Properties

Label 4140.2.a.u.1.5
Level $4140$
Weight $2$
Character 4140.1
Self dual yes
Analytic conductor $33.058$
Analytic rank $0$
Dimension $5$
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] = [4140,2,Mod(1,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, 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("4140.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.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: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.14345904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 13x^{3} + 34x^{2} - 11x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.430705\) of defining polynomial
Character \(\chi\) \(=\) 4140.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +5.21120 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +5.21120 q^{7} +2.67519 q^{11} +0.186219 q^{13} -1.99097 q^{17} -0.675190 q^{19} +1.00000 q^{23} +1.00000 q^{25} +5.02498 q^{29} +7.52757 q^{31} +5.21120 q^{35} +4.34979 q^{37} -6.60401 q^{41} -2.84336 q^{43} -13.5866 q^{47} +20.1566 q^{49} +12.9023 q^{53} +2.67519 q^{55} -6.74779 q^{59} +8.65714 q^{61} +0.186219 q^{65} -12.1226 q^{67} +0.181619 q^{71} +3.81378 q^{73} +13.9409 q^{77} -10.4904 q^{79} +16.8538 q^{83} -1.99097 q^{85} -12.4224 q^{89} +0.970423 q^{91} -0.675190 q^{95} -4.92941 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{5} + 4 q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} + 6 q^{19} + 5 q^{23} + 5 q^{25} + 2 q^{29} + 8 q^{31} + 4 q^{35} + 8 q^{37} + 2 q^{41} + 18 q^{43} - 4 q^{47} + 13 q^{49} - 2 q^{53} + 4 q^{55} + 6 q^{59} + 10 q^{61} + 2 q^{65} + 16 q^{67} + 10 q^{71} + 18 q^{73} - 16 q^{77} + 14 q^{79} + 8 q^{83} + 2 q^{85} - 18 q^{89} + 36 q^{91} + 6 q^{95} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 5.21120 1.96965 0.984823 0.173559i \(-0.0555268\pi\)
0.984823 + 0.173559i \(0.0555268\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.67519 0.806600 0.403300 0.915068i \(-0.367863\pi\)
0.403300 + 0.915068i \(0.367863\pi\)
\(12\) 0 0
\(13\) 0.186219 0.0516478 0.0258239 0.999667i \(-0.491779\pi\)
0.0258239 + 0.999667i \(0.491779\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.99097 −0.482882 −0.241441 0.970415i \(-0.577620\pi\)
−0.241441 + 0.970415i \(0.577620\pi\)
\(18\) 0 0
\(19\) −0.675190 −0.154899 −0.0774496 0.996996i \(-0.524678\pi\)
−0.0774496 + 0.996996i \(0.524678\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.02498 0.933115 0.466557 0.884491i \(-0.345494\pi\)
0.466557 + 0.884491i \(0.345494\pi\)
\(30\) 0 0
\(31\) 7.52757 1.35199 0.675996 0.736905i \(-0.263714\pi\)
0.675996 + 0.736905i \(0.263714\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.21120 0.880853
\(36\) 0 0
\(37\) 4.34979 0.715101 0.357550 0.933894i \(-0.383612\pi\)
0.357550 + 0.933894i \(0.383612\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.60401 −1.03137 −0.515687 0.856777i \(-0.672463\pi\)
−0.515687 + 0.856777i \(0.672463\pi\)
\(42\) 0 0
\(43\) −2.84336 −0.433608 −0.216804 0.976215i \(-0.569563\pi\)
−0.216804 + 0.976215i \(0.569563\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −13.5866 −1.98180 −0.990901 0.134591i \(-0.957028\pi\)
−0.990901 + 0.134591i \(0.957028\pi\)
\(48\) 0 0
\(49\) 20.1566 2.87951
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.9023 1.77227 0.886136 0.463425i \(-0.153380\pi\)
0.886136 + 0.463425i \(0.153380\pi\)
\(54\) 0 0
\(55\) 2.67519 0.360723
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.74779 −0.878488 −0.439244 0.898368i \(-0.644754\pi\)
−0.439244 + 0.898368i \(0.644754\pi\)
\(60\) 0 0
\(61\) 8.65714 1.10843 0.554217 0.832373i \(-0.313018\pi\)
0.554217 + 0.832373i \(0.313018\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.186219 0.0230976
\(66\) 0 0
\(67\) −12.1226 −1.48101 −0.740503 0.672053i \(-0.765413\pi\)
−0.740503 + 0.672053i \(0.765413\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.181619 0.0215542 0.0107771 0.999942i \(-0.496569\pi\)
0.0107771 + 0.999942i \(0.496569\pi\)
\(72\) 0 0
\(73\) 3.81378 0.446369 0.223185 0.974776i \(-0.428355\pi\)
0.223185 + 0.974776i \(0.428355\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 13.9409 1.58872
\(78\) 0 0
\(79\) −10.4904 −1.18026 −0.590131 0.807308i \(-0.700924\pi\)
−0.590131 + 0.807308i \(0.700924\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.8538 1.84995 0.924973 0.380033i \(-0.124087\pi\)
0.924973 + 0.380033i \(0.124087\pi\)
\(84\) 0 0
\(85\) −1.99097 −0.215951
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.4224 −1.31677 −0.658385 0.752681i \(-0.728760\pi\)
−0.658385 + 0.752681i \(0.728760\pi\)
\(90\) 0 0
\(91\) 0.970423 0.101728
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.675190 −0.0692730
\(96\) 0 0
\(97\) −4.92941 −0.500506 −0.250253 0.968180i \(-0.580514\pi\)
−0.250253 + 0.968180i \(0.580514\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −11.5912 −1.15336 −0.576681 0.816969i \(-0.695653\pi\)
−0.576681 + 0.816969i \(0.695653\pi\)
\(102\) 0 0
\(103\) 12.1937 1.20148 0.600742 0.799443i \(-0.294872\pi\)
0.600742 + 0.799443i \(0.294872\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.57001 −0.925168 −0.462584 0.886576i \(-0.653078\pi\)
−0.462584 + 0.886576i \(0.653078\pi\)
\(108\) 0 0
\(109\) −0.675190 −0.0646715 −0.0323357 0.999477i \(-0.510295\pi\)
−0.0323357 + 0.999477i \(0.510295\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.21562 0.302500 0.151250 0.988496i \(-0.451670\pi\)
0.151250 + 0.988496i \(0.451670\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −10.3754 −0.951107
\(120\) 0 0
\(121\) −3.84336 −0.349396
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 12.2542 1.08739 0.543693 0.839284i \(-0.317025\pi\)
0.543693 + 0.839284i \(0.317025\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.42239 −0.561127 −0.280563 0.959835i \(-0.590521\pi\)
−0.280563 + 0.959835i \(0.590521\pi\)
\(132\) 0 0
\(133\) −3.51855 −0.305097
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −15.1885 −1.29764 −0.648822 0.760940i \(-0.724738\pi\)
−0.648822 + 0.760940i \(0.724738\pi\)
\(138\) 0 0
\(139\) −15.1566 −1.28556 −0.642781 0.766050i \(-0.722220\pi\)
−0.642781 + 0.766050i \(0.722220\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.498171 0.0416592
\(144\) 0 0
\(145\) 5.02498 0.417302
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.02557 0.165941 0.0829706 0.996552i \(-0.473559\pi\)
0.0829706 + 0.996552i \(0.473559\pi\)
\(150\) 0 0
\(151\) 3.15664 0.256884 0.128442 0.991717i \(-0.459002\pi\)
0.128442 + 0.991717i \(0.459002\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.52757 0.604629
\(156\) 0 0
\(157\) −10.1891 −0.813182 −0.406591 0.913610i \(-0.633283\pi\)
−0.406591 + 0.913610i \(0.633283\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.21120 0.410700
\(162\) 0 0
\(163\) 11.9819 0.938499 0.469249 0.883066i \(-0.344525\pi\)
0.469249 + 0.883066i \(0.344525\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.3042 1.10689 0.553445 0.832886i \(-0.313313\pi\)
0.553445 + 0.832886i \(0.313313\pi\)
\(168\) 0 0
\(169\) −12.9653 −0.997333
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −21.1051 −1.60459 −0.802296 0.596927i \(-0.796388\pi\)
−0.802296 + 0.596927i \(0.796388\pi\)
\(174\) 0 0
\(175\) 5.21120 0.393929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.2798 0.917835 0.458917 0.888479i \(-0.348237\pi\)
0.458917 + 0.888479i \(0.348237\pi\)
\(180\) 0 0
\(181\) 9.42758 0.700747 0.350373 0.936610i \(-0.386055\pi\)
0.350373 + 0.936610i \(0.386055\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.34979 0.319803
\(186\) 0 0
\(187\) −5.32623 −0.389493
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.2847 0.744175 0.372087 0.928198i \(-0.378642\pi\)
0.372087 + 0.928198i \(0.378642\pi\)
\(192\) 0 0
\(193\) 2.35439 0.169472 0.0847362 0.996403i \(-0.472995\pi\)
0.0847362 + 0.996403i \(0.472995\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.7948 −1.05409 −0.527044 0.849838i \(-0.676700\pi\)
−0.527044 + 0.849838i \(0.676700\pi\)
\(198\) 0 0
\(199\) 25.1220 1.78085 0.890424 0.455131i \(-0.150408\pi\)
0.890424 + 0.455131i \(0.150408\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 26.1861 1.83791
\(204\) 0 0
\(205\) −6.60401 −0.461244
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.80626 −0.124942
\(210\) 0 0
\(211\) 15.9985 1.10138 0.550691 0.834709i \(-0.314364\pi\)
0.550691 + 0.834709i \(0.314364\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.84336 −0.193915
\(216\) 0 0
\(217\) 39.2277 2.66295
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.370757 −0.0249398
\(222\) 0 0
\(223\) 4.01805 0.269069 0.134534 0.990909i \(-0.457046\pi\)
0.134534 + 0.990909i \(0.457046\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −26.5676 −1.76335 −0.881677 0.471854i \(-0.843585\pi\)
−0.881677 + 0.471854i \(0.843585\pi\)
\(228\) 0 0
\(229\) 16.2132 1.07140 0.535700 0.844409i \(-0.320048\pi\)
0.535700 + 0.844409i \(0.320048\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.4723 1.07914 0.539570 0.841941i \(-0.318587\pi\)
0.539570 + 0.841941i \(0.318587\pi\)
\(234\) 0 0
\(235\) −13.5866 −0.886289
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.3407 1.70384 0.851918 0.523674i \(-0.175439\pi\)
0.851918 + 0.523674i \(0.175439\pi\)
\(240\) 0 0
\(241\) −5.01153 −0.322821 −0.161410 0.986887i \(-0.551604\pi\)
−0.161410 + 0.986887i \(0.551604\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 20.1566 1.28776
\(246\) 0 0
\(247\) −0.125733 −0.00800021
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −19.9848 −1.26143 −0.630715 0.776015i \(-0.717238\pi\)
−0.630715 + 0.776015i \(0.717238\pi\)
\(252\) 0 0
\(253\) 2.67519 0.168188
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.04905 0.439708 0.219854 0.975533i \(-0.429442\pi\)
0.219854 + 0.975533i \(0.429442\pi\)
\(258\) 0 0
\(259\) 22.6676 1.40850
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −13.6211 −0.839916 −0.419958 0.907544i \(-0.637955\pi\)
−0.419958 + 0.907544i \(0.637955\pi\)
\(264\) 0 0
\(265\) 12.9023 0.792584
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.23015 −0.135974 −0.0679872 0.997686i \(-0.521658\pi\)
−0.0679872 + 0.997686i \(0.521658\pi\)
\(270\) 0 0
\(271\) 19.0206 1.15542 0.577708 0.816243i \(-0.303947\pi\)
0.577708 + 0.816243i \(0.303947\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.67519 0.161320
\(276\) 0 0
\(277\) 30.6084 1.83908 0.919539 0.392999i \(-0.128562\pi\)
0.919539 + 0.392999i \(0.128562\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.41087 0.322785 0.161393 0.986890i \(-0.448401\pi\)
0.161393 + 0.986890i \(0.448401\pi\)
\(282\) 0 0
\(283\) −20.0572 −1.19227 −0.596137 0.802882i \(-0.703299\pi\)
−0.596137 + 0.802882i \(0.703299\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −34.4148 −2.03144
\(288\) 0 0
\(289\) −13.0360 −0.766825
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −17.6394 −1.03051 −0.515253 0.857038i \(-0.672302\pi\)
−0.515253 + 0.857038i \(0.672302\pi\)
\(294\) 0 0
\(295\) −6.74779 −0.392872
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.186219 0.0107693
\(300\) 0 0
\(301\) −14.8173 −0.854055
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.65714 0.495706
\(306\) 0 0
\(307\) 13.6227 0.777486 0.388743 0.921346i \(-0.372909\pi\)
0.388743 + 0.921346i \(0.372909\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.67752 −0.0951232 −0.0475616 0.998868i \(-0.515145\pi\)
−0.0475616 + 0.998868i \(0.515145\pi\)
\(312\) 0 0
\(313\) −4.15866 −0.235061 −0.117531 0.993069i \(-0.537498\pi\)
−0.117531 + 0.993069i \(0.537498\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.51335 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(318\) 0 0
\(319\) 13.4428 0.752651
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.34429 0.0747981
\(324\) 0 0
\(325\) 0.186219 0.0103296
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −70.8022 −3.90345
\(330\) 0 0
\(331\) −5.25039 −0.288588 −0.144294 0.989535i \(-0.546091\pi\)
−0.144294 + 0.989535i \(0.546091\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.1226 −0.662326
\(336\) 0 0
\(337\) −15.8408 −0.862902 −0.431451 0.902136i \(-0.641998\pi\)
−0.431451 + 0.902136i \(0.641998\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 20.1377 1.09052
\(342\) 0 0
\(343\) 68.5614 3.70197
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −13.7228 −0.736679 −0.368340 0.929691i \(-0.620074\pi\)
−0.368340 + 0.929691i \(0.620074\pi\)
\(348\) 0 0
\(349\) 13.0897 0.700678 0.350339 0.936623i \(-0.386066\pi\)
0.350339 + 0.936623i \(0.386066\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.7085 1.31510 0.657551 0.753410i \(-0.271592\pi\)
0.657551 + 0.753410i \(0.271592\pi\)
\(354\) 0 0
\(355\) 0.181619 0.00963934
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.3148 1.59995 0.799977 0.600031i \(-0.204845\pi\)
0.799977 + 0.600031i \(0.204845\pi\)
\(360\) 0 0
\(361\) −18.5441 −0.976006
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.81378 0.199622
\(366\) 0 0
\(367\) −9.46374 −0.494003 −0.247002 0.969015i \(-0.579445\pi\)
−0.247002 + 0.969015i \(0.579445\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 67.2366 3.49075
\(372\) 0 0
\(373\) 2.70100 0.139852 0.0699262 0.997552i \(-0.477724\pi\)
0.0699262 + 0.997552i \(0.477724\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.935746 0.0481934
\(378\) 0 0
\(379\) 33.1551 1.70306 0.851530 0.524305i \(-0.175675\pi\)
0.851530 + 0.524305i \(0.175675\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.14504 0.211801 0.105901 0.994377i \(-0.466227\pi\)
0.105901 + 0.994377i \(0.466227\pi\)
\(384\) 0 0
\(385\) 13.9409 0.710496
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −14.5084 −0.735607 −0.367804 0.929903i \(-0.619890\pi\)
−0.367804 + 0.929903i \(0.619890\pi\)
\(390\) 0 0
\(391\) −1.99097 −0.100688
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −10.4904 −0.527829
\(396\) 0 0
\(397\) −31.1719 −1.56447 −0.782237 0.622981i \(-0.785921\pi\)
−0.782237 + 0.622981i \(0.785921\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0680 1.20190 0.600949 0.799287i \(-0.294789\pi\)
0.600949 + 0.799287i \(0.294789\pi\)
\(402\) 0 0
\(403\) 1.40178 0.0698275
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.6365 0.576800
\(408\) 0 0
\(409\) 6.01654 0.297499 0.148749 0.988875i \(-0.452475\pi\)
0.148749 + 0.988875i \(0.452475\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −35.1641 −1.73031
\(414\) 0 0
\(415\) 16.8538 0.827321
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −30.7390 −1.50170 −0.750849 0.660474i \(-0.770355\pi\)
−0.750849 + 0.660474i \(0.770355\pi\)
\(420\) 0 0
\(421\) −9.76525 −0.475929 −0.237965 0.971274i \(-0.576480\pi\)
−0.237965 + 0.971274i \(0.576480\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.99097 −0.0965764
\(426\) 0 0
\(427\) 45.1140 2.18322
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 22.1168 1.06533 0.532664 0.846327i \(-0.321191\pi\)
0.532664 + 0.846327i \(0.321191\pi\)
\(432\) 0 0
\(433\) −1.68069 −0.0807688 −0.0403844 0.999184i \(-0.512858\pi\)
−0.0403844 + 0.999184i \(0.512858\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.675190 −0.0322987
\(438\) 0 0
\(439\) −3.91656 −0.186927 −0.0934635 0.995623i \(-0.529794\pi\)
−0.0934635 + 0.995623i \(0.529794\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.136265 0.00647416 0.00323708 0.999995i \(-0.498970\pi\)
0.00323708 + 0.999995i \(0.498970\pi\)
\(444\) 0 0
\(445\) −12.4224 −0.588878
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −25.5334 −1.20500 −0.602498 0.798120i \(-0.705828\pi\)
−0.602498 + 0.798120i \(0.705828\pi\)
\(450\) 0 0
\(451\) −17.6670 −0.831906
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.970423 0.0454941
\(456\) 0 0
\(457\) −16.2166 −0.758582 −0.379291 0.925277i \(-0.623832\pi\)
−0.379291 + 0.925277i \(0.623832\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.1923 0.661002 0.330501 0.943806i \(-0.392782\pi\)
0.330501 + 0.943806i \(0.392782\pi\)
\(462\) 0 0
\(463\) −17.2901 −0.803541 −0.401770 0.915740i \(-0.631605\pi\)
−0.401770 + 0.915740i \(0.631605\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −21.9244 −1.01454 −0.507270 0.861787i \(-0.669345\pi\)
−0.507270 + 0.861787i \(0.669345\pi\)
\(468\) 0 0
\(469\) −63.1730 −2.91706
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −7.60652 −0.349748
\(474\) 0 0
\(475\) −0.675190 −0.0309798
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 14.0244 0.640790 0.320395 0.947284i \(-0.396184\pi\)
0.320395 + 0.947284i \(0.396184\pi\)
\(480\) 0 0
\(481\) 0.810013 0.0369334
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.92941 −0.223833
\(486\) 0 0
\(487\) 0.835839 0.0378755 0.0189377 0.999821i \(-0.493972\pi\)
0.0189377 + 0.999821i \(0.493972\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.84044 0.128187 0.0640936 0.997944i \(-0.479584\pi\)
0.0640936 + 0.997944i \(0.479584\pi\)
\(492\) 0 0
\(493\) −10.0046 −0.450585
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.946452 0.0424542
\(498\) 0 0
\(499\) −15.5944 −0.698101 −0.349050 0.937104i \(-0.613496\pi\)
−0.349050 + 0.937104i \(0.613496\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −20.7666 −0.925935 −0.462968 0.886375i \(-0.653215\pi\)
−0.462968 + 0.886375i \(0.653215\pi\)
\(504\) 0 0
\(505\) −11.5912 −0.515800
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.65104 0.117505 0.0587527 0.998273i \(-0.481288\pi\)
0.0587527 + 0.998273i \(0.481288\pi\)
\(510\) 0 0
\(511\) 19.8744 0.879190
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.1937 0.537320
\(516\) 0 0
\(517\) −36.3466 −1.59852
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 41.8383 1.83297 0.916484 0.400072i \(-0.131015\pi\)
0.916484 + 0.400072i \(0.131015\pi\)
\(522\) 0 0
\(523\) −21.8396 −0.954979 −0.477489 0.878638i \(-0.658453\pi\)
−0.477489 + 0.878638i \(0.658453\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −14.9872 −0.652853
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.22979 −0.0532682
\(534\) 0 0
\(535\) −9.57001 −0.413748
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 53.9226 2.32261
\(540\) 0 0
\(541\) −28.0967 −1.20797 −0.603985 0.796995i \(-0.706421\pi\)
−0.603985 + 0.796995i \(0.706421\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.675190 −0.0289220
\(546\) 0 0
\(547\) 14.5375 0.621579 0.310789 0.950479i \(-0.399407\pi\)
0.310789 + 0.950479i \(0.399407\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.39281 −0.144539
\(552\) 0 0
\(553\) −54.6675 −2.32470
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.6699 0.494470 0.247235 0.968956i \(-0.420478\pi\)
0.247235 + 0.968956i \(0.420478\pi\)
\(558\) 0 0
\(559\) −0.529487 −0.0223949
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −42.8564 −1.80618 −0.903091 0.429448i \(-0.858708\pi\)
−0.903091 + 0.429448i \(0.858708\pi\)
\(564\) 0 0
\(565\) 3.21562 0.135282
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.5544 0.735919 0.367959 0.929842i \(-0.380057\pi\)
0.367959 + 0.929842i \(0.380057\pi\)
\(570\) 0 0
\(571\) 10.6890 0.447322 0.223661 0.974667i \(-0.428199\pi\)
0.223661 + 0.974667i \(0.428199\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −29.5547 −1.23038 −0.615190 0.788379i \(-0.710921\pi\)
−0.615190 + 0.788379i \(0.710921\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 87.8285 3.64374
\(582\) 0 0
\(583\) 34.5162 1.42952
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.8779 0.861722 0.430861 0.902418i \(-0.358210\pi\)
0.430861 + 0.902418i \(0.358210\pi\)
\(588\) 0 0
\(589\) −5.08254 −0.209423
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −23.1002 −0.948611 −0.474306 0.880360i \(-0.657301\pi\)
−0.474306 + 0.880360i \(0.657301\pi\)
\(594\) 0 0
\(595\) −10.3754 −0.425348
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −0.363238 −0.0148415 −0.00742075 0.999972i \(-0.502362\pi\)
−0.00742075 + 0.999972i \(0.502362\pi\)
\(600\) 0 0
\(601\) −9.24000 −0.376908 −0.188454 0.982082i \(-0.560348\pi\)
−0.188454 + 0.982082i \(0.560348\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.84336 −0.156255
\(606\) 0 0
\(607\) 25.8317 1.04848 0.524238 0.851572i \(-0.324350\pi\)
0.524238 + 0.851572i \(0.324350\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.53007 −0.102356
\(612\) 0 0
\(613\) −25.1039 −1.01394 −0.506969 0.861964i \(-0.669234\pi\)
−0.506969 + 0.861964i \(0.669234\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 35.9862 1.44875 0.724374 0.689407i \(-0.242129\pi\)
0.724374 + 0.689407i \(0.242129\pi\)
\(618\) 0 0
\(619\) −7.03591 −0.282797 −0.141399 0.989953i \(-0.545160\pi\)
−0.141399 + 0.989953i \(0.545160\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −64.7355 −2.59357
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −8.66031 −0.345309
\(630\) 0 0
\(631\) 25.4259 1.01219 0.506095 0.862478i \(-0.331089\pi\)
0.506095 + 0.862478i \(0.331089\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.2542 0.486294
\(636\) 0 0
\(637\) 3.75353 0.148720
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −40.6750 −1.60657 −0.803283 0.595597i \(-0.796916\pi\)
−0.803283 + 0.595597i \(0.796916\pi\)
\(642\) 0 0
\(643\) −29.1097 −1.14797 −0.573987 0.818864i \(-0.694604\pi\)
−0.573987 + 0.818864i \(0.694604\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −27.8457 −1.09473 −0.547363 0.836895i \(-0.684368\pi\)
−0.547363 + 0.836895i \(0.684368\pi\)
\(648\) 0 0
\(649\) −18.0516 −0.708589
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.86492 0.229512 0.114756 0.993394i \(-0.463391\pi\)
0.114756 + 0.993394i \(0.463391\pi\)
\(654\) 0 0
\(655\) −6.42239 −0.250944
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −15.3928 −0.599619 −0.299809 0.953999i \(-0.596923\pi\)
−0.299809 + 0.953999i \(0.596923\pi\)
\(660\) 0 0
\(661\) −37.5816 −1.46176 −0.730878 0.682508i \(-0.760889\pi\)
−0.730878 + 0.682508i \(0.760889\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.51855 −0.136443
\(666\) 0 0
\(667\) 5.02498 0.194568
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 23.1595 0.894062
\(672\) 0 0
\(673\) 8.51335 0.328166 0.164083 0.986447i \(-0.447534\pi\)
0.164083 + 0.986447i \(0.447534\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.6432 0.409052 0.204526 0.978861i \(-0.434435\pi\)
0.204526 + 0.978861i \(0.434435\pi\)
\(678\) 0 0
\(679\) −25.6881 −0.985820
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −9.11421 −0.348745 −0.174373 0.984680i \(-0.555790\pi\)
−0.174373 + 0.984680i \(0.555790\pi\)
\(684\) 0 0
\(685\) −15.1885 −0.580324
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.40266 0.0915340
\(690\) 0 0
\(691\) −16.2345 −0.617589 −0.308795 0.951129i \(-0.599926\pi\)
−0.308795 + 0.951129i \(0.599926\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −15.1566 −0.574921
\(696\) 0 0
\(697\) 13.1484 0.498032
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −17.3108 −0.653820 −0.326910 0.945056i \(-0.606007\pi\)
−0.326910 + 0.945056i \(0.606007\pi\)
\(702\) 0 0
\(703\) −2.93693 −0.110769
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −60.4038 −2.27172
\(708\) 0 0
\(709\) −23.8425 −0.895422 −0.447711 0.894178i \(-0.647761\pi\)
−0.447711 + 0.894178i \(0.647761\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.52757 0.281910
\(714\) 0 0
\(715\) 0.498171 0.0186305
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 39.4205 1.47014 0.735068 0.677994i \(-0.237150\pi\)
0.735068 + 0.677994i \(0.237150\pi\)
\(720\) 0 0
\(721\) 63.5440 2.36650
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.02498 0.186623
\(726\) 0 0
\(727\) 11.4523 0.424741 0.212371 0.977189i \(-0.431882\pi\)
0.212371 + 0.977189i \(0.431882\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5.66105 0.209382
\(732\) 0 0
\(733\) −38.0726 −1.40624 −0.703122 0.711069i \(-0.748211\pi\)
−0.703122 + 0.711069i \(0.748211\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −32.4301 −1.19458
\(738\) 0 0
\(739\) 13.2992 0.489217 0.244609 0.969622i \(-0.421340\pi\)
0.244609 + 0.969622i \(0.421340\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −16.7449 −0.614310 −0.307155 0.951660i \(-0.599377\pi\)
−0.307155 + 0.951660i \(0.599377\pi\)
\(744\) 0 0
\(745\) 2.02557 0.0742112
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −49.8712 −1.82225
\(750\) 0 0
\(751\) −31.5791 −1.15234 −0.576169 0.817330i \(-0.695453\pi\)
−0.576169 + 0.817330i \(0.695453\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.15664 0.114882
\(756\) 0 0
\(757\) −17.4896 −0.635669 −0.317835 0.948146i \(-0.602956\pi\)
−0.317835 + 0.948146i \(0.602956\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 32.0162 1.16058 0.580292 0.814408i \(-0.302938\pi\)
0.580292 + 0.814408i \(0.302938\pi\)
\(762\) 0 0
\(763\) −3.51855 −0.127380
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.25657 −0.0453720
\(768\) 0 0
\(769\) −26.3527 −0.950303 −0.475151 0.879904i \(-0.657607\pi\)
−0.475151 + 0.879904i \(0.657607\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 36.3312 1.30674 0.653371 0.757038i \(-0.273354\pi\)
0.653371 + 0.757038i \(0.273354\pi\)
\(774\) 0 0
\(775\) 7.52757 0.270398
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.45896 0.159759
\(780\) 0 0
\(781\) 0.485865 0.0173856
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −10.1891 −0.363666
\(786\) 0 0
\(787\) −39.8105 −1.41909 −0.709545 0.704660i \(-0.751099\pi\)
−0.709545 + 0.704660i \(0.751099\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 16.7572 0.595819
\(792\) 0 0
\(793\) 1.61212 0.0572482
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.00926351 0.000328130 0 0.000164065 1.00000i \(-0.499948\pi\)
0.000164065 1.00000i \(0.499948\pi\)
\(798\) 0 0
\(799\) 27.0505 0.956977
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.2026 0.360042
\(804\) 0 0
\(805\) 5.21120 0.183671
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −15.9905 −0.562196 −0.281098 0.959679i \(-0.590699\pi\)
−0.281098 + 0.959679i \(0.590699\pi\)
\(810\) 0 0
\(811\) −1.70627 −0.0599154 −0.0299577 0.999551i \(-0.509537\pi\)
−0.0299577 + 0.999551i \(0.509537\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 11.9819 0.419709
\(816\) 0 0
\(817\) 1.91981 0.0671655
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 23.9166 0.834694 0.417347 0.908747i \(-0.362960\pi\)
0.417347 + 0.908747i \(0.362960\pi\)
\(822\) 0 0
\(823\) 12.1759 0.424426 0.212213 0.977223i \(-0.431933\pi\)
0.212213 + 0.977223i \(0.431933\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −41.5339 −1.44428 −0.722138 0.691749i \(-0.756840\pi\)
−0.722138 + 0.691749i \(0.756840\pi\)
\(828\) 0 0
\(829\) 26.8628 0.932982 0.466491 0.884526i \(-0.345518\pi\)
0.466491 + 0.884526i \(0.345518\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −40.1312 −1.39046
\(834\) 0 0
\(835\) 14.3042 0.495016
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −40.7058 −1.40532 −0.702659 0.711526i \(-0.748004\pi\)
−0.702659 + 0.711526i \(0.748004\pi\)
\(840\) 0 0
\(841\) −3.74961 −0.129297
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.9653 −0.446021
\(846\) 0 0
\(847\) −20.0285 −0.688187
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.34979 0.149109
\(852\) 0 0
\(853\) 37.6727 1.28989 0.644944 0.764230i \(-0.276881\pi\)
0.644944 + 0.764230i \(0.276881\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 52.9539 1.80887 0.904435 0.426611i \(-0.140293\pi\)
0.904435 + 0.426611i \(0.140293\pi\)
\(858\) 0 0
\(859\) −40.4099 −1.37877 −0.689384 0.724396i \(-0.742119\pi\)
−0.689384 + 0.724396i \(0.742119\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −10.9280 −0.371993 −0.185997 0.982550i \(-0.559551\pi\)
−0.185997 + 0.982550i \(0.559551\pi\)
\(864\) 0 0
\(865\) −21.1051 −0.717595
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −28.0638 −0.951999
\(870\) 0 0
\(871\) −2.25745 −0.0764908
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 5.21120 0.176171
\(876\) 0 0
\(877\) −12.8860 −0.435131 −0.217565 0.976046i \(-0.569812\pi\)
−0.217565 + 0.976046i \(0.569812\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 23.7123 0.798887 0.399444 0.916758i \(-0.369203\pi\)
0.399444 + 0.916758i \(0.369203\pi\)
\(882\) 0 0
\(883\) 51.5545 1.73495 0.867473 0.497484i \(-0.165743\pi\)
0.867473 + 0.497484i \(0.165743\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.07720 0.0697457 0.0348729 0.999392i \(-0.488897\pi\)
0.0348729 + 0.999392i \(0.488897\pi\)
\(888\) 0 0
\(889\) 63.8592 2.14177
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 9.17351 0.306980
\(894\) 0 0
\(895\) 12.2798 0.410468
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 37.8259 1.26156
\(900\) 0 0
\(901\) −25.6882 −0.855799
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 9.42758 0.313384
\(906\) 0 0
\(907\) −17.2916 −0.574159 −0.287080 0.957907i \(-0.592684\pi\)
−0.287080 + 0.957907i \(0.592684\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 53.3171 1.76648 0.883238 0.468926i \(-0.155359\pi\)
0.883238 + 0.468926i \(0.155359\pi\)
\(912\) 0 0
\(913\) 45.0871 1.49217
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −33.4683 −1.10522
\(918\) 0 0
\(919\) 41.5186 1.36957 0.684787 0.728743i \(-0.259895\pi\)
0.684787 + 0.728743i \(0.259895\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0.0338209 0.00111323
\(924\) 0 0
\(925\) 4.34979 0.143020
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −47.4474 −1.55670 −0.778349 0.627832i \(-0.783942\pi\)
−0.778349 + 0.627832i \(0.783942\pi\)
\(930\) 0 0
\(931\) −13.6095 −0.446034
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.32623 −0.174187
\(936\) 0 0
\(937\) −37.0293 −1.20970 −0.604848 0.796341i \(-0.706766\pi\)
−0.604848 + 0.796341i \(0.706766\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30.3469 0.989280 0.494640 0.869098i \(-0.335300\pi\)
0.494640 + 0.869098i \(0.335300\pi\)
\(942\) 0 0
\(943\) −6.60401 −0.215056
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −46.3584 −1.50645 −0.753223 0.657765i \(-0.771502\pi\)
−0.753223 + 0.657765i \(0.771502\pi\)
\(948\) 0 0
\(949\) 0.710198 0.0230540
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −35.9226 −1.16365 −0.581824 0.813315i \(-0.697661\pi\)
−0.581824 + 0.813315i \(0.697661\pi\)
\(954\) 0 0
\(955\) 10.2847 0.332805
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −79.1505 −2.55590
\(960\) 0 0
\(961\) 25.6644 0.827883
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.35439 0.0757904
\(966\) 0 0
\(967\) −45.0531 −1.44881 −0.724405 0.689375i \(-0.757885\pi\)
−0.724405 + 0.689375i \(0.757885\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 26.0231 0.835119 0.417560 0.908649i \(-0.362885\pi\)
0.417560 + 0.908649i \(0.362885\pi\)
\(972\) 0 0
\(973\) −78.9838 −2.53210
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.2710 0.520556 0.260278 0.965534i \(-0.416186\pi\)
0.260278 + 0.965534i \(0.416186\pi\)
\(978\) 0 0
\(979\) −33.2323 −1.06211
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 40.9840 1.30719 0.653593 0.756847i \(-0.273261\pi\)
0.653593 + 0.756847i \(0.273261\pi\)
\(984\) 0 0
\(985\) −14.7948 −0.471402
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.84336 −0.0904135
\(990\) 0 0
\(991\) 36.3250 1.15390 0.576951 0.816779i \(-0.304242\pi\)
0.576951 + 0.816779i \(0.304242\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 25.1220 0.796420
\(996\) 0 0
\(997\) 20.3014 0.642952 0.321476 0.946918i \(-0.395821\pi\)
0.321476 + 0.946918i \(0.395821\pi\)
\(998\) 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 4140.2.a.u.1.5 yes 5
3.2 odd 2 4140.2.a.t.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.a.t.1.5 5 3.2 odd 2
4140.2.a.u.1.5 yes 5 1.1 even 1 trivial