Properties

Label 4140.2.a.u.1.1
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 1.1
Root \(1.84503\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} -3.57723 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -3.57723 q^{7} +4.75505 q^{11} -6.44510 q^{13} +3.46021 q^{17} -2.75505 q^{19} +1.00000 q^{23} +1.00000 q^{25} +2.86787 q^{29} -0.395216 q^{31} -3.57723 q^{35} +0.112824 q^{37} -2.32389 q^{41} +12.6105 q^{43} -6.80074 q^{47} +5.79655 q^{49} -1.41452 q^{53} +4.75505 q^{55} +4.51223 q^{59} -0.165379 q^{61} -6.44510 q^{65} +5.53153 q^{67} +13.4783 q^{71} +10.4451 q^{73} -17.0099 q^{77} +0.498285 q^{79} -12.8487 q^{83} +3.46021 q^{85} +5.15445 q^{89} +23.0556 q^{91} -2.75505 q^{95} -6.96612 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{5} + 4 q^{7}+O(q^{10}) \) 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
\(7\) −3.57723 −1.35206 −0.676032 0.736872i \(-0.736302\pi\)
−0.676032 + 0.736872i \(0.736302\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.75505 1.43370 0.716850 0.697227i \(-0.245583\pi\)
0.716850 + 0.697227i \(0.245583\pi\)
\(12\) 0 0
\(13\) −6.44510 −1.78755 −0.893774 0.448517i \(-0.851952\pi\)
−0.893774 + 0.448517i \(0.851952\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46021 0.839225 0.419612 0.907703i \(-0.362166\pi\)
0.419612 + 0.907703i \(0.362166\pi\)
\(18\) 0 0
\(19\) −2.75505 −0.632051 −0.316026 0.948751i \(-0.602349\pi\)
−0.316026 + 0.948751i \(0.602349\pi\)
\(20\) 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
\(29\) 2.86787 0.532550 0.266275 0.963897i \(-0.414207\pi\)
0.266275 + 0.963897i \(0.414207\pi\)
\(30\) 0 0
\(31\) −0.395216 −0.0709828 −0.0354914 0.999370i \(-0.511300\pi\)
−0.0354914 + 0.999370i \(0.511300\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.57723 −0.604662
\(36\) 0 0
\(37\) 0.112824 0.0185481 0.00927404 0.999957i \(-0.497048\pi\)
0.00927404 + 0.999957i \(0.497048\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.32389 −0.362931 −0.181466 0.983397i \(-0.558084\pi\)
−0.181466 + 0.983397i \(0.558084\pi\)
\(42\) 0 0
\(43\) 12.6105 1.92308 0.961540 0.274666i \(-0.0885672\pi\)
0.961540 + 0.274666i \(0.0885672\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.80074 −0.991990 −0.495995 0.868325i \(-0.665197\pi\)
−0.495995 + 0.868325i \(0.665197\pi\)
\(48\) 0 0
\(49\) 5.79655 0.828079
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.41452 −0.194299 −0.0971497 0.995270i \(-0.530973\pi\)
−0.0971497 + 0.995270i \(0.530973\pi\)
\(54\) 0 0
\(55\) 4.75505 0.641170
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.51223 0.587442 0.293721 0.955891i \(-0.405106\pi\)
0.293721 + 0.955891i \(0.405106\pi\)
\(60\) 0 0
\(61\) −0.165379 −0.0211746 −0.0105873 0.999944i \(-0.503370\pi\)
−0.0105873 + 0.999944i \(0.503370\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.44510 −0.799416
\(66\) 0 0
\(67\) 5.53153 0.675785 0.337892 0.941185i \(-0.390286\pi\)
0.337892 + 0.941185i \(0.390286\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.4783 1.59959 0.799793 0.600276i \(-0.204943\pi\)
0.799793 + 0.600276i \(0.204943\pi\)
\(72\) 0 0
\(73\) 10.4451 1.22251 0.611253 0.791435i \(-0.290666\pi\)
0.611253 + 0.791435i \(0.290666\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −17.0099 −1.93846
\(78\) 0 0
\(79\) 0.498285 0.0560614 0.0280307 0.999607i \(-0.491076\pi\)
0.0280307 + 0.999607i \(0.491076\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.8487 −1.41033 −0.705164 0.709045i \(-0.749126\pi\)
−0.705164 + 0.709045i \(0.749126\pi\)
\(84\) 0 0
\(85\) 3.46021 0.375313
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.15445 0.546371 0.273185 0.961961i \(-0.411923\pi\)
0.273185 + 0.961961i \(0.411923\pi\)
\(90\) 0 0
\(91\) 23.0556 2.41688
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.75505 −0.282662
\(96\) 0 0
\(97\) −6.96612 −0.707302 −0.353651 0.935377i \(-0.615060\pi\)
−0.353651 + 0.935377i \(0.615060\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.1227 1.50477 0.752383 0.658726i \(-0.228904\pi\)
0.752383 + 0.658726i \(0.228904\pi\)
\(102\) 0 0
\(103\) 0.899619 0.0886421 0.0443210 0.999017i \(-0.485888\pi\)
0.0443210 + 0.999017i \(0.485888\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.99581 −0.192942 −0.0964711 0.995336i \(-0.530756\pi\)
−0.0964711 + 0.995336i \(0.530756\pi\)
\(108\) 0 0
\(109\) −2.75505 −0.263886 −0.131943 0.991257i \(-0.542122\pi\)
−0.131943 + 0.991257i \(0.542122\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 19.8064 1.86323 0.931616 0.363444i \(-0.118399\pi\)
0.931616 + 0.363444i \(0.118399\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.3780 −1.13469
\(120\) 0 0
\(121\) 11.6105 1.05550
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 12.2111 1.08356 0.541779 0.840521i \(-0.317751\pi\)
0.541779 + 0.840521i \(0.317751\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 11.1545 0.974569 0.487285 0.873243i \(-0.337987\pi\)
0.487285 + 0.873243i \(0.337987\pi\)
\(132\) 0 0
\(133\) 9.85543 0.854574
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −15.4258 −1.31791 −0.658957 0.752180i \(-0.729002\pi\)
−0.658957 + 0.752180i \(0.729002\pi\)
\(138\) 0 0
\(139\) −0.796550 −0.0675624 −0.0337812 0.999429i \(-0.510755\pi\)
−0.0337812 + 0.999429i \(0.510755\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −30.6467 −2.56281
\(144\) 0 0
\(145\) 2.86787 0.238164
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.26514 0.677107 0.338553 0.940947i \(-0.390062\pi\)
0.338553 + 0.940947i \(0.390062\pi\)
\(150\) 0 0
\(151\) 18.6105 1.51450 0.757250 0.653126i \(-0.226543\pi\)
0.757250 + 0.653126i \(0.226543\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.395216 −0.0317445
\(156\) 0 0
\(157\) −18.8231 −1.50224 −0.751122 0.660163i \(-0.770487\pi\)
−0.751122 + 0.660163i \(0.770487\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.57723 −0.281925
\(162\) 0 0
\(163\) 1.07957 0.0845588 0.0422794 0.999106i \(-0.486538\pi\)
0.0422794 + 0.999106i \(0.486538\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.94681 0.769707 0.384854 0.922978i \(-0.374252\pi\)
0.384854 + 0.922978i \(0.374252\pi\)
\(168\) 0 0
\(169\) 28.5393 2.19533
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.945311 −0.0718706 −0.0359353 0.999354i \(-0.511441\pi\)
−0.0359353 + 0.999354i \(0.511441\pi\)
\(174\) 0 0
\(175\) −3.57723 −0.270413
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.4762 1.38098 0.690488 0.723343i \(-0.257396\pi\)
0.690488 + 0.723343i \(0.257396\pi\)
\(180\) 0 0
\(181\) −19.6806 −1.46285 −0.731425 0.681922i \(-0.761144\pi\)
−0.731425 + 0.681922i \(0.761144\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.112824 0.00829495
\(186\) 0 0
\(187\) 16.4535 1.20320
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.7248 1.06545 0.532725 0.846288i \(-0.321168\pi\)
0.532725 + 0.846288i \(0.321168\pi\)
\(192\) 0 0
\(193\) −21.8106 −1.56996 −0.784981 0.619519i \(-0.787328\pi\)
−0.784981 + 0.619519i \(0.787328\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.0446 1.14313 0.571567 0.820555i \(-0.306336\pi\)
0.571567 + 0.820555i \(0.306336\pi\)
\(198\) 0 0
\(199\) −0.928806 −0.0658413 −0.0329207 0.999458i \(-0.510481\pi\)
−0.0329207 + 0.999458i \(0.510481\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −10.2590 −0.720042
\(204\) 0 0
\(205\) −2.32389 −0.162308
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −13.1004 −0.906172
\(210\) 0 0
\(211\) 5.88450 0.405106 0.202553 0.979271i \(-0.435076\pi\)
0.202553 + 0.979271i \(0.435076\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 12.6105 0.860027
\(216\) 0 0
\(217\) 1.41378 0.0959734
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −22.3014 −1.50016
\(222\) 0 0
\(223\) 14.9204 0.999146 0.499573 0.866272i \(-0.333490\pi\)
0.499573 + 0.866272i \(0.333490\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.6890 1.17406 0.587030 0.809565i \(-0.300297\pi\)
0.587030 + 0.809565i \(0.300297\pi\)
\(228\) 0 0
\(229\) −3.87839 −0.256291 −0.128145 0.991755i \(-0.540902\pi\)
−0.128145 + 0.991755i \(0.540902\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.41871 −0.354992 −0.177496 0.984122i \(-0.556800\pi\)
−0.177496 + 0.984122i \(0.556800\pi\)
\(234\) 0 0
\(235\) −6.80074 −0.443631
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −13.1613 −0.851335 −0.425667 0.904880i \(-0.639961\pi\)
−0.425667 + 0.904880i \(0.639961\pi\)
\(240\) 0 0
\(241\) 27.9760 1.80209 0.901046 0.433723i \(-0.142800\pi\)
0.901046 + 0.433723i \(0.142800\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.79655 0.370328
\(246\) 0 0
\(247\) 17.7565 1.12982
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 30.3173 1.91361 0.956805 0.290730i \(-0.0938982\pi\)
0.956805 + 0.290730i \(0.0938982\pi\)
\(252\) 0 0
\(253\) 4.75505 0.298947
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −23.8336 −1.48670 −0.743349 0.668904i \(-0.766764\pi\)
−0.743349 + 0.668904i \(0.766764\pi\)
\(258\) 0 0
\(259\) −0.403595 −0.0250782
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −18.5261 −1.14237 −0.571184 0.820822i \(-0.693516\pi\)
−0.571184 + 0.820822i \(0.693516\pi\)
\(264\) 0 0
\(265\) −1.41452 −0.0868933
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −30.9125 −1.88477 −0.942385 0.334531i \(-0.891422\pi\)
−0.942385 + 0.334531i \(0.891422\pi\)
\(270\) 0 0
\(271\) −8.51579 −0.517297 −0.258649 0.965971i \(-0.583277\pi\)
−0.258649 + 0.965971i \(0.583277\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.75505 0.286740
\(276\) 0 0
\(277\) 21.8936 1.31546 0.657730 0.753254i \(-0.271517\pi\)
0.657730 + 0.753254i \(0.271517\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.8215 1.24211 0.621055 0.783767i \(-0.286704\pi\)
0.621055 + 0.783767i \(0.286704\pi\)
\(282\) 0 0
\(283\) 7.09159 0.421551 0.210776 0.977534i \(-0.432401\pi\)
0.210776 + 0.977534i \(0.432401\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.31310 0.490707
\(288\) 0 0
\(289\) −5.02693 −0.295702
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.04645 0.177976 0.0889878 0.996033i \(-0.471637\pi\)
0.0889878 + 0.996033i \(0.471637\pi\)
\(294\) 0 0
\(295\) 4.51223 0.262712
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.44510 −0.372730
\(300\) 0 0
\(301\) −45.1105 −2.60013
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.165379 −0.00946956
\(306\) 0 0
\(307\) 28.6416 1.63466 0.817331 0.576168i \(-0.195453\pi\)
0.817331 + 0.576168i \(0.195453\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.6259 −0.602542 −0.301271 0.953539i \(-0.597411\pi\)
−0.301271 + 0.953539i \(0.597411\pi\)
\(312\) 0 0
\(313\) −8.30932 −0.469670 −0.234835 0.972035i \(-0.575455\pi\)
−0.234835 + 0.972035i \(0.575455\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.67074 0.262335 0.131168 0.991360i \(-0.458127\pi\)
0.131168 + 0.991360i \(0.458127\pi\)
\(318\) 0 0
\(319\) 13.6369 0.763518
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −9.53305 −0.530433
\(324\) 0 0
\(325\) −6.44510 −0.357510
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 24.3278 1.34123
\(330\) 0 0
\(331\) 11.7753 0.647230 0.323615 0.946189i \(-0.395102\pi\)
0.323615 + 0.946189i \(0.395102\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.53153 0.302220
\(336\) 0 0
\(337\) −9.01181 −0.490905 −0.245452 0.969409i \(-0.578936\pi\)
−0.245452 + 0.969409i \(0.578936\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.87927 −0.101768
\(342\) 0 0
\(343\) 4.30501 0.232449
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.61990 −0.248009 −0.124005 0.992282i \(-0.539574\pi\)
−0.124005 + 0.992282i \(0.539574\pi\)
\(348\) 0 0
\(349\) 8.93492 0.478276 0.239138 0.970986i \(-0.423135\pi\)
0.239138 + 0.970986i \(0.423135\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.12807 −0.432613 −0.216307 0.976325i \(-0.569401\pi\)
−0.216307 + 0.976325i \(0.569401\pi\)
\(354\) 0 0
\(355\) 13.4783 0.715356
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −33.5985 −1.77326 −0.886631 0.462478i \(-0.846960\pi\)
−0.886631 + 0.462478i \(0.846960\pi\)
\(360\) 0 0
\(361\) −11.4097 −0.600511
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 10.4451 0.546721
\(366\) 0 0
\(367\) −35.8254 −1.87007 −0.935037 0.354551i \(-0.884634\pi\)
−0.935037 + 0.354551i \(0.884634\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5.06006 0.262705
\(372\) 0 0
\(373\) −25.4728 −1.31893 −0.659465 0.751735i \(-0.729217\pi\)
−0.659465 + 0.751735i \(0.729217\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −18.4837 −0.951959
\(378\) 0 0
\(379\) 8.68105 0.445916 0.222958 0.974828i \(-0.428429\pi\)
0.222958 + 0.974828i \(0.428429\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 22.7725 1.16362 0.581811 0.813324i \(-0.302344\pi\)
0.581811 + 0.813324i \(0.302344\pi\)
\(384\) 0 0
\(385\) −17.0099 −0.866904
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −14.4221 −0.731232 −0.365616 0.930766i \(-0.619142\pi\)
−0.365616 + 0.930766i \(0.619142\pi\)
\(390\) 0 0
\(391\) 3.46021 0.174990
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.498285 0.0250714
\(396\) 0 0
\(397\) −0.806936 −0.0404990 −0.0202495 0.999795i \(-0.506446\pi\)
−0.0202495 + 0.999795i \(0.506446\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.6562 1.53090 0.765448 0.643498i \(-0.222517\pi\)
0.765448 + 0.643498i \(0.222517\pi\)
\(402\) 0 0
\(403\) 2.54720 0.126885
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.536481 0.0265924
\(408\) 0 0
\(409\) 6.80493 0.336482 0.168241 0.985746i \(-0.446191\pi\)
0.168241 + 0.985746i \(0.446191\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −16.1413 −0.794260
\(414\) 0 0
\(415\) −12.8487 −0.630718
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.38568 −0.116548 −0.0582740 0.998301i \(-0.518560\pi\)
−0.0582740 + 0.998301i \(0.518560\pi\)
\(420\) 0 0
\(421\) −1.01093 −0.0492695 −0.0246348 0.999697i \(-0.507842\pi\)
−0.0246348 + 0.999697i \(0.507842\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.46021 0.167845
\(426\) 0 0
\(427\) 0.591597 0.0286294
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.59737 0.365952 0.182976 0.983117i \(-0.441427\pi\)
0.182976 + 0.983117i \(0.441427\pi\)
\(432\) 0 0
\(433\) −10.4009 −0.499836 −0.249918 0.968267i \(-0.580404\pi\)
−0.249918 + 0.968267i \(0.580404\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.75505 −0.131792
\(438\) 0 0
\(439\) 16.4805 0.786570 0.393285 0.919417i \(-0.371338\pi\)
0.393285 + 0.919417i \(0.371338\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.18084 −0.103615 −0.0518074 0.998657i \(-0.516498\pi\)
−0.0518074 + 0.998657i \(0.516498\pi\)
\(444\) 0 0
\(445\) 5.15445 0.244345
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −23.2900 −1.09912 −0.549562 0.835453i \(-0.685205\pi\)
−0.549562 + 0.835453i \(0.685205\pi\)
\(450\) 0 0
\(451\) −11.0502 −0.520335
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 23.0556 1.08086
\(456\) 0 0
\(457\) 40.5964 1.89902 0.949509 0.313739i \(-0.101582\pi\)
0.949509 + 0.313739i \(0.101582\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 22.5980 1.05250 0.526248 0.850331i \(-0.323598\pi\)
0.526248 + 0.850331i \(0.323598\pi\)
\(462\) 0 0
\(463\) 26.2943 1.22200 0.611001 0.791630i \(-0.290767\pi\)
0.611001 + 0.791630i \(0.290767\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.81481 0.454175 0.227088 0.973874i \(-0.427080\pi\)
0.227088 + 0.973874i \(0.427080\pi\)
\(468\) 0 0
\(469\) −19.7876 −0.913704
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 59.9634 2.75712
\(474\) 0 0
\(475\) −2.75505 −0.126410
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.47060 0.158576 0.0792879 0.996852i \(-0.474735\pi\)
0.0792879 + 0.996852i \(0.474735\pi\)
\(480\) 0 0
\(481\) −0.727159 −0.0331556
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.96612 −0.316315
\(486\) 0 0
\(487\) −9.95519 −0.451113 −0.225556 0.974230i \(-0.572420\pi\)
−0.225556 + 0.974230i \(0.572420\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −27.8786 −1.25815 −0.629073 0.777347i \(-0.716565\pi\)
−0.629073 + 0.777347i \(0.716565\pi\)
\(492\) 0 0
\(493\) 9.92344 0.446929
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −48.2151 −2.16274
\(498\) 0 0
\(499\) 2.53359 0.113419 0.0567095 0.998391i \(-0.481939\pi\)
0.0567095 + 0.998391i \(0.481939\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 43.2198 1.92708 0.963538 0.267570i \(-0.0862206\pi\)
0.963538 + 0.267570i \(0.0862206\pi\)
\(504\) 0 0
\(505\) 15.1227 0.672952
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −21.2085 −0.940051 −0.470026 0.882653i \(-0.655755\pi\)
−0.470026 + 0.882653i \(0.655755\pi\)
\(510\) 0 0
\(511\) −37.3645 −1.65291
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.899619 0.0396419
\(516\) 0 0
\(517\) −32.3378 −1.42222
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −37.2053 −1.63000 −0.814998 0.579464i \(-0.803262\pi\)
−0.814998 + 0.579464i \(0.803262\pi\)
\(522\) 0 0
\(523\) 1.78273 0.0779535 0.0389767 0.999240i \(-0.487590\pi\)
0.0389767 + 0.999240i \(0.487590\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.36753 −0.0595706
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14.9777 0.648757
\(534\) 0 0
\(535\) −1.99581 −0.0862864
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 27.5629 1.18722
\(540\) 0 0
\(541\) 13.9687 0.600562 0.300281 0.953851i \(-0.402919\pi\)
0.300281 + 0.953851i \(0.402919\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.75505 −0.118013
\(546\) 0 0
\(547\) 38.6343 1.65188 0.825942 0.563755i \(-0.190644\pi\)
0.825942 + 0.563755i \(0.190644\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −7.90112 −0.336599
\(552\) 0 0
\(553\) −1.78248 −0.0757987
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.53271 −0.192057 −0.0960285 0.995379i \(-0.530614\pi\)
−0.0960285 + 0.995379i \(0.530614\pi\)
\(558\) 0 0
\(559\) −81.2757 −3.43760
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.2474 −0.431877 −0.215939 0.976407i \(-0.569281\pi\)
−0.215939 + 0.976407i \(0.569281\pi\)
\(564\) 0 0
\(565\) 19.8064 0.833263
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 41.4786 1.73887 0.869437 0.494044i \(-0.164482\pi\)
0.869437 + 0.494044i \(0.164482\pi\)
\(570\) 0 0
\(571\) −13.3501 −0.558683 −0.279341 0.960192i \(-0.590116\pi\)
−0.279341 + 0.960192i \(0.590116\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 37.1356 1.54598 0.772988 0.634421i \(-0.218761\pi\)
0.772988 + 0.634421i \(0.218761\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 45.9627 1.90685
\(582\) 0 0
\(583\) −6.72612 −0.278567
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.6990 −0.524146 −0.262073 0.965048i \(-0.584406\pi\)
−0.262073 + 0.965048i \(0.584406\pi\)
\(588\) 0 0
\(589\) 1.08884 0.0448648
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4.69671 −0.192871 −0.0964353 0.995339i \(-0.530744\pi\)
−0.0964353 + 0.995339i \(0.530744\pi\)
\(594\) 0 0
\(595\) −12.3780 −0.507447
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −26.9567 −1.10142 −0.550710 0.834696i \(-0.685643\pi\)
−0.550710 + 0.834696i \(0.685643\pi\)
\(600\) 0 0
\(601\) −15.2770 −0.623163 −0.311582 0.950219i \(-0.600859\pi\)
−0.311582 + 0.950219i \(0.600859\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11.6105 0.472033
\(606\) 0 0
\(607\) −16.2623 −0.660068 −0.330034 0.943969i \(-0.607060\pi\)
−0.330034 + 0.943969i \(0.607060\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 43.8314 1.77323
\(612\) 0 0
\(613\) 11.8492 0.478586 0.239293 0.970947i \(-0.423084\pi\)
0.239293 + 0.970947i \(0.423084\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −42.8298 −1.72426 −0.862132 0.506683i \(-0.830871\pi\)
−0.862132 + 0.506683i \(0.830871\pi\)
\(618\) 0 0
\(619\) 36.5054 1.46728 0.733638 0.679541i \(-0.237821\pi\)
0.733638 + 0.679541i \(0.237821\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −18.4386 −0.738729
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.390394 0.0155660
\(630\) 0 0
\(631\) 31.5110 1.25443 0.627216 0.778845i \(-0.284194\pi\)
0.627216 + 0.778845i \(0.284194\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.2111 0.484582
\(636\) 0 0
\(637\) −37.3593 −1.48023
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.8728 0.666436 0.333218 0.942850i \(-0.391866\pi\)
0.333218 + 0.942850i \(0.391866\pi\)
\(642\) 0 0
\(643\) 10.9781 0.432935 0.216468 0.976290i \(-0.430546\pi\)
0.216468 + 0.976290i \(0.430546\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.2604 −0.757205 −0.378602 0.925559i \(-0.623595\pi\)
−0.378602 + 0.925559i \(0.623595\pi\)
\(648\) 0 0
\(649\) 21.4559 0.842216
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.9754 0.977362 0.488681 0.872462i \(-0.337478\pi\)
0.488681 + 0.872462i \(0.337478\pi\)
\(654\) 0 0
\(655\) 11.1545 0.435841
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −19.9011 −0.775238 −0.387619 0.921820i \(-0.626702\pi\)
−0.387619 + 0.921820i \(0.626702\pi\)
\(660\) 0 0
\(661\) −32.5521 −1.26613 −0.633066 0.774098i \(-0.718204\pi\)
−0.633066 + 0.774098i \(0.718204\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 9.85543 0.382177
\(666\) 0 0
\(667\) 2.86787 0.111044
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.786383 −0.0303580
\(672\) 0 0
\(673\) 6.67074 0.257138 0.128569 0.991701i \(-0.458962\pi\)
0.128569 + 0.991701i \(0.458962\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.87420 −0.0720312 −0.0360156 0.999351i \(-0.511467\pi\)
−0.0360156 + 0.999351i \(0.511467\pi\)
\(678\) 0 0
\(679\) 24.9194 0.956318
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −24.2195 −0.926732 −0.463366 0.886167i \(-0.653358\pi\)
−0.463366 + 0.886167i \(0.653358\pi\)
\(684\) 0 0
\(685\) −15.4258 −0.589389
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.11673 0.347320
\(690\) 0 0
\(691\) −40.4823 −1.54002 −0.770009 0.638033i \(-0.779748\pi\)
−0.770009 + 0.638033i \(0.779748\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.796550 −0.0302148
\(696\) 0 0
\(697\) −8.04117 −0.304581
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18.2778 0.690343 0.345171 0.938540i \(-0.387821\pi\)
0.345171 + 0.938540i \(0.387821\pi\)
\(702\) 0 0
\(703\) −0.310834 −0.0117233
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −54.0973 −2.03454
\(708\) 0 0
\(709\) 18.1798 0.682757 0.341378 0.939926i \(-0.389106\pi\)
0.341378 + 0.939926i \(0.389106\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −0.395216 −0.0148009
\(714\) 0 0
\(715\) −30.6467 −1.14612
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −41.9743 −1.56538 −0.782689 0.622413i \(-0.786152\pi\)
−0.782689 + 0.622413i \(0.786152\pi\)
\(720\) 0 0
\(721\) −3.21814 −0.119850
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.86787 0.106510
\(726\) 0 0
\(727\) −10.0380 −0.372288 −0.186144 0.982522i \(-0.559599\pi\)
−0.186144 + 0.982522i \(0.559599\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 43.6349 1.61390
\(732\) 0 0
\(733\) −24.7327 −0.913524 −0.456762 0.889589i \(-0.650991\pi\)
−0.456762 + 0.889589i \(0.650991\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 26.3027 0.968873
\(738\) 0 0
\(739\) −24.8341 −0.913538 −0.456769 0.889585i \(-0.650993\pi\)
−0.456769 + 0.889585i \(0.650993\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.78039 0.358808 0.179404 0.983776i \(-0.442583\pi\)
0.179404 + 0.983776i \(0.442583\pi\)
\(744\) 0 0
\(745\) 8.26514 0.302811
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.13946 0.260870
\(750\) 0 0
\(751\) 45.6650 1.66634 0.833170 0.553017i \(-0.186524\pi\)
0.833170 + 0.553017i \(0.186524\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 18.6105 0.677305
\(756\) 0 0
\(757\) −34.5974 −1.25746 −0.628732 0.777622i \(-0.716426\pi\)
−0.628732 + 0.777622i \(0.716426\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.1810 −0.550312 −0.275156 0.961400i \(-0.588730\pi\)
−0.275156 + 0.961400i \(0.588730\pi\)
\(762\) 0 0
\(763\) 9.85543 0.356790
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −29.0818 −1.05008
\(768\) 0 0
\(769\) −37.3810 −1.34799 −0.673996 0.738735i \(-0.735424\pi\)
−0.673996 + 0.738735i \(0.735424\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.5135 0.665885 0.332943 0.942947i \(-0.391958\pi\)
0.332943 + 0.942947i \(0.391958\pi\)
\(774\) 0 0
\(775\) −0.395216 −0.0141966
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.40244 0.229391
\(780\) 0 0
\(781\) 64.0902 2.29333
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −18.8231 −0.671824
\(786\) 0 0
\(787\) −8.04206 −0.286668 −0.143334 0.989674i \(-0.545782\pi\)
−0.143334 + 0.989674i \(0.545782\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −70.8521 −2.51921
\(792\) 0 0
\(793\) 1.06588 0.0378506
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −31.0328 −1.09924 −0.549618 0.835416i \(-0.685227\pi\)
−0.549618 + 0.835416i \(0.685227\pi\)
\(798\) 0 0
\(799\) −23.5320 −0.832503
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 49.6669 1.75271
\(804\) 0 0
\(805\) −3.57723 −0.126081
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −37.6748 −1.32458 −0.662288 0.749249i \(-0.730415\pi\)
−0.662288 + 0.749249i \(0.730415\pi\)
\(810\) 0 0
\(811\) 8.18503 0.287415 0.143708 0.989620i \(-0.454098\pi\)
0.143708 + 0.989620i \(0.454098\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.07957 0.0378158
\(816\) 0 0
\(817\) −34.7425 −1.21548
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.51952 0.122832 0.0614160 0.998112i \(-0.480438\pi\)
0.0614160 + 0.998112i \(0.480438\pi\)
\(822\) 0 0
\(823\) −46.5138 −1.62137 −0.810684 0.585484i \(-0.800905\pi\)
−0.810684 + 0.585484i \(0.800905\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.1550 −0.422669 −0.211335 0.977414i \(-0.567781\pi\)
−0.211335 + 0.977414i \(0.567781\pi\)
\(828\) 0 0
\(829\) −27.2024 −0.944779 −0.472389 0.881390i \(-0.656608\pi\)
−0.472389 + 0.881390i \(0.656608\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 20.0573 0.694944
\(834\) 0 0
\(835\) 9.94681 0.344224
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 27.8522 0.961566 0.480783 0.876840i \(-0.340352\pi\)
0.480783 + 0.876840i \(0.340352\pi\)
\(840\) 0 0
\(841\) −20.7753 −0.716390
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 28.5393 0.981781
\(846\) 0 0
\(847\) −41.5333 −1.42710
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0.112824 0.00386754
\(852\) 0 0
\(853\) −26.7437 −0.915687 −0.457843 0.889033i \(-0.651378\pi\)
−0.457843 + 0.889033i \(0.651378\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −30.6843 −1.04816 −0.524078 0.851670i \(-0.675590\pi\)
−0.524078 + 0.851670i \(0.675590\pi\)
\(858\) 0 0
\(859\) 0.561699 0.0191649 0.00958247 0.999954i \(-0.496950\pi\)
0.00958247 + 0.999954i \(0.496950\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32.6645 −1.11191 −0.555957 0.831211i \(-0.687648\pi\)
−0.555957 + 0.831211i \(0.687648\pi\)
\(864\) 0 0
\(865\) −0.945311 −0.0321415
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.36937 0.0803753
\(870\) 0 0
\(871\) −35.6513 −1.20800
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.57723 −0.120932
\(876\) 0 0
\(877\) 17.7126 0.598113 0.299056 0.954235i \(-0.403328\pi\)
0.299056 + 0.954235i \(0.403328\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.955810 −0.0322021 −0.0161010 0.999870i \(-0.505125\pi\)
−0.0161010 + 0.999870i \(0.505125\pi\)
\(882\) 0 0
\(883\) 0.357570 0.0120332 0.00601660 0.999982i \(-0.498085\pi\)
0.00601660 + 0.999982i \(0.498085\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −31.1907 −1.04728 −0.523641 0.851939i \(-0.675427\pi\)
−0.523641 + 0.851939i \(0.675427\pi\)
\(888\) 0 0
\(889\) −43.6818 −1.46504
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 18.7364 0.626988
\(894\) 0 0
\(895\) 18.4762 0.617592
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.13343 −0.0378019
\(900\) 0 0
\(901\) −4.89454 −0.163061
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −19.6806 −0.654206
\(906\) 0 0
\(907\) −8.67226 −0.287958 −0.143979 0.989581i \(-0.545990\pi\)
−0.143979 + 0.989581i \(0.545990\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.72762 −0.123501 −0.0617507 0.998092i \(-0.519668\pi\)
−0.0617507 + 0.998092i \(0.519668\pi\)
\(912\) 0 0
\(913\) −61.0961 −2.02199
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −39.9020 −1.31768
\(918\) 0 0
\(919\) −46.9765 −1.54961 −0.774805 0.632200i \(-0.782152\pi\)
−0.774805 + 0.632200i \(0.782152\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −86.8693 −2.85934
\(924\) 0 0
\(925\) 0.112824 0.00370962
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −27.7134 −0.909248 −0.454624 0.890684i \(-0.650226\pi\)
−0.454624 + 0.890684i \(0.650226\pi\)
\(930\) 0 0
\(931\) −15.9698 −0.523388
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 16.4535 0.538086
\(936\) 0 0
\(937\) −30.4376 −0.994353 −0.497177 0.867649i \(-0.665630\pi\)
−0.497177 + 0.867649i \(0.665630\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 10.8447 0.353526 0.176763 0.984253i \(-0.443437\pi\)
0.176763 + 0.984253i \(0.443437\pi\)
\(942\) 0 0
\(943\) −2.32389 −0.0756764
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.412938 0.0134187 0.00670934 0.999977i \(-0.497864\pi\)
0.00670934 + 0.999977i \(0.497864\pi\)
\(948\) 0 0
\(949\) −67.3197 −2.18529
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −9.56287 −0.309772 −0.154886 0.987932i \(-0.549501\pi\)
−0.154886 + 0.987932i \(0.549501\pi\)
\(954\) 0 0
\(955\) 14.7248 0.476484
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 55.1816 1.78191
\(960\) 0 0
\(961\) −30.8438 −0.994961
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −21.8106 −0.702109
\(966\) 0 0
\(967\) 8.43589 0.271280 0.135640 0.990758i \(-0.456691\pi\)
0.135640 + 0.990758i \(0.456691\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −39.9520 −1.28212 −0.641060 0.767490i \(-0.721505\pi\)
−0.641060 + 0.767490i \(0.721505\pi\)
\(972\) 0 0
\(973\) 2.84944 0.0913488
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −19.4770 −0.623124 −0.311562 0.950226i \(-0.600852\pi\)
−0.311562 + 0.950226i \(0.600852\pi\)
\(978\) 0 0
\(979\) 24.5097 0.783332
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 43.6295 1.39157 0.695783 0.718252i \(-0.255058\pi\)
0.695783 + 0.718252i \(0.255058\pi\)
\(984\) 0 0
\(985\) 16.0446 0.511225
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12.6105 0.400990
\(990\) 0 0
\(991\) −5.34375 −0.169750 −0.0848749 0.996392i \(-0.527049\pi\)
−0.0848749 + 0.996392i \(0.527049\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −0.928806 −0.0294451
\(996\) 0 0
\(997\) −19.7774 −0.626355 −0.313178 0.949695i \(-0.601394\pi\)
−0.313178 + 0.949695i \(0.601394\pi\)
\(998\) 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.1.1 yes 5
3.2 odd 2 4140.2.a.t.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.a.t.1.1 5 3.2 odd 2
4140.2.a.u.1.1 yes 5 1.1 even 1 trivial