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Properties

Label 4600.2.a.p.1.1
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
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] = [4600,2,Mod(1,4600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4600, 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("4600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.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: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4600.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +2.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +2.00000 q^{7} +6.00000 q^{9} -1.00000 q^{13} +6.00000 q^{21} -1.00000 q^{23} +9.00000 q^{27} -3.00000 q^{29} +3.00000 q^{31} +8.00000 q^{37} -3.00000 q^{39} +3.00000 q^{41} +2.00000 q^{43} +11.0000 q^{47} -3.00000 q^{49} +14.0000 q^{53} -8.00000 q^{59} -4.00000 q^{61} +12.0000 q^{63} +4.00000 q^{67} -3.00000 q^{69} +7.00000 q^{71} +9.00000 q^{73} +9.00000 q^{81} -4.00000 q^{83} -9.00000 q^{87} -2.00000 q^{89} -2.00000 q^{91} +9.00000 q^{93} -18.0000 q^{97} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 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 \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.0000 1.60451 0.802257 − 0.596978i \(-0.203632\pi\)
0.802257 + 0.596978i \(0.203632\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 14.0000 1.92305 0.961524 − 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 12.0000 1.51186
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 7.00000 0.830747 0.415374 − 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 − 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.00000 −0.964901
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 9.00000 0.933257
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 − 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.0000 1.54678 0.773389 − 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 − 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 24.0000 2.27798
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 9.00000 0.811503
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −11.0000 −0.976092 −0.488046 − 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 − 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.00000 −0.341743 −0.170872 − 0.985293i \(-0.554658\pi\)
−0.170872 + 0.985293i \(0.554658\pi\)
\(138\) 0 0
\(139\) −11.0000 −0.933008 −0.466504 − 0.884519i \(-0.654487\pi\)
−0.466504 + 0.884519i \(0.654487\pi\)
\(140\) 0 0
\(141\) 33.0000 2.77910
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −9.00000 −0.742307
\(148\) 0 0
\(149\) −22.0000 −1.80231 −0.901155 − 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) 7.00000 0.569652 0.284826 − 0.958579i \(-0.408064\pi\)
0.284826 + 0.958579i \(0.408064\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 − 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 0 0
\(159\) 42.0000 3.33082
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) −7.00000 −0.548282 −0.274141 − 0.961689i \(-0.588394\pi\)
−0.274141 + 0.961689i \(0.588394\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −24.0000 −1.80395
\(178\) 0 0
\(179\) 21.0000 1.56961 0.784807 − 0.619740i \(-0.212762\pi\)
0.784807 + 0.619740i \(0.212762\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 − 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 18.0000 1.30931
\(190\) 0 0
\(191\) 2.00000 0.144715 0.0723575 − 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) 0 0
\(193\) 1.00000 0.0719816 0.0359908 − 0.999352i \(-0.488541\pi\)
0.0359908 + 0.999352i \(0.488541\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.00000 −0.213741 −0.106871 − 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 − 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 21.0000 1.43890
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) 27.0000 1.82449
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.00000 −0.132745 −0.0663723 − 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 − 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −21.0000 −1.37576 −0.687878 − 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.00000 −0.0646846 −0.0323423 − 0.999477i \(-0.510297\pi\)
−0.0323423 + 0.999477i \(0.510297\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 − 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.00000 −0.311891 −0.155946 − 0.987766i \(-0.549842\pi\)
−0.155946 + 0.987766i \(0.549842\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) 17.0000 1.03651 0.518254 − 0.855227i \(-0.326582\pi\)
0.518254 + 0.855227i \(0.326582\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 − 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) −6.00000 −0.363137
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 29.0000 1.74244 0.871221 − 0.490892i \(-0.163329\pi\)
0.871221 + 0.490892i \(0.163329\pi\)
\(278\) 0 0
\(279\) 18.0000 1.07763
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −10.0000 −0.594438 −0.297219 − 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −54.0000 −3.16554
\(292\) 0 0
\(293\) 24.0000 1.40209 0.701047 − 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) 54.0000 3.10222
\(304\) 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
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) −29.0000 −1.64444 −0.822220 − 0.569170i \(-0.807264\pi\)
−0.822220 + 0.569170i \(0.807264\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 − 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.0000 0.786318 0.393159 − 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 48.0000 2.67910
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −54.0000 −2.98621
\(328\) 0 0
\(329\) 22.0000 1.21290
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 − 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) 48.0000 2.63038
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 − 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 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 \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −7.00000 −0.374701 −0.187351 − 0.982293i \(-0.559990\pi\)
−0.187351 + 0.982293i \(0.559990\pi\)
\(350\) 0 0
\(351\) −9.00000 −0.480384
\(352\) 0 0
\(353\) −19.0000 −1.01127 −0.505634 − 0.862748i \(-0.668741\pi\)
−0.505634 + 0.862748i \(0.668741\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −33.0000 −1.73205
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) 28.0000 1.45369
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 − 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −33.0000 −1.69064
\(382\) 0 0
\(383\) −30.0000 −1.53293 −0.766464 − 0.642287i \(-0.777986\pi\)
−0.766464 + 0.642287i \(0.777986\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 12.0000 0.609994
\(388\) 0 0
\(389\) −16.0000 −0.811232 −0.405616 − 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −27.0000 −1.36197
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −25.0000 −1.25471 −0.627357 − 0.778732i \(-0.715863\pi\)
−0.627357 + 0.778732i \(0.715863\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 − 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) −3.00000 −0.149441
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −11.0000 −0.543915 −0.271957 − 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 0 0
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −33.0000 −1.61602
\(418\) 0 0
\(419\) 22.0000 1.07477 0.537385 − 0.843337i \(-0.319412\pi\)
0.537385 + 0.843337i \(0.319412\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 66.0000 3.20903
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 − 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 − 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −7.00000 −0.334092 −0.167046 − 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) −33.0000 −1.56788 −0.783939 − 0.620838i \(-0.786792\pi\)
−0.783939 + 0.620838i \(0.786792\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −66.0000 −3.12169
\(448\) 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
\(453\) 21.0000 0.986666
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.00000 0.187112 0.0935561 − 0.995614i \(-0.470177\pi\)
0.0935561 + 0.995614i \(0.470177\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 13.0000 0.605470 0.302735 − 0.953075i \(-0.402100\pi\)
0.302735 + 0.953075i \(0.402100\pi\)
\(462\) 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
\(467\) −42.0000 −1.94353 −0.971764 − 0.235954i \(-0.924178\pi\)
−0.971764 + 0.235954i \(0.924178\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 84.0000 3.84610
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 − 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) −6.00000 −0.273009
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 25.0000 1.13286 0.566429 − 0.824110i \(-0.308325\pi\)
0.566429 + 0.824110i \(0.308325\pi\)
\(488\) 0 0
\(489\) −21.0000 −0.949653
\(490\) 0 0
\(491\) −31.0000 −1.39901 −0.699505 − 0.714628i \(-0.746596\pi\)
−0.699505 + 0.714628i \(0.746596\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.0000 0.627986
\(498\) 0 0
\(499\) −25.0000 −1.11915 −0.559577 − 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 0 0
\(501\) −48.0000 −2.14448
\(502\) 0 0
\(503\) 14.0000 0.624229 0.312115 − 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −36.0000 −1.59882
\(508\) 0 0
\(509\) 21.0000 0.930809 0.465404 − 0.885098i \(-0.345909\pi\)
0.465404 + 0.885098i \(0.345909\pi\)
\(510\) 0 0
\(511\) 18.0000 0.796273
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −42.0000 −1.84360
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 − 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 0 0
\(523\) −42.0000 −1.83653 −0.918266 − 0.395964i \(-0.870410\pi\)
−0.918266 + 0.395964i \(0.870410\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −48.0000 −2.08302
\(532\) 0 0
\(533\) −3.00000 −0.129944
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 63.0000 2.71865
\(538\) 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
\(543\) 36.0000 1.54491
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −35.0000 −1.49649 −0.748246 − 0.663421i \(-0.769104\pi\)
−0.748246 + 0.663421i \(0.769104\pi\)
\(548\) 0 0
\(549\) −24.0000 −1.02430
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14.0000 −0.593199 −0.296600 − 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.0000 0.674320 0.337160 − 0.941447i \(-0.390534\pi\)
0.337160 + 0.941447i \(0.390534\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 18.0000 0.755929
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 − 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 6.00000 0.250654
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 9.00000 0.374675 0.187337 − 0.982296i \(-0.440014\pi\)
0.187337 + 0.982296i \(0.440014\pi\)
\(578\) 0 0
\(579\) 3.00000 0.124676
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −33.0000 −1.36206 −0.681028 − 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −9.00000 −0.370211
\(592\) 0 0
\(593\) −34.0000 −1.39621 −0.698106 − 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −12.0000 −0.491127
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 37.0000 1.50926 0.754631 − 0.656150i \(-0.227816\pi\)
0.754631 + 0.656150i \(0.227816\pi\)
\(602\) 0 0
\(603\) 24.0000 0.977356
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 − 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) −18.0000 −0.729397
\(610\) 0 0
\(611\) −11.0000 −0.445012
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 − 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 48.0000 1.93241 0.966204 − 0.257780i \(-0.0829910\pi\)
0.966204 + 0.257780i \(0.0829910\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) −9.00000 −0.361158
\(622\) 0 0
\(623\) −4.00000 −0.160257
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 − 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 0 0
\(633\) −48.0000 −1.90783
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) 42.0000 1.66149
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 − 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) −34.0000 −1.34083 −0.670415 − 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 39.0000 1.53325 0.766624 − 0.642096i \(-0.221935\pi\)
0.766624 + 0.642096i \(0.221935\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 18.0000 0.705476
\(652\) 0 0
\(653\) 3.00000 0.117399 0.0586995 − 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 54.0000 2.10674
\(658\) 0 0
\(659\) 8.00000 0.311636 0.155818 − 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 − 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.00000 0.116160
\(668\) 0 0
\(669\) 48.0000 1.85579
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −13.0000 −0.501113 −0.250557 − 0.968102i \(-0.580614\pi\)
−0.250557 + 0.968102i \(0.580614\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 46.0000 1.76792 0.883962 − 0.467559i \(-0.154866\pi\)
0.883962 + 0.467559i \(0.154866\pi\)
\(678\) 0 0
\(679\) −36.0000 −1.38155
\(680\) 0 0
\(681\) −6.00000 −0.229920
\(682\) 0 0
\(683\) −35.0000 −1.33924 −0.669619 − 0.742705i \(-0.733543\pi\)
−0.669619 + 0.742705i \(0.733543\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) 0 0
\(689\) −14.0000 −0.533358
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 − 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −63.0000 −2.38288
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 − 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.0000 1.35392
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.00000 −0.112351
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3.00000 −0.112037
\(718\) 0 0
\(719\) 28.0000 1.04422 0.522112 − 0.852877i \(-0.325144\pi\)
0.522112 + 0.852877i \(0.325144\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 6.00000 0.223142
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.00000 0.222528 0.111264 − 0.993791i \(-0.464510\pi\)
0.111264 + 0.993791i \(0.464510\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 8.00000 0.295487 0.147743 − 0.989026i \(-0.452799\pi\)
0.147743 + 0.989026i \(0.452799\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 17.0000 0.625355 0.312678 − 0.949859i \(-0.398774\pi\)
0.312678 + 0.949859i \(0.398774\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −24.0000 −0.878114
\(748\) 0 0
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 − 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) 0 0
\(753\) 48.0000 1.74922
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 29.0000 1.05125 0.525625 − 0.850717i \(-0.323832\pi\)
0.525625 + 0.850717i \(0.323832\pi\)
\(762\) 0 0
\(763\) −36.0000 −1.30329
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 − 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −15.0000 −0.540212
\(772\) 0 0
\(773\) −14.0000 −0.503545 −0.251773 − 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 48.0000 1.72199
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −27.0000 −0.964901
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 − 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 0 0
\(789\) 36.0000 1.28163
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 − 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −12.0000 −0.423999
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 51.0000 1.79529
\(808\) 0 0
\(809\) 26.0000 0.914111 0.457056 − 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) −5.00000 −0.175574 −0.0877869 − 0.996139i \(-0.527979\pi\)
−0.0877869 + 0.996139i \(0.527979\pi\)
\(812\) 0 0
\(813\) −60.0000 −2.10429
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −12.0000 −0.419314
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) −33.0000 −1.15031 −0.575154 − 0.818045i \(-0.695058\pi\)
−0.575154 + 0.818045i \(0.695058\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 − 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 − 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 87.0000 3.01800
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 27.0000 0.933257
\(838\) 0 0
\(839\) 6.00000 0.207143 0.103572 − 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 18.0000 0.619953
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −22.0000 −0.755929
\(848\) 0 0
\(849\) −30.0000 −1.02960
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 − 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −41.0000 −1.40053 −0.700267 − 0.713881i \(-0.746936\pi\)
−0.700267 + 0.713881i \(0.746936\pi\)
\(858\) 0 0
\(859\) −17.0000 −0.580033 −0.290016 − 0.957022i \(-0.593661\pi\)
−0.290016 + 0.957022i \(0.593661\pi\)
\(860\) 0 0
\(861\) 18.0000 0.613438
\(862\) 0 0
\(863\) 17.0000 0.578687 0.289343 − 0.957225i \(-0.406563\pi\)
0.289343 + 0.957225i \(0.406563\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −51.0000 −1.73205
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) −108.000 −3.65525
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 − 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 0 0
\(879\) 72.0000 2.42850
\(880\) 0 0
\(881\) 36.0000 1.21287 0.606435 − 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 15.0000 0.503651 0.251825 − 0.967773i \(-0.418969\pi\)
0.251825 + 0.967773i \(0.418969\pi\)
\(888\) 0 0
\(889\) −22.0000 −0.737856
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.00000 0.100167
\(898\) 0 0
\(899\) −9.00000 −0.300167
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 12.0000 0.399335
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −32.0000 −1.06254 −0.531271 − 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 0 0
\(909\) 108.000 3.58213
\(910\) 0 0
\(911\) −44.0000 −1.45779 −0.728893 − 0.684628i \(-0.759965\pi\)
−0.728893 + 0.684628i \(0.759965\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) −50.0000 −1.64935 −0.824674 − 0.565608i \(-0.808641\pi\)
−0.824674 + 0.565608i \(0.808641\pi\)
\(920\) 0 0
\(921\) 60.0000 1.97707
\(922\) 0 0
\(923\) −7.00000 −0.230408
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 24.0000 0.788263
\(928\) 0 0
\(929\) 19.0000 0.623370 0.311685 − 0.950186i \(-0.399107\pi\)
0.311685 + 0.950186i \(0.399107\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −87.0000 −2.84825
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 44.0000 1.43742 0.718709 − 0.695311i \(-0.244734\pi\)
0.718709 + 0.695311i \(0.244734\pi\)
\(938\) 0 0
\(939\) 60.0000 1.95803
\(940\) 0 0
\(941\) 12.0000 0.391189 0.195594 − 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) 0 0
\(943\) −3.00000 −0.0976934
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 47.0000 1.52729 0.763647 − 0.645634i \(-0.223407\pi\)
0.763647 + 0.645634i \(0.223407\pi\)
\(948\) 0 0
\(949\) −9.00000 −0.292152
\(950\) 0 0
\(951\) 42.0000 1.36194
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 − 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 96.0000 3.09356
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −43.0000 −1.38279 −0.691393 − 0.722478i \(-0.743003\pi\)
−0.691393 + 0.722478i \(0.743003\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −14.0000 −0.449281 −0.224641 − 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) 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
\(981\) −108.000 −3.44817
\(982\) 0 0
\(983\) 14.0000 0.446531 0.223265 − 0.974758i \(-0.428328\pi\)
0.223265 + 0.974758i \(0.428328\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 66.0000 2.10080
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) −21.0000 −0.666415
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −58.0000 −1.83688 −0.918439 − 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) 0 0
\(999\) 72.0000 2.27798
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 4600.2.a.p.1.1 1
4.3 odd 2 9200.2.a.c.1.1 1
5.2 odd 4 4600.2.e.b.4049.1 2
5.3 odd 4 4600.2.e.b.4049.2 2
5.4 even 2 920.2.a.a.1.1 ✓ 1
15.14 odd 2 8280.2.a.d.1.1 1
20.19 odd 2 1840.2.a.i.1.1 1
40.19 odd 2 7360.2.a.a.1.1 1
40.29 even 2 7360.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.a.1.1 ✓ 1 5.4 even 2
1840.2.a.i.1.1 1 20.19 odd 2
4600.2.a.p.1.1 1 1.1 even 1 trivial
4600.2.e.b.4049.1 2 5.2 odd 4
4600.2.e.b.4049.2 2 5.3 odd 4
7360.2.a.a.1.1 1 40.19 odd 2
7360.2.a.ba.1.1 1 40.29 even 2
8280.2.a.d.1.1 1 15.14 odd 2
9200.2.a.c.1.1 1 4.3 odd 2