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Properties

Label 9200.2.a.ba.1.1
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9200,2,Mod(1,9200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.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: \(73.4623698596\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 92)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.00000 q^{7} -2.00000 q^{9} +1.00000 q^{13} +6.00000 q^{17} -2.00000 q^{19} +2.00000 q^{21} -1.00000 q^{23} -5.00000 q^{27} -3.00000 q^{29} -5.00000 q^{31} -8.00000 q^{37} +1.00000 q^{39} +3.00000 q^{41} +8.00000 q^{43} +9.00000 q^{47} -3.00000 q^{49} +6.00000 q^{51} -6.00000 q^{53} -2.00000 q^{57} +12.0000 q^{59} +14.0000 q^{61} -4.00000 q^{63} +8.00000 q^{67} -1.00000 q^{69} +15.0000 q^{71} +7.00000 q^{73} +10.0000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -3.00000 q^{87} +2.00000 q^{91} -5.00000 q^{93} +10.0000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 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
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 − 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 15.0000 1.78017 0.890086 − 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 − 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −5.00000 −0.518476
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 − 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 − 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 − 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 3.00000 0.270501
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.0000 0.976092 0.488046 − 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −21.0000 −1.83478 −0.917389 − 0.397991i \(-0.869707\pi\)
−0.917389 + 0.397991i \(0.869707\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 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
\(139\) −11.0000 −0.933008 −0.466504 − 0.884519i \(-0.654487\pi\)
−0.466504 + 0.884519i \(0.654487\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 7.00000 0.569652 0.284826 − 0.958579i \(-0.408064\pi\)
0.284826 + 0.958579i \(0.408064\pi\)
\(152\) 0 0
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 − 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 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
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 9.00000 0.672692 0.336346 − 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 − 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −10.0000 −0.727393
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 7.00000 0.503871 0.251936 − 0.967744i \(-0.418933\pi\)
0.251936 + 0.967744i \(0.418933\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.0000 1.06871 0.534353 − 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 − 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 − 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) 15.0000 1.02778
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 0 0
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 − 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.00000 0.589610 0.294805 − 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 − 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 − 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 − 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −27.0000 −1.68421 −0.842107 − 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 − 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −15.0000 −0.914566 −0.457283 − 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 − 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 19.0000 1.14160 0.570800 − 0.821089i \(-0.306633\pi\)
0.570800 + 0.821089i \(0.306633\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 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
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) −12.0000 −0.701047 −0.350524 − 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.00000 −0.0578315
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 2.00000 0.113776
\(310\) 0 0
\(311\) 15.0000 0.850572 0.425286 − 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) 0 0
\(313\) 4.00000 0.226093 0.113047 − 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −16.0000 −0.884802
\(328\) 0 0
\(329\) 18.0000 0.992372
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 − 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 0 0
\(333\) 16.0000 0.876795
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 − 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 29.0000 1.55233 0.776167 − 0.630527i \(-0.217161\pi\)
0.776167 + 0.630527i \(0.217161\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) 27.0000 1.43706 0.718532 − 0.695493i \(-0.244814\pi\)
0.718532 + 0.695493i \(0.244814\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 12.0000 0.635107
\(358\) 0 0
\(359\) 30.0000 1.58334 0.791670 − 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 26.0000 1.35719 0.678594 − 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −20.0000 −1.03556 −0.517780 − 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 − 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 11.0000 0.563547
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 − 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −16.0000 −0.813326
\(388\) 0 0
\(389\) −36.0000 −1.82527 −0.912636 − 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) −21.0000 −1.05931
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −11.0000 −0.552074 −0.276037 − 0.961147i \(-0.589021\pi\)
−0.276037 + 0.961147i \(0.589021\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 − 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) −5.00000 −0.249068
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 17.0000 0.840596 0.420298 − 0.907386i \(-0.361926\pi\)
0.420298 + 0.907386i \(0.361926\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −11.0000 −0.538672
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 − 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 − 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) −18.0000 −0.875190
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 28.0000 1.35501
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 − 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 − 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) −35.0000 −1.67046 −0.835229 − 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 21.0000 0.997740 0.498870 − 0.866677i \(-0.333748\pi\)
0.498870 + 0.866677i \(0.333748\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 − 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 7.00000 0.328889
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) −30.0000 −1.40028
\(460\) 0 0
\(461\) −27.0000 −1.25752 −0.628758 − 0.777601i \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.0000 −1.38823 −0.694117 − 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) −42.0000 −1.91903 −0.959514 − 0.281659i \(-0.909115\pi\)
−0.959514 + 0.281659i \(0.909115\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) −2.00000 −0.0910032
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 23.0000 1.04223 0.521115 − 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 0 0
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) −3.00000 −0.135388 −0.0676941 − 0.997706i \(-0.521564\pi\)
−0.0676941 + 0.997706i \(0.521564\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 30.0000 1.34568
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 − 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 − 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) −3.00000 −0.132973 −0.0664863 − 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 0 0
\(513\) 10.0000 0.441511
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −24.0000 −1.05146 −0.525730 − 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 − 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −30.0000 −1.30682
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) 0 0
\(533\) 3.00000 0.129944
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 9.00000 0.388379
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.00000 −0.300954 −0.150477 − 0.988614i \(-0.548081\pi\)
−0.150477 + 0.988614i \(0.548081\pi\)
\(542\) 0 0
\(543\) −16.0000 −0.686626
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 − 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 0 0
\(549\) −28.0000 −1.19501
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −24.0000 −1.01691 −0.508456 − 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 − 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 34.0000 1.42286 0.711428 − 0.702759i \(-0.248049\pi\)
0.711428 + 0.702759i \(0.248049\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 7.00000 0.291414 0.145707 − 0.989328i \(-0.453454\pi\)
0.145707 + 0.989328i \(0.453454\pi\)
\(578\) 0 0
\(579\) 7.00000 0.290910
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.0000 0.866763 0.433381 − 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) 0 0
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 15.0000 0.617018
\(592\) 0 0
\(593\) 42.0000 1.72473 0.862367 − 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 − 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 0 0
\(603\) −16.0000 −0.651570
\(604\) 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
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 9.00000 0.364101
\(612\) 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
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −48.0000 −1.91389
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 16.0000 0.635943
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.00000 −0.118864
\(638\) 0 0
\(639\) −30.0000 −1.18678
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 − 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 33.0000 1.29736 0.648682 − 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −10.0000 −0.391931
\(652\) 0 0
\(653\) −27.0000 −1.05659 −0.528296 − 0.849060i \(-0.677169\pi\)
−0.528296 + 0.849060i \(0.677169\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 − 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 6.00000 0.233021
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.00000 0.116160
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 13.0000 0.501113 0.250557 − 0.968102i \(-0.419386\pi\)
0.250557 + 0.968102i \(0.419386\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) 39.0000 1.49229 0.746147 − 0.665782i \(-0.231902\pi\)
0.746147 + 0.665782i \(0.231902\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) 9.00000 0.340411
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 − 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) −20.0000 −0.750059
\(712\) 0 0
\(713\) 5.00000 0.187251
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −21.0000 −0.784259
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) −4.00000 −0.148762
\(724\) 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
\(731\) 48.0000 1.77534
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 − 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 25.0000 0.919640 0.459820 − 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) 6.00000 0.220119 0.110059 − 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 − 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −20.0000 −0.726912 −0.363456 − 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 − 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) 0 0
\(763\) −32.0000 −1.15848
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) −28.0000 −1.00971 −0.504853 − 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(770\) 0 0
\(771\) −27.0000 −0.972381
\(772\) 0 0
\(773\) −24.0000 −0.863220 −0.431610 − 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −16.0000 −0.573997
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 15.0000 0.536056
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −40.0000 −1.42585 −0.712923 − 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) 0 0
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) −36.0000 −1.28001
\(792\) 0 0
\(793\) 14.0000 0.497155
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 54.0000 1.91038
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −15.0000 −0.528025
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 − 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 7.00000 0.245803 0.122902 − 0.992419i \(-0.460780\pi\)
0.122902 + 0.992419i \(0.460780\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 − 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) 41.0000 1.42917 0.714585 − 0.699549i \(-0.246616\pi\)
0.714585 + 0.699549i \(0.246616\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 − 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 19.0000 0.659103
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 25.0000 0.864126
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 − 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −22.0000 −0.755929
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) −2.00000 −0.0684787 −0.0342393 − 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 33.0000 1.12726 0.563629 − 0.826028i \(-0.309405\pi\)
0.563629 + 0.826028i \(0.309405\pi\)
\(858\) 0 0
\(859\) 19.0000 0.648272 0.324136 − 0.946011i \(-0.394927\pi\)
0.324136 + 0.946011i \(0.394927\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) −9.00000 −0.306364 −0.153182 − 0.988198i \(-0.548952\pi\)
−0.153182 + 0.988198i \(0.548952\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) −20.0000 −0.676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 − 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 0 0
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 − 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 − 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −51.0000 −1.71241 −0.856206 − 0.516634i \(-0.827185\pi\)
−0.856206 + 0.516634i \(0.827185\pi\)
\(888\) 0 0
\(889\) 22.0000 0.737856
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −18.0000 −0.602347
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.00000 −0.0333890
\(898\) 0 0
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 50.0000 1.66022 0.830111 − 0.557598i \(-0.188277\pi\)
0.830111 + 0.557598i \(0.188277\pi\)
\(908\) 0 0
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 − 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −42.0000 −1.38696
\(918\) 0 0
\(919\) −38.0000 −1.25350 −0.626752 − 0.779219i \(-0.715616\pi\)
−0.626752 + 0.779219i \(0.715616\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) 15.0000 0.493731
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) 15.0000 0.492134 0.246067 − 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 15.0000 0.491078
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 28.0000 0.914720 0.457360 − 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(938\) 0 0
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) −3.00000 −0.0976934
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3.00000 −0.0974869 −0.0487435 − 0.998811i \(-0.515522\pi\)
−0.0487435 + 0.998811i \(0.515522\pi\)
\(948\) 0 0
\(949\) 7.00000 0.227230
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 47.0000 1.51142 0.755709 − 0.654907i \(-0.227292\pi\)
0.755709 + 0.654907i \(0.227292\pi\)
\(968\) 0 0
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 − 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 0 0
\(973\) −22.0000 −0.705288
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −48.0000 −1.53566 −0.767828 − 0.640656i \(-0.778662\pi\)
−0.767828 + 0.640656i \(0.778662\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 32.0000 1.02168
\(982\) 0 0
\(983\) −18.0000 −0.574111 −0.287055 − 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 18.0000 0.572946
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 25.0000 0.793351
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −62.0000 −1.96356 −0.981780 − 0.190022i \(-0.939144\pi\)
−0.981780 + 0.190022i \(0.939144\pi\)
\(998\) 0 0
\(999\) 40.0000 1.26554
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 9200.2.a.ba.1.1 1
4.3 odd 2 2300.2.a.c.1.1 1
5.4 even 2 368.2.a.b.1.1 1
15.14 odd 2 3312.2.a.g.1.1 1
20.3 even 4 2300.2.c.f.1749.1 2
20.7 even 4 2300.2.c.f.1749.2 2
20.19 odd 2 92.2.a.b.1.1 ✓ 1
40.19 odd 2 1472.2.a.c.1.1 1
40.29 even 2 1472.2.a.j.1.1 1
60.59 even 2 828.2.a.b.1.1 1
115.114 odd 2 8464.2.a.f.1.1 1
140.139 even 2 4508.2.a.a.1.1 1
460.459 even 2 2116.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
92.2.a.b.1.1 ✓ 1 20.19 odd 2
368.2.a.b.1.1 1 5.4 even 2
828.2.a.b.1.1 1 60.59 even 2
1472.2.a.c.1.1 1 40.19 odd 2
1472.2.a.j.1.1 1 40.29 even 2
2116.2.a.d.1.1 1 460.459 even 2
2300.2.a.c.1.1 1 4.3 odd 2
2300.2.c.f.1749.1 2 20.3 even 4
2300.2.c.f.1749.2 2 20.7 even 4
3312.2.a.g.1.1 1 15.14 odd 2
4508.2.a.a.1.1 1 140.139 even 2
8464.2.a.f.1.1 1 115.114 odd 2
9200.2.a.ba.1.1 1 1.1 even 1 trivial