Properties

Label 4368.2.a.br.1.3
Level $4368$
Weight $2$
Character 4368.1
Self dual yes
Analytic conductor $34.879$
Analytic rank $1$
Dimension $4$
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] = [4368,2,Mod(1,4368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4368.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.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: \(34.8786556029\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.10710\) of defining polynomial
Character \(\chi\) \(=\) 4368.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.926817 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.926817 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.21419 q^{11} +1.00000 q^{13} -0.926817 q^{15} -2.87971 q^{17} +1.28738 q^{19} +1.00000 q^{21} +8.02072 q^{23} -4.14101 q^{25} -1.00000 q^{27} +3.28738 q^{29} +7.04680 q^{31} +4.21419 q^{33} -0.926817 q^{35} +8.57475 q^{37} -1.00000 q^{39} -12.0678 q^{41} -7.14101 q^{43} +0.926817 q^{45} -1.95289 q^{47} +1.00000 q^{49} +2.87971 q^{51} -5.14101 q^{53} -3.90579 q^{55} -1.28738 q^{57} +7.33448 q^{59} +7.75942 q^{61} -1.00000 q^{63} +0.926817 q^{65} -12.0414 q^{67} -8.02072 q^{69} -10.7889 q^{71} -8.32568 q^{73} +4.14101 q^{75} +4.21419 q^{77} -4.47204 q^{79} +1.00000 q^{81} +3.80653 q^{83} -2.66896 q^{85} -3.28738 q^{87} -5.64793 q^{89} -1.00000 q^{91} -7.04680 q^{93} +1.19316 q^{95} +6.90043 q^{97} -4.21419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} - 7 q^{19} + 4 q^{21} - 3 q^{23} + 9 q^{25} - 4 q^{27} + q^{29} - 3 q^{31} - 2 q^{33} + 3 q^{35} + 10 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{43} - 3 q^{45} - 5 q^{47} + 4 q^{49} + 2 q^{51} + 5 q^{53} - 10 q^{55} + 7 q^{57} + 20 q^{59} + 12 q^{61} - 4 q^{63} - 3 q^{65} + 22 q^{67} + 3 q^{69} - 13 q^{73} - 9 q^{75} - 2 q^{77} - 11 q^{79} + 4 q^{81} - q^{83} + 8 q^{85} - q^{87} - 5 q^{89} - 4 q^{91} + 3 q^{93} - 13 q^{95} - 17 q^{97} + 2 q^{99}+O(q^{100}) \) 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
\(5\) 0.926817 0.414485 0.207243 0.978290i \(-0.433551\pi\)
0.207243 + 0.978290i \(0.433551\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.21419 −1.27063 −0.635313 0.772254i \(-0.719129\pi\)
−0.635313 + 0.772254i \(0.719129\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −0.926817 −0.239303
\(16\) 0 0
\(17\) −2.87971 −0.698432 −0.349216 0.937042i \(-0.613552\pi\)
−0.349216 + 0.937042i \(0.613552\pi\)
\(18\) 0 0
\(19\) 1.28738 0.295344 0.147672 0.989036i \(-0.452822\pi\)
0.147672 + 0.989036i \(0.452822\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 8.02072 1.67244 0.836218 0.548397i \(-0.184762\pi\)
0.836218 + 0.548397i \(0.184762\pi\)
\(24\) 0 0
\(25\) −4.14101 −0.828202
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.28738 0.610450 0.305225 0.952280i \(-0.401268\pi\)
0.305225 + 0.952280i \(0.401268\pi\)
\(30\) 0 0
\(31\) 7.04680 1.26564 0.632821 0.774298i \(-0.281897\pi\)
0.632821 + 0.774298i \(0.281897\pi\)
\(32\) 0 0
\(33\) 4.21419 0.733597
\(34\) 0 0
\(35\) −0.926817 −0.156661
\(36\) 0 0
\(37\) 8.57475 1.40968 0.704840 0.709366i \(-0.251019\pi\)
0.704840 + 0.709366i \(0.251019\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −12.0678 −1.88468 −0.942339 0.334660i \(-0.891379\pi\)
−0.942339 + 0.334660i \(0.891379\pi\)
\(42\) 0 0
\(43\) −7.14101 −1.08899 −0.544497 0.838763i \(-0.683279\pi\)
−0.544497 + 0.838763i \(0.683279\pi\)
\(44\) 0 0
\(45\) 0.926817 0.138162
\(46\) 0 0
\(47\) −1.95289 −0.284859 −0.142429 0.989805i \(-0.545491\pi\)
−0.142429 + 0.989805i \(0.545491\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.87971 0.403240
\(52\) 0 0
\(53\) −5.14101 −0.706172 −0.353086 0.935591i \(-0.614868\pi\)
−0.353086 + 0.935591i \(0.614868\pi\)
\(54\) 0 0
\(55\) −3.90579 −0.526656
\(56\) 0 0
\(57\) −1.28738 −0.170517
\(58\) 0 0
\(59\) 7.33448 0.954868 0.477434 0.878668i \(-0.341567\pi\)
0.477434 + 0.878668i \(0.341567\pi\)
\(60\) 0 0
\(61\) 7.75942 0.993492 0.496746 0.867896i \(-0.334528\pi\)
0.496746 + 0.867896i \(0.334528\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0.926817 0.114958
\(66\) 0 0
\(67\) −12.0414 −1.47110 −0.735548 0.677473i \(-0.763075\pi\)
−0.735548 + 0.677473i \(0.763075\pi\)
\(68\) 0 0
\(69\) −8.02072 −0.965581
\(70\) 0 0
\(71\) −10.7889 −1.28041 −0.640206 0.768203i \(-0.721151\pi\)
−0.640206 + 0.768203i \(0.721151\pi\)
\(72\) 0 0
\(73\) −8.32568 −0.974447 −0.487224 0.873277i \(-0.661990\pi\)
−0.487224 + 0.873277i \(0.661990\pi\)
\(74\) 0 0
\(75\) 4.14101 0.478163
\(76\) 0 0
\(77\) 4.21419 0.480252
\(78\) 0 0
\(79\) −4.47204 −0.503144 −0.251572 0.967839i \(-0.580948\pi\)
−0.251572 + 0.967839i \(0.580948\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.80653 0.417821 0.208910 0.977935i \(-0.433008\pi\)
0.208910 + 0.977935i \(0.433008\pi\)
\(84\) 0 0
\(85\) −2.66896 −0.289490
\(86\) 0 0
\(87\) −3.28738 −0.352444
\(88\) 0 0
\(89\) −5.64793 −0.598680 −0.299340 0.954147i \(-0.596766\pi\)
−0.299340 + 0.954147i \(0.596766\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −7.04680 −0.730719
\(94\) 0 0
\(95\) 1.19316 0.122416
\(96\) 0 0
\(97\) 6.90043 0.700633 0.350316 0.936631i \(-0.386074\pi\)
0.350316 + 0.936631i \(0.386074\pi\)
\(98\) 0 0
\(99\) −4.21419 −0.423542
\(100\) 0 0
\(101\) −10.9739 −1.09195 −0.545973 0.837803i \(-0.683840\pi\)
−0.545973 + 0.837803i \(0.683840\pi\)
\(102\) 0 0
\(103\) −16.4284 −1.61874 −0.809368 0.587301i \(-0.800190\pi\)
−0.809368 + 0.587301i \(0.800190\pi\)
\(104\) 0 0
\(105\) 0.926817 0.0904481
\(106\) 0 0
\(107\) 3.45446 0.333955 0.166978 0.985961i \(-0.446599\pi\)
0.166978 + 0.985961i \(0.446599\pi\)
\(108\) 0 0
\(109\) 4.66896 0.447206 0.223603 0.974680i \(-0.428218\pi\)
0.223603 + 0.974680i \(0.428218\pi\)
\(110\) 0 0
\(111\) −8.57475 −0.813879
\(112\) 0 0
\(113\) −3.28738 −0.309250 −0.154625 0.987973i \(-0.549417\pi\)
−0.154625 + 0.987973i \(0.549417\pi\)
\(114\) 0 0
\(115\) 7.43374 0.693200
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 2.87971 0.263983
\(120\) 0 0
\(121\) 6.75942 0.614493
\(122\) 0 0
\(123\) 12.0678 1.08812
\(124\) 0 0
\(125\) −8.47204 −0.757763
\(126\) 0 0
\(127\) 0.428386 0.0380131 0.0190065 0.999819i \(-0.493950\pi\)
0.0190065 + 0.999819i \(0.493950\pi\)
\(128\) 0 0
\(129\) 7.14101 0.628731
\(130\) 0 0
\(131\) −14.1878 −1.23959 −0.619797 0.784762i \(-0.712785\pi\)
−0.619797 + 0.784762i \(0.712785\pi\)
\(132\) 0 0
\(133\) −1.28738 −0.111630
\(134\) 0 0
\(135\) −0.926817 −0.0797677
\(136\) 0 0
\(137\) 14.7070 1.25650 0.628250 0.778011i \(-0.283771\pi\)
0.628250 + 0.778011i \(0.283771\pi\)
\(138\) 0 0
\(139\) −9.85363 −0.835774 −0.417887 0.908499i \(-0.637229\pi\)
−0.417887 + 0.908499i \(0.637229\pi\)
\(140\) 0 0
\(141\) 1.95289 0.164463
\(142\) 0 0
\(143\) −4.21419 −0.352409
\(144\) 0 0
\(145\) 3.04680 0.253023
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −3.38663 −0.277444 −0.138722 0.990331i \(-0.544299\pi\)
−0.138722 + 0.990331i \(0.544299\pi\)
\(150\) 0 0
\(151\) 21.8951 1.78180 0.890898 0.454204i \(-0.150076\pi\)
0.890898 + 0.454204i \(0.150076\pi\)
\(152\) 0 0
\(153\) −2.87971 −0.232811
\(154\) 0 0
\(155\) 6.53109 0.524590
\(156\) 0 0
\(157\) −0.867482 −0.0692326 −0.0346163 0.999401i \(-0.511021\pi\)
−0.0346163 + 0.999401i \(0.511021\pi\)
\(158\) 0 0
\(159\) 5.14101 0.407709
\(160\) 0 0
\(161\) −8.02072 −0.632121
\(162\) 0 0
\(163\) −4.42839 −0.346858 −0.173429 0.984846i \(-0.555485\pi\)
−0.173429 + 0.984846i \(0.555485\pi\)
\(164\) 0 0
\(165\) 3.90579 0.304065
\(166\) 0 0
\(167\) −20.5276 −1.58848 −0.794238 0.607606i \(-0.792130\pi\)
−0.794238 + 0.607606i \(0.792130\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 1.28738 0.0984481
\(172\) 0 0
\(173\) 15.0154 1.14160 0.570799 0.821090i \(-0.306634\pi\)
0.570799 + 0.821090i \(0.306634\pi\)
\(174\) 0 0
\(175\) 4.14101 0.313031
\(176\) 0 0
\(177\) −7.33448 −0.551293
\(178\) 0 0
\(179\) 5.35176 0.400009 0.200004 0.979795i \(-0.435904\pi\)
0.200004 + 0.979795i \(0.435904\pi\)
\(180\) 0 0
\(181\) 11.4667 0.852312 0.426156 0.904650i \(-0.359867\pi\)
0.426156 + 0.904650i \(0.359867\pi\)
\(182\) 0 0
\(183\) −7.75942 −0.573593
\(184\) 0 0
\(185\) 7.94723 0.584292
\(186\) 0 0
\(187\) 12.1357 0.887447
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −20.1756 −1.45985 −0.729927 0.683525i \(-0.760446\pi\)
−0.729927 + 0.683525i \(0.760446\pi\)
\(192\) 0 0
\(193\) −26.3756 −1.89856 −0.949279 0.314435i \(-0.898185\pi\)
−0.949279 + 0.314435i \(0.898185\pi\)
\(194\) 0 0
\(195\) −0.926817 −0.0663708
\(196\) 0 0
\(197\) −20.1322 −1.43436 −0.717180 0.696888i \(-0.754568\pi\)
−0.717180 + 0.696888i \(0.754568\pi\)
\(198\) 0 0
\(199\) −1.75942 −0.124722 −0.0623610 0.998054i \(-0.519863\pi\)
−0.0623610 + 0.998054i \(0.519863\pi\)
\(200\) 0 0
\(201\) 12.0414 0.849338
\(202\) 0 0
\(203\) −3.28738 −0.230729
\(204\) 0 0
\(205\) −11.1847 −0.781171
\(206\) 0 0
\(207\) 8.02072 0.557479
\(208\) 0 0
\(209\) −5.42525 −0.375272
\(210\) 0 0
\(211\) −15.5694 −1.07184 −0.535921 0.844268i \(-0.680035\pi\)
−0.535921 + 0.844268i \(0.680035\pi\)
\(212\) 0 0
\(213\) 10.7889 0.739246
\(214\) 0 0
\(215\) −6.61841 −0.451372
\(216\) 0 0
\(217\) −7.04680 −0.478368
\(218\) 0 0
\(219\) 8.32568 0.562597
\(220\) 0 0
\(221\) −2.87971 −0.193710
\(222\) 0 0
\(223\) 11.5694 0.774744 0.387372 0.921923i \(-0.373383\pi\)
0.387372 + 0.921923i \(0.373383\pi\)
\(224\) 0 0
\(225\) −4.14101 −0.276067
\(226\) 0 0
\(227\) 11.0418 0.732867 0.366433 0.930444i \(-0.380579\pi\)
0.366433 + 0.930444i \(0.380579\pi\)
\(228\) 0 0
\(229\) −13.7073 −0.905802 −0.452901 0.891561i \(-0.649611\pi\)
−0.452901 + 0.891561i \(0.649611\pi\)
\(230\) 0 0
\(231\) −4.21419 −0.277274
\(232\) 0 0
\(233\) 1.23522 0.0809222 0.0404611 0.999181i \(-0.487117\pi\)
0.0404611 + 0.999181i \(0.487117\pi\)
\(234\) 0 0
\(235\) −1.80997 −0.118070
\(236\) 0 0
\(237\) 4.47204 0.290491
\(238\) 0 0
\(239\) −19.1061 −1.23587 −0.617936 0.786228i \(-0.712031\pi\)
−0.617936 + 0.786228i \(0.712031\pi\)
\(240\) 0 0
\(241\) −12.0329 −0.775110 −0.387555 0.921847i \(-0.626680\pi\)
−0.387555 + 0.921847i \(0.626680\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0.926817 0.0592122
\(246\) 0 0
\(247\) 1.28738 0.0819137
\(248\) 0 0
\(249\) −3.80653 −0.241229
\(250\) 0 0
\(251\) 15.0031 0.946990 0.473495 0.880797i \(-0.342992\pi\)
0.473495 + 0.880797i \(0.342992\pi\)
\(252\) 0 0
\(253\) −33.8009 −2.12504
\(254\) 0 0
\(255\) 2.66896 0.167137
\(256\) 0 0
\(257\) −13.0261 −0.812544 −0.406272 0.913752i \(-0.633172\pi\)
−0.406272 + 0.913752i \(0.633172\pi\)
\(258\) 0 0
\(259\) −8.57475 −0.532809
\(260\) 0 0
\(261\) 3.28738 0.203483
\(262\) 0 0
\(263\) 2.59547 0.160044 0.0800218 0.996793i \(-0.474501\pi\)
0.0800218 + 0.996793i \(0.474501\pi\)
\(264\) 0 0
\(265\) −4.76478 −0.292698
\(266\) 0 0
\(267\) 5.64793 0.345648
\(268\) 0 0
\(269\) −23.3081 −1.42112 −0.710560 0.703637i \(-0.751558\pi\)
−0.710560 + 0.703637i \(0.751558\pi\)
\(270\) 0 0
\(271\) 6.05215 0.367642 0.183821 0.982960i \(-0.441153\pi\)
0.183821 + 0.982960i \(0.441153\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) 17.4510 1.05234
\(276\) 0 0
\(277\) 15.1932 0.912869 0.456434 0.889757i \(-0.349126\pi\)
0.456434 + 0.889757i \(0.349126\pi\)
\(278\) 0 0
\(279\) 7.04680 0.421881
\(280\) 0 0
\(281\) −7.57506 −0.451890 −0.225945 0.974140i \(-0.572547\pi\)
−0.225945 + 0.974140i \(0.572547\pi\)
\(282\) 0 0
\(283\) −19.0974 −1.13522 −0.567610 0.823298i \(-0.692132\pi\)
−0.567610 + 0.823298i \(0.692132\pi\)
\(284\) 0 0
\(285\) −1.19316 −0.0706768
\(286\) 0 0
\(287\) 12.0678 0.712341
\(288\) 0 0
\(289\) −8.70727 −0.512192
\(290\) 0 0
\(291\) −6.90043 −0.404510
\(292\) 0 0
\(293\) −3.79430 −0.221665 −0.110833 0.993839i \(-0.535352\pi\)
−0.110833 + 0.993839i \(0.535352\pi\)
\(294\) 0 0
\(295\) 6.79772 0.395779
\(296\) 0 0
\(297\) 4.21419 0.244532
\(298\) 0 0
\(299\) 8.02072 0.463850
\(300\) 0 0
\(301\) 7.14101 0.411601
\(302\) 0 0
\(303\) 10.9739 0.630435
\(304\) 0 0
\(305\) 7.19156 0.411788
\(306\) 0 0
\(307\) 5.19316 0.296389 0.148195 0.988958i \(-0.452654\pi\)
0.148195 + 0.988958i \(0.452654\pi\)
\(308\) 0 0
\(309\) 16.4284 0.934578
\(310\) 0 0
\(311\) 13.8536 0.785568 0.392784 0.919631i \(-0.371512\pi\)
0.392784 + 0.919631i \(0.371512\pi\)
\(312\) 0 0
\(313\) −32.6162 −1.84358 −0.921788 0.387694i \(-0.873272\pi\)
−0.921788 + 0.387694i \(0.873272\pi\)
\(314\) 0 0
\(315\) −0.926817 −0.0522202
\(316\) 0 0
\(317\) 12.0380 0.676121 0.338061 0.941124i \(-0.390229\pi\)
0.338061 + 0.941124i \(0.390229\pi\)
\(318\) 0 0
\(319\) −13.8536 −0.775655
\(320\) 0 0
\(321\) −3.45446 −0.192809
\(322\) 0 0
\(323\) −3.70727 −0.206278
\(324\) 0 0
\(325\) −4.14101 −0.229702
\(326\) 0 0
\(327\) −4.66896 −0.258194
\(328\) 0 0
\(329\) 1.95289 0.107666
\(330\) 0 0
\(331\) 26.2820 1.44459 0.722295 0.691585i \(-0.243087\pi\)
0.722295 + 0.691585i \(0.243087\pi\)
\(332\) 0 0
\(333\) 8.57475 0.469893
\(334\) 0 0
\(335\) −11.1602 −0.609748
\(336\) 0 0
\(337\) 26.2905 1.43214 0.716068 0.698031i \(-0.245940\pi\)
0.716068 + 0.698031i \(0.245940\pi\)
\(338\) 0 0
\(339\) 3.28738 0.178546
\(340\) 0 0
\(341\) −29.6966 −1.60816
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −7.43374 −0.400219
\(346\) 0 0
\(347\) −16.4928 −0.885378 −0.442689 0.896675i \(-0.645975\pi\)
−0.442689 + 0.896675i \(0.645975\pi\)
\(348\) 0 0
\(349\) 14.1793 0.759001 0.379501 0.925191i \(-0.376096\pi\)
0.379501 + 0.925191i \(0.376096\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −29.8272 −1.58754 −0.793772 0.608215i \(-0.791886\pi\)
−0.793772 + 0.608215i \(0.791886\pi\)
\(354\) 0 0
\(355\) −9.99938 −0.530712
\(356\) 0 0
\(357\) −2.87971 −0.152410
\(358\) 0 0
\(359\) −1.34671 −0.0710767 −0.0355383 0.999368i \(-0.511315\pi\)
−0.0355383 + 0.999368i \(0.511315\pi\)
\(360\) 0 0
\(361\) −17.3427 −0.912772
\(362\) 0 0
\(363\) −6.75942 −0.354778
\(364\) 0 0
\(365\) −7.71638 −0.403894
\(366\) 0 0
\(367\) −36.1771 −1.88843 −0.944214 0.329331i \(-0.893177\pi\)
−0.944214 + 0.329331i \(0.893177\pi\)
\(368\) 0 0
\(369\) −12.0678 −0.628226
\(370\) 0 0
\(371\) 5.14101 0.266908
\(372\) 0 0
\(373\) −16.9089 −0.875511 −0.437755 0.899094i \(-0.644226\pi\)
−0.437755 + 0.899094i \(0.644226\pi\)
\(374\) 0 0
\(375\) 8.47204 0.437495
\(376\) 0 0
\(377\) 3.28738 0.169308
\(378\) 0 0
\(379\) 11.6131 0.596523 0.298261 0.954484i \(-0.403593\pi\)
0.298261 + 0.954484i \(0.403593\pi\)
\(380\) 0 0
\(381\) −0.428386 −0.0219469
\(382\) 0 0
\(383\) −14.6134 −0.746708 −0.373354 0.927689i \(-0.621792\pi\)
−0.373354 + 0.927689i \(0.621792\pi\)
\(384\) 0 0
\(385\) 3.90579 0.199057
\(386\) 0 0
\(387\) −7.14101 −0.362998
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −23.0974 −1.16808
\(392\) 0 0
\(393\) 14.1878 0.715680
\(394\) 0 0
\(395\) −4.14477 −0.208546
\(396\) 0 0
\(397\) 25.6982 1.28975 0.644877 0.764287i \(-0.276909\pi\)
0.644877 + 0.764287i \(0.276909\pi\)
\(398\) 0 0
\(399\) 1.28738 0.0644494
\(400\) 0 0
\(401\) −19.5637 −0.976966 −0.488483 0.872573i \(-0.662450\pi\)
−0.488483 + 0.872573i \(0.662450\pi\)
\(402\) 0 0
\(403\) 7.04680 0.351026
\(404\) 0 0
\(405\) 0.926817 0.0460539
\(406\) 0 0
\(407\) −36.1357 −1.79118
\(408\) 0 0
\(409\) 3.19316 0.157892 0.0789458 0.996879i \(-0.474845\pi\)
0.0789458 + 0.996879i \(0.474845\pi\)
\(410\) 0 0
\(411\) −14.7070 −0.725441
\(412\) 0 0
\(413\) −7.33448 −0.360906
\(414\) 0 0
\(415\) 3.52796 0.173181
\(416\) 0 0
\(417\) 9.85363 0.482535
\(418\) 0 0
\(419\) −30.2292 −1.47680 −0.738398 0.674366i \(-0.764417\pi\)
−0.738398 + 0.674366i \(0.764417\pi\)
\(420\) 0 0
\(421\) 26.9510 1.31351 0.656755 0.754104i \(-0.271928\pi\)
0.656755 + 0.754104i \(0.271928\pi\)
\(422\) 0 0
\(423\) −1.95289 −0.0949529
\(424\) 0 0
\(425\) 11.9249 0.578443
\(426\) 0 0
\(427\) −7.75942 −0.375505
\(428\) 0 0
\(429\) 4.21419 0.203463
\(430\) 0 0
\(431\) −16.7713 −0.807847 −0.403923 0.914793i \(-0.632354\pi\)
−0.403923 + 0.914793i \(0.632354\pi\)
\(432\) 0 0
\(433\) 25.8530 1.24242 0.621208 0.783646i \(-0.286642\pi\)
0.621208 + 0.783646i \(0.286642\pi\)
\(434\) 0 0
\(435\) −3.04680 −0.146083
\(436\) 0 0
\(437\) 10.3257 0.493944
\(438\) 0 0
\(439\) −25.0553 −1.19582 −0.597912 0.801562i \(-0.704003\pi\)
−0.597912 + 0.801562i \(0.704003\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −4.64762 −0.220815 −0.110408 0.993886i \(-0.535216\pi\)
−0.110408 + 0.993886i \(0.535216\pi\)
\(444\) 0 0
\(445\) −5.23460 −0.248144
\(446\) 0 0
\(447\) 3.38663 0.160182
\(448\) 0 0
\(449\) 0.494696 0.0233462 0.0116731 0.999932i \(-0.496284\pi\)
0.0116731 + 0.999932i \(0.496284\pi\)
\(450\) 0 0
\(451\) 50.8561 2.39472
\(452\) 0 0
\(453\) −21.8951 −1.02872
\(454\) 0 0
\(455\) −0.926817 −0.0434499
\(456\) 0 0
\(457\) 26.4698 1.23821 0.619103 0.785310i \(-0.287496\pi\)
0.619103 + 0.785310i \(0.287496\pi\)
\(458\) 0 0
\(459\) 2.87971 0.134413
\(460\) 0 0
\(461\) −1.08168 −0.0503786 −0.0251893 0.999683i \(-0.508019\pi\)
−0.0251893 + 0.999683i \(0.508019\pi\)
\(462\) 0 0
\(463\) 32.7098 1.52015 0.760076 0.649834i \(-0.225162\pi\)
0.760076 + 0.649834i \(0.225162\pi\)
\(464\) 0 0
\(465\) −6.53109 −0.302872
\(466\) 0 0
\(467\) 17.0797 0.790356 0.395178 0.918605i \(-0.370683\pi\)
0.395178 + 0.918605i \(0.370683\pi\)
\(468\) 0 0
\(469\) 12.0414 0.556022
\(470\) 0 0
\(471\) 0.867482 0.0399715
\(472\) 0 0
\(473\) 30.0936 1.38370
\(474\) 0 0
\(475\) −5.33104 −0.244605
\(476\) 0 0
\(477\) −5.14101 −0.235391
\(478\) 0 0
\(479\) 17.1790 0.784929 0.392464 0.919767i \(-0.371623\pi\)
0.392464 + 0.919767i \(0.371623\pi\)
\(480\) 0 0
\(481\) 8.57475 0.390975
\(482\) 0 0
\(483\) 8.02072 0.364955
\(484\) 0 0
\(485\) 6.39544 0.290402
\(486\) 0 0
\(487\) 4.28264 0.194065 0.0970325 0.995281i \(-0.469065\pi\)
0.0970325 + 0.995281i \(0.469065\pi\)
\(488\) 0 0
\(489\) 4.42839 0.200259
\(490\) 0 0
\(491\) −27.4545 −1.23900 −0.619501 0.784996i \(-0.712665\pi\)
−0.619501 + 0.784996i \(0.712665\pi\)
\(492\) 0 0
\(493\) −9.46669 −0.426358
\(494\) 0 0
\(495\) −3.90579 −0.175552
\(496\) 0 0
\(497\) 10.7889 0.483950
\(498\) 0 0
\(499\) 37.0452 1.65837 0.829185 0.558974i \(-0.188805\pi\)
0.829185 + 0.558974i \(0.188805\pi\)
\(500\) 0 0
\(501\) 20.5276 0.917108
\(502\) 0 0
\(503\) −3.23744 −0.144350 −0.0721752 0.997392i \(-0.522994\pi\)
−0.0721752 + 0.997392i \(0.522994\pi\)
\(504\) 0 0
\(505\) −10.1708 −0.452596
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −30.4877 −1.35134 −0.675672 0.737202i \(-0.736147\pi\)
−0.675672 + 0.737202i \(0.736147\pi\)
\(510\) 0 0
\(511\) 8.32568 0.368306
\(512\) 0 0
\(513\) −1.28738 −0.0568390
\(514\) 0 0
\(515\) −15.2261 −0.670943
\(516\) 0 0
\(517\) 8.22987 0.361949
\(518\) 0 0
\(519\) −15.0154 −0.659101
\(520\) 0 0
\(521\) −25.3602 −1.11105 −0.555526 0.831499i \(-0.687483\pi\)
−0.555526 + 0.831499i \(0.687483\pi\)
\(522\) 0 0
\(523\) 7.00314 0.306226 0.153113 0.988209i \(-0.451070\pi\)
0.153113 + 0.988209i \(0.451070\pi\)
\(524\) 0 0
\(525\) −4.14101 −0.180728
\(526\) 0 0
\(527\) −20.2927 −0.883965
\(528\) 0 0
\(529\) 41.3320 1.79704
\(530\) 0 0
\(531\) 7.33448 0.318289
\(532\) 0 0
\(533\) −12.0678 −0.522716
\(534\) 0 0
\(535\) 3.20165 0.138420
\(536\) 0 0
\(537\) −5.35176 −0.230945
\(538\) 0 0
\(539\) −4.21419 −0.181518
\(540\) 0 0
\(541\) −1.42463 −0.0612495 −0.0306248 0.999531i \(-0.509750\pi\)
−0.0306248 + 0.999531i \(0.509750\pi\)
\(542\) 0 0
\(543\) −11.4667 −0.492083
\(544\) 0 0
\(545\) 4.32728 0.185360
\(546\) 0 0
\(547\) 22.0499 0.942787 0.471394 0.881923i \(-0.343751\pi\)
0.471394 + 0.881923i \(0.343751\pi\)
\(548\) 0 0
\(549\) 7.75942 0.331164
\(550\) 0 0
\(551\) 4.23209 0.180293
\(552\) 0 0
\(553\) 4.47204 0.190171
\(554\) 0 0
\(555\) −7.94723 −0.337341
\(556\) 0 0
\(557\) −9.08319 −0.384867 −0.192434 0.981310i \(-0.561638\pi\)
−0.192434 + 0.981310i \(0.561638\pi\)
\(558\) 0 0
\(559\) −7.14101 −0.302033
\(560\) 0 0
\(561\) −12.1357 −0.512368
\(562\) 0 0
\(563\) −34.1350 −1.43862 −0.719310 0.694689i \(-0.755542\pi\)
−0.719310 + 0.694689i \(0.755542\pi\)
\(564\) 0 0
\(565\) −3.04680 −0.128180
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 1.23522 0.0517833 0.0258916 0.999665i \(-0.491758\pi\)
0.0258916 + 0.999665i \(0.491758\pi\)
\(570\) 0 0
\(571\) 26.4613 1.10737 0.553686 0.832725i \(-0.313221\pi\)
0.553686 + 0.832725i \(0.313221\pi\)
\(572\) 0 0
\(573\) 20.1756 0.842847
\(574\) 0 0
\(575\) −33.2139 −1.38511
\(576\) 0 0
\(577\) −23.2852 −0.969374 −0.484687 0.874688i \(-0.661066\pi\)
−0.484687 + 0.874688i \(0.661066\pi\)
\(578\) 0 0
\(579\) 26.3756 1.09613
\(580\) 0 0
\(581\) −3.80653 −0.157921
\(582\) 0 0
\(583\) 21.6652 0.897281
\(584\) 0 0
\(585\) 0.926817 0.0383192
\(586\) 0 0
\(587\) 10.7155 0.442274 0.221137 0.975243i \(-0.429023\pi\)
0.221137 + 0.975243i \(0.429023\pi\)
\(588\) 0 0
\(589\) 9.07187 0.373800
\(590\) 0 0
\(591\) 20.1322 0.828128
\(592\) 0 0
\(593\) 14.1008 0.579049 0.289525 0.957171i \(-0.406503\pi\)
0.289525 + 0.957171i \(0.406503\pi\)
\(594\) 0 0
\(595\) 2.66896 0.109417
\(596\) 0 0
\(597\) 1.75942 0.0720083
\(598\) 0 0
\(599\) 38.9296 1.59062 0.795311 0.606202i \(-0.207308\pi\)
0.795311 + 0.606202i \(0.207308\pi\)
\(600\) 0 0
\(601\) −32.6162 −1.33044 −0.665221 0.746646i \(-0.731663\pi\)
−0.665221 + 0.746646i \(0.731663\pi\)
\(602\) 0 0
\(603\) −12.0414 −0.490365
\(604\) 0 0
\(605\) 6.26475 0.254698
\(606\) 0 0
\(607\) −19.3203 −0.784188 −0.392094 0.919925i \(-0.628249\pi\)
−0.392094 + 0.919925i \(0.628249\pi\)
\(608\) 0 0
\(609\) 3.28738 0.133211
\(610\) 0 0
\(611\) −1.95289 −0.0790056
\(612\) 0 0
\(613\) 35.3197 1.42655 0.713275 0.700885i \(-0.247211\pi\)
0.713275 + 0.700885i \(0.247211\pi\)
\(614\) 0 0
\(615\) 11.1847 0.451009
\(616\) 0 0
\(617\) −24.4318 −0.983589 −0.491794 0.870711i \(-0.663659\pi\)
−0.491794 + 0.870711i \(0.663659\pi\)
\(618\) 0 0
\(619\) −2.90892 −0.116919 −0.0584597 0.998290i \(-0.518619\pi\)
−0.0584597 + 0.998290i \(0.518619\pi\)
\(620\) 0 0
\(621\) −8.02072 −0.321860
\(622\) 0 0
\(623\) 5.64793 0.226280
\(624\) 0 0
\(625\) 12.8530 0.514121
\(626\) 0 0
\(627\) 5.42525 0.216664
\(628\) 0 0
\(629\) −24.6928 −0.984566
\(630\) 0 0
\(631\) −5.66521 −0.225528 −0.112764 0.993622i \(-0.535970\pi\)
−0.112764 + 0.993622i \(0.535970\pi\)
\(632\) 0 0
\(633\) 15.5694 0.618828
\(634\) 0 0
\(635\) 0.397035 0.0157559
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −10.7889 −0.426804
\(640\) 0 0
\(641\) 12.2842 0.485198 0.242599 0.970127i \(-0.422000\pi\)
0.242599 + 0.970127i \(0.422000\pi\)
\(642\) 0 0
\(643\) −9.14950 −0.360821 −0.180411 0.983591i \(-0.557743\pi\)
−0.180411 + 0.983591i \(0.557743\pi\)
\(644\) 0 0
\(645\) 6.61841 0.260600
\(646\) 0 0
\(647\) −38.4108 −1.51008 −0.755042 0.655677i \(-0.772383\pi\)
−0.755042 + 0.655677i \(0.772383\pi\)
\(648\) 0 0
\(649\) −30.9089 −1.21328
\(650\) 0 0
\(651\) 7.04680 0.276186
\(652\) 0 0
\(653\) 28.4805 1.11453 0.557265 0.830335i \(-0.311851\pi\)
0.557265 + 0.830335i \(0.311851\pi\)
\(654\) 0 0
\(655\) −13.1495 −0.513794
\(656\) 0 0
\(657\) −8.32568 −0.324816
\(658\) 0 0
\(659\) 30.4384 1.18571 0.592856 0.805309i \(-0.298000\pi\)
0.592856 + 0.805309i \(0.298000\pi\)
\(660\) 0 0
\(661\) 36.7710 1.43023 0.715114 0.699008i \(-0.246375\pi\)
0.715114 + 0.699008i \(0.246375\pi\)
\(662\) 0 0
\(663\) 2.87971 0.111839
\(664\) 0 0
\(665\) −1.19316 −0.0462688
\(666\) 0 0
\(667\) 26.3671 1.02094
\(668\) 0 0
\(669\) −11.5694 −0.447299
\(670\) 0 0
\(671\) −32.6997 −1.26236
\(672\) 0 0
\(673\) 26.7961 1.03291 0.516457 0.856313i \(-0.327250\pi\)
0.516457 + 0.856313i \(0.327250\pi\)
\(674\) 0 0
\(675\) 4.14101 0.159388
\(676\) 0 0
\(677\) −20.3050 −0.780383 −0.390191 0.920734i \(-0.627591\pi\)
−0.390191 + 0.920734i \(0.627591\pi\)
\(678\) 0 0
\(679\) −6.90043 −0.264814
\(680\) 0 0
\(681\) −11.0418 −0.423121
\(682\) 0 0
\(683\) −44.9240 −1.71897 −0.859484 0.511162i \(-0.829215\pi\)
−0.859484 + 0.511162i \(0.829215\pi\)
\(684\) 0 0
\(685\) 13.6307 0.520801
\(686\) 0 0
\(687\) 13.7073 0.522965
\(688\) 0 0
\(689\) −5.14101 −0.195857
\(690\) 0 0
\(691\) −22.0085 −0.837243 −0.418621 0.908161i \(-0.637487\pi\)
−0.418621 + 0.908161i \(0.637487\pi\)
\(692\) 0 0
\(693\) 4.21419 0.160084
\(694\) 0 0
\(695\) −9.13252 −0.346416
\(696\) 0 0
\(697\) 34.7518 1.31632
\(698\) 0 0
\(699\) −1.23522 −0.0467205
\(700\) 0 0
\(701\) 31.4645 1.18840 0.594198 0.804319i \(-0.297469\pi\)
0.594198 + 0.804319i \(0.297469\pi\)
\(702\) 0 0
\(703\) 11.0389 0.416341
\(704\) 0 0
\(705\) 1.80997 0.0681676
\(706\) 0 0
\(707\) 10.9739 0.412717
\(708\) 0 0
\(709\) −25.3373 −0.951563 −0.475781 0.879564i \(-0.657835\pi\)
−0.475781 + 0.879564i \(0.657835\pi\)
\(710\) 0 0
\(711\) −4.47204 −0.167715
\(712\) 0 0
\(713\) 56.5204 2.11670
\(714\) 0 0
\(715\) −3.90579 −0.146068
\(716\) 0 0
\(717\) 19.1061 0.713532
\(718\) 0 0
\(719\) 43.8002 1.63347 0.816737 0.577011i \(-0.195781\pi\)
0.816737 + 0.577011i \(0.195781\pi\)
\(720\) 0 0
\(721\) 16.4284 0.611825
\(722\) 0 0
\(723\) 12.0329 0.447510
\(724\) 0 0
\(725\) −13.6131 −0.505576
\(726\) 0 0
\(727\) −0.944090 −0.0350144 −0.0175072 0.999847i \(-0.505573\pi\)
−0.0175072 + 0.999847i \(0.505573\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 20.5640 0.760588
\(732\) 0 0
\(733\) −5.09957 −0.188357 −0.0941784 0.995555i \(-0.530022\pi\)
−0.0941784 + 0.995555i \(0.530022\pi\)
\(734\) 0 0
\(735\) −0.926817 −0.0341862
\(736\) 0 0
\(737\) 50.7450 1.86921
\(738\) 0 0
\(739\) 8.25129 0.303529 0.151764 0.988417i \(-0.451505\pi\)
0.151764 + 0.988417i \(0.451505\pi\)
\(740\) 0 0
\(741\) −1.28738 −0.0472929
\(742\) 0 0
\(743\) −38.4962 −1.41229 −0.706145 0.708068i \(-0.749567\pi\)
−0.706145 + 0.708068i \(0.749567\pi\)
\(744\) 0 0
\(745\) −3.13879 −0.114996
\(746\) 0 0
\(747\) 3.80653 0.139274
\(748\) 0 0
\(749\) −3.45446 −0.126223
\(750\) 0 0
\(751\) 41.3175 1.50770 0.753848 0.657049i \(-0.228195\pi\)
0.753848 + 0.657049i \(0.228195\pi\)
\(752\) 0 0
\(753\) −15.0031 −0.546745
\(754\) 0 0
\(755\) 20.2927 0.738528
\(756\) 0 0
\(757\) −39.7158 −1.44349 −0.721747 0.692157i \(-0.756661\pi\)
−0.721747 + 0.692157i \(0.756661\pi\)
\(758\) 0 0
\(759\) 33.8009 1.22689
\(760\) 0 0
\(761\) 10.7390 0.389289 0.194644 0.980874i \(-0.437645\pi\)
0.194644 + 0.980874i \(0.437645\pi\)
\(762\) 0 0
\(763\) −4.66896 −0.169028
\(764\) 0 0
\(765\) −2.66896 −0.0964966
\(766\) 0 0
\(767\) 7.33448 0.264833
\(768\) 0 0
\(769\) −50.5549 −1.82306 −0.911529 0.411237i \(-0.865097\pi\)
−0.911529 + 0.411237i \(0.865097\pi\)
\(770\) 0 0
\(771\) 13.0261 0.469123
\(772\) 0 0
\(773\) 8.91770 0.320747 0.160374 0.987056i \(-0.448730\pi\)
0.160374 + 0.987056i \(0.448730\pi\)
\(774\) 0 0
\(775\) −29.1809 −1.04821
\(776\) 0 0
\(777\) 8.57475 0.307617
\(778\) 0 0
\(779\) −15.5358 −0.556629
\(780\) 0 0
\(781\) 45.4667 1.62693
\(782\) 0 0
\(783\) −3.28738 −0.117481
\(784\) 0 0
\(785\) −0.803998 −0.0286959
\(786\) 0 0
\(787\) −22.3671 −0.797302 −0.398651 0.917103i \(-0.630521\pi\)
−0.398651 + 0.917103i \(0.630521\pi\)
\(788\) 0 0
\(789\) −2.59547 −0.0924012
\(790\) 0 0
\(791\) 3.28738 0.116886
\(792\) 0 0
\(793\) 7.75942 0.275545
\(794\) 0 0
\(795\) 4.76478 0.168989
\(796\) 0 0
\(797\) 20.5518 0.727983 0.363991 0.931402i \(-0.381414\pi\)
0.363991 + 0.931402i \(0.381414\pi\)
\(798\) 0 0
\(799\) 5.62377 0.198955
\(800\) 0 0
\(801\) −5.64793 −0.199560
\(802\) 0 0
\(803\) 35.0860 1.23816
\(804\) 0 0
\(805\) −7.43374 −0.262005
\(806\) 0 0
\(807\) 23.3081 0.820484
\(808\) 0 0
\(809\) 7.83443 0.275444 0.137722 0.990471i \(-0.456022\pi\)
0.137722 + 0.990471i \(0.456022\pi\)
\(810\) 0 0
\(811\) 47.5502 1.66971 0.834857 0.550468i \(-0.185551\pi\)
0.834857 + 0.550468i \(0.185551\pi\)
\(812\) 0 0
\(813\) −6.05215 −0.212258
\(814\) 0 0
\(815\) −4.10430 −0.143767
\(816\) 0 0
\(817\) −9.19316 −0.321628
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) 4.42494 0.154431 0.0772157 0.997014i \(-0.475397\pi\)
0.0772157 + 0.997014i \(0.475397\pi\)
\(822\) 0 0
\(823\) 28.7525 1.00225 0.501124 0.865375i \(-0.332920\pi\)
0.501124 + 0.865375i \(0.332920\pi\)
\(824\) 0 0
\(825\) −17.4510 −0.607566
\(826\) 0 0
\(827\) 5.58667 0.194267 0.0971337 0.995271i \(-0.469033\pi\)
0.0971337 + 0.995271i \(0.469033\pi\)
\(828\) 0 0
\(829\) 53.4974 1.85804 0.929021 0.370027i \(-0.120652\pi\)
0.929021 + 0.370027i \(0.120652\pi\)
\(830\) 0 0
\(831\) −15.1932 −0.527045
\(832\) 0 0
\(833\) −2.87971 −0.0997760
\(834\) 0 0
\(835\) −19.0254 −0.658400
\(836\) 0 0
\(837\) −7.04680 −0.243573
\(838\) 0 0
\(839\) 43.0405 1.48592 0.742962 0.669334i \(-0.233420\pi\)
0.742962 + 0.669334i \(0.233420\pi\)
\(840\) 0 0
\(841\) −18.1932 −0.627350
\(842\) 0 0
\(843\) 7.57506 0.260899
\(844\) 0 0
\(845\) 0.926817 0.0318835
\(846\) 0 0
\(847\) −6.75942 −0.232256
\(848\) 0 0
\(849\) 19.0974 0.655419
\(850\) 0 0
\(851\) 68.7757 2.35760
\(852\) 0 0
\(853\) −30.7434 −1.05263 −0.526316 0.850289i \(-0.676427\pi\)
−0.526316 + 0.850289i \(0.676427\pi\)
\(854\) 0 0
\(855\) 1.19316 0.0408053
\(856\) 0 0
\(857\) −24.5449 −0.838438 −0.419219 0.907885i \(-0.637696\pi\)
−0.419219 + 0.907885i \(0.637696\pi\)
\(858\) 0 0
\(859\) 52.1243 1.77846 0.889229 0.457461i \(-0.151241\pi\)
0.889229 + 0.457461i \(0.151241\pi\)
\(860\) 0 0
\(861\) −12.0678 −0.411270
\(862\) 0 0
\(863\) −22.2563 −0.757612 −0.378806 0.925476i \(-0.623665\pi\)
−0.378806 + 0.925476i \(0.623665\pi\)
\(864\) 0 0
\(865\) 13.9165 0.473175
\(866\) 0 0
\(867\) 8.70727 0.295714
\(868\) 0 0
\(869\) 18.8461 0.639309
\(870\) 0 0
\(871\) −12.0414 −0.408009
\(872\) 0 0
\(873\) 6.90043 0.233544
\(874\) 0 0
\(875\) 8.47204 0.286407
\(876\) 0 0
\(877\) 41.0547 1.38632 0.693159 0.720785i \(-0.256218\pi\)
0.693159 + 0.720785i \(0.256218\pi\)
\(878\) 0 0
\(879\) 3.79430 0.127979
\(880\) 0 0
\(881\) −31.0568 −1.04633 −0.523165 0.852231i \(-0.675249\pi\)
−0.523165 + 0.852231i \(0.675249\pi\)
\(882\) 0 0
\(883\) 56.5357 1.90258 0.951289 0.308300i \(-0.0997600\pi\)
0.951289 + 0.308300i \(0.0997600\pi\)
\(884\) 0 0
\(885\) −6.79772 −0.228503
\(886\) 0 0
\(887\) −9.94785 −0.334016 −0.167008 0.985956i \(-0.553411\pi\)
−0.167008 + 0.985956i \(0.553411\pi\)
\(888\) 0 0
\(889\) −0.428386 −0.0143676
\(890\) 0 0
\(891\) −4.21419 −0.141181
\(892\) 0 0
\(893\) −2.51411 −0.0841314
\(894\) 0 0
\(895\) 4.96010 0.165798
\(896\) 0 0
\(897\) −8.02072 −0.267804
\(898\) 0 0
\(899\) 23.1655 0.772612
\(900\) 0 0
\(901\) 14.8046 0.493213
\(902\) 0 0
\(903\) −7.14101 −0.237638
\(904\) 0 0
\(905\) 10.6275 0.353271
\(906\) 0 0
\(907\) 31.2936 1.03909 0.519544 0.854444i \(-0.326102\pi\)
0.519544 + 0.854444i \(0.326102\pi\)
\(908\) 0 0
\(909\) −10.9739 −0.363982
\(910\) 0 0
\(911\) 9.39320 0.311210 0.155605 0.987819i \(-0.450267\pi\)
0.155605 + 0.987819i \(0.450267\pi\)
\(912\) 0 0
\(913\) −16.0414 −0.530894
\(914\) 0 0
\(915\) −7.19156 −0.237746
\(916\) 0 0
\(917\) 14.1878 0.468523
\(918\) 0 0
\(919\) −21.7066 −0.716036 −0.358018 0.933715i \(-0.616547\pi\)
−0.358018 + 0.933715i \(0.616547\pi\)
\(920\) 0 0
\(921\) −5.19316 −0.171120
\(922\) 0 0
\(923\) −10.7889 −0.355122
\(924\) 0 0
\(925\) −35.5081 −1.16750
\(926\) 0 0
\(927\) −16.4284 −0.539579
\(928\) 0 0
\(929\) 35.6787 1.17058 0.585289 0.810824i \(-0.300981\pi\)
0.585289 + 0.810824i \(0.300981\pi\)
\(930\) 0 0
\(931\) 1.28738 0.0421920
\(932\) 0 0
\(933\) −13.8536 −0.453548
\(934\) 0 0
\(935\) 11.2475 0.367834
\(936\) 0 0
\(937\) −47.0866 −1.53825 −0.769127 0.639096i \(-0.779309\pi\)
−0.769127 + 0.639096i \(0.779309\pi\)
\(938\) 0 0
\(939\) 32.6162 1.06439
\(940\) 0 0
\(941\) 45.2923 1.47649 0.738244 0.674534i \(-0.235655\pi\)
0.738244 + 0.674534i \(0.235655\pi\)
\(942\) 0 0
\(943\) −96.7927 −3.15200
\(944\) 0 0
\(945\) 0.926817 0.0301494
\(946\) 0 0
\(947\) −4.77134 −0.155048 −0.0775238 0.996991i \(-0.524701\pi\)
−0.0775238 + 0.996991i \(0.524701\pi\)
\(948\) 0 0
\(949\) −8.32568 −0.270263
\(950\) 0 0
\(951\) −12.0380 −0.390359
\(952\) 0 0
\(953\) −3.95572 −0.128138 −0.0640692 0.997945i \(-0.520408\pi\)
−0.0640692 + 0.997945i \(0.520408\pi\)
\(954\) 0 0
\(955\) −18.6991 −0.605088
\(956\) 0 0
\(957\) 13.8536 0.447824
\(958\) 0 0
\(959\) −14.7070 −0.474912
\(960\) 0 0
\(961\) 18.6573 0.601850
\(962\) 0 0
\(963\) 3.45446 0.111318
\(964\) 0 0
\(965\) −24.4454 −0.786924
\(966\) 0 0
\(967\) −50.2292 −1.61526 −0.807632 0.589687i \(-0.799251\pi\)
−0.807632 + 0.589687i \(0.799251\pi\)
\(968\) 0 0
\(969\) 3.70727 0.119095
\(970\) 0 0
\(971\) −38.6576 −1.24058 −0.620291 0.784372i \(-0.712986\pi\)
−0.620291 + 0.784372i \(0.712986\pi\)
\(972\) 0 0
\(973\) 9.85363 0.315893
\(974\) 0 0
\(975\) 4.14101 0.132618
\(976\) 0 0
\(977\) −20.6134 −0.659480 −0.329740 0.944072i \(-0.606961\pi\)
−0.329740 + 0.944072i \(0.606961\pi\)
\(978\) 0 0
\(979\) 23.8015 0.760699
\(980\) 0 0
\(981\) 4.66896 0.149069
\(982\) 0 0
\(983\) 4.48620 0.143088 0.0715438 0.997437i \(-0.477207\pi\)
0.0715438 + 0.997437i \(0.477207\pi\)
\(984\) 0 0
\(985\) −18.6589 −0.594521
\(986\) 0 0
\(987\) −1.95289 −0.0621613
\(988\) 0 0
\(989\) −57.2760 −1.82127
\(990\) 0 0
\(991\) 32.7971 1.04183 0.520917 0.853607i \(-0.325590\pi\)
0.520917 + 0.853607i \(0.325590\pi\)
\(992\) 0 0
\(993\) −26.2820 −0.834035
\(994\) 0 0
\(995\) −1.63066 −0.0516954
\(996\) 0 0
\(997\) 0.491870 0.0155777 0.00778884 0.999970i \(-0.497521\pi\)
0.00778884 + 0.999970i \(0.497521\pi\)
\(998\) 0 0
\(999\) −8.57475 −0.271293
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 4368.2.a.br.1.3 4
4.3 odd 2 273.2.a.e.1.3 4
12.11 even 2 819.2.a.k.1.2 4
20.19 odd 2 6825.2.a.bg.1.2 4
28.27 even 2 1911.2.a.s.1.3 4
52.51 odd 2 3549.2.a.w.1.2 4
84.83 odd 2 5733.2.a.bf.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.e.1.3 4 4.3 odd 2
819.2.a.k.1.2 4 12.11 even 2
1911.2.a.s.1.3 4 28.27 even 2
3549.2.a.w.1.2 4 52.51 odd 2
4368.2.a.br.1.3 4 1.1 even 1 trivial
5733.2.a.bf.1.2 4 84.83 odd 2
6825.2.a.bg.1.2 4 20.19 odd 2