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Properties

Label 4400.2.a.bd.1.1
Level $4400$
Weight $2$
Character 4400.1
Self dual yes
Analytic conductor $35.134$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4400,2,Mod(1,4400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,3,0,0,0,1,0,6,0,1,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4400.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +1.00000 q^{7} +6.00000 q^{9} +1.00000 q^{11} -5.00000 q^{17} +7.00000 q^{19} +3.00000 q^{21} +8.00000 q^{23} +9.00000 q^{27} +3.00000 q^{29} +5.00000 q^{31} +3.00000 q^{33} -1.00000 q^{37} -8.00000 q^{41} -10.0000 q^{43} -6.00000 q^{49} -15.0000 q^{51} -1.00000 q^{53} +21.0000 q^{57} -12.0000 q^{59} +5.00000 q^{61} +6.00000 q^{63} +4.00000 q^{67} +24.0000 q^{69} +7.00000 q^{71} +2.00000 q^{73} +1.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} +9.00000 q^{87} -7.00000 q^{89} +15.0000 q^{93} +8.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 − 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 − 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 − 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −15.0000 −2.10042
\(52\) 0 0
\(53\) −1.00000 −0.137361 −0.0686803 − 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 21.0000 2.78152
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 − 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 24.0000 2.88926
\(70\) 0 0
\(71\) 7.00000 0.830747 0.415374 − 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) −7.00000 −0.741999 −0.370999 − 0.928633i \(-0.620985\pi\)
−0.370999 + 0.928633i \(0.620985\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 15.0000 1.55543
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.00000 −0.458349
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −24.0000 −2.16401
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) −30.0000 −2.64135
\(130\) 0 0
\(131\) 19.0000 1.66004 0.830019 − 0.557735i \(-0.188330\pi\)
0.830019 + 0.557735i \(0.188330\pi\)
\(132\) 0 0
\(133\) 7.00000 0.606977
\(134\) 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
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −18.0000 −1.48461
\(148\) 0 0
\(149\) −11.0000 −0.901155 −0.450578 − 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 − 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 0 0
\(153\) −30.0000 −2.42536
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.0000 0.877896 0.438948 − 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 − 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.0000 −0.851206 −0.425603 − 0.904910i \(-0.639938\pi\)
−0.425603 + 0.904910i \(0.639938\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 42.0000 3.21182
\(172\) 0 0
\(173\) −26.0000 −1.97674 −0.988372 − 0.152057i \(-0.951410\pi\)
−0.988372 + 0.152057i \(0.951410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −36.0000 −2.70593
\(178\) 0 0
\(179\) 18.0000 1.34538 0.672692 − 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 15.0000 1.10883
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5.00000 −0.365636
\(188\) 0 0
\(189\) 9.00000 0.654654
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −1.00000 −0.0719816 −0.0359908 − 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 19.0000 1.34687 0.673437 − 0.739244i \(-0.264817\pi\)
0.673437 + 0.739244i \(0.264817\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 48.0000 3.33623
\(208\) 0 0
\(209\) 7.00000 0.484200
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 − 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 0 0
\(213\) 21.0000 1.43890
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.00000 0.339422
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 − 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 12.0000 0.792982 0.396491 − 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) 3.00000 0.196537 0.0982683 − 0.995160i \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 12.0000 0.779484
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 − 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −14.0000 −0.883672 −0.441836 − 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.0000 −0.998053 −0.499026 − 0.866587i \(-0.666309\pi\)
−0.499026 + 0.866587i \(0.666309\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) 18.0000 1.11417
\(262\) 0 0
\(263\) −5.00000 −0.308313 −0.154157 − 0.988046i \(-0.549266\pi\)
−0.154157 + 0.988046i \(0.549266\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −21.0000 −1.28518
\(268\) 0 0
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 − 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −16.0000 −0.961347 −0.480673 − 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) 0 0
\(279\) 30.0000 1.79605
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 24.0000 1.40690
\(292\) 0 0
\(293\) −32.0000 −1.86946 −0.934730 − 0.355359i \(-0.884359\pi\)
−0.934730 + 0.355359i \(0.884359\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −10.0000 −0.576390
\(302\) 0 0
\(303\) 18.0000 1.03407
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 − 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 24.0000 1.36531
\(310\) 0 0
\(311\) 17.0000 0.963982 0.481991 − 0.876176i \(-0.339914\pi\)
0.481991 + 0.876176i \(0.339914\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 − 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.00000 −0.393159 −0.196580 − 0.980488i \(-0.562983\pi\)
−0.196580 + 0.980488i \(0.562983\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −35.0000 −1.94745
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 30.0000 1.65900
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −22.0000 −1.20923 −0.604615 − 0.796518i \(-0.706673\pi\)
−0.604615 + 0.796518i \(0.706673\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.00000 0.0544735 0.0272367 − 0.999629i \(-0.491329\pi\)
0.0272367 + 0.999629i \(0.491329\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 5.00000 0.270765
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.00000 −0.107366 −0.0536828 − 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 − 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 − 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −15.0000 −0.793884
\(358\) 0 0
\(359\) 4.00000 0.211112 0.105556 − 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 3.00000 0.157459
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) −48.0000 −2.49878
\(370\) 0 0
\(371\) −1.00000 −0.0519174
\(372\) 0 0
\(373\) 28.0000 1.44979 0.724893 − 0.688862i \(-0.241889\pi\)
0.724893 + 0.688862i \(0.241889\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 − 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) −48.0000 −2.45911
\(382\) 0 0
\(383\) 2.00000 0.102195 0.0510976 − 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −60.0000 −3.04997
\(388\) 0 0
\(389\) −34.0000 −1.72387 −0.861934 − 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) −40.0000 −2.02289
\(392\) 0 0
\(393\) 57.0000 2.87527
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 − 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 21.0000 1.05131
\(400\) 0 0
\(401\) 11.0000 0.549314 0.274657 − 0.961542i \(-0.411436\pi\)
0.274657 + 0.961542i \(0.411436\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.00000 −0.0495682
\(408\) 0 0
\(409\) 16.0000 0.791149 0.395575 − 0.918434i \(-0.370545\pi\)
0.395575 + 0.918434i \(0.370545\pi\)
\(410\) 0 0
\(411\) 36.0000 1.77575
\(412\) 0 0
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −36.0000 −1.76293
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −24.0000 −1.16969 −0.584844 − 0.811146i \(-0.698844\pi\)
−0.584844 + 0.811146i \(0.698844\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.00000 0.241967
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 − 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 − 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 56.0000 2.67884
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 − 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) −36.0000 −1.71429
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −33.0000 −1.56085
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 − 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 18.0000 0.845714
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −27.0000 −1.26301 −0.631503 − 0.775373i \(-0.717562\pi\)
−0.631503 + 0.775373i \(0.717562\pi\)
\(458\) 0 0
\(459\) −45.0000 −2.10042
\(460\) 0 0
\(461\) 29.0000 1.35066 0.675332 − 0.737514i \(-0.264000\pi\)
0.675332 + 0.737514i \(0.264000\pi\)
\(462\) 0 0
\(463\) 12.0000 0.557687 0.278844 − 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.0000 1.06431 0.532157 − 0.846646i \(-0.321382\pi\)
0.532157 + 0.846646i \(0.321382\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 33.0000 1.52056
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −6.00000 −0.274147 −0.137073 − 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 24.0000 1.09204
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −6.00000 −0.271886 −0.135943 − 0.990717i \(-0.543406\pi\)
−0.135943 + 0.990717i \(0.543406\pi\)
\(488\) 0 0
\(489\) −3.00000 −0.135665
\(490\) 0 0
\(491\) −29.0000 −1.30875 −0.654376 − 0.756169i \(-0.727069\pi\)
−0.654376 + 0.756169i \(0.727069\pi\)
\(492\) 0 0
\(493\) −15.0000 −0.675566
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.00000 0.313993
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −33.0000 −1.47433
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −39.0000 −1.73205
\(508\) 0 0
\(509\) −24.0000 −1.06378 −0.531891 − 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) 63.0000 2.78152
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −78.0000 −3.42382
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −44.0000 −1.92399 −0.961993 − 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −25.0000 −1.08902
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −72.0000 −3.12453
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 54.0000 2.33027
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 39.0000 1.67674 0.838370 − 0.545101i \(-0.183509\pi\)
0.838370 + 0.545101i \(0.183509\pi\)
\(542\) 0 0
\(543\) 30.0000 1.28742
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 − 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 0 0
\(549\) 30.0000 1.28037
\(550\) 0 0
\(551\) 21.0000 0.894630
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −32.0000 −1.35588 −0.677942 − 0.735116i \(-0.737128\pi\)
−0.677942 + 0.735116i \(0.737128\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −23.0000 −0.962520 −0.481260 − 0.876578i \(-0.659821\pi\)
−0.481260 + 0.876578i \(0.659821\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 28.0000 1.16566 0.582828 − 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) 0 0
\(579\) −3.00000 −0.124676
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.00000 −0.0414158
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.0000 1.85735 0.928674 − 0.370896i \(-0.120949\pi\)
0.928674 + 0.370896i \(0.120949\pi\)
\(588\) 0 0
\(589\) 35.0000 1.44215
\(590\) 0 0
\(591\) −54.0000 −2.22126
\(592\) 0 0
\(593\) −14.0000 −0.574911 −0.287456 − 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 57.0000 2.33285
\(598\) 0 0
\(599\) 1.00000 0.0408589 0.0204294 − 0.999791i \(-0.493497\pi\)
0.0204294 + 0.999791i \(0.493497\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 24.0000 0.977356
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −21.0000 −0.852364 −0.426182 − 0.904638i \(-0.640142\pi\)
−0.426182 + 0.904638i \(0.640142\pi\)
\(608\) 0 0
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) 0 0
\(612\) 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
\(617\) −8.00000 −0.322068 −0.161034 − 0.986949i \(-0.551483\pi\)
−0.161034 + 0.986949i \(0.551483\pi\)
\(618\) 0 0
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 72.0000 2.88926
\(622\) 0 0
\(623\) −7.00000 −0.280449
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 21.0000 0.838659
\(628\) 0 0
\(629\) 5.00000 0.199363
\(630\) 0 0
\(631\) −9.00000 −0.358284 −0.179142 − 0.983823i \(-0.557332\pi\)
−0.179142 + 0.983823i \(0.557332\pi\)
\(632\) 0 0
\(633\) −3.00000 −0.119239
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 42.0000 1.66149
\(640\) 0 0
\(641\) −17.0000 −0.671460 −0.335730 − 0.941958i \(-0.608983\pi\)
−0.335730 + 0.941958i \(0.608983\pi\)
\(642\) 0 0
\(643\) −29.0000 −1.14365 −0.571824 − 0.820376i \(-0.693764\pi\)
−0.571824 + 0.820376i \(0.693764\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 15.0000 0.587896
\(652\) 0 0
\(653\) −35.0000 −1.36966 −0.684828 − 0.728705i \(-0.740123\pi\)
−0.684828 + 0.728705i \(0.740123\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) 17.0000 0.662226 0.331113 − 0.943591i \(-0.392576\pi\)
0.331113 + 0.943591i \(0.392576\pi\)
\(660\) 0 0
\(661\) 4.00000 0.155582 0.0777910 − 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) 5.00000 0.193023
\(672\) 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
\(676\) 0 0
\(677\) 36.0000 1.38359 0.691796 − 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) 36.0000 1.37952
\(682\) 0 0
\(683\) −35.0000 −1.33924 −0.669619 − 0.742705i \(-0.733543\pi\)
−0.669619 + 0.742705i \(0.733543\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 36.0000 1.37349
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 − 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 40.0000 1.51511
\(698\) 0 0
\(699\) 9.00000 0.340411
\(700\) 0 0
\(701\) −35.0000 −1.32193 −0.660966 − 0.750416i \(-0.729853\pi\)
−0.660966 + 0.750416i \(0.729853\pi\)
\(702\) 0 0
\(703\) −7.00000 −0.264010
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 − 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) 24.0000 0.900070
\(712\) 0 0
\(713\) 40.0000 1.49801
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −72.0000 −2.68889
\(718\) 0 0
\(719\) 21.0000 0.783168 0.391584 − 0.920142i \(-0.371927\pi\)
0.391584 + 0.920142i \(0.371927\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 30.0000 1.11264 0.556319 − 0.830969i \(-0.312213\pi\)
0.556319 + 0.830969i \(0.312213\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 50.0000 1.84932
\(732\) 0 0
\(733\) 30.0000 1.10808 0.554038 − 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 − 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −51.0000 −1.87101 −0.935504 − 0.353315i \(-0.885054\pi\)
−0.935504 + 0.353315i \(0.885054\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 25.0000 0.912263 0.456131 − 0.889912i \(-0.349235\pi\)
0.456131 + 0.889912i \(0.349235\pi\)
\(752\) 0 0
\(753\) −42.0000 −1.53057
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 − 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) −28.0000 −1.01500 −0.507500 − 0.861652i \(-0.669430\pi\)
−0.507500 + 0.861652i \(0.669430\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 − 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) −48.0000 −1.72868
\(772\) 0 0
\(773\) 51.0000 1.83434 0.917171 − 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −3.00000 −0.107624
\(778\) 0 0
\(779\) −56.0000 −2.00641
\(780\) 0 0
\(781\) 7.00000 0.250480
\(782\) 0 0
\(783\) 27.0000 0.964901
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −18.0000 −0.641631 −0.320815 − 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) 0 0
\(789\) −15.0000 −0.534014
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10.0000 −0.354218 −0.177109 − 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −42.0000 −1.48400
\(802\) 0 0
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −38.0000 −1.33601 −0.668004 − 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) 49.0000 1.72062 0.860311 − 0.509769i \(-0.170269\pi\)
0.860311 + 0.509769i \(0.170269\pi\)
\(812\) 0 0
\(813\) −6.00000 −0.210429
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −70.0000 −2.44899
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.0000 0.907406 0.453703 − 0.891153i \(-0.350103\pi\)
0.453703 + 0.891153i \(0.350103\pi\)
\(822\) 0 0
\(823\) 50.0000 1.74289 0.871445 − 0.490493i \(-0.163183\pi\)
0.871445 + 0.490493i \(0.163183\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 18.0000 0.625921 0.312961 − 0.949766i \(-0.398679\pi\)
0.312961 + 0.949766i \(0.398679\pi\)
\(828\) 0 0
\(829\) 44.0000 1.52818 0.764092 − 0.645108i \(-0.223188\pi\)
0.764092 + 0.645108i \(0.223188\pi\)
\(830\) 0 0
\(831\) −48.0000 −1.66510
\(832\) 0 0
\(833\) 30.0000 1.03944
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 45.0000 1.55543
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) −40.0000 −1.36957 −0.684787 − 0.728743i \(-0.740105\pi\)
−0.684787 + 0.728743i \(0.740105\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −17.0000 −0.580709 −0.290354 − 0.956919i \(-0.593773\pi\)
−0.290354 + 0.956919i \(0.593773\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 − 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) 36.0000 1.22545 0.612727 − 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 24.0000 0.815083
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 48.0000 1.62455
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 0 0
\(879\) −96.0000 −3.23800
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 − 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 29.0000 0.975928 0.487964 − 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.0000 0.537227 0.268614 − 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 9.00000 0.301511
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) 5.00000 0.166574
\(902\) 0 0
\(903\) −30.0000 −0.998337
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 11.0000 0.365249 0.182625 − 0.983183i \(-0.441541\pi\)
0.182625 + 0.983183i \(0.441541\pi\)
\(908\) 0 0
\(909\) 36.0000 1.19404
\(910\) 0 0
\(911\) −57.0000 −1.88849 −0.944247 − 0.329238i \(-0.893208\pi\)
−0.944247 + 0.329238i \(0.893208\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 19.0000 0.627435
\(918\) 0 0
\(919\) −22.0000 −0.725713 −0.362857 − 0.931845i \(-0.618198\pi\)
−0.362857 + 0.931845i \(0.618198\pi\)
\(920\) 0 0
\(921\) −48.0000 −1.58165
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 48.0000 1.57653
\(928\) 0 0
\(929\) 47.0000 1.54202 0.771010 − 0.636823i \(-0.219752\pi\)
0.771010 + 0.636823i \(0.219752\pi\)
\(930\) 0 0
\(931\) −42.0000 −1.37649
\(932\) 0 0
\(933\) 51.0000 1.66967
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 − 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) −18.0000 −0.587408
\(940\) 0 0
\(941\) 31.0000 1.01057 0.505286 − 0.862952i \(-0.331387\pi\)
0.505286 + 0.862952i \(0.331387\pi\)
\(942\) 0 0
\(943\) −64.0000 −2.08413
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −15.0000 −0.487435 −0.243717 − 0.969846i \(-0.578367\pi\)
−0.243717 + 0.969846i \(0.578367\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −21.0000 −0.680972
\(952\) 0 0
\(953\) 29.0000 0.939402 0.469701 − 0.882826i \(-0.344362\pi\)
0.469701 + 0.882826i \(0.344362\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 9.00000 0.290929
\(958\) 0 0
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −49.0000 −1.57573 −0.787867 − 0.615846i \(-0.788815\pi\)
−0.787867 + 0.615846i \(0.788815\pi\)
\(968\) 0 0
\(969\) −105.000 −3.37309
\(970\) 0 0
\(971\) 16.0000 0.513464 0.256732 − 0.966483i \(-0.417354\pi\)
0.256732 + 0.966483i \(0.417354\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8.00000 −0.255943 −0.127971 − 0.991778i \(-0.540847\pi\)
−0.127971 + 0.991778i \(0.540847\pi\)
\(978\) 0 0
\(979\) −7.00000 −0.223721
\(980\) 0 0
\(981\) 60.0000 1.91565
\(982\) 0 0
\(983\) −46.0000 −1.46717 −0.733586 − 0.679597i \(-0.762155\pi\)
−0.733586 + 0.679597i \(0.762155\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −80.0000 −2.54385
\(990\) 0 0
\(991\) 48.0000 1.52477 0.762385 − 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) 0 0
\(993\) −66.0000 −2.09445
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 46.0000 1.45683 0.728417 − 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) 0 0
\(999\) −9.00000 −0.284747
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 4400.2.a.bd.1.1 1
4.3 odd 2 550.2.a.h.1.1 1
5.2 odd 4 880.2.b.f.529.1 2
5.3 odd 4 880.2.b.f.529.2 2
5.4 even 2 4400.2.a.b.1.1 1
12.11 even 2 4950.2.a.h.1.1 1
20.3 even 4 110.2.b.c.89.1 ✓ 2
20.7 even 4 110.2.b.c.89.2 yes 2
20.19 odd 2 550.2.a.g.1.1 1
44.43 even 2 6050.2.a.b.1.1 1
60.23 odd 4 990.2.c.b.199.2 2
60.47 odd 4 990.2.c.b.199.1 2
60.59 even 2 4950.2.a.bn.1.1 1
220.43 odd 4 1210.2.b.e.969.2 2
220.87 odd 4 1210.2.b.e.969.1 2
220.219 even 2 6050.2.a.bo.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.b.c.89.1 ✓ 2 20.3 even 4
110.2.b.c.89.2 yes 2 20.7 even 4
550.2.a.g.1.1 1 20.19 odd 2
550.2.a.h.1.1 1 4.3 odd 2
880.2.b.f.529.1 2 5.2 odd 4
880.2.b.f.529.2 2 5.3 odd 4
990.2.c.b.199.1 2 60.47 odd 4
990.2.c.b.199.2 2 60.23 odd 4
1210.2.b.e.969.1 2 220.87 odd 4
1210.2.b.e.969.2 2 220.43 odd 4
4400.2.a.b.1.1 1 5.4 even 2
4400.2.a.bd.1.1 1 1.1 even 1 trivial
4950.2.a.h.1.1 1 12.11 even 2
4950.2.a.bn.1.1 1 60.59 even 2
6050.2.a.b.1.1 1 44.43 even 2
6050.2.a.bo.1.1 1 220.219 even 2