Properties

Label 1456.2.a.t.1.2
Level $1456$
Weight $2$
Character 1456.1
Self dual yes
Analytic conductor $11.626$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1456,2,Mod(1,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, 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("1456.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 1456.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.14637 q^{3} -1.34292 q^{5} +1.00000 q^{7} -1.68585 q^{9} +O(q^{10})\) \(q+1.14637 q^{3} -1.34292 q^{5} +1.00000 q^{7} -1.68585 q^{9} -1.14637 q^{11} +1.00000 q^{13} -1.53948 q^{15} +5.83221 q^{17} +3.34292 q^{19} +1.14637 q^{21} +3.17513 q^{23} -3.19656 q^{25} -5.37169 q^{27} +10.4893 q^{29} -1.63565 q^{31} -1.31415 q^{33} -1.34292 q^{35} +8.51806 q^{37} +1.14637 q^{39} -0.292731 q^{41} +8.15371 q^{43} +2.26396 q^{45} +10.6142 q^{47} +1.00000 q^{49} +6.68585 q^{51} -0.782020 q^{53} +1.53948 q^{55} +3.83221 q^{57} -12.6430 q^{59} -2.00000 q^{61} -1.68585 q^{63} -1.34292 q^{65} +6.10038 q^{67} +3.63986 q^{69} -1.53948 q^{71} -15.3001 q^{73} -3.66442 q^{75} -1.14637 q^{77} -0.882404 q^{79} -1.10038 q^{81} +12.1292 q^{83} -7.83221 q^{85} +12.0246 q^{87} +5.73604 q^{89} +1.00000 q^{91} -1.87506 q^{93} -4.48929 q^{95} -5.34292 q^{97} +1.93260 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 2 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} + 2 q^{5} + 3 q^{7} + 7 q^{9} - 2 q^{11} + 3 q^{13} + 6 q^{15} + 4 q^{17} + 4 q^{19} + 2 q^{21} - 10 q^{23} - 5 q^{25} + 8 q^{27} + 24 q^{29} + 4 q^{31} - 16 q^{33} + 2 q^{35} + 2 q^{39} + 2 q^{41} - 10 q^{43} + 22 q^{45} + 8 q^{47} + 3 q^{49} + 8 q^{51} + 8 q^{53} - 6 q^{55} - 2 q^{57} + 4 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} + 12 q^{67} - 6 q^{69} + 6 q^{71} - 10 q^{73} + 16 q^{75} - 2 q^{77} + 14 q^{79} + 3 q^{81} + 12 q^{83} - 10 q^{85} + 26 q^{87} + 2 q^{89} + 3 q^{91} - 22 q^{93} - 6 q^{95} - 10 q^{97} - 14 q^{99}+O(q^{100}) \) 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
\(3\) 1.14637 0.661854 0.330927 0.943656i \(-0.392639\pi\)
0.330927 + 0.943656i \(0.392639\pi\)
\(4\) 0 0
\(5\) −1.34292 −0.600573 −0.300287 0.953849i \(-0.597082\pi\)
−0.300287 + 0.953849i \(0.597082\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.68585 −0.561949
\(10\) 0 0
\(11\) −1.14637 −0.345642 −0.172821 0.984953i \(-0.555288\pi\)
−0.172821 + 0.984953i \(0.555288\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.53948 −0.397492
\(16\) 0 0
\(17\) 5.83221 1.41452 0.707260 0.706954i \(-0.249931\pi\)
0.707260 + 0.706954i \(0.249931\pi\)
\(18\) 0 0
\(19\) 3.34292 0.766919 0.383460 0.923558i \(-0.374733\pi\)
0.383460 + 0.923558i \(0.374733\pi\)
\(20\) 0 0
\(21\) 1.14637 0.250157
\(22\) 0 0
\(23\) 3.17513 0.662061 0.331031 0.943620i \(-0.392604\pi\)
0.331031 + 0.943620i \(0.392604\pi\)
\(24\) 0 0
\(25\) −3.19656 −0.639312
\(26\) 0 0
\(27\) −5.37169 −1.03378
\(28\) 0 0
\(29\) 10.4893 1.94781 0.973906 0.226952i \(-0.0728760\pi\)
0.973906 + 0.226952i \(0.0728760\pi\)
\(30\) 0 0
\(31\) −1.63565 −0.293772 −0.146886 0.989153i \(-0.546925\pi\)
−0.146886 + 0.989153i \(0.546925\pi\)
\(32\) 0 0
\(33\) −1.31415 −0.228765
\(34\) 0 0
\(35\) −1.34292 −0.226995
\(36\) 0 0
\(37\) 8.51806 1.40036 0.700180 0.713966i \(-0.253103\pi\)
0.700180 + 0.713966i \(0.253103\pi\)
\(38\) 0 0
\(39\) 1.14637 0.183565
\(40\) 0 0
\(41\) −0.292731 −0.0457169 −0.0228584 0.999739i \(-0.507277\pi\)
−0.0228584 + 0.999739i \(0.507277\pi\)
\(42\) 0 0
\(43\) 8.15371 1.24343 0.621715 0.783244i \(-0.286436\pi\)
0.621715 + 0.783244i \(0.286436\pi\)
\(44\) 0 0
\(45\) 2.26396 0.337491
\(46\) 0 0
\(47\) 10.6142 1.54824 0.774122 0.633036i \(-0.218191\pi\)
0.774122 + 0.633036i \(0.218191\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.68585 0.936206
\(52\) 0 0
\(53\) −0.782020 −0.107419 −0.0537093 0.998557i \(-0.517104\pi\)
−0.0537093 + 0.998557i \(0.517104\pi\)
\(54\) 0 0
\(55\) 1.53948 0.207584
\(56\) 0 0
\(57\) 3.83221 0.507589
\(58\) 0 0
\(59\) −12.6430 −1.64598 −0.822989 0.568057i \(-0.807695\pi\)
−0.822989 + 0.568057i \(0.807695\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −1.68585 −0.212397
\(64\) 0 0
\(65\) −1.34292 −0.166569
\(66\) 0 0
\(67\) 6.10038 0.745281 0.372640 0.927976i \(-0.378453\pi\)
0.372640 + 0.927976i \(0.378453\pi\)
\(68\) 0 0
\(69\) 3.63986 0.438188
\(70\) 0 0
\(71\) −1.53948 −0.182703 −0.0913514 0.995819i \(-0.529119\pi\)
−0.0913514 + 0.995819i \(0.529119\pi\)
\(72\) 0 0
\(73\) −15.3001 −1.79074 −0.895369 0.445324i \(-0.853088\pi\)
−0.895369 + 0.445324i \(0.853088\pi\)
\(74\) 0 0
\(75\) −3.66442 −0.423131
\(76\) 0 0
\(77\) −1.14637 −0.130640
\(78\) 0 0
\(79\) −0.882404 −0.0992782 −0.0496391 0.998767i \(-0.515807\pi\)
−0.0496391 + 0.998767i \(0.515807\pi\)
\(80\) 0 0
\(81\) −1.10038 −0.122265
\(82\) 0 0
\(83\) 12.1292 1.33135 0.665674 0.746243i \(-0.268144\pi\)
0.665674 + 0.746243i \(0.268144\pi\)
\(84\) 0 0
\(85\) −7.83221 −0.849523
\(86\) 0 0
\(87\) 12.0246 1.28917
\(88\) 0 0
\(89\) 5.73604 0.608019 0.304009 0.952669i \(-0.401675\pi\)
0.304009 + 0.952669i \(0.401675\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −1.87506 −0.194434
\(94\) 0 0
\(95\) −4.48929 −0.460591
\(96\) 0 0
\(97\) −5.34292 −0.542492 −0.271246 0.962510i \(-0.587436\pi\)
−0.271246 + 0.962510i \(0.587436\pi\)
\(98\) 0 0
\(99\) 1.93260 0.194233
\(100\) 0 0
\(101\) 11.1464 1.10910 0.554552 0.832149i \(-0.312889\pi\)
0.554552 + 0.832149i \(0.312889\pi\)
\(102\) 0 0
\(103\) 3.41454 0.336444 0.168222 0.985749i \(-0.446197\pi\)
0.168222 + 0.985749i \(0.446197\pi\)
\(104\) 0 0
\(105\) −1.53948 −0.150238
\(106\) 0 0
\(107\) −4.97858 −0.481297 −0.240649 0.970612i \(-0.577360\pi\)
−0.240649 + 0.970612i \(0.577360\pi\)
\(108\) 0 0
\(109\) −13.4966 −1.29274 −0.646372 0.763023i \(-0.723714\pi\)
−0.646372 + 0.763023i \(0.723714\pi\)
\(110\) 0 0
\(111\) 9.76481 0.926835
\(112\) 0 0
\(113\) 16.4464 1.54715 0.773576 0.633704i \(-0.218466\pi\)
0.773576 + 0.633704i \(0.218466\pi\)
\(114\) 0 0
\(115\) −4.26396 −0.397616
\(116\) 0 0
\(117\) −1.68585 −0.155857
\(118\) 0 0
\(119\) 5.83221 0.534638
\(120\) 0 0
\(121\) −9.68585 −0.880531
\(122\) 0 0
\(123\) −0.335577 −0.0302579
\(124\) 0 0
\(125\) 11.0073 0.984527
\(126\) 0 0
\(127\) −12.0575 −1.06993 −0.534967 0.844873i \(-0.679676\pi\)
−0.534967 + 0.844873i \(0.679676\pi\)
\(128\) 0 0
\(129\) 9.34713 0.822969
\(130\) 0 0
\(131\) 3.66442 0.320162 0.160081 0.987104i \(-0.448824\pi\)
0.160081 + 0.987104i \(0.448824\pi\)
\(132\) 0 0
\(133\) 3.34292 0.289868
\(134\) 0 0
\(135\) 7.21377 0.620862
\(136\) 0 0
\(137\) −13.1035 −1.11951 −0.559755 0.828658i \(-0.689105\pi\)
−0.559755 + 0.828658i \(0.689105\pi\)
\(138\) 0 0
\(139\) −7.49663 −0.635856 −0.317928 0.948115i \(-0.602987\pi\)
−0.317928 + 0.948115i \(0.602987\pi\)
\(140\) 0 0
\(141\) 12.1678 1.02471
\(142\) 0 0
\(143\) −1.14637 −0.0958639
\(144\) 0 0
\(145\) −14.0863 −1.16980
\(146\) 0 0
\(147\) 1.14637 0.0945506
\(148\) 0 0
\(149\) 2.16779 0.177592 0.0887961 0.996050i \(-0.471698\pi\)
0.0887961 + 0.996050i \(0.471698\pi\)
\(150\) 0 0
\(151\) −14.9112 −1.21345 −0.606727 0.794910i \(-0.707518\pi\)
−0.606727 + 0.794910i \(0.707518\pi\)
\(152\) 0 0
\(153\) −9.83221 −0.794887
\(154\) 0 0
\(155\) 2.19656 0.176432
\(156\) 0 0
\(157\) 22.8683 1.82509 0.912546 0.408975i \(-0.134114\pi\)
0.912546 + 0.408975i \(0.134114\pi\)
\(158\) 0 0
\(159\) −0.896480 −0.0710955
\(160\) 0 0
\(161\) 3.17513 0.250236
\(162\) 0 0
\(163\) −7.07896 −0.554467 −0.277234 0.960803i \(-0.589418\pi\)
−0.277234 + 0.960803i \(0.589418\pi\)
\(164\) 0 0
\(165\) 1.76481 0.137390
\(166\) 0 0
\(167\) −2.61423 −0.202295 −0.101148 0.994871i \(-0.532251\pi\)
−0.101148 + 0.994871i \(0.532251\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −5.63565 −0.430969
\(172\) 0 0
\(173\) 11.0031 0.836553 0.418276 0.908320i \(-0.362634\pi\)
0.418276 + 0.908320i \(0.362634\pi\)
\(174\) 0 0
\(175\) −3.19656 −0.241637
\(176\) 0 0
\(177\) −14.4935 −1.08940
\(178\) 0 0
\(179\) −23.9614 −1.79096 −0.895478 0.445105i \(-0.853166\pi\)
−0.895478 + 0.445105i \(0.853166\pi\)
\(180\) 0 0
\(181\) 6.56090 0.487668 0.243834 0.969817i \(-0.421595\pi\)
0.243834 + 0.969817i \(0.421595\pi\)
\(182\) 0 0
\(183\) −2.29273 −0.169484
\(184\) 0 0
\(185\) −11.4391 −0.841019
\(186\) 0 0
\(187\) −6.68585 −0.488917
\(188\) 0 0
\(189\) −5.37169 −0.390733
\(190\) 0 0
\(191\) 4.39312 0.317875 0.158937 0.987289i \(-0.449193\pi\)
0.158937 + 0.987289i \(0.449193\pi\)
\(192\) 0 0
\(193\) −8.29273 −0.596924 −0.298462 0.954422i \(-0.596474\pi\)
−0.298462 + 0.954422i \(0.596474\pi\)
\(194\) 0 0
\(195\) −1.53948 −0.110245
\(196\) 0 0
\(197\) −3.17092 −0.225919 −0.112959 0.993600i \(-0.536033\pi\)
−0.112959 + 0.993600i \(0.536033\pi\)
\(198\) 0 0
\(199\) 13.5970 0.963867 0.481934 0.876208i \(-0.339935\pi\)
0.481934 + 0.876208i \(0.339935\pi\)
\(200\) 0 0
\(201\) 6.99327 0.493267
\(202\) 0 0
\(203\) 10.4893 0.736204
\(204\) 0 0
\(205\) 0.393115 0.0274564
\(206\) 0 0
\(207\) −5.35279 −0.372045
\(208\) 0 0
\(209\) −3.83221 −0.265080
\(210\) 0 0
\(211\) −9.27552 −0.638553 −0.319277 0.947662i \(-0.603440\pi\)
−0.319277 + 0.947662i \(0.603440\pi\)
\(212\) 0 0
\(213\) −1.76481 −0.120923
\(214\) 0 0
\(215\) −10.9498 −0.746771
\(216\) 0 0
\(217\) −1.63565 −0.111035
\(218\) 0 0
\(219\) −17.5395 −1.18521
\(220\) 0 0
\(221\) 5.83221 0.392317
\(222\) 0 0
\(223\) 19.5928 1.31203 0.656016 0.754747i \(-0.272240\pi\)
0.656016 + 0.754747i \(0.272240\pi\)
\(224\) 0 0
\(225\) 5.38890 0.359260
\(226\) 0 0
\(227\) 19.6644 1.30517 0.652587 0.757714i \(-0.273684\pi\)
0.652587 + 0.757714i \(0.273684\pi\)
\(228\) 0 0
\(229\) 7.76481 0.513113 0.256556 0.966529i \(-0.417412\pi\)
0.256556 + 0.966529i \(0.417412\pi\)
\(230\) 0 0
\(231\) −1.31415 −0.0864650
\(232\) 0 0
\(233\) 12.1966 0.799023 0.399512 0.916728i \(-0.369180\pi\)
0.399512 + 0.916728i \(0.369180\pi\)
\(234\) 0 0
\(235\) −14.2541 −0.929835
\(236\) 0 0
\(237\) −1.01156 −0.0657077
\(238\) 0 0
\(239\) 10.2927 0.665781 0.332891 0.942965i \(-0.391976\pi\)
0.332891 + 0.942965i \(0.391976\pi\)
\(240\) 0 0
\(241\) −4.02877 −0.259516 −0.129758 0.991546i \(-0.541420\pi\)
−0.129758 + 0.991546i \(0.541420\pi\)
\(242\) 0 0
\(243\) 14.8536 0.952861
\(244\) 0 0
\(245\) −1.34292 −0.0857962
\(246\) 0 0
\(247\) 3.34292 0.212705
\(248\) 0 0
\(249\) 13.9044 0.881158
\(250\) 0 0
\(251\) −2.91117 −0.183752 −0.0918758 0.995770i \(-0.529286\pi\)
−0.0918758 + 0.995770i \(0.529286\pi\)
\(252\) 0 0
\(253\) −3.63986 −0.228836
\(254\) 0 0
\(255\) −8.97858 −0.562260
\(256\) 0 0
\(257\) −19.5970 −1.22243 −0.611214 0.791465i \(-0.709319\pi\)
−0.611214 + 0.791465i \(0.709319\pi\)
\(258\) 0 0
\(259\) 8.51806 0.529286
\(260\) 0 0
\(261\) −17.6833 −1.09457
\(262\) 0 0
\(263\) −7.56825 −0.466678 −0.233339 0.972395i \(-0.574965\pi\)
−0.233339 + 0.972395i \(0.574965\pi\)
\(264\) 0 0
\(265\) 1.05019 0.0645128
\(266\) 0 0
\(267\) 6.57560 0.402420
\(268\) 0 0
\(269\) −9.47208 −0.577523 −0.288761 0.957401i \(-0.593243\pi\)
−0.288761 + 0.957401i \(0.593243\pi\)
\(270\) 0 0
\(271\) −29.3717 −1.78420 −0.892102 0.451835i \(-0.850770\pi\)
−0.892102 + 0.451835i \(0.850770\pi\)
\(272\) 0 0
\(273\) 1.14637 0.0693812
\(274\) 0 0
\(275\) 3.66442 0.220973
\(276\) 0 0
\(277\) −1.90383 −0.114390 −0.0571949 0.998363i \(-0.518216\pi\)
−0.0571949 + 0.998363i \(0.518216\pi\)
\(278\) 0 0
\(279\) 2.75746 0.165085
\(280\) 0 0
\(281\) −20.5756 −1.22744 −0.613719 0.789525i \(-0.710327\pi\)
−0.613719 + 0.789525i \(0.710327\pi\)
\(282\) 0 0
\(283\) 26.9933 1.60458 0.802292 0.596932i \(-0.203614\pi\)
0.802292 + 0.596932i \(0.203614\pi\)
\(284\) 0 0
\(285\) −5.14637 −0.304844
\(286\) 0 0
\(287\) −0.292731 −0.0172794
\(288\) 0 0
\(289\) 17.0147 1.00086
\(290\) 0 0
\(291\) −6.12494 −0.359050
\(292\) 0 0
\(293\) −14.9070 −0.870874 −0.435437 0.900219i \(-0.643406\pi\)
−0.435437 + 0.900219i \(0.643406\pi\)
\(294\) 0 0
\(295\) 16.9786 0.988531
\(296\) 0 0
\(297\) 6.15792 0.357319
\(298\) 0 0
\(299\) 3.17513 0.183623
\(300\) 0 0
\(301\) 8.15371 0.469972
\(302\) 0 0
\(303\) 12.7778 0.734066
\(304\) 0 0
\(305\) 2.68585 0.153791
\(306\) 0 0
\(307\) −26.0288 −1.48554 −0.742770 0.669546i \(-0.766488\pi\)
−0.742770 + 0.669546i \(0.766488\pi\)
\(308\) 0 0
\(309\) 3.91431 0.222677
\(310\) 0 0
\(311\) −19.4966 −1.10555 −0.552776 0.833330i \(-0.686432\pi\)
−0.552776 + 0.833330i \(0.686432\pi\)
\(312\) 0 0
\(313\) −3.48194 −0.196811 −0.0984055 0.995146i \(-0.531374\pi\)
−0.0984055 + 0.995146i \(0.531374\pi\)
\(314\) 0 0
\(315\) 2.26396 0.127560
\(316\) 0 0
\(317\) 5.02142 0.282031 0.141016 0.990007i \(-0.454963\pi\)
0.141016 + 0.990007i \(0.454963\pi\)
\(318\) 0 0
\(319\) −12.0246 −0.673246
\(320\) 0 0
\(321\) −5.70727 −0.318549
\(322\) 0 0
\(323\) 19.4966 1.08482
\(324\) 0 0
\(325\) −3.19656 −0.177313
\(326\) 0 0
\(327\) −15.4721 −0.855608
\(328\) 0 0
\(329\) 10.6142 0.585182
\(330\) 0 0
\(331\) 6.14950 0.338007 0.169004 0.985615i \(-0.445945\pi\)
0.169004 + 0.985615i \(0.445945\pi\)
\(332\) 0 0
\(333\) −14.3601 −0.786931
\(334\) 0 0
\(335\) −8.19235 −0.447596
\(336\) 0 0
\(337\) −25.6258 −1.39593 −0.697963 0.716134i \(-0.745910\pi\)
−0.697963 + 0.716134i \(0.745910\pi\)
\(338\) 0 0
\(339\) 18.8536 1.02399
\(340\) 0 0
\(341\) 1.87506 0.101540
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −4.88806 −0.263164
\(346\) 0 0
\(347\) 16.7005 0.896532 0.448266 0.893900i \(-0.352042\pi\)
0.448266 + 0.893900i \(0.352042\pi\)
\(348\) 0 0
\(349\) −23.5500 −1.26060 −0.630300 0.776351i \(-0.717068\pi\)
−0.630300 + 0.776351i \(0.717068\pi\)
\(350\) 0 0
\(351\) −5.37169 −0.286720
\(352\) 0 0
\(353\) −7.64973 −0.407154 −0.203577 0.979059i \(-0.565257\pi\)
−0.203577 + 0.979059i \(0.565257\pi\)
\(354\) 0 0
\(355\) 2.06740 0.109726
\(356\) 0 0
\(357\) 6.68585 0.353853
\(358\) 0 0
\(359\) −18.3748 −0.969786 −0.484893 0.874573i \(-0.661142\pi\)
−0.484893 + 0.874573i \(0.661142\pi\)
\(360\) 0 0
\(361\) −7.82487 −0.411835
\(362\) 0 0
\(363\) −11.1035 −0.582784
\(364\) 0 0
\(365\) 20.5468 1.07547
\(366\) 0 0
\(367\) −5.33871 −0.278679 −0.139339 0.990245i \(-0.544498\pi\)
−0.139339 + 0.990245i \(0.544498\pi\)
\(368\) 0 0
\(369\) 0.493499 0.0256906
\(370\) 0 0
\(371\) −0.782020 −0.0406004
\(372\) 0 0
\(373\) 21.5212 1.11433 0.557163 0.830403i \(-0.311890\pi\)
0.557163 + 0.830403i \(0.311890\pi\)
\(374\) 0 0
\(375\) 12.6184 0.651614
\(376\) 0 0
\(377\) 10.4893 0.540226
\(378\) 0 0
\(379\) −4.61002 −0.236801 −0.118400 0.992966i \(-0.537777\pi\)
−0.118400 + 0.992966i \(0.537777\pi\)
\(380\) 0 0
\(381\) −13.8223 −0.708140
\(382\) 0 0
\(383\) 8.33558 0.425928 0.212964 0.977060i \(-0.431688\pi\)
0.212964 + 0.977060i \(0.431688\pi\)
\(384\) 0 0
\(385\) 1.53948 0.0784592
\(386\) 0 0
\(387\) −13.7459 −0.698744
\(388\) 0 0
\(389\) −6.44223 −0.326634 −0.163317 0.986574i \(-0.552219\pi\)
−0.163317 + 0.986574i \(0.552219\pi\)
\(390\) 0 0
\(391\) 18.5181 0.936498
\(392\) 0 0
\(393\) 4.20077 0.211901
\(394\) 0 0
\(395\) 1.18500 0.0596238
\(396\) 0 0
\(397\) 1.40046 0.0702872 0.0351436 0.999382i \(-0.488811\pi\)
0.0351436 + 0.999382i \(0.488811\pi\)
\(398\) 0 0
\(399\) 3.83221 0.191851
\(400\) 0 0
\(401\) −6.97858 −0.348494 −0.174247 0.984702i \(-0.555749\pi\)
−0.174247 + 0.984702i \(0.555749\pi\)
\(402\) 0 0
\(403\) −1.63565 −0.0814777
\(404\) 0 0
\(405\) 1.47773 0.0734291
\(406\) 0 0
\(407\) −9.76481 −0.484024
\(408\) 0 0
\(409\) −18.3790 −0.908785 −0.454392 0.890802i \(-0.650144\pi\)
−0.454392 + 0.890802i \(0.650144\pi\)
\(410\) 0 0
\(411\) −15.0214 −0.740952
\(412\) 0 0
\(413\) −12.6430 −0.622121
\(414\) 0 0
\(415\) −16.2885 −0.799572
\(416\) 0 0
\(417\) −8.59388 −0.420844
\(418\) 0 0
\(419\) 30.0393 1.46751 0.733757 0.679412i \(-0.237765\pi\)
0.733757 + 0.679412i \(0.237765\pi\)
\(420\) 0 0
\(421\) −8.31729 −0.405360 −0.202680 0.979245i \(-0.564965\pi\)
−0.202680 + 0.979245i \(0.564965\pi\)
\(422\) 0 0
\(423\) −17.8940 −0.870034
\(424\) 0 0
\(425\) −18.6430 −0.904318
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) −1.31415 −0.0634479
\(430\) 0 0
\(431\) −9.64973 −0.464811 −0.232406 0.972619i \(-0.574660\pi\)
−0.232406 + 0.972619i \(0.574660\pi\)
\(432\) 0 0
\(433\) 26.3074 1.26425 0.632127 0.774865i \(-0.282182\pi\)
0.632127 + 0.774865i \(0.282182\pi\)
\(434\) 0 0
\(435\) −16.1481 −0.774240
\(436\) 0 0
\(437\) 10.6142 0.507748
\(438\) 0 0
\(439\) −33.8139 −1.61385 −0.806925 0.590654i \(-0.798870\pi\)
−0.806925 + 0.590654i \(0.798870\pi\)
\(440\) 0 0
\(441\) −1.68585 −0.0802784
\(442\) 0 0
\(443\) −26.4464 −1.25651 −0.628254 0.778008i \(-0.716230\pi\)
−0.628254 + 0.778008i \(0.716230\pi\)
\(444\) 0 0
\(445\) −7.70306 −0.365160
\(446\) 0 0
\(447\) 2.48508 0.117540
\(448\) 0 0
\(449\) 2.64300 0.124731 0.0623655 0.998053i \(-0.480136\pi\)
0.0623655 + 0.998053i \(0.480136\pi\)
\(450\) 0 0
\(451\) 0.335577 0.0158017
\(452\) 0 0
\(453\) −17.0937 −0.803130
\(454\) 0 0
\(455\) −1.34292 −0.0629572
\(456\) 0 0
\(457\) −33.6890 −1.57590 −0.787952 0.615737i \(-0.788859\pi\)
−0.787952 + 0.615737i \(0.788859\pi\)
\(458\) 0 0
\(459\) −31.3288 −1.46231
\(460\) 0 0
\(461\) 33.0790 1.54064 0.770320 0.637657i \(-0.220096\pi\)
0.770320 + 0.637657i \(0.220096\pi\)
\(462\) 0 0
\(463\) 2.51806 0.117024 0.0585120 0.998287i \(-0.481364\pi\)
0.0585120 + 0.998287i \(0.481364\pi\)
\(464\) 0 0
\(465\) 2.51806 0.116772
\(466\) 0 0
\(467\) −2.57560 −0.119184 −0.0595922 0.998223i \(-0.518980\pi\)
−0.0595922 + 0.998223i \(0.518980\pi\)
\(468\) 0 0
\(469\) 6.10038 0.281690
\(470\) 0 0
\(471\) 26.2155 1.20794
\(472\) 0 0
\(473\) −9.34713 −0.429782
\(474\) 0 0
\(475\) −10.6858 −0.490300
\(476\) 0 0
\(477\) 1.31836 0.0603638
\(478\) 0 0
\(479\) −0.513847 −0.0234783 −0.0117391 0.999931i \(-0.503737\pi\)
−0.0117391 + 0.999931i \(0.503737\pi\)
\(480\) 0 0
\(481\) 8.51806 0.388390
\(482\) 0 0
\(483\) 3.63986 0.165620
\(484\) 0 0
\(485\) 7.17513 0.325806
\(486\) 0 0
\(487\) −36.0575 −1.63392 −0.816962 0.576692i \(-0.804343\pi\)
−0.816962 + 0.576692i \(0.804343\pi\)
\(488\) 0 0
\(489\) −8.11508 −0.366976
\(490\) 0 0
\(491\) 9.22846 0.416475 0.208237 0.978078i \(-0.433227\pi\)
0.208237 + 0.978078i \(0.433227\pi\)
\(492\) 0 0
\(493\) 61.1758 2.75522
\(494\) 0 0
\(495\) −2.59533 −0.116651
\(496\) 0 0
\(497\) −1.53948 −0.0690551
\(498\) 0 0
\(499\) −1.00314 −0.0449065 −0.0224533 0.999748i \(-0.507148\pi\)
−0.0224533 + 0.999748i \(0.507148\pi\)
\(500\) 0 0
\(501\) −2.99686 −0.133890
\(502\) 0 0
\(503\) 30.3503 1.35325 0.676626 0.736327i \(-0.263441\pi\)
0.676626 + 0.736327i \(0.263441\pi\)
\(504\) 0 0
\(505\) −14.9687 −0.666099
\(506\) 0 0
\(507\) 1.14637 0.0509119
\(508\) 0 0
\(509\) 10.5995 0.469816 0.234908 0.972018i \(-0.424521\pi\)
0.234908 + 0.972018i \(0.424521\pi\)
\(510\) 0 0
\(511\) −15.3001 −0.676836
\(512\) 0 0
\(513\) −17.9572 −0.792828
\(514\) 0 0
\(515\) −4.58546 −0.202060
\(516\) 0 0
\(517\) −12.1678 −0.535139
\(518\) 0 0
\(519\) 12.6136 0.553676
\(520\) 0 0
\(521\) −16.2646 −0.712564 −0.356282 0.934378i \(-0.615956\pi\)
−0.356282 + 0.934378i \(0.615956\pi\)
\(522\) 0 0
\(523\) −7.22219 −0.315804 −0.157902 0.987455i \(-0.550473\pi\)
−0.157902 + 0.987455i \(0.550473\pi\)
\(524\) 0 0
\(525\) −3.66442 −0.159929
\(526\) 0 0
\(527\) −9.53948 −0.415546
\(528\) 0 0
\(529\) −12.9185 −0.561675
\(530\) 0 0
\(531\) 21.3142 0.924955
\(532\) 0 0
\(533\) −0.292731 −0.0126796
\(534\) 0 0
\(535\) 6.68585 0.289054
\(536\) 0 0
\(537\) −27.4685 −1.18535
\(538\) 0 0
\(539\) −1.14637 −0.0493775
\(540\) 0 0
\(541\) 17.3534 0.746081 0.373041 0.927815i \(-0.378315\pi\)
0.373041 + 0.927815i \(0.378315\pi\)
\(542\) 0 0
\(543\) 7.52119 0.322765
\(544\) 0 0
\(545\) 18.1249 0.776387
\(546\) 0 0
\(547\) 34.1109 1.45848 0.729238 0.684261i \(-0.239875\pi\)
0.729238 + 0.684261i \(0.239875\pi\)
\(548\) 0 0
\(549\) 3.37169 0.143900
\(550\) 0 0
\(551\) 35.0649 1.49381
\(552\) 0 0
\(553\) −0.882404 −0.0375236
\(554\) 0 0
\(555\) −13.1134 −0.556632
\(556\) 0 0
\(557\) −13.2222 −0.560242 −0.280121 0.959965i \(-0.590375\pi\)
−0.280121 + 0.959965i \(0.590375\pi\)
\(558\) 0 0
\(559\) 8.15371 0.344865
\(560\) 0 0
\(561\) −7.66442 −0.323592
\(562\) 0 0
\(563\) 41.3717 1.74361 0.871804 0.489854i \(-0.162950\pi\)
0.871804 + 0.489854i \(0.162950\pi\)
\(564\) 0 0
\(565\) −22.0863 −0.929178
\(566\) 0 0
\(567\) −1.10038 −0.0462118
\(568\) 0 0
\(569\) −2.68164 −0.112420 −0.0562100 0.998419i \(-0.517902\pi\)
−0.0562100 + 0.998419i \(0.517902\pi\)
\(570\) 0 0
\(571\) 28.9315 1.21075 0.605373 0.795942i \(-0.293024\pi\)
0.605373 + 0.795942i \(0.293024\pi\)
\(572\) 0 0
\(573\) 5.03612 0.210387
\(574\) 0 0
\(575\) −10.1495 −0.423263
\(576\) 0 0
\(577\) 37.5296 1.56238 0.781189 0.624294i \(-0.214613\pi\)
0.781189 + 0.624294i \(0.214613\pi\)
\(578\) 0 0
\(579\) −9.50650 −0.395077
\(580\) 0 0
\(581\) 12.1292 0.503202
\(582\) 0 0
\(583\) 0.896480 0.0371284
\(584\) 0 0
\(585\) 2.26396 0.0936033
\(586\) 0 0
\(587\) −23.0649 −0.951990 −0.475995 0.879448i \(-0.657912\pi\)
−0.475995 + 0.879448i \(0.657912\pi\)
\(588\) 0 0
\(589\) −5.46787 −0.225299
\(590\) 0 0
\(591\) −3.63504 −0.149525
\(592\) 0 0
\(593\) −11.0502 −0.453777 −0.226889 0.973921i \(-0.572855\pi\)
−0.226889 + 0.973921i \(0.572855\pi\)
\(594\) 0 0
\(595\) −7.83221 −0.321089
\(596\) 0 0
\(597\) 15.5872 0.637940
\(598\) 0 0
\(599\) 44.6044 1.82248 0.911242 0.411870i \(-0.135124\pi\)
0.911242 + 0.411870i \(0.135124\pi\)
\(600\) 0 0
\(601\) 47.8715 1.95272 0.976359 0.216156i \(-0.0693520\pi\)
0.976359 + 0.216156i \(0.0693520\pi\)
\(602\) 0 0
\(603\) −10.2843 −0.418809
\(604\) 0 0
\(605\) 13.0073 0.528824
\(606\) 0 0
\(607\) −45.0691 −1.82930 −0.914649 0.404249i \(-0.867533\pi\)
−0.914649 + 0.404249i \(0.867533\pi\)
\(608\) 0 0
\(609\) 12.0246 0.487260
\(610\) 0 0
\(611\) 10.6142 0.429406
\(612\) 0 0
\(613\) −25.5212 −1.03079 −0.515396 0.856952i \(-0.672355\pi\)
−0.515396 + 0.856952i \(0.672355\pi\)
\(614\) 0 0
\(615\) 0.450654 0.0181721
\(616\) 0 0
\(617\) 29.2432 1.17729 0.588643 0.808393i \(-0.299663\pi\)
0.588643 + 0.808393i \(0.299663\pi\)
\(618\) 0 0
\(619\) −4.78623 −0.192375 −0.0961874 0.995363i \(-0.530665\pi\)
−0.0961874 + 0.995363i \(0.530665\pi\)
\(620\) 0 0
\(621\) −17.0558 −0.684428
\(622\) 0 0
\(623\) 5.73604 0.229810
\(624\) 0 0
\(625\) 1.20077 0.0480307
\(626\) 0 0
\(627\) −4.39312 −0.175444
\(628\) 0 0
\(629\) 49.6791 1.98084
\(630\) 0 0
\(631\) 28.3931 1.13031 0.565156 0.824984i \(-0.308816\pi\)
0.565156 + 0.824984i \(0.308816\pi\)
\(632\) 0 0
\(633\) −10.6331 −0.422629
\(634\) 0 0
\(635\) 16.1923 0.642574
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 2.59533 0.102670
\(640\) 0 0
\(641\) 5.96137 0.235460 0.117730 0.993046i \(-0.462438\pi\)
0.117730 + 0.993046i \(0.462438\pi\)
\(642\) 0 0
\(643\) −31.1940 −1.23017 −0.615086 0.788460i \(-0.710879\pi\)
−0.615086 + 0.788460i \(0.710879\pi\)
\(644\) 0 0
\(645\) −12.5525 −0.494253
\(646\) 0 0
\(647\) −14.9112 −0.586219 −0.293109 0.956079i \(-0.594690\pi\)
−0.293109 + 0.956079i \(0.594690\pi\)
\(648\) 0 0
\(649\) 14.4935 0.568920
\(650\) 0 0
\(651\) −1.87506 −0.0734893
\(652\) 0 0
\(653\) −3.57246 −0.139801 −0.0699006 0.997554i \(-0.522268\pi\)
−0.0699006 + 0.997554i \(0.522268\pi\)
\(654\) 0 0
\(655\) −4.92104 −0.192281
\(656\) 0 0
\(657\) 25.7936 1.00630
\(658\) 0 0
\(659\) 3.90383 0.152071 0.0760357 0.997105i \(-0.475774\pi\)
0.0760357 + 0.997105i \(0.475774\pi\)
\(660\) 0 0
\(661\) 13.7936 0.536508 0.268254 0.963348i \(-0.413553\pi\)
0.268254 + 0.963348i \(0.413553\pi\)
\(662\) 0 0
\(663\) 6.68585 0.259657
\(664\) 0 0
\(665\) −4.48929 −0.174087
\(666\) 0 0
\(667\) 33.3049 1.28957
\(668\) 0 0
\(669\) 22.4605 0.868374
\(670\) 0 0
\(671\) 2.29273 0.0885099
\(672\) 0 0
\(673\) 5.70306 0.219837 0.109918 0.993941i \(-0.464941\pi\)
0.109918 + 0.993941i \(0.464941\pi\)
\(674\) 0 0
\(675\) 17.1709 0.660909
\(676\) 0 0
\(677\) 35.2614 1.35521 0.677604 0.735427i \(-0.263019\pi\)
0.677604 + 0.735427i \(0.263019\pi\)
\(678\) 0 0
\(679\) −5.34292 −0.205043
\(680\) 0 0
\(681\) 22.5426 0.863835
\(682\) 0 0
\(683\) −1.03612 −0.0396459 −0.0198229 0.999804i \(-0.506310\pi\)
−0.0198229 + 0.999804i \(0.506310\pi\)
\(684\) 0 0
\(685\) 17.5970 0.672348
\(686\) 0 0
\(687\) 8.90131 0.339606
\(688\) 0 0
\(689\) −0.782020 −0.0297926
\(690\) 0 0
\(691\) −3.67850 −0.139937 −0.0699684 0.997549i \(-0.522290\pi\)
−0.0699684 + 0.997549i \(0.522290\pi\)
\(692\) 0 0
\(693\) 1.93260 0.0734132
\(694\) 0 0
\(695\) 10.0674 0.381878
\(696\) 0 0
\(697\) −1.70727 −0.0646674
\(698\) 0 0
\(699\) 13.9817 0.528837
\(700\) 0 0
\(701\) −0.0617493 −0.00233224 −0.00116612 0.999999i \(-0.500371\pi\)
−0.00116612 + 0.999999i \(0.500371\pi\)
\(702\) 0 0
\(703\) 28.4752 1.07396
\(704\) 0 0
\(705\) −16.3404 −0.615415
\(706\) 0 0
\(707\) 11.1464 0.419202
\(708\) 0 0
\(709\) −42.9834 −1.61428 −0.807138 0.590363i \(-0.798985\pi\)
−0.807138 + 0.590363i \(0.798985\pi\)
\(710\) 0 0
\(711\) 1.48760 0.0557892
\(712\) 0 0
\(713\) −5.19342 −0.194495
\(714\) 0 0
\(715\) 1.53948 0.0575733
\(716\) 0 0
\(717\) 11.7992 0.440650
\(718\) 0 0
\(719\) −17.6546 −0.658404 −0.329202 0.944260i \(-0.606780\pi\)
−0.329202 + 0.944260i \(0.606780\pi\)
\(720\) 0 0
\(721\) 3.41454 0.127164
\(722\) 0 0
\(723\) −4.61844 −0.171762
\(724\) 0 0
\(725\) −33.5296 −1.24526
\(726\) 0 0
\(727\) 23.8077 0.882977 0.441488 0.897267i \(-0.354451\pi\)
0.441488 + 0.897267i \(0.354451\pi\)
\(728\) 0 0
\(729\) 20.3288 0.752920
\(730\) 0 0
\(731\) 47.5542 1.75885
\(732\) 0 0
\(733\) 31.3492 1.15791 0.578954 0.815360i \(-0.303461\pi\)
0.578954 + 0.815360i \(0.303461\pi\)
\(734\) 0 0
\(735\) −1.53948 −0.0567846
\(736\) 0 0
\(737\) −6.99327 −0.257600
\(738\) 0 0
\(739\) 24.8108 0.912680 0.456340 0.889806i \(-0.349160\pi\)
0.456340 + 0.889806i \(0.349160\pi\)
\(740\) 0 0
\(741\) 3.83221 0.140780
\(742\) 0 0
\(743\) −21.3717 −0.784051 −0.392026 0.919954i \(-0.628226\pi\)
−0.392026 + 0.919954i \(0.628226\pi\)
\(744\) 0 0
\(745\) −2.91117 −0.106657
\(746\) 0 0
\(747\) −20.4479 −0.748149
\(748\) 0 0
\(749\) −4.97858 −0.181913
\(750\) 0 0
\(751\) −19.2243 −0.701503 −0.350751 0.936469i \(-0.614074\pi\)
−0.350751 + 0.936469i \(0.614074\pi\)
\(752\) 0 0
\(753\) −3.33727 −0.121617
\(754\) 0 0
\(755\) 20.0246 0.728768
\(756\) 0 0
\(757\) −19.8610 −0.721860 −0.360930 0.932593i \(-0.617541\pi\)
−0.360930 + 0.932593i \(0.617541\pi\)
\(758\) 0 0
\(759\) −4.17262 −0.151456
\(760\) 0 0
\(761\) 6.12073 0.221876 0.110938 0.993827i \(-0.464614\pi\)
0.110938 + 0.993827i \(0.464614\pi\)
\(762\) 0 0
\(763\) −13.4966 −0.488611
\(764\) 0 0
\(765\) 13.2039 0.477388
\(766\) 0 0
\(767\) −12.6430 −0.456512
\(768\) 0 0
\(769\) 3.82800 0.138041 0.0690206 0.997615i \(-0.478013\pi\)
0.0690206 + 0.997615i \(0.478013\pi\)
\(770\) 0 0
\(771\) −22.4653 −0.809070
\(772\) 0 0
\(773\) −6.53635 −0.235096 −0.117548 0.993067i \(-0.537503\pi\)
−0.117548 + 0.993067i \(0.537503\pi\)
\(774\) 0 0
\(775\) 5.22846 0.187812
\(776\) 0 0
\(777\) 9.76481 0.350311
\(778\) 0 0
\(779\) −0.978577 −0.0350612
\(780\) 0 0
\(781\) 1.76481 0.0631498
\(782\) 0 0
\(783\) −56.3452 −2.01361
\(784\) 0 0
\(785\) −30.7104 −1.09610
\(786\) 0 0
\(787\) −30.8066 −1.09814 −0.549068 0.835778i \(-0.685017\pi\)
−0.549068 + 0.835778i \(0.685017\pi\)
\(788\) 0 0
\(789\) −8.67598 −0.308873
\(790\) 0 0
\(791\) 16.4464 0.584768
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) 1.20390 0.0426981
\(796\) 0 0
\(797\) −38.8156 −1.37492 −0.687460 0.726222i \(-0.741274\pi\)
−0.687460 + 0.726222i \(0.741274\pi\)
\(798\) 0 0
\(799\) 61.9044 2.19002
\(800\) 0 0
\(801\) −9.67008 −0.341675
\(802\) 0 0
\(803\) 17.5395 0.618955
\(804\) 0 0
\(805\) −4.26396 −0.150285
\(806\) 0 0
\(807\) −10.8585 −0.382236
\(808\) 0 0
\(809\) 1.04033 0.0365759 0.0182880 0.999833i \(-0.494178\pi\)
0.0182880 + 0.999833i \(0.494178\pi\)
\(810\) 0 0
\(811\) −12.6712 −0.444944 −0.222472 0.974939i \(-0.571413\pi\)
−0.222472 + 0.974939i \(0.571413\pi\)
\(812\) 0 0
\(813\) −33.6707 −1.18088
\(814\) 0 0
\(815\) 9.50650 0.332998
\(816\) 0 0
\(817\) 27.2572 0.953610
\(818\) 0 0
\(819\) −1.68585 −0.0589082
\(820\) 0 0
\(821\) 27.0361 0.943567 0.471783 0.881714i \(-0.343610\pi\)
0.471783 + 0.881714i \(0.343610\pi\)
\(822\) 0 0
\(823\) 31.6363 1.10277 0.551386 0.834251i \(-0.314099\pi\)
0.551386 + 0.834251i \(0.314099\pi\)
\(824\) 0 0
\(825\) 4.20077 0.146252
\(826\) 0 0
\(827\) −56.4800 −1.96400 −0.982002 0.188872i \(-0.939517\pi\)
−0.982002 + 0.188872i \(0.939517\pi\)
\(828\) 0 0
\(829\) −42.6760 −1.48220 −0.741099 0.671396i \(-0.765695\pi\)
−0.741099 + 0.671396i \(0.765695\pi\)
\(830\) 0 0
\(831\) −2.18248 −0.0757094
\(832\) 0 0
\(833\) 5.83221 0.202074
\(834\) 0 0
\(835\) 3.51071 0.121493
\(836\) 0 0
\(837\) 8.78623 0.303697
\(838\) 0 0
\(839\) 40.1642 1.38662 0.693311 0.720639i \(-0.256151\pi\)
0.693311 + 0.720639i \(0.256151\pi\)
\(840\) 0 0
\(841\) 81.0252 2.79397
\(842\) 0 0
\(843\) −23.5872 −0.812385
\(844\) 0 0
\(845\) −1.34292 −0.0461980
\(846\) 0 0
\(847\) −9.68585 −0.332810
\(848\) 0 0
\(849\) 30.9442 1.06200
\(850\) 0 0
\(851\) 27.0460 0.927124
\(852\) 0 0
\(853\) 19.6932 0.674282 0.337141 0.941454i \(-0.390540\pi\)
0.337141 + 0.941454i \(0.390540\pi\)
\(854\) 0 0
\(855\) 7.56825 0.258829
\(856\) 0 0
\(857\) 1.66442 0.0568556 0.0284278 0.999596i \(-0.490950\pi\)
0.0284278 + 0.999596i \(0.490950\pi\)
\(858\) 0 0
\(859\) 41.2944 1.40895 0.704474 0.709730i \(-0.251183\pi\)
0.704474 + 0.709730i \(0.251183\pi\)
\(860\) 0 0
\(861\) −0.335577 −0.0114364
\(862\) 0 0
\(863\) 31.3288 1.06645 0.533223 0.845975i \(-0.320981\pi\)
0.533223 + 0.845975i \(0.320981\pi\)
\(864\) 0 0
\(865\) −14.7764 −0.502411
\(866\) 0 0
\(867\) 19.5051 0.662426
\(868\) 0 0
\(869\) 1.01156 0.0343147
\(870\) 0 0
\(871\) 6.10038 0.206704
\(872\) 0 0
\(873\) 9.00735 0.304852
\(874\) 0 0
\(875\) 11.0073 0.372116
\(876\) 0 0
\(877\) −53.8041 −1.81683 −0.908417 0.418065i \(-0.862708\pi\)
−0.908417 + 0.418065i \(0.862708\pi\)
\(878\) 0 0
\(879\) −17.0888 −0.576392
\(880\) 0 0
\(881\) 8.09196 0.272625 0.136313 0.990666i \(-0.456475\pi\)
0.136313 + 0.990666i \(0.456475\pi\)
\(882\) 0 0
\(883\) 27.0705 0.910996 0.455498 0.890237i \(-0.349461\pi\)
0.455498 + 0.890237i \(0.349461\pi\)
\(884\) 0 0
\(885\) 19.4637 0.654264
\(886\) 0 0
\(887\) 38.4935 1.29249 0.646243 0.763132i \(-0.276339\pi\)
0.646243 + 0.763132i \(0.276339\pi\)
\(888\) 0 0
\(889\) −12.0575 −0.404397
\(890\) 0 0
\(891\) 1.26144 0.0422599
\(892\) 0 0
\(893\) 35.4826 1.18738
\(894\) 0 0
\(895\) 32.1783 1.07560
\(896\) 0 0
\(897\) 3.63986 0.121532
\(898\) 0 0
\(899\) −17.1568 −0.572213
\(900\) 0 0
\(901\) −4.56090 −0.151946
\(902\) 0 0
\(903\) 9.34713 0.311053
\(904\) 0 0
\(905\) −8.81079 −0.292881
\(906\) 0 0
\(907\) −15.7031 −0.521411 −0.260706 0.965418i \(-0.583955\pi\)
−0.260706 + 0.965418i \(0.583955\pi\)
\(908\) 0 0
\(909\) −18.7911 −0.623260
\(910\) 0 0
\(911\) −44.9399 −1.48893 −0.744463 0.667663i \(-0.767295\pi\)
−0.744463 + 0.667663i \(0.767295\pi\)
\(912\) 0 0
\(913\) −13.9044 −0.460170
\(914\) 0 0
\(915\) 3.07896 0.101787
\(916\) 0 0
\(917\) 3.66442 0.121010
\(918\) 0 0
\(919\) 27.2432 0.898669 0.449334 0.893364i \(-0.351661\pi\)
0.449334 + 0.893364i \(0.351661\pi\)
\(920\) 0 0
\(921\) −29.8385 −0.983211
\(922\) 0 0
\(923\) −1.53948 −0.0506726
\(924\) 0 0
\(925\) −27.2285 −0.895266
\(926\) 0 0
\(927\) −5.75639 −0.189065
\(928\) 0 0
\(929\) −17.1422 −0.562416 −0.281208 0.959647i \(-0.590735\pi\)
−0.281208 + 0.959647i \(0.590735\pi\)
\(930\) 0 0
\(931\) 3.34292 0.109560
\(932\) 0 0
\(933\) −22.3503 −0.731715
\(934\) 0 0
\(935\) 8.97858 0.293631
\(936\) 0 0
\(937\) −51.5197 −1.68308 −0.841538 0.540197i \(-0.818350\pi\)
−0.841538 + 0.540197i \(0.818350\pi\)
\(938\) 0 0
\(939\) −3.99158 −0.130260
\(940\) 0 0
\(941\) 32.7575 1.06786 0.533931 0.845528i \(-0.320714\pi\)
0.533931 + 0.845528i \(0.320714\pi\)
\(942\) 0 0
\(943\) −0.929460 −0.0302674
\(944\) 0 0
\(945\) 7.21377 0.234664
\(946\) 0 0
\(947\) −20.9295 −0.680116 −0.340058 0.940404i \(-0.610447\pi\)
−0.340058 + 0.940404i \(0.610447\pi\)
\(948\) 0 0
\(949\) −15.3001 −0.496662
\(950\) 0 0
\(951\) 5.75639 0.186664
\(952\) 0 0
\(953\) 46.4120 1.50343 0.751716 0.659487i \(-0.229226\pi\)
0.751716 + 0.659487i \(0.229226\pi\)
\(954\) 0 0
\(955\) −5.89962 −0.190907
\(956\) 0 0
\(957\) −13.7845 −0.445591
\(958\) 0 0
\(959\) −13.1035 −0.423135
\(960\) 0 0
\(961\) −28.3246 −0.913698
\(962\) 0 0
\(963\) 8.39312 0.270464
\(964\) 0 0
\(965\) 11.1365 0.358497
\(966\) 0 0
\(967\) 23.2186 0.746660 0.373330 0.927699i \(-0.378216\pi\)
0.373330 + 0.927699i \(0.378216\pi\)
\(968\) 0 0
\(969\) 22.3503 0.717994
\(970\) 0 0
\(971\) 14.1004 0.452503 0.226251 0.974069i \(-0.427353\pi\)
0.226251 + 0.974069i \(0.427353\pi\)
\(972\) 0 0
\(973\) −7.49663 −0.240331
\(974\) 0 0
\(975\) −3.66442 −0.117355
\(976\) 0 0
\(977\) −2.95402 −0.0945074 −0.0472537 0.998883i \(-0.515047\pi\)
−0.0472537 + 0.998883i \(0.515047\pi\)
\(978\) 0 0
\(979\) −6.57560 −0.210157
\(980\) 0 0
\(981\) 22.7533 0.726455
\(982\) 0 0
\(983\) 35.0367 1.11750 0.558749 0.829337i \(-0.311281\pi\)
0.558749 + 0.829337i \(0.311281\pi\)
\(984\) 0 0
\(985\) 4.25831 0.135681
\(986\) 0 0
\(987\) 12.1678 0.387305
\(988\) 0 0
\(989\) 25.8891 0.823227
\(990\) 0 0
\(991\) −47.9718 −1.52388 −0.761938 0.647650i \(-0.775752\pi\)
−0.761938 + 0.647650i \(0.775752\pi\)
\(992\) 0 0
\(993\) 7.04958 0.223712
\(994\) 0 0
\(995\) −18.2598 −0.578873
\(996\) 0 0
\(997\) −38.4422 −1.21748 −0.608739 0.793371i \(-0.708324\pi\)
−0.608739 + 0.793371i \(0.708324\pi\)
\(998\) 0 0
\(999\) −45.7564 −1.44767
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 1456.2.a.t.1.2 3
4.3 odd 2 91.2.a.d.1.3 3
8.3 odd 2 5824.2.a.by.1.2 3
8.5 even 2 5824.2.a.bs.1.2 3
12.11 even 2 819.2.a.i.1.1 3
20.19 odd 2 2275.2.a.m.1.1 3
28.3 even 6 637.2.e.i.79.1 6
28.11 odd 6 637.2.e.j.79.1 6
28.19 even 6 637.2.e.i.508.1 6
28.23 odd 6 637.2.e.j.508.1 6
28.27 even 2 637.2.a.j.1.3 3
52.31 even 4 1183.2.c.f.337.1 6
52.47 even 4 1183.2.c.f.337.6 6
52.51 odd 2 1183.2.a.i.1.1 3
84.83 odd 2 5733.2.a.x.1.1 3
364.363 even 2 8281.2.a.bg.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.3 3 4.3 odd 2
637.2.a.j.1.3 3 28.27 even 2
637.2.e.i.79.1 6 28.3 even 6
637.2.e.i.508.1 6 28.19 even 6
637.2.e.j.79.1 6 28.11 odd 6
637.2.e.j.508.1 6 28.23 odd 6
819.2.a.i.1.1 3 12.11 even 2
1183.2.a.i.1.1 3 52.51 odd 2
1183.2.c.f.337.1 6 52.31 even 4
1183.2.c.f.337.6 6 52.47 even 4
1456.2.a.t.1.2 3 1.1 even 1 trivial
2275.2.a.m.1.1 3 20.19 odd 2
5733.2.a.x.1.1 3 84.83 odd 2
5824.2.a.bs.1.2 3 8.5 even 2
5824.2.a.by.1.2 3 8.3 odd 2
8281.2.a.bg.1.1 3 364.363 even 2