00000 −0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.0000 0.674320 0.337160 − 0.941447i \(-0.390534\pi\)
0.337160 + 0.941447i \(0.390534\pi\)
\(564\) 0 0
\(565\) −64.0000 −2.69250
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 13.0000 0.544033 0.272017 − 0.962293i \(-0.412309\pi\)
0.272017 + 0.962293i \(0.412309\pi\)
\(572\) −10.0000 −0.418121
\(573\) 0 0
\(574\) 0 0
\(575\) −88.0000 −3.66985
\(576\) 0 0
\(577\) −11.0000 −0.457936 −0.228968 − 0.973434i \(-0.573535\pi\)
−0.228968 + 0.973434i \(0.573535\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −32.0000 −1.32873
\(581\) 0 0
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) 0 0
\(592\) 28.0000 1.15079
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.00000 0.327693
\(597\) 0 0
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 − 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 − 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 24.0000 0.976546
\(605\) 4.00000 0.162623
\(606\) 0 0
\(607\) 37.0000 1.50178 0.750892 − 0.660425i \(-0.229624\pi\)
0.750892 + 0.660425i \(0.229624\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 40.0000 1.61823
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 − 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 1.00000 0.0401934 0.0200967 − 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) 28.0000 1.11643
\(630\) 0 0
\(631\) 12.0000 0.477712 0.238856 − 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −28.0000 −1.11115
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −48.0000 −1.89589 −0.947943 − 0.318440i \(-0.896841\pi\)
−0.947943 + 0.318440i \(0.896841\pi\)
\(642\) 0 0
\(643\) −25.0000 −0.985904 −0.492952 − 0.870057i \(-0.664082\pi\)
−0.492952 + 0.870057i \(0.664082\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 4.00000 0.156532 0.0782660 − 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) 0 0
\(655\) −48.0000 −1.87552
\(656\) −16.0000 −0.624695
\(657\) 0 0
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 − 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 − 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −32.0000 −1.23904
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −41.0000 −1.58043 −0.790217 − 0.612827i \(-0.790032\pi\)
−0.790217 + 0.612827i \(0.790032\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) −32.0000 −1.22986 −0.614930 − 0.788582i \(-0.710816\pi\)
−0.614930 + 0.788582i \(0.710816\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 − 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 48.0000 1.83399
\(686\) 0 0
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) −15.0000 −0.570627 −0.285313 − 0.958434i \(-0.592098\pi\)
−0.285313 + 0.958434i \(0.592098\pi\)
\(692\) −40.0000 −1.52057
\(693\) 0 0
\(694\) 0 0
\(695\) 76.0000 2.88284
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 − 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 21.0000 0.792030
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 − 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 20.0000 0.747958
\(716\) −8.00000 −0.298974
\(717\) 0 0
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −14.0000 −0.520306
\(725\) 44.0000 1.63412
\(726\) 0 0
\(727\) −23.0000 −0.853023 −0.426511 − 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) 19.0000 0.701781 0.350891 − 0.936416i \(-0.385879\pi\)
0.350891 + 0.936416i \(0.385879\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.00000 −0.110506
\(738\) 0 0
\(739\) −15.0000 −0.551784 −0.275892 − 0.961189i \(-0.588973\pi\)
−0.275892 + 0.961189i \(0.588973\pi\)
\(740\) −56.0000 −2.05860
\(741\) 0 0
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 − 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) −16.0000 −0.586195
\(746\) 0 0
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 0 0
\(751\) −47.0000 −1.71505 −0.857527 − 0.514439i \(-0.828000\pi\)
−0.857527 + 0.514439i \(0.828000\pi\)
\(752\) −32.0000 −1.16692
\(753\) 0 0
\(754\) 0 0
\(755\) −48.0000 −1.74690
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 − 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 − 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 − 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) 48.0000 1.72644 0.863220 − 0.504828i \(-0.168444\pi\)
0.863220 + 0.504828i \(0.168444\pi\)
\(774\) 0 0
\(775\) −11.0000 −0.395132
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 8.00000 0.285532
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 − 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 48.0000 1.70993
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −8.00000 −0.283375 −0.141687 − 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.0000 −0.388182
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 24.0000 0.843795 0.421898 − 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 − 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −32.0000 −1.12091
\(816\) 0 0
\(817\) 3.00000 0.104957
\(818\) 0 0
\(819\) 0 0
\(820\) 32.0000 1.11749
\(821\) 12.0000 0.418803 0.209401 − 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −16.0000 −0.556375 −0.278187 − 0.960527i \(-0.589734\pi\)
−0.278187 + 0.960527i \(0.589734\pi\)
\(828\) 0 0
\(829\) −25.0000 −0.868286 −0.434143 − 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 40.0000 1.38675
\(833\) 0 0
\(834\) 0 0
\(835\) 32.0000 1.10741
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 0 0
\(844\) 16.0000 0.550743
\(845\) 48.0000 1.65125
\(846\) 0 0
\(847\) 0 0
\(848\) 48.0000 1.64833
\(849\) 0 0
\(850\) 0 0
\(851\) −56.0000 −1.91966
\(852\) 0 0
\(853\) 49.0000 1.67773 0.838864 − 0.544341i \(-0.183220\pi\)
0.838864 + 0.544341i \(0.183220\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −32.0000 −1.09310 −0.546550 − 0.837427i \(-0.684059\pi\)
−0.546550 + 0.837427i \(0.684059\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 − 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 0 0
\(863\) −20.0000 −0.680808 −0.340404 − 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(864\) 0 0
\(865\) 80.0000 2.72008
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −15.0000 −0.508840
\(870\) 0 0
\(871\) −15.0000 −0.508256
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 − 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −16.0000 −0.539360
\(881\) −12.0000 −0.404290 −0.202145 − 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) 0 0
\(883\) −41.0000 −1.37976 −0.689880 − 0.723924i \(-0.742337\pi\)
−0.689880 + 0.723924i \(0.742337\pi\)
\(884\) 40.0000 1.34535
\(885\) 0 0
\(886\) 0 0
\(887\) 8.00000 0.268614 0.134307 − 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) 16.0000 0.534821
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 48.0000 1.59911
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 28.0000 0.930751
\(906\) 0 0
\(907\) −37.0000 −1.22856 −0.614282 − 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) 32.0000 1.06196
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 34.0000 1.12339
\(917\) 0 0
\(918\) 0 0
\(919\) −45.0000 −1.48441 −0.742207 − 0.670171i \(-0.766221\pi\)
−0.742207 + 0.670171i \(0.766221\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −20.0000 −0.658308
\(924\) 0 0
\(925\) 77.0000 2.53174
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 24.0000 0.786146
\(933\) 0 0
\(934\) 0 0
\(935\) −16.0000 −0.523256
\(936\) 0 0
\(937\) −15.0000 −0.490029 −0.245014 − 0.969519i \(-0.578793\pi\)
−0.245014 + 0.969519i \(0.578793\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 64.0000 2.08745
\(941\) 8.00000 0.260793 0.130396 − 0.991462i \(-0.458375\pi\)
0.130396 + 0.991462i \(0.458375\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 8.00000 0.259965 0.129983 − 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) −55.0000 −1.78538
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 20.0000 0.647864 0.323932 − 0.946080i \(-0.394995\pi\)
0.323932 + 0.946080i \(0.394995\pi\)
\(954\) 0 0
\(955\) 32.0000 1.03550
\(956\) −56.0000 −1.81117
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 0 0
\(964\) −36.0000 −1.15948
\(965\) 44.0000 1.41641
\(966\) 0 0
\(967\) −49.0000 −1.57573 −0.787867 − 0.615846i \(-0.788815\pi\)
−0.787867 + 0.615846i \(0.788815\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 40.0000 1.28366 0.641831 − 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) 20.0000 0.639857 0.319928 − 0.947442i \(-0.396341\pi\)
0.319928 + 0.947442i \(0.396341\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.0000 0.382741 0.191370 − 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) −96.0000 −3.05881
\(986\) 0 0
\(987\) 0 0
\(988\) 30.0000 0.954427
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −23.0000 −0.730619 −0.365310 − 0.930886i \(-0.619037\pi\)
−0.365310 + 0.930886i \(0.619037\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 21.0000 0.665077 0.332538 − 0.943090i \(-0.392095\pi\)
0.332538 + 0.943090i \(0.392095\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4851.2.a.l.1.1 1
3.2 odd 2 4851.2.a.i.1.1 1
7.3 odd 6 693.2.i.d.100.1 yes 2
7.5 odd 6 693.2.i.d.298.1 yes 2
7.6 odd 2 4851.2.a.h.1.1 1
21.5 even 6 693.2.i.c.298.1 yes 2
21.17 even 6 693.2.i.c.100.1 ✓ 2
21.20 even 2 4851.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.i.c.100.1 ✓ 2 21.17 even 6
693.2.i.c.298.1 yes 2 21.5 even 6
693.2.i.d.100.1 yes 2 7.3 odd 6
693.2.i.d.298.1 yes 2 7.5 odd 6
4851.2.a.h.1.1 1 7.6 odd 2
4851.2.a.i.1.1 1 3.2 odd 2
4851.2.a.l.1.1 1 1.1 even 1 trivial
4851.2.a.m.1.1 1 21.20 even 2