↑

Properties

Label 1600.2.a.a.1.1
Level $1600$
Weight $2$
Character 1600.1
Self dual yes
Analytic conductor $12.776$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1600,2,Mod(1,1600)] 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("1600.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,0,0,-2,0,6,0,-1,0,4] 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: \(12.7760643234\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1600.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -2.00000 q^{7} +6.00000 q^{9} -1.00000 q^{11} +4.00000 q^{13} -5.00000 q^{17} -1.00000 q^{19} +6.00000 q^{21} +2.00000 q^{23} -9.00000 q^{27} +8.00000 q^{29} +10.0000 q^{31} +3.00000 q^{33} -6.00000 q^{37} -12.0000 q^{39} -3.00000 q^{41} +4.00000 q^{43} -4.00000 q^{47} -3.00000 q^{49} +15.0000 q^{51} +6.00000 q^{53} +3.00000 q^{57} -8.00000 q^{59} -10.0000 q^{61} -12.0000 q^{63} -1.00000 q^{67} -6.00000 q^{69} -12.0000 q^{71} -3.00000 q^{73} +2.00000 q^{77} +6.00000 q^{79} +9.00000 q^{81} -13.0000 q^{83} -24.0000 q^{87} -9.00000 q^{89} -8.00000 q^{91} -30.0000 q^{93} +14.0000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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 \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 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
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 − 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −12.0000 −1.92154
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 15.0000 2.10042
\(52\) 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
\(57\) 3.00000 0.397360
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −12.0000 −1.51186
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 −0.122169 −0.0610847 − 0.998133i \(-0.519456\pi\)
−0.0610847 + 0.998133i \(0.519456\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −3.00000 −0.351123 −0.175562 − 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −13.0000 −1.42694 −0.713468 − 0.700688i \(-0.752876\pi\)
−0.713468 + 0.700688i \(0.752876\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −24.0000 −2.57307
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) −30.0000 −3.11086
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.0000 −1.45010 −0.725052 − 0.688694i \(-0.758184\pi\)
−0.725052 + 0.688694i \(0.758184\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 18.0000 1.70848
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 − 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 24.0000 2.21880
\(118\) 0 0
\(119\) 10.0000 0.916698
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 9.00000 0.811503
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 − 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 0 0
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.0000 −1.11066 −0.555332 − 0.831628i \(-0.687409\pi\)
−0.555332 + 0.831628i \(0.687409\pi\)
\(138\) 0 0
\(139\) −9.00000 −0.763370 −0.381685 − 0.924292i \(-0.624656\pi\)
−0.381685 + 0.924292i \(0.624656\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.00000 0.742307
\(148\) 0 0
\(149\) −8.00000 −0.655386 −0.327693 − 0.944784i \(-0.606271\pi\)
−0.327693 + 0.944784i \(0.606271\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 − 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) −30.0000 −2.42536
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) −18.0000 −1.42749
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 7.00000 0.548282 0.274141 − 0.961689i \(-0.411606\pi\)
0.274141 + 0.961689i \(0.411606\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 0 0
\(173\) 8.00000 0.608229 0.304114 − 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 24.0000 1.80395
\(178\) 0 0
\(179\) 19.0000 1.42013 0.710063 − 0.704138i \(-0.248666\pi\)
0.710063 + 0.704138i \(0.248666\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 30.0000 2.21766
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.00000 0.365636
\(188\) 0 0
\(189\) 18.0000 1.30931
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 − 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 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
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −28.0000 −1.98487 −0.992434 − 0.122782i \(-0.960818\pi\)
−0.992434 + 0.122782i \(0.960818\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) 0 0
\(203\) −16.0000 −1.12298
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 12.0000 0.834058
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 1.00000 0.0688428 0.0344214 − 0.999407i \(-0.489041\pi\)
0.0344214 + 0.999407i \(0.489041\pi\)
\(212\) 0 0
\(213\) 36.0000 2.46668
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −20.0000 −1.35769
\(218\) 0 0
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) −20.0000 −1.34535
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 − 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 − 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −18.0000 −1.16923
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −7.00000 −0.450910 −0.225455 − 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 39.0000 2.47152
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 − 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −30.0000 −1.87135 −0.935674 − 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 48.0000 2.97113
\(262\) 0 0
\(263\) −30.0000 −1.84988 −0.924940 − 0.380114i \(-0.875885\pi\)
−0.924940 + 0.380114i \(0.875885\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 27.0000 1.65237
\(268\) 0 0
\(269\) 16.0000 0.975537 0.487769 − 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 0 0
\(271\) −6.00000 −0.364474 −0.182237 − 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) 0 0
\(273\) 24.0000 1.45255
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 − 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) 60.0000 3.59211
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 − 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −1.00000 −0.0594438 −0.0297219 − 0.999558i \(-0.509462\pi\)
−0.0297219 + 0.999558i \(0.509462\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −42.0000 −2.46208
\(292\) 0 0
\(293\) −2.00000 −0.116841 −0.0584206 − 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 0 0
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 18.0000 1.03407
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 13.0000 0.741949 0.370975 − 0.928643i \(-0.379024\pi\)
0.370975 + 0.928643i \(0.379024\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 14.0000 0.793867 0.396934 − 0.917847i \(-0.370074\pi\)
0.396934 + 0.917847i \(0.370074\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 − 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 − 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 45.0000 2.51166
\(322\) 0 0
\(323\) 5.00000 0.278207
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 42.0000 2.32261
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 27.0000 1.48405 0.742027 − 0.670370i \(-0.233865\pi\)
0.742027 + 0.670370i \(0.233865\pi\)
\(332\) 0 0
\(333\) −36.0000 −1.97279
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −19.0000 −1.03500 −0.517498 − 0.855684i \(-0.673136\pi\)
−0.517498 + 0.855684i \(0.673136\pi\)
\(338\) 0 0
\(339\) −27.0000 −1.46644
\(340\) 0 0
\(341\) −10.0000 −0.541530
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.0000 0.912608 0.456304 − 0.889824i \(-0.349173\pi\)
0.456304 + 0.889824i \(0.349173\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 − 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) −36.0000 −1.92154
\(352\) 0 0
\(353\) 10.0000 0.532246 0.266123 − 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −30.0000 −1.58777
\(358\) 0 0
\(359\) −18.0000 −0.950004 −0.475002 − 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 30.0000 1.57459
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −20.0000 −1.04399 −0.521996 − 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 32.0000 1.64808
\(378\) 0 0
\(379\) 5.00000 0.256833 0.128416 − 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 − 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 24.0000 1.21999
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −10.0000 −0.505722
\(392\) 0 0
\(393\) −36.0000 −1.81596
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.00000 0.200754 0.100377 − 0.994949i \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) 21.0000 1.04869 0.524345 − 0.851506i \(-0.324310\pi\)
0.524345 + 0.851506i \(0.324310\pi\)
\(402\) 0 0
\(403\) 40.0000 1.99254
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) −11.0000 −0.543915 −0.271957 − 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) 0 0
\(411\) 39.0000 1.92373
\(412\) 0 0
\(413\) 16.0000 0.787309
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 27.0000 1.32220
\(418\) 0 0
\(419\) 3.00000 0.146560 0.0732798 − 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 0 0
\(421\) 12.0000 0.584844 0.292422 − 0.956289i \(-0.405539\pi\)
0.292422 + 0.956289i \(0.405539\pi\)
\(422\) 0 0
\(423\) −24.0000 −1.16692
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) 0 0
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) −34.0000 −1.63772 −0.818861 − 0.573992i \(-0.805394\pi\)
−0.818861 + 0.573992i \(0.805394\pi\)
\(432\) 0 0
\(433\) −5.00000 −0.240285 −0.120142 − 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.00000 −0.0956730
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 − 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) −21.0000 −0.997740 −0.498870 − 0.866677i \(-0.666252\pi\)
−0.498870 + 0.866677i \(0.666252\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 24.0000 1.13516
\(448\) 0 0
\(449\) −39.0000 −1.84052 −0.920262 − 0.391303i \(-0.872024\pi\)
−0.920262 + 0.391303i \(0.872024\pi\)
\(450\) 0 0
\(451\) 3.00000 0.141264
\(452\) 0 0
\(453\) −6.00000 −0.281905
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 −0.0467780 −0.0233890 − 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) 0 0
\(459\) 45.0000 2.10042
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 − 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 42.0000 1.93526
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 36.0000 1.64833
\(478\) 0 0
\(479\) 22.0000 1.00521 0.502603 − 0.864517i \(-0.332376\pi\)
0.502603 + 0.864517i \(0.332376\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 0 0
\(483\) 12.0000 0.546019
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 38.0000 1.72194 0.860972 − 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) 0 0
\(489\) −21.0000 −0.949653
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 − 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) −40.0000 −1.80151
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −36.0000 −1.60836
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 − 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −34.0000 −1.50702 −0.753512 − 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 9.00000 0.397360
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −35.0000 −1.53338 −0.766689 − 0.642019i \(-0.778097\pi\)
−0.766689 + 0.642019i \(0.778097\pi\)
\(522\) 0 0
\(523\) −35.0000 −1.53044 −0.765222 − 0.643767i \(-0.777371\pi\)
−0.765222 + 0.643767i \(0.777371\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −50.0000 −2.17803
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) −48.0000 −2.08302
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −57.0000 −2.45973
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 − 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) 0 0
\(543\) 6.00000 0.257485
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −19.0000 −0.812381 −0.406191 − 0.913788i \(-0.633143\pi\)
−0.406191 + 0.913788i \(0.633143\pi\)
\(548\) 0 0
\(549\) −60.0000 −2.56074
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −28.0000 −1.18640 −0.593199 − 0.805056i \(-0.702135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) −32.0000 −1.34864 −0.674320 − 0.738440i \(-0.735563\pi\)
−0.674320 + 0.738440i \(0.735563\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −18.0000 −0.755929
\(568\) 0 0
\(569\) 17.0000 0.712677 0.356339 − 0.934357i \(-0.384025\pi\)
0.356339 + 0.934357i \(0.384025\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 30.0000 1.25327
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13.0000 0.541197 0.270599 − 0.962692i \(-0.412778\pi\)
0.270599 + 0.962692i \(0.412778\pi\)
\(578\) 0 0
\(579\) −33.0000 −1.37143
\(580\) 0 0
\(581\) 26.0000 1.07866
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.00000 0.123823 0.0619116 − 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) −39.0000 −1.60154 −0.800769 − 0.598973i \(-0.795576\pi\)
−0.800769 + 0.598973i \(0.795576\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 84.0000 3.43789
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 − 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 43.0000 1.75401 0.877003 − 0.480484i \(-0.159539\pi\)
0.877003 + 0.480484i \(0.159539\pi\)
\(602\) 0 0
\(603\) −6.00000 −0.244339
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 48.0000 1.94506
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 30.0000 1.21169 0.605844 − 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −44.0000 −1.76851 −0.884255 − 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) −18.0000 −0.722315
\(622\) 0 0
\(623\) 18.0000 0.721155
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −3.00000 −0.119808
\(628\) 0 0
\(629\) 30.0000 1.19618
\(630\) 0 0
\(631\) 10.0000 0.398094 0.199047 − 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(632\) 0 0
\(633\) −3.00000 −0.119239
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) −72.0000 −2.84828
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 − 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 − 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 60.0000 2.35159
\(652\) 0 0
\(653\) −34.0000 −1.33052 −0.665261 − 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −18.0000 −0.702247
\(658\) 0 0
\(659\) 21.0000 0.818044 0.409022 − 0.912525i \(-0.365870\pi\)
0.409022 + 0.912525i \(0.365870\pi\)
\(660\) 0 0
\(661\) −24.0000 −0.933492 −0.466746 − 0.884391i \(-0.654574\pi\)
−0.466746 + 0.884391i \(0.654574\pi\)
\(662\) 0 0
\(663\) 60.0000 2.33021
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 0 0
\(669\) −12.0000 −0.463947
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) 6.00000 0.231283 0.115642 − 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −32.0000 −1.22986 −0.614930 − 0.788582i \(-0.710816\pi\)
−0.614930 + 0.788582i \(0.710816\pi\)
\(678\) 0 0
\(679\) −28.0000 −1.07454
\(680\) 0 0
\(681\) 36.0000 1.37952
\(682\) 0 0
\(683\) −3.00000 −0.114792 −0.0573959 − 0.998351i \(-0.518280\pi\)
−0.0573959 + 0.998351i \(0.518280\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) 0 0
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 19.0000 0.722794 0.361397 − 0.932412i \(-0.382300\pi\)
0.361397 + 0.932412i \(0.382300\pi\)
\(692\) 0 0
\(693\) 12.0000 0.455842
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 15.0000 0.568166
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 28.0000 1.05755 0.528773 − 0.848763i \(-0.322652\pi\)
0.528773 + 0.848763i \(0.322652\pi\)
\(702\) 0 0
\(703\) 6.00000 0.226294
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) 28.0000 1.05156 0.525781 − 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) 0 0
\(711\) 36.0000 1.35011
\(712\) 0 0
\(713\) 20.0000 0.749006
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −72.0000 −2.68889
\(718\) 0 0
\(719\) −14.0000 −0.522112 −0.261056 − 0.965324i \(-0.584071\pi\)
−0.261056 + 0.965324i \(0.584071\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 21.0000 0.780998
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) 0 0
\(733\) 36.0000 1.32969 0.664845 − 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.00000 0.0368355
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 − 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 12.0000 0.440831
\(742\) 0 0
\(743\) −30.0000 −1.10059 −0.550297 − 0.834969i \(-0.685485\pi\)
−0.550297 + 0.834969i \(0.685485\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −78.0000 −2.85387
\(748\) 0 0
\(749\) 30.0000 1.09618
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 − 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) 45.0000 1.63989
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 32.0000 1.16306 0.581530 − 0.813525i \(-0.302454\pi\)
0.581530 + 0.813525i \(0.302454\pi\)
\(758\) 0 0
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) −3.00000 −0.108750 −0.0543750 − 0.998521i \(-0.517317\pi\)
−0.0543750 + 0.998521i \(0.517317\pi\)
\(762\) 0 0
\(763\) 28.0000 1.01367
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −32.0000 −1.15545
\(768\) 0 0
\(769\) 13.0000 0.468792 0.234396 − 0.972141i \(-0.424689\pi\)
0.234396 + 0.972141i \(0.424689\pi\)
\(770\) 0 0
\(771\) 90.0000 3.24127
\(772\) 0 0
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −36.0000 −1.29149
\(778\) 0 0
\(779\) 3.00000 0.107486
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) −72.0000 −2.57307
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −52.0000 −1.85360 −0.926800 − 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) 0 0
\(789\) 90.0000 3.20408
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −40.0000 −1.42044
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.0000 1.48772 0.743858 − 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 20.0000 0.707549
\(800\) 0 0
\(801\) −54.0000 −1.90800
\(802\) 0 0
\(803\) 3.00000 0.105868
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −48.0000 −1.68968
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 − 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) 18.0000 0.631288
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 0 0
\(819\) −48.0000 −1.67726
\(820\) 0 0
\(821\) 38.0000 1.32621 0.663105 − 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 0 0
\(823\) −28.0000 −0.976019 −0.488009 − 0.872838i \(-0.662277\pi\)
−0.488009 + 0.872838i \(0.662277\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.00000 0.104320 0.0521601 − 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) 0 0
\(829\) 20.0000 0.694629 0.347314 − 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) −24.0000 −0.832551
\(832\) 0 0
\(833\) 15.0000 0.519719
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −90.0000 −3.11086
\(838\) 0 0
\(839\) −16.0000 −0.552381 −0.276191 − 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 6.00000 0.206651
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 20.0000 0.687208
\(848\) 0 0
\(849\) 3.00000 0.102960
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) −22.0000 −0.753266 −0.376633 − 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.00000 0.102478 0.0512390 − 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) −41.0000 −1.39890 −0.699451 − 0.714681i \(-0.746572\pi\)
−0.699451 + 0.714681i \(0.746572\pi\)
\(860\) 0 0
\(861\) −18.0000 −0.613438
\(862\) 0 0
\(863\) −48.0000 −1.63394 −0.816970 − 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −24.0000 −0.815083
\(868\) 0 0
\(869\) −6.00000 −0.203536
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) 84.0000 2.84297
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4.00000 0.135070 0.0675352 − 0.997717i \(-0.478487\pi\)
0.0675352 + 0.997717i \(0.478487\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 − 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 5.00000 0.168263 0.0841317 − 0.996455i \(-0.473188\pi\)
0.0841317 + 0.996455i \(0.473188\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −28.0000 −0.940148 −0.470074 − 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) 0 0
\(893\) 4.00000 0.133855
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 0 0
\(899\) 80.0000 2.66815
\(900\) 0 0
\(901\) −30.0000 −0.999445
\(902\) 0 0
\(903\) 24.0000 0.798670
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 0 0
\(909\) −36.0000 −1.19404
\(910\) 0 0
\(911\) 28.0000 0.927681 0.463841 − 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 0 0
\(913\) 13.0000 0.430237
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −24.0000 −0.792550
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 − 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) −39.0000 −1.28509
\(922\) 0 0
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −24.0000 −0.788263
\(928\) 0 0
\(929\) −2.00000 −0.0656179 −0.0328089 − 0.999462i \(-0.510445\pi\)
−0.0328089 + 0.999462i \(0.510445\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 0 0
\(933\) −42.0000 −1.37502
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.00000 0.163343 0.0816714 − 0.996659i \(-0.473974\pi\)
0.0816714 + 0.996659i \(0.473974\pi\)
\(938\) 0 0
\(939\) −66.0000 −2.15383
\(940\) 0 0
\(941\) −4.00000 −0.130396 −0.0651981 − 0.997872i \(-0.520768\pi\)
−0.0651981 + 0.997872i \(0.520768\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40.0000 1.29983 0.649913 − 0.760009i \(-0.274805\pi\)
0.649913 + 0.760009i \(0.274805\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) −36.0000 −1.16738
\(952\) 0 0
\(953\) 39.0000 1.26333 0.631667 − 0.775240i \(-0.282371\pi\)
0.631667 + 0.775240i \(0.282371\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 24.0000 0.775810
\(958\) 0 0
\(959\) 26.0000 0.839584
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) −90.0000 −2.90021
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) −15.0000 −0.481869
\(970\) 0 0
\(971\) 31.0000 0.994837 0.497419 − 0.867511i \(-0.334281\pi\)
0.497419 + 0.867511i \(0.334281\pi\)
\(972\) 0 0
\(973\) 18.0000 0.577054
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −17.0000 −0.543878 −0.271939 − 0.962314i \(-0.587665\pi\)
−0.271939 + 0.962314i \(0.587665\pi\)
\(978\) 0 0
\(979\) 9.00000 0.287641
\(980\) 0 0
\(981\) −84.0000 −2.68191
\(982\) 0 0
\(983\) −6.00000 −0.191370 −0.0956851 − 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 38.0000 1.20711 0.603555 − 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) 0 0
\(993\) −81.0000 −2.57046
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −12.0000 −0.380044 −0.190022 − 0.981780i \(-0.560856\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) 0 0
\(999\) 54.0000 1.70848
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 1600.2.a.a.1.1 1
4.3 odd 2 1600.2.a.y.1.1 1
5.2 odd 4 1600.2.c.a.449.2 2
5.3 odd 4 1600.2.c.a.449.1 2
5.4 even 2 1600.2.a.x.1.1 1
8.3 odd 2 400.2.a.a.1.1 1
8.5 even 2 200.2.a.e.1.1 yes 1
20.3 even 4 1600.2.c.b.449.2 2
20.7 even 4 1600.2.c.b.449.1 2
20.19 odd 2 1600.2.a.b.1.1 1
24.5 odd 2 1800.2.a.h.1.1 1
24.11 even 2 3600.2.a.bf.1.1 1
40.3 even 4 400.2.c.a.49.1 2
40.13 odd 4 200.2.c.a.49.2 2
40.19 odd 2 400.2.a.h.1.1 1
40.27 even 4 400.2.c.a.49.2 2
40.29 even 2 200.2.a.a.1.1 ✓ 1
40.37 odd 4 200.2.c.a.49.1 2
56.13 odd 2 9800.2.a.c.1.1 1
120.29 odd 2 1800.2.a.r.1.1 1
120.53 even 4 1800.2.f.f.649.2 2
120.59 even 2 3600.2.a.m.1.1 1
120.77 even 4 1800.2.f.f.649.1 2
120.83 odd 4 3600.2.f.n.2449.1 2
120.107 odd 4 3600.2.f.n.2449.2 2
280.69 odd 2 9800.2.a.bq.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.a.a.1.1 ✓ 1 40.29 even 2
200.2.a.e.1.1 yes 1 8.5 even 2
200.2.c.a.49.1 2 40.37 odd 4
200.2.c.a.49.2 2 40.13 odd 4
400.2.a.a.1.1 1 8.3 odd 2
400.2.a.h.1.1 1 40.19 odd 2
400.2.c.a.49.1 2 40.3 even 4
400.2.c.a.49.2 2 40.27 even 4
1600.2.a.a.1.1 1 1.1 even 1 trivial
1600.2.a.b.1.1 1 20.19 odd 2
1600.2.a.x.1.1 1 5.4 even 2
1600.2.a.y.1.1 1 4.3 odd 2
1600.2.c.a.449.1 2 5.3 odd 4
1600.2.c.a.449.2 2 5.2 odd 4
1600.2.c.b.449.1 2 20.7 even 4
1600.2.c.b.449.2 2 20.3 even 4
1800.2.a.h.1.1 1 24.5 odd 2
1800.2.a.r.1.1 1 120.29 odd 2
1800.2.f.f.649.1 2 120.77 even 4
1800.2.f.f.649.2 2 120.53 even 4
3600.2.a.m.1.1 1 120.59 even 2
3600.2.a.bf.1.1 1 24.11 even 2
3600.2.f.n.2449.1 2 120.83 odd 4
3600.2.f.n.2449.2 2 120.107 odd 4
9800.2.a.c.1.1 1 56.13 odd 2
9800.2.a.bq.1.1 1 280.69 odd 2