Properties

Label 3840.2.a.bo.1.3
Level $3840$
Weight $2$
Character 3840.1
Self dual yes
Analytic conductor $30.663$
Analytic rank $0$
Dimension $3$
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] = [3840,2,Mod(1,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, 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("3840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.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: \(30.6625543762\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 3840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +4.68585 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +4.68585 q^{7} +1.00000 q^{9} -2.29273 q^{11} +4.97858 q^{13} +1.00000 q^{15} -2.97858 q^{17} +2.68585 q^{19} -4.68585 q^{21} +2.68585 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} +6.97858 q^{31} +2.29273 q^{33} -4.68585 q^{35} +4.39312 q^{37} -4.97858 q^{39} +11.3717 q^{41} -9.37169 q^{43} -1.00000 q^{45} -7.27131 q^{47} +14.9572 q^{49} +2.97858 q^{51} -2.00000 q^{53} +2.29273 q^{55} -2.68585 q^{57} -1.70727 q^{59} -4.58546 q^{61} +4.68585 q^{63} -4.97858 q^{65} +4.00000 q^{67} -2.68585 q^{69} +0.585462 q^{71} +6.00000 q^{73} -1.00000 q^{75} -10.7434 q^{77} -1.02142 q^{79} +1.00000 q^{81} -13.3717 q^{83} +2.97858 q^{85} +2.00000 q^{87} -3.37169 q^{89} +23.3288 q^{91} -6.97858 q^{93} -2.68585 q^{95} -3.95715 q^{97} -2.29273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 4 q^{11} + 3 q^{15} + 6 q^{17} - 4 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} + 6 q^{31} + 4 q^{33} - 2 q^{35} + 4 q^{37} + 10 q^{41} - 4 q^{43} - 3 q^{45} - 4 q^{47} + 15 q^{49} - 6 q^{51} - 6 q^{53} + 4 q^{55} + 4 q^{57} - 8 q^{59} - 8 q^{61} + 2 q^{63} + 12 q^{67} + 4 q^{69} - 4 q^{71} + 18 q^{73} - 3 q^{75} + 16 q^{77} - 18 q^{79} + 3 q^{81} - 16 q^{83} - 6 q^{85} + 6 q^{87} + 14 q^{89} + 16 q^{91} - 6 q^{93} + 4 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.68585 1.77108 0.885542 0.464560i \(-0.153787\pi\)
0.885542 + 0.464560i \(0.153787\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.29273 −0.691284 −0.345642 0.938366i \(-0.612339\pi\)
−0.345642 + 0.938366i \(0.612339\pi\)
\(12\) 0 0
\(13\) 4.97858 1.38081 0.690404 0.723424i \(-0.257433\pi\)
0.690404 + 0.723424i \(0.257433\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −2.97858 −0.722411 −0.361206 0.932486i \(-0.617635\pi\)
−0.361206 + 0.932486i \(0.617635\pi\)
\(18\) 0 0
\(19\) 2.68585 0.616175 0.308088 0.951358i \(-0.400311\pi\)
0.308088 + 0.951358i \(0.400311\pi\)
\(20\) 0 0
\(21\) −4.68585 −1.02254
\(22\) 0 0
\(23\) 2.68585 0.560038 0.280019 0.959995i \(-0.409659\pi\)
0.280019 + 0.959995i \(0.409659\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 6.97858 1.25339 0.626695 0.779265i \(-0.284407\pi\)
0.626695 + 0.779265i \(0.284407\pi\)
\(32\) 0 0
\(33\) 2.29273 0.399113
\(34\) 0 0
\(35\) −4.68585 −0.792053
\(36\) 0 0
\(37\) 4.39312 0.722224 0.361112 0.932523i \(-0.382397\pi\)
0.361112 + 0.932523i \(0.382397\pi\)
\(38\) 0 0
\(39\) −4.97858 −0.797210
\(40\) 0 0
\(41\) 11.3717 1.77596 0.887980 0.459882i \(-0.152108\pi\)
0.887980 + 0.459882i \(0.152108\pi\)
\(42\) 0 0
\(43\) −9.37169 −1.42917 −0.714585 0.699549i \(-0.753384\pi\)
−0.714585 + 0.699549i \(0.753384\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −7.27131 −1.06063 −0.530315 0.847801i \(-0.677926\pi\)
−0.530315 + 0.847801i \(0.677926\pi\)
\(48\) 0 0
\(49\) 14.9572 2.13674
\(50\) 0 0
\(51\) 2.97858 0.417084
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 2.29273 0.309152
\(56\) 0 0
\(57\) −2.68585 −0.355749
\(58\) 0 0
\(59\) −1.70727 −0.222267 −0.111134 0.993805i \(-0.535448\pi\)
−0.111134 + 0.993805i \(0.535448\pi\)
\(60\) 0 0
\(61\) −4.58546 −0.587108 −0.293554 0.955942i \(-0.594838\pi\)
−0.293554 + 0.955942i \(0.594838\pi\)
\(62\) 0 0
\(63\) 4.68585 0.590361
\(64\) 0 0
\(65\) −4.97858 −0.617516
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −2.68585 −0.323338
\(70\) 0 0
\(71\) 0.585462 0.0694816 0.0347408 0.999396i \(-0.488939\pi\)
0.0347408 + 0.999396i \(0.488939\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −10.7434 −1.22432
\(78\) 0 0
\(79\) −1.02142 −0.114919 −0.0574595 0.998348i \(-0.518300\pi\)
−0.0574595 + 0.998348i \(0.518300\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.3717 −1.46773 −0.733867 0.679293i \(-0.762286\pi\)
−0.733867 + 0.679293i \(0.762286\pi\)
\(84\) 0 0
\(85\) 2.97858 0.323072
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −3.37169 −0.357399 −0.178699 0.983904i \(-0.557189\pi\)
−0.178699 + 0.983904i \(0.557189\pi\)
\(90\) 0 0
\(91\) 23.3288 2.44553
\(92\) 0 0
\(93\) −6.97858 −0.723645
\(94\) 0 0
\(95\) −2.68585 −0.275562
\(96\) 0 0
\(97\) −3.95715 −0.401788 −0.200894 0.979613i \(-0.564385\pi\)
−0.200894 + 0.979613i \(0.564385\pi\)
\(98\) 0 0
\(99\) −2.29273 −0.230428
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −14.6430 −1.44282 −0.721409 0.692509i \(-0.756505\pi\)
−0.721409 + 0.692509i \(0.756505\pi\)
\(104\) 0 0
\(105\) 4.68585 0.457292
\(106\) 0 0
\(107\) 11.3288 1.09520 0.547600 0.836740i \(-0.315541\pi\)
0.547600 + 0.836740i \(0.315541\pi\)
\(108\) 0 0
\(109\) 9.37169 0.897645 0.448823 0.893621i \(-0.351843\pi\)
0.448823 + 0.893621i \(0.351843\pi\)
\(110\) 0 0
\(111\) −4.39312 −0.416976
\(112\) 0 0
\(113\) 19.7648 1.85932 0.929658 0.368423i \(-0.120102\pi\)
0.929658 + 0.368423i \(0.120102\pi\)
\(114\) 0 0
\(115\) −2.68585 −0.250456
\(116\) 0 0
\(117\) 4.97858 0.460270
\(118\) 0 0
\(119\) −13.9572 −1.27945
\(120\) 0 0
\(121\) −5.74338 −0.522126
\(122\) 0 0
\(123\) −11.3717 −1.02535
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.64300 −0.589471 −0.294735 0.955579i \(-0.595232\pi\)
−0.294735 + 0.955579i \(0.595232\pi\)
\(128\) 0 0
\(129\) 9.37169 0.825132
\(130\) 0 0
\(131\) 7.07896 0.618492 0.309246 0.950982i \(-0.399923\pi\)
0.309246 + 0.950982i \(0.399923\pi\)
\(132\) 0 0
\(133\) 12.5855 1.09130
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 14.9786 1.27971 0.639853 0.768497i \(-0.278995\pi\)
0.639853 + 0.768497i \(0.278995\pi\)
\(138\) 0 0
\(139\) −4.64300 −0.393814 −0.196907 0.980422i \(-0.563090\pi\)
−0.196907 + 0.980422i \(0.563090\pi\)
\(140\) 0 0
\(141\) 7.27131 0.612355
\(142\) 0 0
\(143\) −11.4145 −0.954532
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) −14.9572 −1.23365
\(148\) 0 0
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) −8.35027 −0.679535 −0.339768 0.940509i \(-0.610348\pi\)
−0.339768 + 0.940509i \(0.610348\pi\)
\(152\) 0 0
\(153\) −2.97858 −0.240804
\(154\) 0 0
\(155\) −6.97858 −0.560533
\(156\) 0 0
\(157\) 22.3503 1.78375 0.891873 0.452286i \(-0.149391\pi\)
0.891873 + 0.452286i \(0.149391\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 12.5855 0.991873
\(162\) 0 0
\(163\) −1.37169 −0.107439 −0.0537196 0.998556i \(-0.517108\pi\)
−0.0537196 + 0.998556i \(0.517108\pi\)
\(164\) 0 0
\(165\) −2.29273 −0.178489
\(166\) 0 0
\(167\) 11.2713 0.872200 0.436100 0.899898i \(-0.356359\pi\)
0.436100 + 0.899898i \(0.356359\pi\)
\(168\) 0 0
\(169\) 11.7862 0.906633
\(170\) 0 0
\(171\) 2.68585 0.205392
\(172\) 0 0
\(173\) 10.7862 0.820062 0.410031 0.912072i \(-0.365518\pi\)
0.410031 + 0.912072i \(0.365518\pi\)
\(174\) 0 0
\(175\) 4.68585 0.354217
\(176\) 0 0
\(177\) 1.70727 0.128326
\(178\) 0 0
\(179\) 3.66442 0.273892 0.136946 0.990579i \(-0.456271\pi\)
0.136946 + 0.990579i \(0.456271\pi\)
\(180\) 0 0
\(181\) −6.62831 −0.492678 −0.246339 0.969184i \(-0.579228\pi\)
−0.246339 + 0.969184i \(0.579228\pi\)
\(182\) 0 0
\(183\) 4.58546 0.338967
\(184\) 0 0
\(185\) −4.39312 −0.322988
\(186\) 0 0
\(187\) 6.82908 0.499392
\(188\) 0 0
\(189\) −4.68585 −0.340845
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 1.21377 0.0873690 0.0436845 0.999045i \(-0.486090\pi\)
0.0436845 + 0.999045i \(0.486090\pi\)
\(194\) 0 0
\(195\) 4.97858 0.356523
\(196\) 0 0
\(197\) 23.9572 1.70688 0.853438 0.521194i \(-0.174513\pi\)
0.853438 + 0.521194i \(0.174513\pi\)
\(198\) 0 0
\(199\) 0.350269 0.0248299 0.0124150 0.999923i \(-0.496048\pi\)
0.0124150 + 0.999923i \(0.496048\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) −9.37169 −0.657764
\(204\) 0 0
\(205\) −11.3717 −0.794233
\(206\) 0 0
\(207\) 2.68585 0.186679
\(208\) 0 0
\(209\) −6.15792 −0.425952
\(210\) 0 0
\(211\) 14.1004 0.970710 0.485355 0.874317i \(-0.338690\pi\)
0.485355 + 0.874317i \(0.338690\pi\)
\(212\) 0 0
\(213\) −0.585462 −0.0401152
\(214\) 0 0
\(215\) 9.37169 0.639144
\(216\) 0 0
\(217\) 32.7005 2.21986
\(218\) 0 0
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) −14.8291 −0.997512
\(222\) 0 0
\(223\) 6.72869 0.450587 0.225293 0.974291i \(-0.427666\pi\)
0.225293 + 0.974291i \(0.427666\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 9.95715 0.660880 0.330440 0.943827i \(-0.392803\pi\)
0.330440 + 0.943827i \(0.392803\pi\)
\(228\) 0 0
\(229\) −11.3288 −0.748631 −0.374316 0.927301i \(-0.622122\pi\)
−0.374316 + 0.927301i \(0.622122\pi\)
\(230\) 0 0
\(231\) 10.7434 0.706863
\(232\) 0 0
\(233\) 18.9786 1.24333 0.621664 0.783284i \(-0.286457\pi\)
0.621664 + 0.783284i \(0.286457\pi\)
\(234\) 0 0
\(235\) 7.27131 0.474328
\(236\) 0 0
\(237\) 1.02142 0.0663485
\(238\) 0 0
\(239\) −2.62831 −0.170011 −0.0850055 0.996380i \(-0.527091\pi\)
−0.0850055 + 0.996380i \(0.527091\pi\)
\(240\) 0 0
\(241\) 10.7862 0.694802 0.347401 0.937717i \(-0.387064\pi\)
0.347401 + 0.937717i \(0.387064\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −14.9572 −0.955578
\(246\) 0 0
\(247\) 13.3717 0.850820
\(248\) 0 0
\(249\) 13.3717 0.847397
\(250\) 0 0
\(251\) −30.9933 −1.95628 −0.978139 0.207952i \(-0.933320\pi\)
−0.978139 + 0.207952i \(0.933320\pi\)
\(252\) 0 0
\(253\) −6.15792 −0.387145
\(254\) 0 0
\(255\) −2.97858 −0.186526
\(256\) 0 0
\(257\) 20.9357 1.30594 0.652968 0.757386i \(-0.273524\pi\)
0.652968 + 0.757386i \(0.273524\pi\)
\(258\) 0 0
\(259\) 20.5855 1.27912
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −19.2713 −1.18832 −0.594160 0.804347i \(-0.702515\pi\)
−0.594160 + 0.804347i \(0.702515\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 3.37169 0.206344
\(268\) 0 0
\(269\) 24.7434 1.50863 0.754315 0.656512i \(-0.227969\pi\)
0.754315 + 0.656512i \(0.227969\pi\)
\(270\) 0 0
\(271\) −27.5640 −1.67440 −0.837198 0.546900i \(-0.815808\pi\)
−0.837198 + 0.546900i \(0.815808\pi\)
\(272\) 0 0
\(273\) −23.3288 −1.41193
\(274\) 0 0
\(275\) −2.29273 −0.138257
\(276\) 0 0
\(277\) −20.3074 −1.22015 −0.610077 0.792342i \(-0.708862\pi\)
−0.610077 + 0.792342i \(0.708862\pi\)
\(278\) 0 0
\(279\) 6.97858 0.417796
\(280\) 0 0
\(281\) 10.7862 0.643453 0.321726 0.946833i \(-0.395737\pi\)
0.321726 + 0.946833i \(0.395737\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 2.68585 0.159096
\(286\) 0 0
\(287\) 53.2860 3.14537
\(288\) 0 0
\(289\) −8.12808 −0.478122
\(290\) 0 0
\(291\) 3.95715 0.231972
\(292\) 0 0
\(293\) −21.9143 −1.28025 −0.640124 0.768272i \(-0.721117\pi\)
−0.640124 + 0.768272i \(0.721117\pi\)
\(294\) 0 0
\(295\) 1.70727 0.0994010
\(296\) 0 0
\(297\) 2.29273 0.133038
\(298\) 0 0
\(299\) 13.3717 0.773305
\(300\) 0 0
\(301\) −43.9143 −2.53118
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) 4.58546 0.262563
\(306\) 0 0
\(307\) 26.5426 1.51487 0.757434 0.652912i \(-0.226453\pi\)
0.757434 + 0.652912i \(0.226453\pi\)
\(308\) 0 0
\(309\) 14.6430 0.833011
\(310\) 0 0
\(311\) 12.2008 0.691842 0.345921 0.938264i \(-0.387566\pi\)
0.345921 + 0.938264i \(0.387566\pi\)
\(312\) 0 0
\(313\) −15.9572 −0.901952 −0.450976 0.892536i \(-0.648924\pi\)
−0.450976 + 0.892536i \(0.648924\pi\)
\(314\) 0 0
\(315\) −4.68585 −0.264018
\(316\) 0 0
\(317\) 33.5296 1.88321 0.941605 0.336718i \(-0.109317\pi\)
0.941605 + 0.336718i \(0.109317\pi\)
\(318\) 0 0
\(319\) 4.58546 0.256737
\(320\) 0 0
\(321\) −11.3288 −0.632315
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 4.97858 0.276162
\(326\) 0 0
\(327\) −9.37169 −0.518256
\(328\) 0 0
\(329\) −34.0722 −1.87846
\(330\) 0 0
\(331\) 19.8568 1.09143 0.545713 0.837972i \(-0.316259\pi\)
0.545713 + 0.837972i \(0.316259\pi\)
\(332\) 0 0
\(333\) 4.39312 0.240741
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 7.17092 0.390625 0.195313 0.980741i \(-0.437428\pi\)
0.195313 + 0.980741i \(0.437428\pi\)
\(338\) 0 0
\(339\) −19.7648 −1.07348
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 37.2860 2.01325
\(344\) 0 0
\(345\) 2.68585 0.144601
\(346\) 0 0
\(347\) −0.786230 −0.0422071 −0.0211035 0.999777i \(-0.506718\pi\)
−0.0211035 + 0.999777i \(0.506718\pi\)
\(348\) 0 0
\(349\) −6.15792 −0.329626 −0.164813 0.986325i \(-0.552702\pi\)
−0.164813 + 0.986325i \(0.552702\pi\)
\(350\) 0 0
\(351\) −4.97858 −0.265737
\(352\) 0 0
\(353\) 21.7220 1.15614 0.578072 0.815986i \(-0.303805\pi\)
0.578072 + 0.815986i \(0.303805\pi\)
\(354\) 0 0
\(355\) −0.585462 −0.0310731
\(356\) 0 0
\(357\) 13.9572 0.738691
\(358\) 0 0
\(359\) 0.585462 0.0308995 0.0154498 0.999881i \(-0.495082\pi\)
0.0154498 + 0.999881i \(0.495082\pi\)
\(360\) 0 0
\(361\) −11.7862 −0.620328
\(362\) 0 0
\(363\) 5.74338 0.301450
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 0.485078 0.0253209 0.0126604 0.999920i \(-0.495970\pi\)
0.0126604 + 0.999920i \(0.495970\pi\)
\(368\) 0 0
\(369\) 11.3717 0.591987
\(370\) 0 0
\(371\) −9.37169 −0.486554
\(372\) 0 0
\(373\) −12.3931 −0.641691 −0.320846 0.947132i \(-0.603967\pi\)
−0.320846 + 0.947132i \(0.603967\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −9.95715 −0.512820
\(378\) 0 0
\(379\) 26.0147 1.33629 0.668143 0.744033i \(-0.267090\pi\)
0.668143 + 0.744033i \(0.267090\pi\)
\(380\) 0 0
\(381\) 6.64300 0.340331
\(382\) 0 0
\(383\) −6.68585 −0.341631 −0.170815 0.985303i \(-0.554640\pi\)
−0.170815 + 0.985303i \(0.554640\pi\)
\(384\) 0 0
\(385\) 10.7434 0.547534
\(386\) 0 0
\(387\) −9.37169 −0.476390
\(388\) 0 0
\(389\) 29.9143 1.51672 0.758358 0.651838i \(-0.226002\pi\)
0.758358 + 0.651838i \(0.226002\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) −7.07896 −0.357086
\(394\) 0 0
\(395\) 1.02142 0.0513934
\(396\) 0 0
\(397\) −9.76481 −0.490082 −0.245041 0.969513i \(-0.578801\pi\)
−0.245041 + 0.969513i \(0.578801\pi\)
\(398\) 0 0
\(399\) −12.5855 −0.630061
\(400\) 0 0
\(401\) −6.58546 −0.328862 −0.164431 0.986389i \(-0.552579\pi\)
−0.164431 + 0.986389i \(0.552579\pi\)
\(402\) 0 0
\(403\) 34.7434 1.73069
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −10.0722 −0.499262
\(408\) 0 0
\(409\) 25.9143 1.28138 0.640690 0.767800i \(-0.278648\pi\)
0.640690 + 0.767800i \(0.278648\pi\)
\(410\) 0 0
\(411\) −14.9786 −0.738839
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 13.3717 0.656391
\(416\) 0 0
\(417\) 4.64300 0.227369
\(418\) 0 0
\(419\) −12.2499 −0.598446 −0.299223 0.954183i \(-0.596727\pi\)
−0.299223 + 0.954183i \(0.596727\pi\)
\(420\) 0 0
\(421\) −4.67115 −0.227658 −0.113829 0.993500i \(-0.536312\pi\)
−0.113829 + 0.993500i \(0.536312\pi\)
\(422\) 0 0
\(423\) −7.27131 −0.353543
\(424\) 0 0
\(425\) −2.97858 −0.144482
\(426\) 0 0
\(427\) −21.4868 −1.03982
\(428\) 0 0
\(429\) 11.4145 0.551099
\(430\) 0 0
\(431\) −0.585462 −0.0282007 −0.0141004 0.999901i \(-0.504488\pi\)
−0.0141004 + 0.999901i \(0.504488\pi\)
\(432\) 0 0
\(433\) −21.9143 −1.05313 −0.526567 0.850133i \(-0.676521\pi\)
−0.526567 + 0.850133i \(0.676521\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 0 0
\(437\) 7.21377 0.345081
\(438\) 0 0
\(439\) −2.39312 −0.114217 −0.0571086 0.998368i \(-0.518188\pi\)
−0.0571086 + 0.998368i \(0.518188\pi\)
\(440\) 0 0
\(441\) 14.9572 0.712245
\(442\) 0 0
\(443\) 20.7005 0.983512 0.491756 0.870733i \(-0.336355\pi\)
0.491756 + 0.870733i \(0.336355\pi\)
\(444\) 0 0
\(445\) 3.37169 0.159834
\(446\) 0 0
\(447\) −2.00000 −0.0945968
\(448\) 0 0
\(449\) −37.9143 −1.78929 −0.894643 0.446781i \(-0.852570\pi\)
−0.894643 + 0.446781i \(0.852570\pi\)
\(450\) 0 0
\(451\) −26.0722 −1.22769
\(452\) 0 0
\(453\) 8.35027 0.392330
\(454\) 0 0
\(455\) −23.3288 −1.09367
\(456\) 0 0
\(457\) −38.7005 −1.81033 −0.905167 0.425055i \(-0.860255\pi\)
−0.905167 + 0.425055i \(0.860255\pi\)
\(458\) 0 0
\(459\) 2.97858 0.139028
\(460\) 0 0
\(461\) 4.74338 0.220921 0.110461 0.993880i \(-0.464767\pi\)
0.110461 + 0.993880i \(0.464767\pi\)
\(462\) 0 0
\(463\) −15.3142 −0.711709 −0.355855 0.934541i \(-0.615810\pi\)
−0.355855 + 0.934541i \(0.615810\pi\)
\(464\) 0 0
\(465\) 6.97858 0.323624
\(466\) 0 0
\(467\) −30.5426 −1.41334 −0.706672 0.707541i \(-0.749804\pi\)
−0.706672 + 0.707541i \(0.749804\pi\)
\(468\) 0 0
\(469\) 18.7434 0.865489
\(470\) 0 0
\(471\) −22.3503 −1.02985
\(472\) 0 0
\(473\) 21.4868 0.987963
\(474\) 0 0
\(475\) 2.68585 0.123235
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −3.32885 −0.152099 −0.0760494 0.997104i \(-0.524231\pi\)
−0.0760494 + 0.997104i \(0.524231\pi\)
\(480\) 0 0
\(481\) 21.8715 0.997253
\(482\) 0 0
\(483\) −12.5855 −0.572658
\(484\) 0 0
\(485\) 3.95715 0.179685
\(486\) 0 0
\(487\) −12.1004 −0.548321 −0.274160 0.961684i \(-0.588400\pi\)
−0.274160 + 0.961684i \(0.588400\pi\)
\(488\) 0 0
\(489\) 1.37169 0.0620301
\(490\) 0 0
\(491\) −14.2927 −0.645022 −0.322511 0.946566i \(-0.604527\pi\)
−0.322511 + 0.946566i \(0.604527\pi\)
\(492\) 0 0
\(493\) 5.95715 0.268297
\(494\) 0 0
\(495\) 2.29273 0.103051
\(496\) 0 0
\(497\) 2.74338 0.123058
\(498\) 0 0
\(499\) −9.22846 −0.413123 −0.206561 0.978434i \(-0.566227\pi\)
−0.206561 + 0.978434i \(0.566227\pi\)
\(500\) 0 0
\(501\) −11.2713 −0.503565
\(502\) 0 0
\(503\) −14.1004 −0.628705 −0.314353 0.949306i \(-0.601787\pi\)
−0.314353 + 0.949306i \(0.601787\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) −11.7862 −0.523445
\(508\) 0 0
\(509\) −43.4011 −1.92372 −0.961859 0.273544i \(-0.911804\pi\)
−0.961859 + 0.273544i \(0.911804\pi\)
\(510\) 0 0
\(511\) 28.1151 1.24374
\(512\) 0 0
\(513\) −2.68585 −0.118583
\(514\) 0 0
\(515\) 14.6430 0.645248
\(516\) 0 0
\(517\) 16.6712 0.733196
\(518\) 0 0
\(519\) −10.7862 −0.473463
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 13.5725 0.593482 0.296741 0.954958i \(-0.404100\pi\)
0.296741 + 0.954958i \(0.404100\pi\)
\(524\) 0 0
\(525\) −4.68585 −0.204507
\(526\) 0 0
\(527\) −20.7862 −0.905462
\(528\) 0 0
\(529\) −15.7862 −0.686358
\(530\) 0 0
\(531\) −1.70727 −0.0740892
\(532\) 0 0
\(533\) 56.6148 2.45226
\(534\) 0 0
\(535\) −11.3288 −0.489789
\(536\) 0 0
\(537\) −3.66442 −0.158132
\(538\) 0 0
\(539\) −34.2927 −1.47709
\(540\) 0 0
\(541\) −37.2860 −1.60305 −0.801525 0.597961i \(-0.795978\pi\)
−0.801525 + 0.597961i \(0.795978\pi\)
\(542\) 0 0
\(543\) 6.62831 0.284448
\(544\) 0 0
\(545\) −9.37169 −0.401439
\(546\) 0 0
\(547\) 0.200768 0.00858424 0.00429212 0.999991i \(-0.498634\pi\)
0.00429212 + 0.999991i \(0.498634\pi\)
\(548\) 0 0
\(549\) −4.58546 −0.195703
\(550\) 0 0
\(551\) −5.37169 −0.228842
\(552\) 0 0
\(553\) −4.78623 −0.203531
\(554\) 0 0
\(555\) 4.39312 0.186477
\(556\) 0 0
\(557\) 9.21377 0.390400 0.195200 0.980763i \(-0.437464\pi\)
0.195200 + 0.980763i \(0.437464\pi\)
\(558\) 0 0
\(559\) −46.6577 −1.97341
\(560\) 0 0
\(561\) −6.82908 −0.288324
\(562\) 0 0
\(563\) 36.7005 1.54674 0.773372 0.633953i \(-0.218569\pi\)
0.773372 + 0.633953i \(0.218569\pi\)
\(564\) 0 0
\(565\) −19.7648 −0.831512
\(566\) 0 0
\(567\) 4.68585 0.196787
\(568\) 0 0
\(569\) −13.4145 −0.562367 −0.281183 0.959654i \(-0.590727\pi\)
−0.281183 + 0.959654i \(0.590727\pi\)
\(570\) 0 0
\(571\) 18.6858 0.781978 0.390989 0.920395i \(-0.372133\pi\)
0.390989 + 0.920395i \(0.372133\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 2.68585 0.112008
\(576\) 0 0
\(577\) −2.78623 −0.115992 −0.0579961 0.998317i \(-0.518471\pi\)
−0.0579961 + 0.998317i \(0.518471\pi\)
\(578\) 0 0
\(579\) −1.21377 −0.0504425
\(580\) 0 0
\(581\) −62.6577 −2.59948
\(582\) 0 0
\(583\) 4.58546 0.189910
\(584\) 0 0
\(585\) −4.97858 −0.205839
\(586\) 0 0
\(587\) −27.3288 −1.12798 −0.563991 0.825781i \(-0.690735\pi\)
−0.563991 + 0.825781i \(0.690735\pi\)
\(588\) 0 0
\(589\) 18.7434 0.772308
\(590\) 0 0
\(591\) −23.9572 −0.985466
\(592\) 0 0
\(593\) −6.97858 −0.286576 −0.143288 0.989681i \(-0.545767\pi\)
−0.143288 + 0.989681i \(0.545767\pi\)
\(594\) 0 0
\(595\) 13.9572 0.572188
\(596\) 0 0
\(597\) −0.350269 −0.0143356
\(598\) 0 0
\(599\) 36.4998 1.49134 0.745670 0.666315i \(-0.232130\pi\)
0.745670 + 0.666315i \(0.232130\pi\)
\(600\) 0 0
\(601\) 15.5725 0.635214 0.317607 0.948222i \(-0.397121\pi\)
0.317607 + 0.948222i \(0.397121\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 5.74338 0.233502
\(606\) 0 0
\(607\) 31.2285 1.26752 0.633762 0.773528i \(-0.281510\pi\)
0.633762 + 0.773528i \(0.281510\pi\)
\(608\) 0 0
\(609\) 9.37169 0.379760
\(610\) 0 0
\(611\) −36.2008 −1.46453
\(612\) 0 0
\(613\) 0.978577 0.0395244 0.0197622 0.999805i \(-0.493709\pi\)
0.0197622 + 0.999805i \(0.493709\pi\)
\(614\) 0 0
\(615\) 11.3717 0.458551
\(616\) 0 0
\(617\) −32.9357 −1.32594 −0.662971 0.748645i \(-0.730705\pi\)
−0.662971 + 0.748645i \(0.730705\pi\)
\(618\) 0 0
\(619\) 3.35700 0.134929 0.0674646 0.997722i \(-0.478509\pi\)
0.0674646 + 0.997722i \(0.478509\pi\)
\(620\) 0 0
\(621\) −2.68585 −0.107779
\(622\) 0 0
\(623\) −15.7992 −0.632983
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.15792 0.245924
\(628\) 0 0
\(629\) −13.0852 −0.521742
\(630\) 0 0
\(631\) −27.7648 −1.10530 −0.552650 0.833414i \(-0.686383\pi\)
−0.552650 + 0.833414i \(0.686383\pi\)
\(632\) 0 0
\(633\) −14.1004 −0.560440
\(634\) 0 0
\(635\) 6.64300 0.263619
\(636\) 0 0
\(637\) 74.4653 2.95042
\(638\) 0 0
\(639\) 0.585462 0.0231605
\(640\) 0 0
\(641\) −21.1281 −0.834509 −0.417254 0.908790i \(-0.637008\pi\)
−0.417254 + 0.908790i \(0.637008\pi\)
\(642\) 0 0
\(643\) 29.2860 1.15493 0.577464 0.816416i \(-0.304043\pi\)
0.577464 + 0.816416i \(0.304043\pi\)
\(644\) 0 0
\(645\) −9.37169 −0.369010
\(646\) 0 0
\(647\) 15.6728 0.616163 0.308082 0.951360i \(-0.400313\pi\)
0.308082 + 0.951360i \(0.400313\pi\)
\(648\) 0 0
\(649\) 3.91431 0.153650
\(650\) 0 0
\(651\) −32.7005 −1.28164
\(652\) 0 0
\(653\) −17.5296 −0.685987 −0.342993 0.939338i \(-0.611441\pi\)
−0.342993 + 0.939338i \(0.611441\pi\)
\(654\) 0 0
\(655\) −7.07896 −0.276598
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) −23.8652 −0.929656 −0.464828 0.885401i \(-0.653884\pi\)
−0.464828 + 0.885401i \(0.653884\pi\)
\(660\) 0 0
\(661\) −30.1579 −1.17301 −0.586504 0.809947i \(-0.699496\pi\)
−0.586504 + 0.809947i \(0.699496\pi\)
\(662\) 0 0
\(663\) 14.8291 0.575914
\(664\) 0 0
\(665\) −12.5855 −0.488043
\(666\) 0 0
\(667\) −5.37169 −0.207993
\(668\) 0 0
\(669\) −6.72869 −0.260146
\(670\) 0 0
\(671\) 10.5132 0.405859
\(672\) 0 0
\(673\) −18.0000 −0.693849 −0.346925 0.937893i \(-0.612774\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 9.61531 0.369546 0.184773 0.982781i \(-0.440845\pi\)
0.184773 + 0.982781i \(0.440845\pi\)
\(678\) 0 0
\(679\) −18.5426 −0.711600
\(680\) 0 0
\(681\) −9.95715 −0.381559
\(682\) 0 0
\(683\) −18.6283 −0.712792 −0.356396 0.934335i \(-0.615995\pi\)
−0.356396 + 0.934335i \(0.615995\pi\)
\(684\) 0 0
\(685\) −14.9786 −0.572302
\(686\) 0 0
\(687\) 11.3288 0.432222
\(688\) 0 0
\(689\) −9.95715 −0.379337
\(690\) 0 0
\(691\) −13.4292 −0.510872 −0.255436 0.966826i \(-0.582219\pi\)
−0.255436 + 0.966826i \(0.582219\pi\)
\(692\) 0 0
\(693\) −10.7434 −0.408107
\(694\) 0 0
\(695\) 4.64300 0.176119
\(696\) 0 0
\(697\) −33.8715 −1.28297
\(698\) 0 0
\(699\) −18.9786 −0.717836
\(700\) 0 0
\(701\) −19.1709 −0.724076 −0.362038 0.932163i \(-0.617919\pi\)
−0.362038 + 0.932163i \(0.617919\pi\)
\(702\) 0 0
\(703\) 11.7992 0.445016
\(704\) 0 0
\(705\) −7.27131 −0.273853
\(706\) 0 0
\(707\) −9.37169 −0.352459
\(708\) 0 0
\(709\) 15.4145 0.578905 0.289453 0.957192i \(-0.406527\pi\)
0.289453 + 0.957192i \(0.406527\pi\)
\(710\) 0 0
\(711\) −1.02142 −0.0383063
\(712\) 0 0
\(713\) 18.7434 0.701945
\(714\) 0 0
\(715\) 11.4145 0.426880
\(716\) 0 0
\(717\) 2.62831 0.0981559
\(718\) 0 0
\(719\) −20.7862 −0.775196 −0.387598 0.921829i \(-0.626695\pi\)
−0.387598 + 0.921829i \(0.626695\pi\)
\(720\) 0 0
\(721\) −68.6148 −2.55535
\(722\) 0 0
\(723\) −10.7862 −0.401144
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 12.3012 0.456224 0.228112 0.973635i \(-0.426745\pi\)
0.228112 + 0.973635i \(0.426745\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 27.9143 1.03245
\(732\) 0 0
\(733\) 35.9227 1.32684 0.663418 0.748249i \(-0.269105\pi\)
0.663418 + 0.748249i \(0.269105\pi\)
\(734\) 0 0
\(735\) 14.9572 0.551703
\(736\) 0 0
\(737\) −9.17092 −0.337815
\(738\) 0 0
\(739\) 29.0277 1.06780 0.533900 0.845547i \(-0.320726\pi\)
0.533900 + 0.845547i \(0.320726\pi\)
\(740\) 0 0
\(741\) −13.3717 −0.491221
\(742\) 0 0
\(743\) 2.60015 0.0953904 0.0476952 0.998862i \(-0.484812\pi\)
0.0476952 + 0.998862i \(0.484812\pi\)
\(744\) 0 0
\(745\) −2.00000 −0.0732743
\(746\) 0 0
\(747\) −13.3717 −0.489245
\(748\) 0 0
\(749\) 53.0852 1.93969
\(750\) 0 0
\(751\) 10.8929 0.397487 0.198744 0.980052i \(-0.436314\pi\)
0.198744 + 0.980052i \(0.436314\pi\)
\(752\) 0 0
\(753\) 30.9933 1.12946
\(754\) 0 0
\(755\) 8.35027 0.303897
\(756\) 0 0
\(757\) 34.3503 1.24848 0.624241 0.781232i \(-0.285408\pi\)
0.624241 + 0.781232i \(0.285408\pi\)
\(758\) 0 0
\(759\) 6.15792 0.223518
\(760\) 0 0
\(761\) 19.0852 0.691839 0.345920 0.938264i \(-0.387567\pi\)
0.345920 + 0.938264i \(0.387567\pi\)
\(762\) 0 0
\(763\) 43.9143 1.58980
\(764\) 0 0
\(765\) 2.97858 0.107691
\(766\) 0 0
\(767\) −8.49977 −0.306909
\(768\) 0 0
\(769\) 31.8715 1.14931 0.574657 0.818394i \(-0.305135\pi\)
0.574657 + 0.818394i \(0.305135\pi\)
\(770\) 0 0
\(771\) −20.9357 −0.753982
\(772\) 0 0
\(773\) −11.9572 −0.430069 −0.215034 0.976606i \(-0.568986\pi\)
−0.215034 + 0.976606i \(0.568986\pi\)
\(774\) 0 0
\(775\) 6.97858 0.250678
\(776\) 0 0
\(777\) −20.5855 −0.738499
\(778\) 0 0
\(779\) 30.5426 1.09430
\(780\) 0 0
\(781\) −1.34231 −0.0480315
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) −22.3503 −0.797715
\(786\) 0 0
\(787\) 33.0852 1.17936 0.589681 0.807637i \(-0.299254\pi\)
0.589681 + 0.807637i \(0.299254\pi\)
\(788\) 0 0
\(789\) 19.2713 0.686077
\(790\) 0 0
\(791\) 92.6148 3.29300
\(792\) 0 0
\(793\) −22.8291 −0.810684
\(794\) 0 0
\(795\) −2.00000 −0.0709327
\(796\) 0 0
\(797\) 10.0000 0.354218 0.177109 0.984191i \(-0.443325\pi\)
0.177109 + 0.984191i \(0.443325\pi\)
\(798\) 0 0
\(799\) 21.6582 0.766210
\(800\) 0 0
\(801\) −3.37169 −0.119133
\(802\) 0 0
\(803\) −13.7564 −0.485452
\(804\) 0 0
\(805\) −12.5855 −0.443579
\(806\) 0 0
\(807\) −24.7434 −0.871008
\(808\) 0 0
\(809\) 30.6148 1.07636 0.538180 0.842830i \(-0.319112\pi\)
0.538180 + 0.842830i \(0.319112\pi\)
\(810\) 0 0
\(811\) −53.9290 −1.89370 −0.946852 0.321670i \(-0.895756\pi\)
−0.946852 + 0.321670i \(0.895756\pi\)
\(812\) 0 0
\(813\) 27.5640 0.966713
\(814\) 0 0
\(815\) 1.37169 0.0480483
\(816\) 0 0
\(817\) −25.1709 −0.880619
\(818\) 0 0
\(819\) 23.3288 0.815176
\(820\) 0 0
\(821\) 12.6577 0.441757 0.220878 0.975301i \(-0.429108\pi\)
0.220878 + 0.975301i \(0.429108\pi\)
\(822\) 0 0
\(823\) −19.8139 −0.690670 −0.345335 0.938479i \(-0.612235\pi\)
−0.345335 + 0.938479i \(0.612235\pi\)
\(824\) 0 0
\(825\) 2.29273 0.0798226
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) 41.3717 1.43690 0.718449 0.695580i \(-0.244852\pi\)
0.718449 + 0.695580i \(0.244852\pi\)
\(830\) 0 0
\(831\) 20.3074 0.704457
\(832\) 0 0
\(833\) −44.5510 −1.54360
\(834\) 0 0
\(835\) −11.2713 −0.390060
\(836\) 0 0
\(837\) −6.97858 −0.241215
\(838\) 0 0
\(839\) 20.9013 0.721593 0.360797 0.932645i \(-0.382505\pi\)
0.360797 + 0.932645i \(0.382505\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −10.7862 −0.371498
\(844\) 0 0
\(845\) −11.7862 −0.405459
\(846\) 0 0
\(847\) −26.9126 −0.924728
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 11.7992 0.404472
\(852\) 0 0
\(853\) 6.63673 0.227237 0.113619 0.993524i \(-0.463756\pi\)
0.113619 + 0.993524i \(0.463756\pi\)
\(854\) 0 0
\(855\) −2.68585 −0.0918540
\(856\) 0 0
\(857\) −9.80765 −0.335023 −0.167512 0.985870i \(-0.553573\pi\)
−0.167512 + 0.985870i \(0.553573\pi\)
\(858\) 0 0
\(859\) −39.3864 −1.34385 −0.671923 0.740621i \(-0.734531\pi\)
−0.671923 + 0.740621i \(0.734531\pi\)
\(860\) 0 0
\(861\) −53.2860 −1.81598
\(862\) 0 0
\(863\) −7.07054 −0.240684 −0.120342 0.992732i \(-0.538399\pi\)
−0.120342 + 0.992732i \(0.538399\pi\)
\(864\) 0 0
\(865\) −10.7862 −0.366743
\(866\) 0 0
\(867\) 8.12808 0.276044
\(868\) 0 0
\(869\) 2.34185 0.0794417
\(870\) 0 0
\(871\) 19.9143 0.674771
\(872\) 0 0
\(873\) −3.95715 −0.133929
\(874\) 0 0
\(875\) −4.68585 −0.158411
\(876\) 0 0
\(877\) −23.1365 −0.781264 −0.390632 0.920547i \(-0.627744\pi\)
−0.390632 + 0.920547i \(0.627744\pi\)
\(878\) 0 0
\(879\) 21.9143 0.739151
\(880\) 0 0
\(881\) 28.4569 0.958738 0.479369 0.877613i \(-0.340866\pi\)
0.479369 + 0.877613i \(0.340866\pi\)
\(882\) 0 0
\(883\) −41.2003 −1.38650 −0.693250 0.720697i \(-0.743822\pi\)
−0.693250 + 0.720697i \(0.743822\pi\)
\(884\) 0 0
\(885\) −1.70727 −0.0573892
\(886\) 0 0
\(887\) −3.55777 −0.119458 −0.0597291 0.998215i \(-0.519024\pi\)
−0.0597291 + 0.998215i \(0.519024\pi\)
\(888\) 0 0
\(889\) −31.1281 −1.04400
\(890\) 0 0
\(891\) −2.29273 −0.0768094
\(892\) 0 0
\(893\) −19.5296 −0.653534
\(894\) 0 0
\(895\) −3.66442 −0.122488
\(896\) 0 0
\(897\) −13.3717 −0.446468
\(898\) 0 0
\(899\) −13.9572 −0.465497
\(900\) 0 0
\(901\) 5.95715 0.198462
\(902\) 0 0
\(903\) 43.9143 1.46138
\(904\) 0 0
\(905\) 6.62831 0.220332
\(906\) 0 0
\(907\) 50.6577 1.68206 0.841031 0.540988i \(-0.181949\pi\)
0.841031 + 0.540988i \(0.181949\pi\)
\(908\) 0 0
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 26.4569 0.876557 0.438279 0.898839i \(-0.355588\pi\)
0.438279 + 0.898839i \(0.355588\pi\)
\(912\) 0 0
\(913\) 30.6577 1.01462
\(914\) 0 0
\(915\) −4.58546 −0.151591
\(916\) 0 0
\(917\) 33.1709 1.09540
\(918\) 0 0
\(919\) 29.8077 0.983264 0.491632 0.870803i \(-0.336401\pi\)
0.491632 + 0.870803i \(0.336401\pi\)
\(920\) 0 0
\(921\) −26.5426 −0.874609
\(922\) 0 0
\(923\) 2.91477 0.0959407
\(924\) 0 0
\(925\) 4.39312 0.144445
\(926\) 0 0
\(927\) −14.6430 −0.480939
\(928\) 0 0
\(929\) −8.82908 −0.289673 −0.144836 0.989456i \(-0.546266\pi\)
−0.144836 + 0.989456i \(0.546266\pi\)
\(930\) 0 0
\(931\) 40.1726 1.31660
\(932\) 0 0
\(933\) −12.2008 −0.399435
\(934\) 0 0
\(935\) −6.82908 −0.223335
\(936\) 0 0
\(937\) 42.2302 1.37960 0.689799 0.724000i \(-0.257699\pi\)
0.689799 + 0.724000i \(0.257699\pi\)
\(938\) 0 0
\(939\) 15.9572 0.520742
\(940\) 0 0
\(941\) −32.7434 −1.06740 −0.533702 0.845673i \(-0.679200\pi\)
−0.533702 + 0.845673i \(0.679200\pi\)
\(942\) 0 0
\(943\) 30.5426 0.994604
\(944\) 0 0
\(945\) 4.68585 0.152431
\(946\) 0 0
\(947\) −15.9143 −0.517146 −0.258573 0.965992i \(-0.583252\pi\)
−0.258573 + 0.965992i \(0.583252\pi\)
\(948\) 0 0
\(949\) 29.8715 0.969669
\(950\) 0 0
\(951\) −33.5296 −1.08727
\(952\) 0 0
\(953\) −55.6791 −1.80362 −0.901812 0.432129i \(-0.857762\pi\)
−0.901812 + 0.432129i \(0.857762\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 0 0
\(957\) −4.58546 −0.148227
\(958\) 0 0
\(959\) 70.1873 2.26647
\(960\) 0 0
\(961\) 17.7005 0.570985
\(962\) 0 0
\(963\) 11.3288 0.365067
\(964\) 0 0
\(965\) −1.21377 −0.0390726
\(966\) 0 0
\(967\) −54.7581 −1.76090 −0.880451 0.474138i \(-0.842760\pi\)
−0.880451 + 0.474138i \(0.842760\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 30.3221 0.973083 0.486542 0.873657i \(-0.338258\pi\)
0.486542 + 0.873657i \(0.338258\pi\)
\(972\) 0 0
\(973\) −21.7564 −0.697478
\(974\) 0 0
\(975\) −4.97858 −0.159442
\(976\) 0 0
\(977\) −29.7220 −0.950890 −0.475445 0.879746i \(-0.657713\pi\)
−0.475445 + 0.879746i \(0.657713\pi\)
\(978\) 0 0
\(979\) 7.73038 0.247064
\(980\) 0 0
\(981\) 9.37169 0.299215
\(982\) 0 0
\(983\) −49.5443 −1.58022 −0.790109 0.612966i \(-0.789976\pi\)
−0.790109 + 0.612966i \(0.789976\pi\)
\(984\) 0 0
\(985\) −23.9572 −0.763338
\(986\) 0 0
\(987\) 34.0722 1.08453
\(988\) 0 0
\(989\) −25.1709 −0.800389
\(990\) 0 0
\(991\) −19.0937 −0.606530 −0.303265 0.952906i \(-0.598077\pi\)
−0.303265 + 0.952906i \(0.598077\pi\)
\(992\) 0 0
\(993\) −19.8568 −0.630136
\(994\) 0 0
\(995\) −0.350269 −0.0111043
\(996\) 0 0
\(997\) 38.8500 1.23039 0.615197 0.788374i \(-0.289077\pi\)
0.615197 + 0.788374i \(0.289077\pi\)
\(998\) 0 0
\(999\) −4.39312 −0.138992
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 3840.2.a.bo.1.3 3
4.3 odd 2 3840.2.a.bq.1.1 3
8.3 odd 2 3840.2.a.bp.1.1 3
8.5 even 2 3840.2.a.br.1.3 3
16.3 odd 4 120.2.k.b.61.1 6
16.5 even 4 480.2.k.b.241.4 6
16.11 odd 4 120.2.k.b.61.2 yes 6
16.13 even 4 480.2.k.b.241.1 6
48.5 odd 4 1440.2.k.f.721.4 6
48.11 even 4 360.2.k.f.181.5 6
48.29 odd 4 1440.2.k.f.721.1 6
48.35 even 4 360.2.k.f.181.6 6
80.3 even 4 600.2.d.e.349.2 6
80.13 odd 4 2400.2.d.f.49.6 6
80.19 odd 4 600.2.k.c.301.6 6
80.27 even 4 600.2.d.e.349.1 6
80.29 even 4 2400.2.k.c.1201.6 6
80.37 odd 4 2400.2.d.f.49.1 6
80.43 even 4 600.2.d.f.349.6 6
80.53 odd 4 2400.2.d.e.49.6 6
80.59 odd 4 600.2.k.c.301.5 6
80.67 even 4 600.2.d.f.349.5 6
80.69 even 4 2400.2.k.c.1201.3 6
80.77 odd 4 2400.2.d.e.49.1 6
240.29 odd 4 7200.2.k.p.3601.5 6
240.53 even 4 7200.2.d.r.2449.6 6
240.59 even 4 1800.2.k.p.901.2 6
240.77 even 4 7200.2.d.r.2449.1 6
240.83 odd 4 1800.2.d.q.1549.5 6
240.107 odd 4 1800.2.d.q.1549.6 6
240.149 odd 4 7200.2.k.p.3601.6 6
240.173 even 4 7200.2.d.q.2449.6 6
240.179 even 4 1800.2.k.p.901.1 6
240.197 even 4 7200.2.d.q.2449.1 6
240.203 odd 4 1800.2.d.r.1549.1 6
240.227 odd 4 1800.2.d.r.1549.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.k.b.61.1 6 16.3 odd 4
120.2.k.b.61.2 yes 6 16.11 odd 4
360.2.k.f.181.5 6 48.11 even 4
360.2.k.f.181.6 6 48.35 even 4
480.2.k.b.241.1 6 16.13 even 4
480.2.k.b.241.4 6 16.5 even 4
600.2.d.e.349.1 6 80.27 even 4
600.2.d.e.349.2 6 80.3 even 4
600.2.d.f.349.5 6 80.67 even 4
600.2.d.f.349.6 6 80.43 even 4
600.2.k.c.301.5 6 80.59 odd 4
600.2.k.c.301.6 6 80.19 odd 4
1440.2.k.f.721.1 6 48.29 odd 4
1440.2.k.f.721.4 6 48.5 odd 4
1800.2.d.q.1549.5 6 240.83 odd 4
1800.2.d.q.1549.6 6 240.107 odd 4
1800.2.d.r.1549.1 6 240.203 odd 4
1800.2.d.r.1549.2 6 240.227 odd 4
1800.2.k.p.901.1 6 240.179 even 4
1800.2.k.p.901.2 6 240.59 even 4
2400.2.d.e.49.1 6 80.77 odd 4
2400.2.d.e.49.6 6 80.53 odd 4
2400.2.d.f.49.1 6 80.37 odd 4
2400.2.d.f.49.6 6 80.13 odd 4
2400.2.k.c.1201.3 6 80.69 even 4
2400.2.k.c.1201.6 6 80.29 even 4
3840.2.a.bo.1.3 3 1.1 even 1 trivial
3840.2.a.bp.1.1 3 8.3 odd 2
3840.2.a.bq.1.1 3 4.3 odd 2
3840.2.a.br.1.3 3 8.5 even 2
7200.2.d.q.2449.1 6 240.197 even 4
7200.2.d.q.2449.6 6 240.173 even 4
7200.2.d.r.2449.1 6 240.77 even 4
7200.2.d.r.2449.6 6 240.53 even 4
7200.2.k.p.3601.5 6 240.29 odd 4
7200.2.k.p.3601.6 6 240.149 odd 4