Properties

Label 5040.2.k.a.1889.3
Level $5040$
Weight $2$
Character 5040.1889
Analytic conductor $40.245$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5040,2,Mod(1889,5040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5040.1889");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.k (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.3
Root \(-1.58114 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 5040.1889
Dual form 5040.2.k.a.1889.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607i q^{5} +(-1.58114 + 2.12132i) q^{7} +O(q^{10})\) \(q+2.23607i q^{5} +(-1.58114 + 2.12132i) q^{7} +1.41421i q^{11} +3.16228 q^{13} +4.47214i q^{17} -6.00000 q^{23} -5.00000 q^{25} +2.82843i q^{29} +(-4.74342 - 3.53553i) q^{35} +4.24264i q^{37} -9.48683 q^{41} +8.48528i q^{43} -4.47214i q^{47} +(-2.00000 - 6.70820i) q^{49} +6.00000 q^{53} -3.16228 q^{55} +9.48683 q^{59} -13.4164i q^{61} +7.07107i q^{65} +5.65685i q^{71} -6.32456 q^{73} +(-3.00000 - 2.23607i) q^{77} +4.00000 q^{79} +8.94427i q^{83} -10.0000 q^{85} +9.48683 q^{89} +(-5.00000 + 6.70820i) q^{91} +12.6491 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 24 q^{23} - 20 q^{25} - 8 q^{49} + 24 q^{53} - 12 q^{77} + 16 q^{79} - 40 q^{85} - 20 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times\).

\(n\) \(2017\) \(2801\) \(3151\) \(3601\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) −1.58114 + 2.12132i −0.597614 + 0.801784i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421i 0.426401i 0.977008 + 0.213201i \(0.0683888\pi\)
−0.977008 + 0.213201i \(0.931611\pi\)
\(12\) 0 0
\(13\) 3.16228 0.877058 0.438529 0.898717i \(-0.355500\pi\)
0.438529 + 0.898717i \(0.355500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.47214i 1.08465i 0.840168 + 0.542326i \(0.182456\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.82843i 0.525226i 0.964901 + 0.262613i \(0.0845842\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.74342 3.53553i −0.801784 0.597614i
\(36\) 0 0
\(37\) 4.24264i 0.697486i 0.937218 + 0.348743i \(0.113391\pi\)
−0.937218 + 0.348743i \(0.886609\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.48683 −1.48159 −0.740797 0.671729i \(-0.765552\pi\)
−0.740797 + 0.671729i \(0.765552\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.47214i 0.652328i −0.945313 0.326164i \(-0.894244\pi\)
0.945313 0.326164i \(-0.105756\pi\)
\(48\) 0 0
\(49\) −2.00000 6.70820i −0.285714 0.958315i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −3.16228 −0.426401
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.48683 1.23508 0.617540 0.786539i \(-0.288129\pi\)
0.617540 + 0.786539i \(0.288129\pi\)
\(60\) 0 0
\(61\) 13.4164i 1.71780i −0.512148 0.858898i \(-0.671150\pi\)
0.512148 0.858898i \(-0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.07107i 0.877058i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.65685i 0.671345i 0.941979 + 0.335673i \(0.108964\pi\)
−0.941979 + 0.335673i \(0.891036\pi\)
\(72\) 0 0
\(73\) −6.32456 −0.740233 −0.370117 0.928985i \(-0.620682\pi\)
−0.370117 + 0.928985i \(0.620682\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.00000 2.23607i −0.341882 0.254824i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.94427i 0.981761i 0.871227 + 0.490881i \(0.163325\pi\)
−0.871227 + 0.490881i \(0.836675\pi\)
\(84\) 0 0
\(85\) −10.0000 −1.08465
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.48683 1.00560 0.502801 0.864402i \(-0.332303\pi\)
0.502801 + 0.864402i \(0.332303\pi\)
\(90\) 0 0
\(91\) −5.00000 + 6.70820i −0.524142 + 0.703211i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.6491 1.28432 0.642161 0.766570i \(-0.278038\pi\)
0.642161 + 0.766570i \(0.278038\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −18.9737 −1.88795 −0.943975 0.330017i \(-0.892946\pi\)
−0.943975 + 0.330017i \(0.892946\pi\)
\(102\) 0 0
\(103\) 15.8114 1.55794 0.778971 0.627060i \(-0.215742\pi\)
0.778971 + 0.627060i \(0.215742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 13.4164i 1.25109i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.48683 7.07107i −0.869657 0.648204i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 21.2132i 1.88237i −0.337895 0.941184i \(-0.609715\pi\)
0.337895 0.941184i \(-0.390285\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.48683 −0.828868 −0.414434 0.910079i \(-0.636021\pi\)
−0.414434 + 0.910079i \(0.636021\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 13.4164i 1.13796i −0.822350 0.568982i \(-0.807337\pi\)
0.822350 0.568982i \(-0.192663\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.47214i 0.373979i
\(144\) 0 0
\(145\) −6.32456 −0.525226
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 11.3137i 0.926855i 0.886135 + 0.463428i \(0.153381\pi\)
−0.886135 + 0.463428i \(0.846619\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −15.8114 −1.26189 −0.630943 0.775829i \(-0.717332\pi\)
−0.630943 + 0.775829i \(0.717332\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 9.48683 12.7279i 0.747667 1.00310i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.94427i 0.692129i 0.938211 + 0.346064i \(0.112482\pi\)
−0.938211 + 0.346064i \(0.887518\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.94427i 0.680020i −0.940422 0.340010i \(-0.889569\pi\)
0.940422 0.340010i \(-0.110431\pi\)
\(174\) 0 0
\(175\) 7.90569 10.6066i 0.597614 0.801784i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.3848i 1.37414i 0.726590 + 0.687071i \(0.241104\pi\)
−0.726590 + 0.687071i \(0.758896\pi\)
\(180\) 0 0
\(181\) 13.4164i 0.997234i −0.866822 0.498617i \(-0.833841\pi\)
0.866822 0.498617i \(-0.166159\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.48683 −0.697486
\(186\) 0 0
\(187\) −6.32456 −0.462497
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.6274i 1.63726i 0.574320 + 0.818631i \(0.305267\pi\)
−0.574320 + 0.818631i \(0.694733\pi\)
\(192\) 0 0
\(193\) 8.48528i 0.610784i 0.952227 + 0.305392i \(0.0987875\pi\)
−0.952227 + 0.305392i \(0.901213\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 26.8328i 1.90213i −0.308994 0.951064i \(-0.599992\pi\)
0.308994 0.951064i \(-0.400008\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.00000 4.47214i −0.421117 0.313882i
\(204\) 0 0
\(205\) 21.2132i 1.48159i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −18.9737 −1.29399
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 14.1421i 0.951303i
\(222\) 0 0
\(223\) −22.1359 −1.48233 −0.741166 0.671322i \(-0.765727\pi\)
−0.741166 + 0.671322i \(0.765727\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.8885i 1.18730i −0.804722 0.593652i \(-0.797686\pi\)
0.804722 0.593652i \(-0.202314\pi\)
\(228\) 0 0
\(229\) 13.4164i 0.886581i −0.896378 0.443291i \(-0.853811\pi\)
0.896378 0.443291i \(-0.146189\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 10.0000 0.652328
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.3137i 0.731823i −0.930650 0.365911i \(-0.880757\pi\)
0.930650 0.365911i \(-0.119243\pi\)
\(240\) 0 0
\(241\) 13.4164i 0.864227i 0.901819 + 0.432113i \(0.142232\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 15.0000 4.47214i 0.958315 0.285714i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.48683 −0.598804 −0.299402 0.954127i \(-0.596787\pi\)
−0.299402 + 0.954127i \(0.596787\pi\)
\(252\) 0 0
\(253\) 8.48528i 0.533465i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.47214i 0.278964i 0.990225 + 0.139482i \(0.0445438\pi\)
−0.990225 + 0.139482i \(0.955456\pi\)
\(258\) 0 0
\(259\) −9.00000 6.70820i −0.559233 0.416828i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) 13.4164i 0.824163i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.9737 −1.15684 −0.578422 0.815737i \(-0.696331\pi\)
−0.578422 + 0.815737i \(0.696331\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.07107i 0.426401i
\(276\) 0 0
\(277\) 21.2132i 1.27458i −0.770625 0.637289i \(-0.780056\pi\)
0.770625 0.637289i \(-0.219944\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.41421i 0.0843649i −0.999110 0.0421825i \(-0.986569\pi\)
0.999110 0.0421825i \(-0.0134311\pi\)
\(282\) 0 0
\(283\) 6.32456 0.375956 0.187978 0.982173i \(-0.439807\pi\)
0.187978 + 0.982173i \(0.439807\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 15.0000 20.1246i 0.885422 1.18792i
\(288\) 0 0
\(289\) −3.00000 −0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.47214i 0.261265i 0.991431 + 0.130632i \(0.0417008\pi\)
−0.991431 + 0.130632i \(0.958299\pi\)
\(294\) 0 0
\(295\) 21.2132i 1.23508i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −18.9737 −1.09728
\(300\) 0 0
\(301\) −18.0000 13.4164i −1.03750 0.773309i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) −12.6491 −0.721923 −0.360961 0.932581i \(-0.617551\pi\)
−0.360961 + 0.932581i \(0.617551\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 18.9737 1.07590 0.537949 0.842977i \(-0.319199\pi\)
0.537949 + 0.842977i \(0.319199\pi\)
\(312\) 0 0
\(313\) 31.6228 1.78743 0.893713 0.448640i \(-0.148091\pi\)
0.893713 + 0.448640i \(0.148091\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −15.8114 −0.877058
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.48683 + 7.07107i 0.523026 + 0.389841i
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 25.4558i 1.38667i 0.720616 + 0.693334i \(0.243859\pi\)
−0.720616 + 0.693334i \(0.756141\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 17.3925 + 6.36396i 0.939108 + 0.343622i
\(344\) 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
\(349\) 13.4164i 0.718164i −0.933306 0.359082i \(-0.883090\pi\)
0.933306 0.359082i \(-0.116910\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 31.3050i 1.66619i 0.553127 + 0.833097i \(0.313435\pi\)
−0.553127 + 0.833097i \(0.686565\pi\)
\(354\) 0 0
\(355\) −12.6491 −0.671345
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 31.1127i 1.64207i 0.570881 + 0.821033i \(0.306602\pi\)
−0.570881 + 0.821033i \(0.693398\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 14.1421i 0.740233i
\(366\) 0 0
\(367\) −22.1359 −1.15549 −0.577743 0.816218i \(-0.696067\pi\)
−0.577743 + 0.816218i \(0.696067\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −9.48683 + 12.7279i −0.492532 + 0.660801i
\(372\) 0 0
\(373\) 21.2132i 1.09838i 0.835698 + 0.549189i \(0.185063\pi\)
−0.835698 + 0.549189i \(0.814937\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.94427i 0.460653i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.47214i 0.228515i −0.993451 0.114258i \(-0.963551\pi\)
0.993451 0.114258i \(-0.0364490\pi\)
\(384\) 0 0
\(385\) 5.00000 6.70820i 0.254824 0.341882i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11.3137i 0.573628i 0.957986 + 0.286814i \(0.0925961\pi\)
−0.957986 + 0.286814i \(0.907404\pi\)
\(390\) 0 0
\(391\) 26.8328i 1.35699i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.94427i 0.450035i
\(396\) 0 0
\(397\) 22.1359 1.11097 0.555486 0.831526i \(-0.312532\pi\)
0.555486 + 0.831526i \(0.312532\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.41421i 0.0706225i −0.999376 0.0353112i \(-0.988758\pi\)
0.999376 0.0353112i \(-0.0112422\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −15.0000 + 20.1246i −0.738102 + 0.990267i
\(414\) 0 0
\(415\) −20.0000 −0.981761
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.48683 0.463462 0.231731 0.972780i \(-0.425561\pi\)
0.231731 + 0.972780i \(0.425561\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 22.3607i 1.08465i
\(426\) 0 0
\(427\) 28.4605 + 21.2132i 1.37730 + 1.02658i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.7696i 1.77113i −0.464518 0.885564i \(-0.653773\pi\)
0.464518 0.885564i \(-0.346227\pi\)
\(432\) 0 0
\(433\) −6.32456 −0.303939 −0.151969 0.988385i \(-0.548562\pi\)
−0.151969 + 0.988385i \(0.548562\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 21.2132i 1.00560i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.3848i 0.867631i −0.901002 0.433816i \(-0.857167\pi\)
0.901002 0.433816i \(-0.142833\pi\)
\(450\) 0 0
\(451\) 13.4164i 0.631754i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −15.0000 11.1803i −0.703211 0.524142i
\(456\) 0 0
\(457\) 16.9706i 0.793849i 0.917851 + 0.396925i \(0.129923\pi\)
−0.917851 + 0.396925i \(0.870077\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 21.2132i 0.985861i 0.870069 + 0.492931i \(0.164074\pi\)
−0.870069 + 0.492931i \(0.835926\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.94427i 0.413892i 0.978352 + 0.206946i \(0.0663524\pi\)
−0.978352 + 0.206946i \(0.933648\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −18.9737 −0.866929 −0.433464 0.901171i \(-0.642709\pi\)
−0.433464 + 0.901171i \(0.642709\pi\)
\(480\) 0 0
\(481\) 13.4164i 0.611736i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 28.2843i 1.28432i
\(486\) 0 0
\(487\) 4.24264i 0.192252i 0.995369 + 0.0961262i \(0.0306452\pi\)
−0.995369 + 0.0961262i \(0.969355\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.5563i 0.702048i −0.936366 0.351024i \(-0.885834\pi\)
0.936366 0.351024i \(-0.114166\pi\)
\(492\) 0 0
\(493\) −12.6491 −0.569687
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 8.94427i −0.538274 0.401205i
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 22.3607i 0.997013i 0.866886 + 0.498507i \(0.166118\pi\)
−0.866886 + 0.498507i \(0.833882\pi\)
\(504\) 0 0
\(505\) 42.4264i 1.88795i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.9737 −0.840993 −0.420496 0.907294i \(-0.638144\pi\)
−0.420496 + 0.907294i \(0.638144\pi\)
\(510\) 0 0
\(511\) 10.0000 13.4164i 0.442374 0.593507i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 35.3553i 1.55794i
\(516\) 0 0
\(517\) 6.32456 0.278154
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 28.4605 1.24688 0.623439 0.781872i \(-0.285735\pi\)
0.623439 + 0.781872i \(0.285735\pi\)
\(522\) 0 0
\(523\) 6.32456 0.276553 0.138277 0.990394i \(-0.455844\pi\)
0.138277 + 0.990394i \(0.455844\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −30.0000 −1.29944
\(534\) 0 0
\(535\) 26.8328i 1.16008i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 9.48683 2.82843i 0.408627 0.121829i
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 22.3607i 0.957826i
\(546\) 0 0
\(547\) 42.4264i 1.81402i −0.421107 0.907011i \(-0.638358\pi\)
0.421107 0.907011i \(-0.361642\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −6.32456 + 8.48528i −0.268947 + 0.360831i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 26.8328i 1.13491i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 35.7771i 1.50782i 0.656975 + 0.753912i \(0.271836\pi\)
−0.656975 + 0.753912i \(0.728164\pi\)
\(564\) 0 0
\(565\) 13.4164i 0.564433i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.89949i 0.415008i −0.978234 0.207504i \(-0.933466\pi\)
0.978234 0.207504i \(-0.0665341\pi\)
\(570\) 0 0
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 30.0000 1.25109
\(576\) 0 0
\(577\) −25.2982 −1.05318 −0.526589 0.850120i \(-0.676529\pi\)
−0.526589 + 0.850120i \(0.676529\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −18.9737 14.1421i −0.787160 0.586715i
\(582\) 0 0
\(583\) 8.48528i 0.351424i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.94427i 0.369170i 0.982817 + 0.184585i \(0.0590940\pi\)
−0.982817 + 0.184585i \(0.940906\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.3607i 0.918243i −0.888373 0.459122i \(-0.848164\pi\)
0.888373 0.459122i \(-0.151836\pi\)
\(594\) 0 0
\(595\) 15.8114 21.2132i 0.648204 0.869657i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.82843i 0.115566i −0.998329 0.0577832i \(-0.981597\pi\)
0.998329 0.0577832i \(-0.0184032\pi\)
\(600\) 0 0
\(601\) 13.4164i 0.547267i 0.961834 + 0.273633i \(0.0882255\pi\)
−0.961834 + 0.273633i \(0.911775\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 20.1246i 0.818182i
\(606\) 0 0
\(607\) 15.8114 0.641764 0.320882 0.947119i \(-0.396021\pi\)
0.320882 + 0.947119i \(0.396021\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 14.1421i 0.572130i
\(612\) 0 0
\(613\) 21.2132i 0.856793i 0.903591 + 0.428397i \(0.140921\pi\)
−0.903591 + 0.428397i \(0.859079\pi\)
\(614\) 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
\(619\) 40.2492i 1.61775i 0.587979 + 0.808876i \(0.299924\pi\)
−0.587979 + 0.808876i \(0.700076\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −15.0000 + 20.1246i −0.600962 + 0.806276i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18.9737 −0.756530
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 47.4342 1.88237
\(636\) 0 0
\(637\) −6.32456 21.2132i −0.250588 0.840498i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.5563i 0.614439i 0.951639 + 0.307219i \(0.0993986\pi\)
−0.951639 + 0.307219i \(0.900601\pi\)
\(642\) 0 0
\(643\) 6.32456 0.249416 0.124708 0.992193i \(-0.460201\pi\)
0.124708 + 0.992193i \(0.460201\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22.3607i 0.879089i 0.898221 + 0.439545i \(0.144860\pi\)
−0.898221 + 0.439545i \(0.855140\pi\)
\(648\) 0 0
\(649\) 13.4164i 0.526640i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) 21.2132i 0.828868i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 32.5269i 1.26707i −0.773715 0.633534i \(-0.781604\pi\)
0.773715 0.633534i \(-0.218396\pi\)
\(660\) 0 0
\(661\) 40.2492i 1.56551i −0.622328 0.782757i \(-0.713813\pi\)
0.622328 0.782757i \(-0.286187\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.9706i 0.657103i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 18.9737 0.732470
\(672\) 0 0
\(673\) 8.48528i 0.327084i 0.986536 + 0.163542i \(0.0522919\pi\)
−0.986536 + 0.163542i \(0.947708\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.3607i 0.859391i −0.902974 0.429695i \(-0.858621\pi\)
0.902974 0.429695i \(-0.141379\pi\)
\(678\) 0 0
\(679\) −20.0000 + 26.8328i −0.767530 + 1.02975i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 40.2492i 1.53784i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 18.9737 0.722839
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 30.0000 1.13796
\(696\) 0 0
\(697\) 42.4264i 1.60701i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.7696i 1.38877i 0.719605 + 0.694383i \(0.244323\pi\)
−0.719605 + 0.694383i \(0.755677\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 30.0000 40.2492i 1.12827 1.51373i
\(708\) 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
\(713\) 0 0
\(714\) 0 0
\(715\) −10.0000 −0.373979
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 37.9473 1.41520 0.707598 0.706615i \(-0.249779\pi\)
0.707598 + 0.706615i \(0.249779\pi\)
\(720\) 0 0
\(721\) −25.0000 + 33.5410i −0.931049 + 1.24913i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 14.1421i 0.525226i
\(726\) 0 0
\(727\) −3.16228 −0.117282 −0.0586412 0.998279i \(-0.518677\pi\)
−0.0586412 + 0.998279i \(0.518677\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −37.9473 −1.40353
\(732\) 0 0
\(733\) 41.1096 1.51842 0.759209 0.650847i \(-0.225586\pi\)
0.759209 + 0.650847i \(0.225586\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −25.2982 −0.926855
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −18.9737 + 25.4558i −0.693283 + 0.930136i
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 44.7214i 1.62758i
\(756\) 0 0
\(757\) 46.6690i 1.69622i 0.529824 + 0.848108i \(0.322258\pi\)
−0.529824 + 0.848108i \(0.677742\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.48683 0.343897 0.171949 0.985106i \(-0.444994\pi\)
0.171949 + 0.985106i \(0.444994\pi\)
\(762\) 0 0
\(763\) 15.8114 21.2132i 0.572411 0.767970i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 30.0000 1.08324
\(768\) 0 0
\(769\) 40.2492i 1.45142i 0.687999 + 0.725712i \(0.258490\pi\)
−0.687999 + 0.725712i \(0.741510\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 31.3050i 1.12596i 0.826470 + 0.562980i \(0.190345\pi\)
−0.826470 + 0.562980i \(0.809655\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 35.3553i 1.26189i
\(786\) 0 0
\(787\) −31.6228 −1.12723 −0.563615 0.826038i \(-0.690590\pi\)
−0.563615 + 0.826038i \(0.690590\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.48683 12.7279i 0.337313 0.452553i
\(792\) 0 0
\(793\) 42.4264i 1.50661i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.94427i 0.316822i −0.987373 0.158411i \(-0.949363\pi\)
0.987373 0.158411i \(-0.0506372\pi\)
\(798\) 0 0
\(799\) 20.0000 0.707549
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8.94427i 0.315637i
\(804\) 0 0
\(805\) 28.4605 + 21.2132i 1.00310 + 0.747667i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 52.3259i 1.83968i −0.392293 0.919840i \(-0.628318\pi\)
0.392293 0.919840i \(-0.371682\pi\)
\(810\) 0 0
\(811\) 40.2492i 1.41334i 0.707543 + 0.706671i \(0.249804\pi\)
−0.707543 + 0.706671i \(0.750196\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19.7990i 0.690990i 0.938421 + 0.345495i \(0.112289\pi\)
−0.938421 + 0.345495i \(0.887711\pi\)
\(822\) 0 0
\(823\) 21.2132i 0.739446i 0.929142 + 0.369723i \(0.120547\pi\)
−0.929142 + 0.369723i \(0.879453\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −48.0000 −1.66912 −0.834562 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 0 0
\(829\) 13.4164i 0.465971i −0.972480 0.232986i \(-0.925151\pi\)
0.972480 0.232986i \(-0.0748495\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 30.0000 8.94427i 1.03944 0.309901i
\(834\) 0 0
\(835\) −20.0000 −0.692129
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −37.9473 −1.31009 −0.655044 0.755591i \(-0.727350\pi\)
−0.655044 + 0.755591i \(0.727350\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.70820i 0.230769i
\(846\) 0 0
\(847\) −14.2302 + 19.0919i −0.488957 + 0.656005i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 25.4558i 0.872615i
\(852\) 0 0
\(853\) 3.16228 0.108274 0.0541372 0.998534i \(-0.482759\pi\)
0.0541372 + 0.998534i \(0.482759\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 31.3050i 1.06936i 0.845056 + 0.534678i \(0.179567\pi\)
−0.845056 + 0.534678i \(0.820433\pi\)
\(858\) 0 0
\(859\) 13.4164i 0.457762i −0.973454 0.228881i \(-0.926493\pi\)
0.973454 0.228881i \(-0.0735067\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) 20.0000 0.680020
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 5.65685i 0.191896i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 23.7171 + 17.6777i 0.801784 + 0.597614i
\(876\) 0 0
\(877\) 21.2132i 0.716319i −0.933660 0.358159i \(-0.883404\pi\)
0.933660 0.358159i \(-0.116596\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 47.4342 1.59810 0.799049 0.601266i \(-0.205337\pi\)
0.799049 + 0.601266i \(0.205337\pi\)
\(882\) 0 0
\(883\) 33.9411i 1.14221i 0.820877 + 0.571105i \(0.193485\pi\)
−0.820877 + 0.571105i \(0.806515\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 44.7214i 1.50160i −0.660532 0.750798i \(-0.729669\pi\)
0.660532 0.750798i \(-0.270331\pi\)
\(888\) 0 0
\(889\) 45.0000 + 33.5410i 1.50925 + 1.12493i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −41.1096 −1.37414
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 26.8328i 0.893931i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 30.0000 0.997234
\(906\) 0 0
\(907\) 25.4558i 0.845247i 0.906305 + 0.422624i \(0.138891\pi\)
−0.906305 + 0.422624i \(0.861109\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 48.0833i 1.59307i 0.604593 + 0.796535i \(0.293336\pi\)
−0.604593 + 0.796535i \(0.706664\pi\)
\(912\) 0 0
\(913\) −12.6491 −0.418624
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15.0000 20.1246i 0.495344 0.664573i
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 17.8885i 0.588809i
\(924\) 0 0
\(925\) 21.2132i 0.697486i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −9.48683 −0.311253 −0.155626 0.987816i \(-0.549740\pi\)
−0.155626 + 0.987816i \(0.549740\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 14.1421i 0.462497i
\(936\) 0 0
\(937\) 12.6491 0.413228 0.206614 0.978422i \(-0.433755\pi\)
0.206614 + 0.978422i \(0.433755\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 37.9473 1.23705 0.618524 0.785766i \(-0.287731\pi\)
0.618524 + 0.785766i \(0.287731\pi\)
\(942\) 0 0
\(943\) 56.9210 1.85360
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −50.5964 −1.63726
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 28.4605 38.1838i 0.919037 1.23302i
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −18.9737 −0.610784
\(966\) 0 0
\(967\) 21.2132i 0.682171i −0.940032 0.341085i \(-0.889205\pi\)
0.940032 0.341085i \(-0.110795\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 28.4605 0.913341 0.456670 0.889636i \(-0.349042\pi\)
0.456670 + 0.889636i \(0.349042\pi\)
\(972\) 0 0
\(973\) 28.4605 + 21.2132i 0.912402 + 0.680064i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 13.4164i 0.428790i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.47214i 0.142639i −0.997454 0.0713195i \(-0.977279\pi\)
0.997454 0.0713195i \(-0.0227210\pi\)
\(984\) 0 0
\(985\) 26.8328i 0.854965i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 50.9117i 1.61890i
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 60.0000 1.90213
\(996\) 0 0
\(997\) −53.7587 −1.70256 −0.851278 0.524715i \(-0.824172\pi\)
−0.851278 + 0.524715i \(0.824172\pi\)
\(998\) 0 0
\(999\) 0 0
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 5040.2.k.a.1889.3 4
3.2 odd 2 5040.2.k.d.1889.1 4
4.3 odd 2 630.2.d.d.629.4 yes 4
5.4 even 2 5040.2.k.d.1889.2 4
7.6 odd 2 inner 5040.2.k.a.1889.2 4
12.11 even 2 630.2.d.a.629.2 yes 4
15.14 odd 2 inner 5040.2.k.a.1889.4 4
20.3 even 4 3150.2.b.c.251.1 8
20.7 even 4 3150.2.b.c.251.8 8
20.19 odd 2 630.2.d.a.629.1 4
21.20 even 2 5040.2.k.d.1889.4 4
28.27 even 2 630.2.d.d.629.1 yes 4
35.34 odd 2 5040.2.k.d.1889.3 4
60.23 odd 4 3150.2.b.c.251.5 8
60.47 odd 4 3150.2.b.c.251.4 8
60.59 even 2 630.2.d.d.629.3 yes 4
84.83 odd 2 630.2.d.a.629.3 yes 4
105.104 even 2 inner 5040.2.k.a.1889.1 4
140.27 odd 4 3150.2.b.c.251.7 8
140.83 odd 4 3150.2.b.c.251.2 8
140.139 even 2 630.2.d.a.629.4 yes 4
420.83 even 4 3150.2.b.c.251.6 8
420.167 even 4 3150.2.b.c.251.3 8
420.419 odd 2 630.2.d.d.629.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.d.a.629.1 4 20.19 odd 2
630.2.d.a.629.2 yes 4 12.11 even 2
630.2.d.a.629.3 yes 4 84.83 odd 2
630.2.d.a.629.4 yes 4 140.139 even 2
630.2.d.d.629.1 yes 4 28.27 even 2
630.2.d.d.629.2 yes 4 420.419 odd 2
630.2.d.d.629.3 yes 4 60.59 even 2
630.2.d.d.629.4 yes 4 4.3 odd 2
3150.2.b.c.251.1 8 20.3 even 4
3150.2.b.c.251.2 8 140.83 odd 4
3150.2.b.c.251.3 8 420.167 even 4
3150.2.b.c.251.4 8 60.47 odd 4
3150.2.b.c.251.5 8 60.23 odd 4
3150.2.b.c.251.6 8 420.83 even 4
3150.2.b.c.251.7 8 140.27 odd 4
3150.2.b.c.251.8 8 20.7 even 4
5040.2.k.a.1889.1 4 105.104 even 2 inner
5040.2.k.a.1889.2 4 7.6 odd 2 inner
5040.2.k.a.1889.3 4 1.1 even 1 trivial
5040.2.k.a.1889.4 4 15.14 odd 2 inner
5040.2.k.d.1889.1 4 3.2 odd 2
5040.2.k.d.1889.2 4 5.4 even 2
5040.2.k.d.1889.3 4 35.34 odd 2
5040.2.k.d.1889.4 4 21.20 even 2