Properties

Label 4056.2.c.p.337.8
Level $4056$
Weight $2$
Character 4056.337
Analytic conductor $32.387$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4056,2,Mod(337,4056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4056.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.649638144.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.8
Root \(-2.34138 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 4056.337
Dual form 4056.2.c.p.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +4.20740i q^{5} +3.55539i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +4.20740i q^{5} +3.55539i q^{7} +1.00000 q^{9} -5.08003i q^{11} +4.20740i q^{15} -3.25670 q^{17} -3.33477i q^{19} +3.55539i q^{21} -0.384069 q^{23} -12.7022 q^{25} +1.00000 q^{27} -7.28744 q^{29} +6.47535i q^{31} -5.08003i q^{33} -14.9589 q^{35} -3.12737i q^{37} +6.77404i q^{41} -4.15811 q^{43} +4.20740i q^{45} +5.49484i q^{47} -5.64077 q^{49} -3.25670 q^{51} +0.613974 q^{53} +21.3737 q^{55} -3.33477i q^{57} +7.87891i q^{59} -8.31227 q^{61} +3.55539i q^{63} +2.06878i q^{67} -0.384069 q^{69} -2.82529i q^{71} -3.21865i q^{73} -12.7022 q^{75} +18.0615 q^{77} -2.64077 q^{79} +1.00000 q^{81} +1.96926i q^{83} -13.7022i q^{85} -7.28744 q^{87} -14.1601i q^{89} +6.47535i q^{93} +14.0307 q^{95} +8.86766i q^{97} -5.08003i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 8 q^{9} - 24 q^{17} + 4 q^{23} - 4 q^{25} + 8 q^{27} - 12 q^{29} - 20 q^{35} + 16 q^{43} - 36 q^{49} - 24 q^{51} + 4 q^{53} + 20 q^{55} - 4 q^{61} + 4 q^{69} - 4 q^{75} + 56 q^{77} - 12 q^{79} + 8 q^{81} - 12 q^{87} + 68 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times\).

\(n\) \(1015\) \(2029\) \(2705\) \(3889\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 4.20740i 1.88161i 0.338951 + 0.940804i \(0.389928\pi\)
−0.338951 + 0.940804i \(0.610072\pi\)
\(6\) 0 0
\(7\) 3.55539i 1.34381i 0.740638 + 0.671905i \(0.234524\pi\)
−0.740638 + 0.671905i \(0.765476\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) − 5.08003i − 1.53169i −0.643027 0.765844i \(-0.722322\pi\)
0.643027 0.765844i \(-0.277678\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 4.20740i 1.08635i
\(16\) 0 0
\(17\) −3.25670 −0.789865 −0.394933 0.918710i \(-0.629232\pi\)
−0.394933 + 0.918710i \(0.629232\pi\)
\(18\) 0 0
\(19\) − 3.33477i − 0.765050i −0.923945 0.382525i \(-0.875055\pi\)
0.923945 0.382525i \(-0.124945\pi\)
\(20\) 0 0
\(21\) 3.55539i 0.775849i
\(22\) 0 0
\(23\) −0.384069 −0.0800838 −0.0400419 0.999198i \(-0.512749\pi\)
−0.0400419 + 0.999198i \(0.512749\pi\)
\(24\) 0 0
\(25\) −12.7022 −2.54045
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.28744 −1.35324 −0.676621 0.736331i \(-0.736557\pi\)
−0.676621 + 0.736331i \(0.736557\pi\)
\(30\) 0 0
\(31\) 6.47535i 1.16301i 0.813544 + 0.581504i \(0.197535\pi\)
−0.813544 + 0.581504i \(0.802465\pi\)
\(32\) 0 0
\(33\) − 5.08003i − 0.884320i
\(34\) 0 0
\(35\) −14.9589 −2.52852
\(36\) 0 0
\(37\) − 3.12737i − 0.514137i −0.966393 0.257068i \(-0.917243\pi\)
0.966393 0.257068i \(-0.0827565\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.77404i 1.05793i 0.848644 + 0.528964i \(0.177419\pi\)
−0.848644 + 0.528964i \(0.822581\pi\)
\(42\) 0 0
\(43\) −4.15811 −0.634106 −0.317053 0.948408i \(-0.602693\pi\)
−0.317053 + 0.948408i \(0.602693\pi\)
\(44\) 0 0
\(45\) 4.20740i 0.627203i
\(46\) 0 0
\(47\) 5.49484i 0.801505i 0.916186 + 0.400752i \(0.131251\pi\)
−0.916186 + 0.400752i \(0.868749\pi\)
\(48\) 0 0
\(49\) −5.64077 −0.805824
\(50\) 0 0
\(51\) −3.25670 −0.456029
\(52\) 0 0
\(53\) 0.613974 0.0843358 0.0421679 0.999111i \(-0.486574\pi\)
0.0421679 + 0.999111i \(0.486574\pi\)
\(54\) 0 0
\(55\) 21.3737 2.88204
\(56\) 0 0
\(57\) − 3.33477i − 0.441702i
\(58\) 0 0
\(59\) 7.87891i 1.02575i 0.858464 + 0.512873i \(0.171419\pi\)
−0.858464 + 0.512873i \(0.828581\pi\)
\(60\) 0 0
\(61\) −8.31227 −1.06428 −0.532139 0.846657i \(-0.678612\pi\)
−0.532139 + 0.846657i \(0.678612\pi\)
\(62\) 0 0
\(63\) 3.55539i 0.447936i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.06878i 0.252742i 0.991983 + 0.126371i \(0.0403330\pi\)
−0.991983 + 0.126371i \(0.959667\pi\)
\(68\) 0 0
\(69\) −0.384069 −0.0462364
\(70\) 0 0
\(71\) − 2.82529i − 0.335301i −0.985847 0.167650i \(-0.946382\pi\)
0.985847 0.167650i \(-0.0536179\pi\)
\(72\) 0 0
\(73\) − 3.21865i − 0.376715i −0.982101 0.188357i \(-0.939684\pi\)
0.982101 0.188357i \(-0.0603164\pi\)
\(74\) 0 0
\(75\) −12.7022 −1.46673
\(76\) 0 0
\(77\) 18.0615 2.05830
\(78\) 0 0
\(79\) −2.64077 −0.297109 −0.148555 0.988904i \(-0.547462\pi\)
−0.148555 + 0.988904i \(0.547462\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.96926i 0.216155i 0.994142 + 0.108077i \(0.0344694\pi\)
−0.994142 + 0.108077i \(0.965531\pi\)
\(84\) 0 0
\(85\) − 13.7022i − 1.48622i
\(86\) 0 0
\(87\) −7.28744 −0.781295
\(88\) 0 0
\(89\) − 14.1601i − 1.50096i −0.660891 0.750482i \(-0.729821\pi\)
0.660891 0.750482i \(-0.270179\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.47535i 0.671463i
\(94\) 0 0
\(95\) 14.0307 1.43952
\(96\) 0 0
\(97\) 8.86766i 0.900374i 0.892934 + 0.450187i \(0.148643\pi\)
−0.892934 + 0.450187i \(0.851357\pi\)
\(98\) 0 0
\(99\) − 5.08003i − 0.510563i
\(100\) 0 0
\(101\) −9.44750 −0.940062 −0.470031 0.882650i \(-0.655757\pi\)
−0.470031 + 0.882650i \(0.655757\pi\)
\(102\) 0 0
\(103\) 2.15811 0.212645 0.106322 0.994332i \(-0.466092\pi\)
0.106322 + 0.994332i \(0.466092\pi\)
\(104\) 0 0
\(105\) −14.9589 −1.45984
\(106\) 0 0
\(107\) −18.7003 −1.80782 −0.903912 0.427718i \(-0.859318\pi\)
−0.903912 + 0.427718i \(0.859318\pi\)
\(108\) 0 0
\(109\) 14.4754i 1.38649i 0.720703 + 0.693244i \(0.243819\pi\)
−0.720703 + 0.693244i \(0.756181\pi\)
\(110\) 0 0
\(111\) − 3.12737i − 0.296837i
\(112\) 0 0
\(113\) 2.74330 0.258068 0.129034 0.991640i \(-0.458812\pi\)
0.129034 + 0.991640i \(0.458812\pi\)
\(114\) 0 0
\(115\) − 1.61593i − 0.150686i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 11.5788i − 1.06143i
\(120\) 0 0
\(121\) −14.8067 −1.34607
\(122\) 0 0
\(123\) 6.77404i 0.610795i
\(124\) 0 0
\(125\) − 32.4064i − 2.89852i
\(126\) 0 0
\(127\) 10.1274 0.898659 0.449329 0.893366i \(-0.351663\pi\)
0.449329 + 0.893366i \(0.351663\pi\)
\(128\) 0 0
\(129\) −4.15811 −0.366101
\(130\) 0 0
\(131\) −13.3430 −1.16578 −0.582892 0.812550i \(-0.698079\pi\)
−0.582892 + 0.812550i \(0.698079\pi\)
\(132\) 0 0
\(133\) 11.8564 1.02808
\(134\) 0 0
\(135\) 4.20740i 0.362116i
\(136\) 0 0
\(137\) − 11.8955i − 1.01630i −0.861268 0.508151i \(-0.830329\pi\)
0.861268 0.508151i \(-0.169671\pi\)
\(138\) 0 0
\(139\) 13.4207 1.13833 0.569165 0.822223i \(-0.307267\pi\)
0.569165 + 0.822223i \(0.307267\pi\)
\(140\) 0 0
\(141\) 5.49484i 0.462749i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) − 30.6612i − 2.54627i
\(146\) 0 0
\(147\) −5.64077 −0.465243
\(148\) 0 0
\(149\) − 5.51144i − 0.451515i −0.974184 0.225757i \(-0.927514\pi\)
0.974184 0.225757i \(-0.0724856\pi\)
\(150\) 0 0
\(151\) 5.08003i 0.413407i 0.978404 + 0.206704i \(0.0662736\pi\)
−0.978404 + 0.206704i \(0.933726\pi\)
\(152\) 0 0
\(153\) −3.25670 −0.263288
\(154\) 0 0
\(155\) −27.2444 −2.18832
\(156\) 0 0
\(157\) −12.5670 −1.00296 −0.501478 0.865170i \(-0.667210\pi\)
−0.501478 + 0.865170i \(0.667210\pi\)
\(158\) 0 0
\(159\) 0.613974 0.0486913
\(160\) 0 0
\(161\) − 1.36551i − 0.107617i
\(162\) 0 0
\(163\) − 8.81799i − 0.690678i −0.938478 0.345339i \(-0.887764\pi\)
0.938478 0.345339i \(-0.112236\pi\)
\(164\) 0 0
\(165\) 21.3737 1.66394
\(166\) 0 0
\(167\) − 14.6695i − 1.13516i −0.823317 0.567582i \(-0.807879\pi\)
0.823317 0.567582i \(-0.192121\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) − 3.33477i − 0.255017i
\(172\) 0 0
\(173\) 2.41481 0.183594 0.0917972 0.995778i \(-0.470739\pi\)
0.0917972 + 0.995778i \(0.470739\pi\)
\(174\) 0 0
\(175\) − 45.1614i − 3.41388i
\(176\) 0 0
\(177\) 7.87891i 0.592215i
\(178\) 0 0
\(179\) 5.35728 0.400422 0.200211 0.979753i \(-0.435837\pi\)
0.200211 + 0.979753i \(0.435837\pi\)
\(180\) 0 0
\(181\) 17.6715 1.31351 0.656756 0.754103i \(-0.271928\pi\)
0.656756 + 0.754103i \(0.271928\pi\)
\(182\) 0 0
\(183\) −8.31227 −0.614461
\(184\) 0 0
\(185\) 13.1581 0.967403
\(186\) 0 0
\(187\) 16.5441i 1.20983i
\(188\) 0 0
\(189\) 3.55539i 0.258616i
\(190\) 0 0
\(191\) 4.57487 0.331026 0.165513 0.986208i \(-0.447072\pi\)
0.165513 + 0.986208i \(0.447072\pi\)
\(192\) 0 0
\(193\) 16.8428i 1.21237i 0.795323 + 0.606186i \(0.207301\pi\)
−0.795323 + 0.606186i \(0.792699\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1.85641i − 0.132263i −0.997811 0.0661317i \(-0.978934\pi\)
0.997811 0.0661317i \(-0.0210658\pi\)
\(198\) 0 0
\(199\) 18.9877 1.34600 0.673002 0.739641i \(-0.265005\pi\)
0.673002 + 0.739641i \(0.265005\pi\)
\(200\) 0 0
\(201\) 2.06878i 0.145921i
\(202\) 0 0
\(203\) − 25.9096i − 1.81850i
\(204\) 0 0
\(205\) −28.5011 −1.99060
\(206\) 0 0
\(207\) −0.384069 −0.0266946
\(208\) 0 0
\(209\) −16.9408 −1.17182
\(210\) 0 0
\(211\) 12.2424 0.842804 0.421402 0.906874i \(-0.361538\pi\)
0.421402 + 0.906874i \(0.361538\pi\)
\(212\) 0 0
\(213\) − 2.82529i − 0.193586i
\(214\) 0 0
\(215\) − 17.4948i − 1.19314i
\(216\) 0 0
\(217\) −23.0224 −1.56286
\(218\) 0 0
\(219\) − 3.21865i − 0.217497i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 20.7085i 1.38675i 0.720579 + 0.693373i \(0.243876\pi\)
−0.720579 + 0.693373i \(0.756124\pi\)
\(224\) 0 0
\(225\) −12.7022 −0.846816
\(226\) 0 0
\(227\) − 8.63881i − 0.573378i −0.958024 0.286689i \(-0.907445\pi\)
0.958024 0.286689i \(-0.0925546\pi\)
\(228\) 0 0
\(229\) − 20.1111i − 1.32898i −0.747296 0.664491i \(-0.768648\pi\)
0.747296 0.664491i \(-0.231352\pi\)
\(230\) 0 0
\(231\) 18.0615 1.18836
\(232\) 0 0
\(233\) 4.41089 0.288967 0.144484 0.989507i \(-0.453848\pi\)
0.144484 + 0.989507i \(0.453848\pi\)
\(234\) 0 0
\(235\) −23.1190 −1.50812
\(236\) 0 0
\(237\) −2.64077 −0.171536
\(238\) 0 0
\(239\) 3.51734i 0.227518i 0.993508 + 0.113759i \(0.0362892\pi\)
−0.993508 + 0.113759i \(0.963711\pi\)
\(240\) 0 0
\(241\) 4.74330i 0.305543i 0.988262 + 0.152771i \(0.0488198\pi\)
−0.988262 + 0.152771i \(0.951180\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) − 23.7330i − 1.51624i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 1.96926i 0.124797i
\(250\) 0 0
\(251\) −6.98968 −0.441185 −0.220592 0.975366i \(-0.570799\pi\)
−0.220592 + 0.975366i \(0.570799\pi\)
\(252\) 0 0
\(253\) 1.95108i 0.122663i
\(254\) 0 0
\(255\) − 13.7022i − 0.858068i
\(256\) 0 0
\(257\) 4.71010 0.293808 0.146904 0.989151i \(-0.453069\pi\)
0.146904 + 0.989151i \(0.453069\pi\)
\(258\) 0 0
\(259\) 11.1190 0.690902
\(260\) 0 0
\(261\) −7.28744 −0.451081
\(262\) 0 0
\(263\) −4.12933 −0.254625 −0.127313 0.991863i \(-0.540635\pi\)
−0.127313 + 0.991863i \(0.540635\pi\)
\(264\) 0 0
\(265\) 2.58324i 0.158687i
\(266\) 0 0
\(267\) − 14.1601i − 0.866582i
\(268\) 0 0
\(269\) 16.2587 0.991308 0.495654 0.868520i \(-0.334928\pi\)
0.495654 + 0.868520i \(0.334928\pi\)
\(270\) 0 0
\(271\) 32.3171i 1.96313i 0.191137 + 0.981563i \(0.438783\pi\)
−0.191137 + 0.981563i \(0.561217\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 64.5278i 3.89117i
\(276\) 0 0
\(277\) 23.7369 1.42621 0.713106 0.701056i \(-0.247288\pi\)
0.713106 + 0.701056i \(0.247288\pi\)
\(278\) 0 0
\(279\) 6.47535i 0.387669i
\(280\) 0 0
\(281\) 25.5812i 1.52604i 0.646373 + 0.763022i \(0.276285\pi\)
−0.646373 + 0.763022i \(0.723715\pi\)
\(282\) 0 0
\(283\) 11.6486 0.692439 0.346220 0.938154i \(-0.387465\pi\)
0.346220 + 0.938154i \(0.387465\pi\)
\(284\) 0 0
\(285\) 14.0307 0.831109
\(286\) 0 0
\(287\) −24.0843 −1.42165
\(288\) 0 0
\(289\) −6.39392 −0.376113
\(290\) 0 0
\(291\) 8.86766i 0.519831i
\(292\) 0 0
\(293\) 9.85016i 0.575452i 0.957713 + 0.287726i \(0.0928993\pi\)
−0.957713 + 0.287726i \(0.907101\pi\)
\(294\) 0 0
\(295\) −33.1497 −1.93005
\(296\) 0 0
\(297\) − 5.08003i − 0.294773i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 14.7837i − 0.852117i
\(302\) 0 0
\(303\) −9.44750 −0.542745
\(304\) 0 0
\(305\) − 34.9731i − 2.00255i
\(306\) 0 0
\(307\) 12.4182i 0.708744i 0.935105 + 0.354372i \(0.115305\pi\)
−0.935105 + 0.354372i \(0.884695\pi\)
\(308\) 0 0
\(309\) 2.15811 0.122771
\(310\) 0 0
\(311\) −7.42267 −0.420901 −0.210450 0.977605i \(-0.567493\pi\)
−0.210450 + 0.977605i \(0.567493\pi\)
\(312\) 0 0
\(313\) 27.9799 1.58152 0.790758 0.612129i \(-0.209687\pi\)
0.790758 + 0.612129i \(0.209687\pi\)
\(314\) 0 0
\(315\) −14.9589 −0.842841
\(316\) 0 0
\(317\) − 8.02484i − 0.450720i −0.974276 0.225360i \(-0.927644\pi\)
0.974276 0.225360i \(-0.0723558\pi\)
\(318\) 0 0
\(319\) 37.0204i 2.07275i
\(320\) 0 0
\(321\) −18.7003 −1.04375
\(322\) 0 0
\(323\) 10.8604i 0.604286i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 14.4754i 0.800489i
\(328\) 0 0
\(329\) −19.5363 −1.07707
\(330\) 0 0
\(331\) 18.9252i 1.04022i 0.854098 + 0.520112i \(0.174110\pi\)
−0.854098 + 0.520112i \(0.825890\pi\)
\(332\) 0 0
\(333\) − 3.12737i − 0.171379i
\(334\) 0 0
\(335\) −8.70420 −0.475561
\(336\) 0 0
\(337\) −13.8296 −0.753347 −0.376674 0.926346i \(-0.622932\pi\)
−0.376674 + 0.926346i \(0.622932\pi\)
\(338\) 0 0
\(339\) 2.74330 0.148996
\(340\) 0 0
\(341\) 32.8950 1.78136
\(342\) 0 0
\(343\) 4.83260i 0.260936i
\(344\) 0 0
\(345\) − 1.61593i − 0.0869988i
\(346\) 0 0
\(347\) −16.9093 −0.907737 −0.453869 0.891069i \(-0.649956\pi\)
−0.453869 + 0.891069i \(0.649956\pi\)
\(348\) 0 0
\(349\) − 15.7462i − 0.842874i −0.906858 0.421437i \(-0.861526\pi\)
0.906858 0.421437i \(-0.138474\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 21.9960i − 1.17073i −0.810771 0.585363i \(-0.800952\pi\)
0.810771 0.585363i \(-0.199048\pi\)
\(354\) 0 0
\(355\) 11.8871 0.630904
\(356\) 0 0
\(357\) − 11.5788i − 0.612816i
\(358\) 0 0
\(359\) 24.5606i 1.29626i 0.761530 + 0.648130i \(0.224449\pi\)
−0.761530 + 0.648130i \(0.775551\pi\)
\(360\) 0 0
\(361\) 7.87928 0.414699
\(362\) 0 0
\(363\) −14.8067 −0.777152
\(364\) 0 0
\(365\) 13.5422 0.708830
\(366\) 0 0
\(367\) 8.20778 0.428443 0.214221 0.976785i \(-0.431279\pi\)
0.214221 + 0.976785i \(0.431279\pi\)
\(368\) 0 0
\(369\) 6.77404i 0.352642i
\(370\) 0 0
\(371\) 2.18291i 0.113331i
\(372\) 0 0
\(373\) −10.1190 −0.523942 −0.261971 0.965076i \(-0.584373\pi\)
−0.261971 + 0.965076i \(0.584373\pi\)
\(374\) 0 0
\(375\) − 32.4064i − 1.67346i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 21.1614i 1.08699i 0.839413 + 0.543493i \(0.182899\pi\)
−0.839413 + 0.543493i \(0.817101\pi\)
\(380\) 0 0
\(381\) 10.1274 0.518841
\(382\) 0 0
\(383\) 26.1151i 1.33442i 0.744871 + 0.667209i \(0.232511\pi\)
−0.744871 + 0.667209i \(0.767489\pi\)
\(384\) 0 0
\(385\) 75.9919i 3.87291i
\(386\) 0 0
\(387\) −4.15811 −0.211369
\(388\) 0 0
\(389\) −9.18493 −0.465695 −0.232847 0.972513i \(-0.574804\pi\)
−0.232847 + 0.972513i \(0.574804\pi\)
\(390\) 0 0
\(391\) 1.25080 0.0632554
\(392\) 0 0
\(393\) −13.3430 −0.673066
\(394\) 0 0
\(395\) − 11.1108i − 0.559044i
\(396\) 0 0
\(397\) 11.7912i 0.591783i 0.955222 + 0.295892i \(0.0956167\pi\)
−0.955222 + 0.295892i \(0.904383\pi\)
\(398\) 0 0
\(399\) 11.8564 0.593563
\(400\) 0 0
\(401\) 30.4822i 1.52221i 0.648630 + 0.761104i \(0.275342\pi\)
−0.648630 + 0.761104i \(0.724658\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 4.20740i 0.209068i
\(406\) 0 0
\(407\) −15.8871 −0.787497
\(408\) 0 0
\(409\) 16.8096i 0.831182i 0.909552 + 0.415591i \(0.136425\pi\)
−0.909552 + 0.415591i \(0.863575\pi\)
\(410\) 0 0
\(411\) − 11.8955i − 0.586762i
\(412\) 0 0
\(413\) −28.0126 −1.37841
\(414\) 0 0
\(415\) −8.28548 −0.406718
\(416\) 0 0
\(417\) 13.4207 0.657215
\(418\) 0 0
\(419\) −24.5788 −1.20075 −0.600376 0.799718i \(-0.704982\pi\)
−0.600376 + 0.799718i \(0.704982\pi\)
\(420\) 0 0
\(421\) 11.1157i 0.541748i 0.962615 + 0.270874i \(0.0873127\pi\)
−0.962615 + 0.270874i \(0.912687\pi\)
\(422\) 0 0
\(423\) 5.49484i 0.267168i
\(424\) 0 0
\(425\) 41.3674 2.00661
\(426\) 0 0
\(427\) − 29.5533i − 1.43019i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.5606i 1.76106i 0.473988 + 0.880531i \(0.342814\pi\)
−0.473988 + 0.880531i \(0.657186\pi\)
\(432\) 0 0
\(433\) 19.1765 0.921566 0.460783 0.887513i \(-0.347569\pi\)
0.460783 + 0.887513i \(0.347569\pi\)
\(434\) 0 0
\(435\) − 30.6612i − 1.47009i
\(436\) 0 0
\(437\) 1.28078i 0.0612681i
\(438\) 0 0
\(439\) −23.8213 −1.13693 −0.568463 0.822709i \(-0.692462\pi\)
−0.568463 + 0.822709i \(0.692462\pi\)
\(440\) 0 0
\(441\) −5.64077 −0.268608
\(442\) 0 0
\(443\) 6.48020 0.307884 0.153942 0.988080i \(-0.450803\pi\)
0.153942 + 0.988080i \(0.450803\pi\)
\(444\) 0 0
\(445\) 59.5771 2.82423
\(446\) 0 0
\(447\) − 5.51144i − 0.260682i
\(448\) 0 0
\(449\) − 7.11077i − 0.335578i −0.985823 0.167789i \(-0.946337\pi\)
0.985823 0.167789i \(-0.0536627\pi\)
\(450\) 0 0
\(451\) 34.4123 1.62041
\(452\) 0 0
\(453\) 5.08003i 0.238681i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 1.71558i − 0.0802514i −0.999195 0.0401257i \(-0.987224\pi\)
0.999195 0.0401257i \(-0.0127758\pi\)
\(458\) 0 0
\(459\) −3.25670 −0.152010
\(460\) 0 0
\(461\) 32.5129i 1.51428i 0.653254 + 0.757139i \(0.273403\pi\)
−0.653254 + 0.757139i \(0.726597\pi\)
\(462\) 0 0
\(463\) − 3.37282i − 0.156748i −0.996924 0.0783741i \(-0.975027\pi\)
0.996924 0.0783741i \(-0.0249728\pi\)
\(464\) 0 0
\(465\) −27.2444 −1.26343
\(466\) 0 0
\(467\) 19.3241 0.894212 0.447106 0.894481i \(-0.352455\pi\)
0.447106 + 0.894481i \(0.352455\pi\)
\(468\) 0 0
\(469\) −7.35532 −0.339637
\(470\) 0 0
\(471\) −12.5670 −0.579057
\(472\) 0 0
\(473\) 21.1233i 0.971252i
\(474\) 0 0
\(475\) 42.3591i 1.94357i
\(476\) 0 0
\(477\) 0.613974 0.0281119
\(478\) 0 0
\(479\) 8.72313i 0.398570i 0.979942 + 0.199285i \(0.0638620\pi\)
−0.979942 + 0.199285i \(0.936138\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 1.36551i − 0.0621329i
\(484\) 0 0
\(485\) −37.3098 −1.69415
\(486\) 0 0
\(487\) − 11.6431i − 0.527599i −0.964578 0.263800i \(-0.915024\pi\)
0.964578 0.263800i \(-0.0849758\pi\)
\(488\) 0 0
\(489\) − 8.81799i − 0.398763i
\(490\) 0 0
\(491\) −7.77208 −0.350749 −0.175375 0.984502i \(-0.556114\pi\)
−0.175375 + 0.984502i \(0.556114\pi\)
\(492\) 0 0
\(493\) 23.7330 1.06888
\(494\) 0 0
\(495\) 21.3737 0.960679
\(496\) 0 0
\(497\) 10.0450 0.450580
\(498\) 0 0
\(499\) 20.3434i 0.910695i 0.890314 + 0.455348i \(0.150485\pi\)
−0.890314 + 0.455348i \(0.849515\pi\)
\(500\) 0 0
\(501\) − 14.6695i − 0.655387i
\(502\) 0 0
\(503\) −21.1522 −0.943130 −0.471565 0.881831i \(-0.656311\pi\)
−0.471565 + 0.881831i \(0.656311\pi\)
\(504\) 0 0
\(505\) − 39.7495i − 1.76883i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 38.0803i − 1.68788i −0.536438 0.843939i \(-0.680231\pi\)
0.536438 0.843939i \(-0.319769\pi\)
\(510\) 0 0
\(511\) 11.4436 0.506233
\(512\) 0 0
\(513\) − 3.33477i − 0.147234i
\(514\) 0 0
\(515\) 9.08003i 0.400114i
\(516\) 0 0
\(517\) 27.9140 1.22765
\(518\) 0 0
\(519\) 2.41481 0.105998
\(520\) 0 0
\(521\) 20.7433 0.908781 0.454390 0.890803i \(-0.349857\pi\)
0.454390 + 0.890803i \(0.349857\pi\)
\(522\) 0 0
\(523\) −4.94247 −0.216119 −0.108060 0.994144i \(-0.534464\pi\)
−0.108060 + 0.994144i \(0.534464\pi\)
\(524\) 0 0
\(525\) − 45.1614i − 1.97100i
\(526\) 0 0
\(527\) − 21.0883i − 0.918619i
\(528\) 0 0
\(529\) −22.8525 −0.993587
\(530\) 0 0
\(531\) 7.87891i 0.341915i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 78.6796i − 3.40162i
\(536\) 0 0
\(537\) 5.35728 0.231184
\(538\) 0 0
\(539\) 28.6553i 1.23427i
\(540\) 0 0
\(541\) − 43.7196i − 1.87965i −0.341650 0.939827i \(-0.610986\pi\)
0.341650 0.939827i \(-0.389014\pi\)
\(542\) 0 0
\(543\) 17.6715 0.758357
\(544\) 0 0
\(545\) −60.9036 −2.60883
\(546\) 0 0
\(547\) 39.1028 1.67191 0.835957 0.548795i \(-0.184913\pi\)
0.835957 + 0.548795i \(0.184913\pi\)
\(548\) 0 0
\(549\) −8.31227 −0.354759
\(550\) 0 0
\(551\) 24.3020i 1.03530i
\(552\) 0 0
\(553\) − 9.38894i − 0.399259i
\(554\) 0 0
\(555\) 13.1581 0.558531
\(556\) 0 0
\(557\) 23.9916i 1.01656i 0.861192 + 0.508279i \(0.169718\pi\)
−0.861192 + 0.508279i \(0.830282\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 16.5441i 0.698494i
\(562\) 0 0
\(563\) −10.4763 −0.441523 −0.220761 0.975328i \(-0.570854\pi\)
−0.220761 + 0.975328i \(0.570854\pi\)
\(564\) 0 0
\(565\) 11.5422i 0.485583i
\(566\) 0 0
\(567\) 3.55539i 0.149312i
\(568\) 0 0
\(569\) −30.7964 −1.29105 −0.645526 0.763738i \(-0.723362\pi\)
−0.645526 + 0.763738i \(0.723362\pi\)
\(570\) 0 0
\(571\) −9.52126 −0.398452 −0.199226 0.979954i \(-0.563843\pi\)
−0.199226 + 0.979954i \(0.563843\pi\)
\(572\) 0 0
\(573\) 4.57487 0.191118
\(574\) 0 0
\(575\) 4.87853 0.203449
\(576\) 0 0
\(577\) − 47.7495i − 1.98784i −0.110124 0.993918i \(-0.535125\pi\)
0.110124 0.993918i \(-0.464875\pi\)
\(578\) 0 0
\(579\) 16.8428i 0.699964i
\(580\) 0 0
\(581\) −7.00149 −0.290471
\(582\) 0 0
\(583\) − 3.11901i − 0.129176i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 39.4845i − 1.62970i −0.579671 0.814851i \(-0.696819\pi\)
0.579671 0.814851i \(-0.303181\pi\)
\(588\) 0 0
\(589\) 21.5938 0.889758
\(590\) 0 0
\(591\) − 1.85641i − 0.0763624i
\(592\) 0 0
\(593\) − 20.0166i − 0.821983i −0.911639 0.410992i \(-0.865183\pi\)
0.911639 0.410992i \(-0.134817\pi\)
\(594\) 0 0
\(595\) 48.7168 1.99719
\(596\) 0 0
\(597\) 18.9877 0.777116
\(598\) 0 0
\(599\) −19.8517 −0.811120 −0.405560 0.914068i \(-0.632923\pi\)
−0.405560 + 0.914068i \(0.632923\pi\)
\(600\) 0 0
\(601\) 11.7022 0.477344 0.238672 0.971100i \(-0.423288\pi\)
0.238672 + 0.971100i \(0.423288\pi\)
\(602\) 0 0
\(603\) 2.06878i 0.0842473i
\(604\) 0 0
\(605\) − 62.2979i − 2.53277i
\(606\) 0 0
\(607\) −43.1955 −1.75325 −0.876626 0.481173i \(-0.840211\pi\)
−0.876626 + 0.481173i \(0.840211\pi\)
\(608\) 0 0
\(609\) − 25.9096i − 1.04991i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 31.6417i − 1.27800i −0.769208 0.638998i \(-0.779349\pi\)
0.769208 0.638998i \(-0.220651\pi\)
\(614\) 0 0
\(615\) −28.5011 −1.14928
\(616\) 0 0
\(617\) − 15.8027i − 0.636193i −0.948058 0.318096i \(-0.896956\pi\)
0.948058 0.318096i \(-0.103044\pi\)
\(618\) 0 0
\(619\) 36.0663i 1.44963i 0.688945 + 0.724814i \(0.258074\pi\)
−0.688945 + 0.724814i \(0.741926\pi\)
\(620\) 0 0
\(621\) −0.384069 −0.0154121
\(622\) 0 0
\(623\) 50.3445 2.01701
\(624\) 0 0
\(625\) 72.8358 2.91343
\(626\) 0 0
\(627\) −16.9408 −0.676549
\(628\) 0 0
\(629\) 10.1849i 0.406099i
\(630\) 0 0
\(631\) − 17.9063i − 0.712837i −0.934326 0.356418i \(-0.883998\pi\)
0.934326 0.356418i \(-0.116002\pi\)
\(632\) 0 0
\(633\) 12.2424 0.486593
\(634\) 0 0
\(635\) 42.6099i 1.69092i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 2.82529i − 0.111767i
\(640\) 0 0
\(641\) −34.7101 −1.37097 −0.685483 0.728088i \(-0.740409\pi\)
−0.685483 + 0.728088i \(0.740409\pi\)
\(642\) 0 0
\(643\) − 45.5430i − 1.79604i −0.439955 0.898020i \(-0.645006\pi\)
0.439955 0.898020i \(-0.354994\pi\)
\(644\) 0 0
\(645\) − 17.4948i − 0.688859i
\(646\) 0 0
\(647\) 17.8603 0.702162 0.351081 0.936345i \(-0.385814\pi\)
0.351081 + 0.936345i \(0.385814\pi\)
\(648\) 0 0
\(649\) 40.0251 1.57112
\(650\) 0 0
\(651\) −23.0224 −0.902318
\(652\) 0 0
\(653\) −37.6757 −1.47436 −0.737182 0.675694i \(-0.763844\pi\)
−0.737182 + 0.675694i \(0.763844\pi\)
\(654\) 0 0
\(655\) − 56.1394i − 2.19355i
\(656\) 0 0
\(657\) − 3.21865i − 0.125572i
\(658\) 0 0
\(659\) 34.7475 1.35357 0.676785 0.736181i \(-0.263373\pi\)
0.676785 + 0.736181i \(0.263373\pi\)
\(660\) 0 0
\(661\) 39.3234i 1.52950i 0.644325 + 0.764752i \(0.277139\pi\)
−0.644325 + 0.764752i \(0.722861\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 49.8847i 1.93445i
\(666\) 0 0
\(667\) 2.79888 0.108373
\(668\) 0 0
\(669\) 20.7085i 0.800638i
\(670\) 0 0
\(671\) 42.2266i 1.63014i
\(672\) 0 0
\(673\) −11.1436 −0.429554 −0.214777 0.976663i \(-0.568902\pi\)
−0.214777 + 0.976663i \(0.568902\pi\)
\(674\) 0 0
\(675\) −12.7022 −0.488910
\(676\) 0 0
\(677\) −12.3037 −0.472868 −0.236434 0.971648i \(-0.575979\pi\)
−0.236434 + 0.971648i \(0.575979\pi\)
\(678\) 0 0
\(679\) −31.5279 −1.20993
\(680\) 0 0
\(681\) − 8.63881i − 0.331040i
\(682\) 0 0
\(683\) 17.5381i 0.671078i 0.942026 + 0.335539i \(0.108918\pi\)
−0.942026 + 0.335539i \(0.891082\pi\)
\(684\) 0 0
\(685\) 50.0492 1.91228
\(686\) 0 0
\(687\) − 20.1111i − 0.767288i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 13.1181i − 0.499035i −0.968370 0.249518i \(-0.919728\pi\)
0.968370 0.249518i \(-0.0802720\pi\)
\(692\) 0 0
\(693\) 18.0615 0.686099
\(694\) 0 0
\(695\) 56.4663i 2.14189i
\(696\) 0 0
\(697\) − 22.0610i − 0.835620i
\(698\) 0 0
\(699\) 4.41089 0.166835
\(700\) 0 0
\(701\) 0.768137 0.0290121 0.0145061 0.999895i \(-0.495382\pi\)
0.0145061 + 0.999895i \(0.495382\pi\)
\(702\) 0 0
\(703\) −10.4291 −0.393340
\(704\) 0 0
\(705\) −23.1190 −0.870712
\(706\) 0 0
\(707\) − 33.5895i − 1.26326i
\(708\) 0 0
\(709\) 32.0897i 1.20515i 0.798061 + 0.602577i \(0.205859\pi\)
−0.798061 + 0.602577i \(0.794141\pi\)
\(710\) 0 0
\(711\) −2.64077 −0.0990365
\(712\) 0 0
\(713\) − 2.48698i − 0.0931381i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.51734i 0.131358i
\(718\) 0 0
\(719\) 10.9771 0.409378 0.204689 0.978827i \(-0.434382\pi\)
0.204689 + 0.978827i \(0.434382\pi\)
\(720\) 0 0
\(721\) 7.67291i 0.285754i
\(722\) 0 0
\(723\) 4.74330i 0.176405i
\(724\) 0 0
\(725\) 92.5668 3.43784
\(726\) 0 0
\(727\) −40.9056 −1.51710 −0.758552 0.651612i \(-0.774093\pi\)
−0.758552 + 0.651612i \(0.774093\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 13.5417 0.500858
\(732\) 0 0
\(733\) − 6.18829i − 0.228570i −0.993448 0.114285i \(-0.963542\pi\)
0.993448 0.114285i \(-0.0364577\pi\)
\(734\) 0 0
\(735\) − 23.7330i − 0.875404i
\(736\) 0 0
\(737\) 10.5095 0.387122
\(738\) 0 0
\(739\) 53.3488i 1.96247i 0.192824 + 0.981233i \(0.438235\pi\)
−0.192824 + 0.981233i \(0.561765\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 44.5728i 1.63522i 0.575774 + 0.817609i \(0.304701\pi\)
−0.575774 + 0.817609i \(0.695299\pi\)
\(744\) 0 0
\(745\) 23.1888 0.849574
\(746\) 0 0
\(747\) 1.96926i 0.0720515i
\(748\) 0 0
\(749\) − 66.4867i − 2.42937i
\(750\) 0 0
\(751\) 1.50945 0.0550807 0.0275403 0.999621i \(-0.491233\pi\)
0.0275403 + 0.999621i \(0.491233\pi\)
\(752\) 0 0
\(753\) −6.98968 −0.254718
\(754\) 0 0
\(755\) −21.3737 −0.777870
\(756\) 0 0
\(757\) −51.9794 −1.88922 −0.944611 0.328192i \(-0.893561\pi\)
−0.944611 + 0.328192i \(0.893561\pi\)
\(758\) 0 0
\(759\) 1.95108i 0.0708198i
\(760\) 0 0
\(761\) 10.6695i 0.386771i 0.981123 + 0.193385i \(0.0619468\pi\)
−0.981123 + 0.193385i \(0.938053\pi\)
\(762\) 0 0
\(763\) −51.4655 −1.86317
\(764\) 0 0
\(765\) − 13.7022i − 0.495406i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.01256i 0.216818i 0.994106 + 0.108409i \(0.0345757\pi\)
−0.994106 + 0.108409i \(0.965424\pi\)
\(770\) 0 0
\(771\) 4.71010 0.169630
\(772\) 0 0
\(773\) 39.8907i 1.43477i 0.696677 + 0.717385i \(0.254661\pi\)
−0.696677 + 0.717385i \(0.745339\pi\)
\(774\) 0 0
\(775\) − 82.2515i − 2.95456i
\(776\) 0 0
\(777\) 11.1190 0.398892
\(778\) 0 0
\(779\) 22.5899 0.809367
\(780\) 0 0
\(781\) −14.3526 −0.513576
\(782\) 0 0
\(783\) −7.28744 −0.260432
\(784\) 0 0
\(785\) − 52.8745i − 1.88717i
\(786\) 0 0
\(787\) − 3.50215i − 0.124838i −0.998050 0.0624190i \(-0.980118\pi\)
0.998050 0.0624190i \(-0.0198815\pi\)
\(788\) 0 0
\(789\) −4.12933 −0.147008
\(790\) 0 0
\(791\) 9.75350i 0.346794i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 2.58324i 0.0916179i
\(796\) 0 0
\(797\) 24.2869 0.860287 0.430144 0.902760i \(-0.358463\pi\)
0.430144 + 0.902760i \(0.358463\pi\)
\(798\) 0 0
\(799\) − 17.8950i − 0.633081i
\(800\) 0 0
\(801\) − 14.1601i − 0.500321i
\(802\) 0 0
\(803\) −16.3509 −0.577010
\(804\) 0 0
\(805\) 5.74526 0.202494
\(806\) 0 0
\(807\) 16.2587 0.572332
\(808\) 0 0
\(809\) −19.1288 −0.672534 −0.336267 0.941767i \(-0.609164\pi\)
−0.336267 + 0.941767i \(0.609164\pi\)
\(810\) 0 0
\(811\) 55.1857i 1.93783i 0.247387 + 0.968917i \(0.420428\pi\)
−0.247387 + 0.968917i \(0.579572\pi\)
\(812\) 0 0
\(813\) 32.3171i 1.13341i
\(814\) 0 0
\(815\) 37.1008 1.29959
\(816\) 0 0
\(817\) 13.8664i 0.485122i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 23.3362i − 0.814440i −0.913330 0.407220i \(-0.866498\pi\)
0.913330 0.407220i \(-0.133502\pi\)
\(822\) 0 0
\(823\) 2.05359 0.0715835 0.0357918 0.999359i \(-0.488605\pi\)
0.0357918 + 0.999359i \(0.488605\pi\)
\(824\) 0 0
\(825\) 64.5278i 2.24657i
\(826\) 0 0
\(827\) − 5.96289i − 0.207350i −0.994611 0.103675i \(-0.966940\pi\)
0.994611 0.103675i \(-0.0330602\pi\)
\(828\) 0 0
\(829\) −14.5631 −0.505797 −0.252899 0.967493i \(-0.581384\pi\)
−0.252899 + 0.967493i \(0.581384\pi\)
\(830\) 0 0
\(831\) 23.7369 0.823424
\(832\) 0 0
\(833\) 18.3703 0.636492
\(834\) 0 0
\(835\) 61.7207 2.13593
\(836\) 0 0
\(837\) 6.47535i 0.223821i
\(838\) 0 0
\(839\) 49.0737i 1.69421i 0.531425 + 0.847105i \(0.321657\pi\)
−0.531425 + 0.847105i \(0.678343\pi\)
\(840\) 0 0
\(841\) 24.1067 0.831267
\(842\) 0 0
\(843\) 25.5812i 0.881062i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 52.6437i − 1.80886i
\(848\) 0 0
\(849\) 11.6486 0.399780
\(850\) 0 0
\(851\) 1.20112i 0.0411740i
\(852\) 0 0
\(853\) 54.6929i 1.87265i 0.351138 + 0.936324i \(0.385795\pi\)
−0.351138 + 0.936324i \(0.614205\pi\)
\(854\) 0 0
\(855\) 14.0307 0.479841
\(856\) 0 0
\(857\) 23.2811 0.795266 0.397633 0.917545i \(-0.369832\pi\)
0.397633 + 0.917545i \(0.369832\pi\)
\(858\) 0 0
\(859\) 43.4229 1.48157 0.740785 0.671742i \(-0.234454\pi\)
0.740785 + 0.671742i \(0.234454\pi\)
\(860\) 0 0
\(861\) −24.0843 −0.820792
\(862\) 0 0
\(863\) − 7.28977i − 0.248147i −0.992273 0.124073i \(-0.960404\pi\)
0.992273 0.124073i \(-0.0395958\pi\)
\(864\) 0 0
\(865\) 10.1601i 0.345453i
\(866\) 0 0
\(867\) −6.39392 −0.217149
\(868\) 0 0
\(869\) 13.4152i 0.455079i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 8.86766i 0.300125i
\(874\) 0 0
\(875\) 115.217 3.89506
\(876\) 0 0
\(877\) − 33.2464i − 1.12265i −0.827595 0.561326i \(-0.810292\pi\)
0.827595 0.561326i \(-0.189708\pi\)
\(878\) 0 0
\(879\) 9.85016i 0.332238i
\(880\) 0 0
\(881\) −16.1888 −0.545415 −0.272708 0.962097i \(-0.587919\pi\)
−0.272708 + 0.962097i \(0.587919\pi\)
\(882\) 0 0
\(883\) 20.0492 0.674708 0.337354 0.941378i \(-0.390468\pi\)
0.337354 + 0.941378i \(0.390468\pi\)
\(884\) 0 0
\(885\) −33.1497 −1.11432
\(886\) 0 0
\(887\) −54.3120 −1.82362 −0.911810 0.410612i \(-0.865315\pi\)
−0.911810 + 0.410612i \(0.865315\pi\)
\(888\) 0 0
\(889\) 36.0067i 1.20763i
\(890\) 0 0
\(891\) − 5.08003i − 0.170188i
\(892\) 0 0
\(893\) 18.3240 0.613191
\(894\) 0 0
\(895\) 22.5402i 0.753436i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 47.1887i − 1.57383i
\(900\) 0 0
\(901\) −1.99953 −0.0666139
\(902\) 0 0
\(903\) − 14.7837i − 0.491970i
\(904\) 0 0
\(905\) 74.3511i 2.47152i
\(906\) 0 0
\(907\) 51.8972 1.72322 0.861610 0.507571i \(-0.169457\pi\)
0.861610 + 0.507571i \(0.169457\pi\)
\(908\) 0 0
\(909\) −9.44750 −0.313354
\(910\) 0 0
\(911\) 17.0433 0.564669 0.282334 0.959316i \(-0.408891\pi\)
0.282334 + 0.959316i \(0.408891\pi\)
\(912\) 0 0
\(913\) 10.0039 0.331081
\(914\) 0 0
\(915\) − 34.9731i − 1.15617i
\(916\) 0 0
\(917\) − 47.4395i − 1.56659i
\(918\) 0 0
\(919\) −16.0693 −0.530078 −0.265039 0.964238i \(-0.585385\pi\)
−0.265039 + 0.964238i \(0.585385\pi\)
\(920\) 0 0
\(921\) 12.4182i 0.409194i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 39.7246i 1.30614i
\(926\) 0 0
\(927\) 2.15811 0.0708816
\(928\) 0 0
\(929\) − 11.4025i − 0.374104i −0.982350 0.187052i \(-0.940107\pi\)
0.982350 0.187052i \(-0.0598933\pi\)
\(930\) 0 0
\(931\) 18.8107i 0.616495i
\(932\) 0 0
\(933\) −7.42267 −0.243007
\(934\) 0 0
\(935\) −69.6078 −2.27642
\(936\) 0 0
\(937\) −33.8213 −1.10489 −0.552447 0.833548i \(-0.686306\pi\)
−0.552447 + 0.833548i \(0.686306\pi\)
\(938\) 0 0
\(939\) 27.9799 0.913088
\(940\) 0 0
\(941\) − 48.0429i − 1.56615i −0.621925 0.783077i \(-0.713649\pi\)
0.621925 0.783077i \(-0.286351\pi\)
\(942\) 0 0
\(943\) − 2.60170i − 0.0847229i
\(944\) 0 0
\(945\) −14.9589 −0.486614
\(946\) 0 0
\(947\) − 12.0779i − 0.392481i −0.980556 0.196240i \(-0.937127\pi\)
0.980556 0.196240i \(-0.0628733\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) − 8.02484i − 0.260223i
\(952\) 0 0
\(953\) −15.4251 −0.499669 −0.249834 0.968289i \(-0.580376\pi\)
−0.249834 + 0.968289i \(0.580376\pi\)
\(954\) 0 0
\(955\) 19.2483i 0.622862i
\(956\) 0 0
\(957\) 37.0204i 1.19670i
\(958\) 0 0
\(959\) 42.2931 1.36572
\(960\) 0 0
\(961\) −10.9302 −0.352587
\(962\) 0 0
\(963\) −18.7003 −0.602608
\(964\) 0 0
\(965\) −70.8645 −2.28121
\(966\) 0 0
\(967\) 49.7289i 1.59917i 0.600550 + 0.799587i \(0.294948\pi\)
−0.600550 + 0.799587i \(0.705052\pi\)
\(968\) 0 0
\(969\) 10.8604i 0.348885i
\(970\) 0 0
\(971\) −30.9897 −0.994506 −0.497253 0.867606i \(-0.665658\pi\)
−0.497253 + 0.867606i \(0.665658\pi\)
\(972\) 0 0
\(973\) 47.7158i 1.52970i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 22.8765i − 0.731886i −0.930637 0.365943i \(-0.880747\pi\)
0.930637 0.365943i \(-0.119253\pi\)
\(978\) 0 0
\(979\) −71.9336 −2.29901
\(980\) 0 0
\(981\) 14.4754i 0.462162i
\(982\) 0 0
\(983\) − 28.0019i − 0.893121i −0.894754 0.446560i \(-0.852649\pi\)
0.894754 0.446560i \(-0.147351\pi\)
\(984\) 0 0
\(985\) 7.81065 0.248868
\(986\) 0 0
\(987\) −19.5363 −0.621846
\(988\) 0 0
\(989\) 1.59700 0.0507816
\(990\) 0 0
\(991\) −16.2855 −0.517325 −0.258663 0.965968i \(-0.583282\pi\)
−0.258663 + 0.965968i \(0.583282\pi\)
\(992\) 0 0
\(993\) 18.9252i 0.600574i
\(994\) 0 0
\(995\) 79.8890i 2.53265i
\(996\) 0 0
\(997\) 46.3034 1.46644 0.733222 0.679989i \(-0.238015\pi\)
0.733222 + 0.679989i \(0.238015\pi\)
\(998\) 0 0
\(999\) − 3.12737i − 0.0989456i
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 4056.2.c.p.337.8 8
13.5 odd 4 4056.2.a.be.1.4 4
13.8 odd 4 4056.2.a.bd.1.1 4
13.9 even 3 312.2.bf.b.49.4 8
13.10 even 6 312.2.bf.b.121.1 yes 8
13.12 even 2 inner 4056.2.c.p.337.1 8
39.23 odd 6 936.2.bi.c.433.4 8
39.35 odd 6 936.2.bi.c.361.1 8
52.23 odd 6 624.2.bv.g.433.1 8
52.31 even 4 8112.2.a.cs.1.4 4
52.35 odd 6 624.2.bv.g.49.4 8
52.47 even 4 8112.2.a.cq.1.1 4
156.23 even 6 1872.2.by.m.433.4 8
156.35 even 6 1872.2.by.m.1297.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.bf.b.49.4 8 13.9 even 3
312.2.bf.b.121.1 yes 8 13.10 even 6
624.2.bv.g.49.4 8 52.35 odd 6
624.2.bv.g.433.1 8 52.23 odd 6
936.2.bi.c.361.1 8 39.35 odd 6
936.2.bi.c.433.4 8 39.23 odd 6
1872.2.by.m.433.4 8 156.23 even 6
1872.2.by.m.1297.1 8 156.35 even 6
4056.2.a.bd.1.1 4 13.8 odd 4
4056.2.a.be.1.4 4 13.5 odd 4
4056.2.c.p.337.1 8 13.12 even 2 inner
4056.2.c.p.337.8 8 1.1 even 1 trivial
8112.2.a.cq.1.1 4 52.47 even 4
8112.2.a.cs.1.4 4 52.31 even 4