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Properties

Label 1600.2.a.r.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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(1,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.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: \(12.7760643234\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 800)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.00000 q^{7} -2.00000 q^{9} -5.00000 q^{11} -5.00000 q^{17} -5.00000 q^{19} +2.00000 q^{21} -6.00000 q^{23} -5.00000 q^{27} -4.00000 q^{29} +10.0000 q^{31} -5.00000 q^{33} -10.0000 q^{37} +5.00000 q^{41} +4.00000 q^{43} +8.00000 q^{47} -3.00000 q^{49} -5.00000 q^{51} -10.0000 q^{53} -5.00000 q^{57} +10.0000 q^{61} -4.00000 q^{63} +3.00000 q^{67} -6.00000 q^{69} +5.00000 q^{73} -10.0000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -1.00000 q^{83} -4.00000 q^{87} -9.00000 q^{89} +10.0000 q^{93} -10.0000 q^{97} +10.0000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 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
\(19\) −5.00000 −1.14708 −0.573539 − 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 − 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 − 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 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
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.00000 0.366508 0.183254 − 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 5.00000 0.585206 0.292603 − 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.0000 −1.13961
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.00000 −0.109764 −0.0548821 − 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 10.0000 1.03695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 10.0000 1.00504
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 − 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.00000 −0.290021 −0.145010 − 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 − 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −10.0000 −0.916698
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 5.00000 0.450835
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −18.0000 −1.59724 −0.798621 − 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 − 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) −10.0000 −0.867110
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.00000 −0.427179 −0.213589 − 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 − 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 0 0
\(149\) 20.0000 1.63846 0.819232 − 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 − 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 10.0000 0.808452
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −21.0000 −1.64485 −0.822423 − 0.568876i \(-0.807379\pi\)
−0.822423 + 0.568876i \(0.807379\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 10.0000 0.764719
\(172\) 0 0
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.0000 1.12115 0.560576 − 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 − 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 25.0000 1.82818
\(188\) 0 0
\(189\) −10.0000 −0.727393
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 − 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) −5.00000 −0.359908 −0.179954 − 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 − 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) 0 0
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 12.0000 0.834058
\(208\) 0 0
\(209\) 25.0000 1.72929
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 − 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 20.0000 1.35769
\(218\) 0 0
\(219\) 5.00000 0.337869
\(220\) 0 0
\(221\) 0 0
\(222\) 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 \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 − 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) −10.0000 −0.657952
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 − 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 − 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −15.0000 −0.966235 −0.483117 − 0.875556i \(-0.660496\pi\)
−0.483117 + 0.875556i \(0.660496\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1.00000 −0.0633724
\(250\) 0 0
\(251\) 5.00000 0.315597 0.157799 − 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) 0 0
\(253\) 30.0000 1.88608
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.0000 0.623783 0.311891 − 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) 0 0
\(259\) −20.0000 −1.24274
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) 0 0
\(263\) −14.0000 −0.863277 −0.431638 − 0.902047i \(-0.642064\pi\)
−0.431638 + 0.902047i \(0.642064\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −9.00000 −0.550791
\(268\) 0 0
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 10.0000 0.607457 0.303728 − 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 20.0000 1.20168 0.600842 − 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) 0 0
\(279\) −20.0000 −1.19737
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 − 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −29.0000 −1.72387 −0.861936 − 0.507018i \(-0.830748\pi\)
−0.861936 + 0.507018i \(0.830748\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.0000 0.590281
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) −30.0000 −1.75262 −0.876309 − 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 25.0000 1.45065
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) −2.00000 −0.114897
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 − 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 10.0000 0.567048 0.283524 − 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 0 0
\(313\) 30.0000 1.69570 0.847850 − 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.0000 1.12331 0.561656 − 0.827371i \(-0.310164\pi\)
0.561656 + 0.827371i \(0.310164\pi\)
\(318\) 0 0
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 25.0000 1.39104
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −10.0000 −0.553001
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 − 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) 0 0
\(333\) 20.0000 1.09599
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5.00000 0.272367 0.136184 − 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) 0 0
\(339\) −15.0000 −0.814688
\(340\) 0 0
\(341\) −50.0000 −2.70765
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −27.0000 −1.44944 −0.724718 − 0.689046i \(-0.758030\pi\)
−0.724718 + 0.689046i \(0.758030\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 − 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −10.0000 −0.529256
\(358\) 0 0
\(359\) 10.0000 0.527780 0.263890 − 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −12.0000 −0.626395 −0.313197 − 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −20.0000 −1.03835
\(372\) 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
\(377\) 0 0
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 − 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) 14.0000 0.715367 0.357683 − 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) 10.0000 0.507020 0.253510 − 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 30.0000 1.51717
\(392\) 0 0
\(393\) 20.0000 1.00887
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.0000 1.00377 0.501886 − 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) 0 0
\(399\) −10.0000 −0.500626
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 − 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 50.0000 2.47841
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 − 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) −5.00000 −0.246632
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 − 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 − 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) −16.0000 −0.777947
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.0000 0.481683 0.240842 − 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 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
\(437\) 30.0000 1.43509
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 39.0000 1.85295 0.926473 − 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 20.0000 0.945968
\(448\) 0 0
\(449\) 25.0000 1.17982 0.589911 − 0.807468i \(-0.299163\pi\)
0.589911 + 0.807468i \(0.299163\pi\)
\(450\) 0 0
\(451\) −25.0000 −1.17720
\(452\) 0 0
\(453\) −10.0000 −0.469841
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 15.0000 0.701670 0.350835 − 0.936437i \(-0.385898\pi\)
0.350835 + 0.936437i \(0.385898\pi\)
\(458\) 0 0
\(459\) 25.0000 1.16690
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 − 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 − 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) 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
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) −20.0000 −0.919601
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 20.0000 0.915737
\(478\) 0 0
\(479\) 30.0000 1.37073 0.685367 − 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 − 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) −21.0000 −0.949653
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 0 0
\(503\) −4.00000 −0.178351 −0.0891756 − 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 − 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 0 0
\(513\) 25.0000 1.10378
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −40.0000 −1.75920
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 − 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) 1.00000 0.0437269 0.0218635 − 0.999761i \(-0.493040\pi\)
0.0218635 + 0.999761i \(0.493040\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −50.0000 −2.17803
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.0000 0.647298
\(538\) 0 0
\(539\) 15.0000 0.646096
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 − 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 0 0
\(543\) −18.0000 −0.772454
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −23.0000 −0.983409 −0.491704 − 0.870762i \(-0.663626\pi\)
−0.491704 + 0.870762i \(0.663626\pi\)
\(548\) 0 0
\(549\) −20.0000 −0.853579
\(550\) 0 0
\(551\) 20.0000 0.852029
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −40.0000 −1.69485 −0.847427 − 0.530912i \(-0.821850\pi\)
−0.847427 + 0.530912i \(0.821850\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 25.0000 1.05550
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) −15.0000 −0.628833 −0.314416 − 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) −10.0000 −0.417756
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 5.00000 0.208153 0.104076 − 0.994569i \(-0.466811\pi\)
0.104076 + 0.994569i \(0.466811\pi\)
\(578\) 0 0
\(579\) −5.00000 −0.207793
\(580\) 0 0
\(581\) −2.00000 −0.0829740
\(582\) 0 0
\(583\) 50.0000 2.07079
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.00000 0.288921 0.144460 − 0.989511i \(-0.453855\pi\)
0.144460 + 0.989511i \(0.453855\pi\)
\(588\) 0 0
\(589\) −50.0000 −2.06021
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 0 0
\(593\) −15.0000 −0.615976 −0.307988 − 0.951390i \(-0.599656\pi\)
−0.307988 + 0.951390i \(0.599656\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 − 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −5.00000 −0.203954 −0.101977 − 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) 0 0
\(603\) −6.00000 −0.244339
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 − 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 − 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 30.0000 1.20386
\(622\) 0 0
\(623\) −18.0000 −0.721155
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 25.0000 0.998404
\(628\) 0 0
\(629\) 50.0000 1.99363
\(630\) 0 0
\(631\) −30.0000 −1.19428 −0.597141 − 0.802137i \(-0.703697\pi\)
−0.597141 + 0.802137i \(0.703697\pi\)
\(632\) 0 0
\(633\) 5.00000 0.198732
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 36.0000 1.41970 0.709851 − 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 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
\(649\) 0 0
\(650\) 0 0
\(651\) 20.0000 0.783862
\(652\) 0 0
\(653\) −30.0000 −1.17399 −0.586995 − 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 25.0000 0.973862 0.486931 − 0.873441i \(-0.338116\pi\)
0.486931 + 0.873441i \(0.338116\pi\)
\(660\) 0 0
\(661\) 20.0000 0.777910 0.388955 − 0.921257i \(-0.372836\pi\)
0.388955 + 0.921257i \(0.372836\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) −50.0000 −1.93023
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 − 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −40.0000 −1.53732 −0.768662 − 0.639655i \(-0.779077\pi\)
−0.768662 + 0.639655i \(0.779077\pi\)
\(678\) 0 0
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −31.0000 −1.18618 −0.593091 − 0.805135i \(-0.702093\pi\)
−0.593091 + 0.805135i \(0.702093\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4.00000 0.152610
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 15.0000 0.570627 0.285313 − 0.958434i \(-0.407902\pi\)
0.285313 + 0.958434i \(0.407902\pi\)
\(692\) 0 0
\(693\) 20.0000 0.759737
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −25.0000 −0.946943
\(698\) 0 0
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 50.0000 1.88579
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) −44.0000 −1.65245 −0.826227 − 0.563337i \(-0.809517\pi\)
−0.826227 + 0.563337i \(0.809517\pi\)
\(710\) 0 0
\(711\) −20.0000 −0.750059
\(712\) 0 0
\(713\) −60.0000 −2.24702
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −20.0000 −0.746914
\(718\) 0 0
\(719\) −10.0000 −0.372937 −0.186469 − 0.982461i \(-0.559704\pi\)
−0.186469 + 0.982461i \(0.559704\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 0 0
\(723\) −15.0000 −0.557856
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 − 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) 0 0
\(733\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −15.0000 −0.552532
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 − 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.00000 0.0731762
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 − 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 0 0
\(753\) 5.00000 0.182210
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 40.0000 1.45382 0.726912 − 0.686730i \(-0.240955\pi\)
0.726912 + 0.686730i \(0.240955\pi\)
\(758\) 0 0
\(759\) 30.0000 1.08893
\(760\) 0 0
\(761\) 37.0000 1.34125 0.670624 − 0.741797i \(-0.266026\pi\)
0.670624 + 0.741797i \(0.266026\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −51.0000 −1.83911 −0.919554 − 0.392965i \(-0.871449\pi\)
−0.919554 + 0.392965i \(0.871449\pi\)
\(770\) 0 0
\(771\) 10.0000 0.360141
\(772\) 0 0
\(773\) −20.0000 −0.719350 −0.359675 − 0.933078i \(-0.617112\pi\)
−0.359675 + 0.933078i \(0.617112\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −20.0000 −0.717496
\(778\) 0 0
\(779\) −25.0000 −0.895718
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 20.0000 0.714742
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 − 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 0 0
\(789\) −14.0000 −0.498413
\(790\) 0 0
\(791\) −30.0000 −1.06668
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) −25.0000 −0.882231
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 10.0000 0.350715
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −20.0000 −0.699711
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 − 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) −36.0000 −1.25488 −0.627441 − 0.778664i \(-0.715897\pi\)
−0.627441 + 0.778664i \(0.715897\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 23.0000 0.799788 0.399894 − 0.916561i \(-0.369047\pi\)
0.399894 + 0.916561i \(0.369047\pi\)
\(828\) 0 0
\(829\) 40.0000 1.38926 0.694629 − 0.719368i \(-0.255569\pi\)
0.694629 + 0.719368i \(0.255569\pi\)
\(830\) 0 0
\(831\) 20.0000 0.693792
\(832\) 0 0
\(833\) 15.0000 0.519719
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −50.0000 −1.72825
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −10.0000 −0.344418
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 28.0000 0.962091
\(848\) 0 0
\(849\) −29.0000 −0.995277
\(850\) 0 0
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 − 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −45.0000 −1.53717 −0.768585 − 0.639747i \(-0.779039\pi\)
−0.768585 + 0.639747i \(0.779039\pi\)
\(858\) 0 0
\(859\) −5.00000 −0.170598 −0.0852989 − 0.996355i \(-0.527185\pi\)
−0.0852989 + 0.996355i \(0.527185\pi\)
\(860\) 0 0
\(861\) 10.0000 0.340799
\(862\) 0 0
\(863\) 4.00000 0.136162 0.0680808 − 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −50.0000 −1.69613
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 20.0000 0.676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20.0000 −0.675352 −0.337676 − 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) 0 0
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) −10.0000 −0.336909 −0.168454 − 0.985709i \(-0.553878\pi\)
−0.168454 + 0.985709i \(0.553878\pi\)
\(882\) 0 0
\(883\) 1.00000 0.0336527 0.0168263 − 0.999858i \(-0.494644\pi\)
0.0168263 + 0.999858i \(0.494644\pi\)
\(884\) 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
\(889\) −36.0000 −1.20740
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) −40.0000 −1.33855
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −40.0000 −1.33407
\(900\) 0 0
\(901\) 50.0000 1.66574
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 − 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 0 0
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 − 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 0 0
\(913\) 5.00000 0.165476
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 40.0000 1.32092
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 − 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −32.0000 −1.05102
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 − 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 15.0000 0.491605
\(932\) 0 0
\(933\) 10.0000 0.327385
\(934\) 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
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 − 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 0 0
\(943\) −30.0000 −0.976934
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −32.0000 −1.03986 −0.519930 − 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 20.0000 0.648544
\(952\) 0 0
\(953\) −25.0000 −0.809829 −0.404915 − 0.914354i \(-0.632699\pi\)
−0.404915 + 0.914354i \(0.632699\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.0000 0.646508
\(958\) 0 0
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 0 0
\(969\) 25.0000 0.803116
\(970\) 0 0
\(971\) −45.0000 −1.44412 −0.722059 − 0.691831i \(-0.756804\pi\)
−0.722059 + 0.691831i \(0.756804\pi\)
\(972\) 0 0
\(973\) −10.0000 −0.320585
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.0000 0.479893 0.239946 − 0.970786i \(-0.422870\pi\)
0.239946 + 0.970786i \(0.422870\pi\)
\(978\) 0 0
\(979\) 45.0000 1.43821
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) 0 0
\(983\) 46.0000 1.46717 0.733586 − 0.679597i \(-0.237845\pi\)
0.733586 + 0.679597i \(0.237845\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 16.0000 0.509286
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 10.0000 0.317660 0.158830 − 0.987306i \(-0.449228\pi\)
0.158830 + 0.987306i \(0.449228\pi\)
\(992\) 0 0
\(993\) −25.0000 −0.793351
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −60.0000 −1.90022 −0.950110 − 0.311916i \(-0.899029\pi\)
−0.950110 + 0.311916i \(0.899029\pi\)
\(998\) 0 0
\(999\) 50.0000 1.58193
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.r.1.1 1
4.3 odd 2 1600.2.a.h.1.1 1
5.2 odd 4 1600.2.c.g.449.1 2
5.3 odd 4 1600.2.c.g.449.2 2
5.4 even 2 1600.2.a.g.1.1 1
8.3 odd 2 800.2.a.g.1.1 yes 1
8.5 even 2 800.2.a.c.1.1 yes 1
20.3 even 4 1600.2.c.j.449.1 2
20.7 even 4 1600.2.c.j.449.2 2
20.19 odd 2 1600.2.a.s.1.1 1
24.5 odd 2 7200.2.a.bm.1.1 1
24.11 even 2 7200.2.a.o.1.1 1
40.3 even 4 800.2.c.c.449.2 2
40.13 odd 4 800.2.c.d.449.1 2
40.19 odd 2 800.2.a.b.1.1 ✓ 1
40.27 even 4 800.2.c.c.449.1 2
40.29 even 2 800.2.a.h.1.1 yes 1
40.37 odd 4 800.2.c.d.449.2 2
120.29 odd 2 7200.2.a.k.1.1 1
120.53 even 4 7200.2.f.a.6049.1 2
120.59 even 2 7200.2.a.bq.1.1 1
120.77 even 4 7200.2.f.a.6049.2 2
120.83 odd 4 7200.2.f.bc.6049.2 2
120.107 odd 4 7200.2.f.bc.6049.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.2.a.b.1.1 ✓ 1 40.19 odd 2
800.2.a.c.1.1 yes 1 8.5 even 2
800.2.a.g.1.1 yes 1 8.3 odd 2
800.2.a.h.1.1 yes 1 40.29 even 2
800.2.c.c.449.1 2 40.27 even 4
800.2.c.c.449.2 2 40.3 even 4
800.2.c.d.449.1 2 40.13 odd 4
800.2.c.d.449.2 2 40.37 odd 4
1600.2.a.g.1.1 1 5.4 even 2
1600.2.a.h.1.1 1 4.3 odd 2
1600.2.a.r.1.1 1 1.1 even 1 trivial
1600.2.a.s.1.1 1 20.19 odd 2
1600.2.c.g.449.1 2 5.2 odd 4
1600.2.c.g.449.2 2 5.3 odd 4
1600.2.c.j.449.1 2 20.3 even 4
1600.2.c.j.449.2 2 20.7 even 4
7200.2.a.k.1.1 1 120.29 odd 2
7200.2.a.o.1.1 1 24.11 even 2
7200.2.a.bm.1.1 1 24.5 odd 2
7200.2.a.bq.1.1 1 120.59 even 2
7200.2.f.a.6049.1 2 120.53 even 4
7200.2.f.a.6049.2 2 120.77 even 4
7200.2.f.bc.6049.1 2 120.107 odd 4
7200.2.f.bc.6049.2 2 120.83 odd 4