Properties

Label 3640.2.a.r.1.1
Level $3640$
Weight $2$
Character 3640.1
Self dual yes
Analytic conductor $29.066$
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] = [3640,2,Mod(1,3640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3640, 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("3640.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.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: \(29.0655463357\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.13676.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.48425\) of defining polynomial
Character \(\chi\) \(=\) 3640.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.17148 q^{3} -1.00000 q^{5} -1.00000 q^{7} +7.05826 q^{9} +O(q^{10})\) \(q-3.17148 q^{3} -1.00000 q^{5} -1.00000 q^{7} +7.05826 q^{9} +1.82047 q^{11} -1.00000 q^{13} +3.17148 q^{15} -3.53577 q^{17} -2.05826 q^{19} +3.17148 q^{21} -4.00000 q^{23} +1.00000 q^{25} -12.8707 q^{27} +1.53054 q^{29} +2.64899 q^{31} -5.77356 q^{33} +1.00000 q^{35} +10.5860 q^{37} +3.17148 q^{39} +2.23779 q^{41} +0.522485 q^{43} -7.05826 q^{45} -12.1165 q^{47} +1.00000 q^{49} +11.2136 q^{51} +12.8654 q^{53} -1.82047 q^{55} +6.52771 q^{57} +14.9289 q^{59} +0.226436 q^{61} -7.05826 q^{63} +1.00000 q^{65} +4.40121 q^{67} +12.6859 q^{69} -2.70202 q^{71} -3.65705 q^{73} -3.17148 q^{75} -1.82047 q^{77} -5.69919 q^{79} +19.6442 q^{81} +2.34295 q^{83} +3.53577 q^{85} -4.85408 q^{87} +0.765506 q^{89} +1.00000 q^{91} -8.40121 q^{93} +2.05826 q^{95} +7.41450 q^{97} +12.8493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - 4 q^{5} - 4 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - 4 q^{5} - 4 q^{7} + 11 q^{9} - 2 q^{11} - 4 q^{13} + q^{15} - 11 q^{17} + 9 q^{19} + q^{21} - 16 q^{23} + 4 q^{25} - 4 q^{27} + 13 q^{29} + 13 q^{31} - 12 q^{33} + 4 q^{35} + q^{37} + q^{39} + q^{41} - 12 q^{43} - 11 q^{45} - 14 q^{47} + 4 q^{49} - 26 q^{51} + 14 q^{53} + 2 q^{55} + 2 q^{57} - 5 q^{59} + 12 q^{61} - 11 q^{63} + 4 q^{65} - 23 q^{67} + 4 q^{69} - 6 q^{71} - 38 q^{73} - q^{75} + 2 q^{77} + 13 q^{79} + 20 q^{81} - 14 q^{83} + 11 q^{85} - q^{87} - 29 q^{89} + 4 q^{91} + 7 q^{93} - 9 q^{95} - 28 q^{99}+O(q^{100}) \) 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
\(3\) −3.17148 −1.83105 −0.915526 0.402258i \(-0.868225\pi\)
−0.915526 + 0.402258i \(0.868225\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 7.05826 2.35275
\(10\) 0 0
\(11\) 1.82047 0.548891 0.274446 0.961603i \(-0.411506\pi\)
0.274446 + 0.961603i \(0.411506\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.17148 0.818871
\(16\) 0 0
\(17\) −3.53577 −0.857551 −0.428775 0.903411i \(-0.641055\pi\)
−0.428775 + 0.903411i \(0.641055\pi\)
\(18\) 0 0
\(19\) −2.05826 −0.472197 −0.236098 0.971729i \(-0.575869\pi\)
−0.236098 + 0.971729i \(0.575869\pi\)
\(20\) 0 0
\(21\) 3.17148 0.692073
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −12.8707 −2.47696
\(28\) 0 0
\(29\) 1.53054 0.284215 0.142107 0.989851i \(-0.454612\pi\)
0.142107 + 0.989851i \(0.454612\pi\)
\(30\) 0 0
\(31\) 2.64899 0.475773 0.237886 0.971293i \(-0.423545\pi\)
0.237886 + 0.971293i \(0.423545\pi\)
\(32\) 0 0
\(33\) −5.77356 −1.00505
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 10.5860 1.74032 0.870162 0.492766i \(-0.164014\pi\)
0.870162 + 0.492766i \(0.164014\pi\)
\(38\) 0 0
\(39\) 3.17148 0.507843
\(40\) 0 0
\(41\) 2.23779 0.349484 0.174742 0.984614i \(-0.444091\pi\)
0.174742 + 0.984614i \(0.444091\pi\)
\(42\) 0 0
\(43\) 0.522485 0.0796782 0.0398391 0.999206i \(-0.487315\pi\)
0.0398391 + 0.999206i \(0.487315\pi\)
\(44\) 0 0
\(45\) −7.05826 −1.05218
\(46\) 0 0
\(47\) −12.1165 −1.76738 −0.883688 0.468077i \(-0.844947\pi\)
−0.883688 + 0.468077i \(0.844947\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 11.2136 1.57022
\(52\) 0 0
\(53\) 12.8654 1.76720 0.883602 0.468239i \(-0.155111\pi\)
0.883602 + 0.468239i \(0.155111\pi\)
\(54\) 0 0
\(55\) −1.82047 −0.245472
\(56\) 0 0
\(57\) 6.52771 0.864617
\(58\) 0 0
\(59\) 14.9289 1.94358 0.971790 0.235847i \(-0.0757865\pi\)
0.971790 + 0.235847i \(0.0757865\pi\)
\(60\) 0 0
\(61\) 0.226436 0.0289921 0.0144961 0.999895i \(-0.495386\pi\)
0.0144961 + 0.999895i \(0.495386\pi\)
\(62\) 0 0
\(63\) −7.05826 −0.889257
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 4.40121 0.537693 0.268847 0.963183i \(-0.413358\pi\)
0.268847 + 0.963183i \(0.413358\pi\)
\(68\) 0 0
\(69\) 12.6859 1.52720
\(70\) 0 0
\(71\) −2.70202 −0.320671 −0.160335 0.987063i \(-0.551258\pi\)
−0.160335 + 0.987063i \(0.551258\pi\)
\(72\) 0 0
\(73\) −3.65705 −0.428025 −0.214013 0.976831i \(-0.568653\pi\)
−0.214013 + 0.976831i \(0.568653\pi\)
\(74\) 0 0
\(75\) −3.17148 −0.366210
\(76\) 0 0
\(77\) −1.82047 −0.207461
\(78\) 0 0
\(79\) −5.69919 −0.641209 −0.320605 0.947213i \(-0.603886\pi\)
−0.320605 + 0.947213i \(0.603886\pi\)
\(80\) 0 0
\(81\) 19.6442 2.18269
\(82\) 0 0
\(83\) 2.34295 0.257172 0.128586 0.991698i \(-0.458956\pi\)
0.128586 + 0.991698i \(0.458956\pi\)
\(84\) 0 0
\(85\) 3.53577 0.383508
\(86\) 0 0
\(87\) −4.85408 −0.520412
\(88\) 0 0
\(89\) 0.765506 0.0811435 0.0405718 0.999177i \(-0.487082\pi\)
0.0405718 + 0.999177i \(0.487082\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −8.40121 −0.871165
\(94\) 0 0
\(95\) 2.05826 0.211173
\(96\) 0 0
\(97\) 7.41450 0.752828 0.376414 0.926452i \(-0.377157\pi\)
0.376414 + 0.926452i \(0.377157\pi\)
\(98\) 0 0
\(99\) 12.8493 1.29141
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 5.65229 0.556937 0.278468 0.960445i \(-0.410173\pi\)
0.278468 + 0.960445i \(0.410173\pi\)
\(104\) 0 0
\(105\) −3.17148 −0.309504
\(106\) 0 0
\(107\) −12.1634 −1.17588 −0.587941 0.808904i \(-0.700061\pi\)
−0.587941 + 0.808904i \(0.700061\pi\)
\(108\) 0 0
\(109\) 14.5225 1.39100 0.695501 0.718525i \(-0.255183\pi\)
0.695501 + 0.718525i \(0.255183\pi\)
\(110\) 0 0
\(111\) −33.5732 −3.18662
\(112\) 0 0
\(113\) 2.56939 0.241708 0.120854 0.992670i \(-0.461437\pi\)
0.120854 + 0.992670i \(0.461437\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) −7.05826 −0.652536
\(118\) 0 0
\(119\) 3.53577 0.324124
\(120\) 0 0
\(121\) −7.68590 −0.698718
\(122\) 0 0
\(123\) −7.09710 −0.639924
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −20.8024 −1.84592 −0.922958 0.384901i \(-0.874236\pi\)
−0.922958 + 0.384901i \(0.874236\pi\)
\(128\) 0 0
\(129\) −1.65705 −0.145895
\(130\) 0 0
\(131\) −15.6409 −1.36655 −0.683277 0.730159i \(-0.739446\pi\)
−0.683277 + 0.730159i \(0.739446\pi\)
\(132\) 0 0
\(133\) 2.05826 0.178474
\(134\) 0 0
\(135\) 12.8707 1.10773
\(136\) 0 0
\(137\) −22.2269 −1.89897 −0.949486 0.313808i \(-0.898395\pi\)
−0.949486 + 0.313808i \(0.898395\pi\)
\(138\) 0 0
\(139\) 1.77356 0.150432 0.0752159 0.997167i \(-0.476035\pi\)
0.0752159 + 0.997167i \(0.476035\pi\)
\(140\) 0 0
\(141\) 38.4272 3.23616
\(142\) 0 0
\(143\) −1.82047 −0.152235
\(144\) 0 0
\(145\) −1.53054 −0.127105
\(146\) 0 0
\(147\) −3.17148 −0.261579
\(148\) 0 0
\(149\) −1.47751 −0.121043 −0.0605214 0.998167i \(-0.519276\pi\)
−0.0605214 + 0.998167i \(0.519276\pi\)
\(150\) 0 0
\(151\) −10.4595 −0.851179 −0.425590 0.904916i \(-0.639933\pi\)
−0.425590 + 0.904916i \(0.639933\pi\)
\(152\) 0 0
\(153\) −24.9564 −2.01761
\(154\) 0 0
\(155\) −2.64899 −0.212772
\(156\) 0 0
\(157\) 9.64706 0.769919 0.384960 0.922933i \(-0.374215\pi\)
0.384960 + 0.922933i \(0.374215\pi\)
\(158\) 0 0
\(159\) −40.8024 −3.23584
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −6.81241 −0.533589 −0.266794 0.963753i \(-0.585964\pi\)
−0.266794 + 0.963753i \(0.585964\pi\)
\(164\) 0 0
\(165\) 5.77356 0.449471
\(166\) 0 0
\(167\) 9.41450 0.728516 0.364258 0.931298i \(-0.381323\pi\)
0.364258 + 0.931298i \(0.381323\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −14.5277 −1.11096
\(172\) 0 0
\(173\) 10.5860 0.804837 0.402418 0.915456i \(-0.368170\pi\)
0.402418 + 0.915456i \(0.368170\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −47.3467 −3.55880
\(178\) 0 0
\(179\) −14.6276 −1.09332 −0.546661 0.837354i \(-0.684101\pi\)
−0.546661 + 0.837354i \(0.684101\pi\)
\(180\) 0 0
\(181\) −15.6248 −1.16138 −0.580692 0.814124i \(-0.697218\pi\)
−0.580692 + 0.814124i \(0.697218\pi\)
\(182\) 0 0
\(183\) −0.718135 −0.0530861
\(184\) 0 0
\(185\) −10.5860 −0.778296
\(186\) 0 0
\(187\) −6.43676 −0.470702
\(188\) 0 0
\(189\) 12.8707 0.936203
\(190\) 0 0
\(191\) 4.94504 0.357811 0.178905 0.983866i \(-0.442744\pi\)
0.178905 + 0.983866i \(0.442744\pi\)
\(192\) 0 0
\(193\) −5.47275 −0.393938 −0.196969 0.980410i \(-0.563110\pi\)
−0.196969 + 0.980410i \(0.563110\pi\)
\(194\) 0 0
\(195\) −3.17148 −0.227114
\(196\) 0 0
\(197\) −11.9900 −0.854253 −0.427126 0.904192i \(-0.640474\pi\)
−0.427126 + 0.904192i \(0.640474\pi\)
\(198\) 0 0
\(199\) 13.7309 0.973356 0.486678 0.873582i \(-0.338209\pi\)
0.486678 + 0.873582i \(0.338209\pi\)
\(200\) 0 0
\(201\) −13.9583 −0.984545
\(202\) 0 0
\(203\) −1.53054 −0.107423
\(204\) 0 0
\(205\) −2.23779 −0.156294
\(206\) 0 0
\(207\) −28.2330 −1.96233
\(208\) 0 0
\(209\) −3.74699 −0.259185
\(210\) 0 0
\(211\) −13.2463 −0.911914 −0.455957 0.890002i \(-0.650703\pi\)
−0.455957 + 0.890002i \(0.650703\pi\)
\(212\) 0 0
\(213\) 8.56939 0.587165
\(214\) 0 0
\(215\) −0.522485 −0.0356332
\(216\) 0 0
\(217\) −2.64899 −0.179825
\(218\) 0 0
\(219\) 11.5982 0.783736
\(220\) 0 0
\(221\) 3.53577 0.237842
\(222\) 0 0
\(223\) −4.35907 −0.291905 −0.145952 0.989292i \(-0.546625\pi\)
−0.145952 + 0.989292i \(0.546625\pi\)
\(224\) 0 0
\(225\) 7.05826 0.470551
\(226\) 0 0
\(227\) 5.98388 0.397164 0.198582 0.980084i \(-0.436366\pi\)
0.198582 + 0.980084i \(0.436366\pi\)
\(228\) 0 0
\(229\) −24.3851 −1.61141 −0.805706 0.592315i \(-0.798214\pi\)
−0.805706 + 0.592315i \(0.798214\pi\)
\(230\) 0 0
\(231\) 5.77356 0.379873
\(232\) 0 0
\(233\) 2.84511 0.186389 0.0931947 0.995648i \(-0.470292\pi\)
0.0931947 + 0.995648i \(0.470292\pi\)
\(234\) 0 0
\(235\) 12.1165 0.790394
\(236\) 0 0
\(237\) 18.0748 1.17409
\(238\) 0 0
\(239\) 29.2885 1.89451 0.947257 0.320476i \(-0.103843\pi\)
0.947257 + 0.320476i \(0.103843\pi\)
\(240\) 0 0
\(241\) 1.99524 0.128525 0.0642623 0.997933i \(-0.479531\pi\)
0.0642623 + 0.997933i \(0.479531\pi\)
\(242\) 0 0
\(243\) −23.6892 −1.51966
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 2.05826 0.130964
\(248\) 0 0
\(249\) −7.43061 −0.470896
\(250\) 0 0
\(251\) −7.87396 −0.497000 −0.248500 0.968632i \(-0.579938\pi\)
−0.248500 + 0.968632i \(0.579938\pi\)
\(252\) 0 0
\(253\) −7.28186 −0.457807
\(254\) 0 0
\(255\) −11.2136 −0.702224
\(256\) 0 0
\(257\) −30.0269 −1.87303 −0.936514 0.350630i \(-0.885967\pi\)
−0.936514 + 0.350630i \(0.885967\pi\)
\(258\) 0 0
\(259\) −10.5860 −0.657780
\(260\) 0 0
\(261\) 10.8030 0.668687
\(262\) 0 0
\(263\) 13.7840 0.849959 0.424980 0.905203i \(-0.360281\pi\)
0.424980 + 0.905203i \(0.360281\pi\)
\(264\) 0 0
\(265\) −12.8654 −0.790317
\(266\) 0 0
\(267\) −2.42778 −0.148578
\(268\) 0 0
\(269\) −19.8901 −1.21272 −0.606360 0.795190i \(-0.707371\pi\)
−0.606360 + 0.795190i \(0.707371\pi\)
\(270\) 0 0
\(271\) −25.5779 −1.55375 −0.776874 0.629656i \(-0.783196\pi\)
−0.776874 + 0.629656i \(0.783196\pi\)
\(272\) 0 0
\(273\) −3.17148 −0.191946
\(274\) 0 0
\(275\) 1.82047 0.109778
\(276\) 0 0
\(277\) 18.1634 1.09133 0.545667 0.838002i \(-0.316276\pi\)
0.545667 + 0.838002i \(0.316276\pi\)
\(278\) 0 0
\(279\) 18.6973 1.11938
\(280\) 0 0
\(281\) 17.9573 1.07124 0.535622 0.844458i \(-0.320077\pi\)
0.535622 + 0.844458i \(0.320077\pi\)
\(282\) 0 0
\(283\) 6.42778 0.382092 0.191046 0.981581i \(-0.438812\pi\)
0.191046 + 0.981581i \(0.438812\pi\)
\(284\) 0 0
\(285\) −6.52771 −0.386668
\(286\) 0 0
\(287\) −2.23779 −0.132093
\(288\) 0 0
\(289\) −4.49831 −0.264607
\(290\) 0 0
\(291\) −23.5149 −1.37847
\(292\) 0 0
\(293\) −25.5982 −1.49547 −0.747733 0.664000i \(-0.768857\pi\)
−0.747733 + 0.664000i \(0.768857\pi\)
\(294\) 0 0
\(295\) −14.9289 −0.869196
\(296\) 0 0
\(297\) −23.4306 −1.35958
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −0.522485 −0.0301155
\(302\) 0 0
\(303\) 6.34295 0.364393
\(304\) 0 0
\(305\) −0.226436 −0.0129657
\(306\) 0 0
\(307\) −17.8796 −1.02044 −0.510222 0.860043i \(-0.670437\pi\)
−0.510222 + 0.860043i \(0.670437\pi\)
\(308\) 0 0
\(309\) −17.9261 −1.01978
\(310\) 0 0
\(311\) 15.6636 0.888204 0.444102 0.895976i \(-0.353523\pi\)
0.444102 + 0.895976i \(0.353523\pi\)
\(312\) 0 0
\(313\) 8.39598 0.474569 0.237284 0.971440i \(-0.423743\pi\)
0.237284 + 0.971440i \(0.423743\pi\)
\(314\) 0 0
\(315\) 7.05826 0.397688
\(316\) 0 0
\(317\) −23.0450 −1.29433 −0.647167 0.762348i \(-0.724046\pi\)
−0.647167 + 0.762348i \(0.724046\pi\)
\(318\) 0 0
\(319\) 2.78630 0.156003
\(320\) 0 0
\(321\) 38.5760 2.15310
\(322\) 0 0
\(323\) 7.27753 0.404933
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −46.0577 −2.54700
\(328\) 0 0
\(329\) 12.1165 0.668005
\(330\) 0 0
\(331\) 25.9474 1.42620 0.713100 0.701062i \(-0.247290\pi\)
0.713100 + 0.701062i \(0.247290\pi\)
\(332\) 0 0
\(333\) 74.7185 4.09455
\(334\) 0 0
\(335\) −4.40121 −0.240464
\(336\) 0 0
\(337\) 1.18806 0.0647178 0.0323589 0.999476i \(-0.489698\pi\)
0.0323589 + 0.999476i \(0.489698\pi\)
\(338\) 0 0
\(339\) −8.14875 −0.442579
\(340\) 0 0
\(341\) 4.82240 0.261147
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −12.6859 −0.682986
\(346\) 0 0
\(347\) −10.1795 −0.546466 −0.273233 0.961948i \(-0.588093\pi\)
−0.273233 + 0.961948i \(0.588093\pi\)
\(348\) 0 0
\(349\) −33.3723 −1.78638 −0.893188 0.449683i \(-0.851537\pi\)
−0.893188 + 0.449683i \(0.851537\pi\)
\(350\) 0 0
\(351\) 12.8707 0.686985
\(352\) 0 0
\(353\) −18.3268 −0.975439 −0.487720 0.873000i \(-0.662171\pi\)
−0.487720 + 0.873000i \(0.662171\pi\)
\(354\) 0 0
\(355\) 2.70202 0.143408
\(356\) 0 0
\(357\) −11.2136 −0.593488
\(358\) 0 0
\(359\) −5.29798 −0.279617 −0.139808 0.990179i \(-0.544649\pi\)
−0.139808 + 0.990179i \(0.544649\pi\)
\(360\) 0 0
\(361\) −14.7636 −0.777030
\(362\) 0 0
\(363\) 24.3757 1.27939
\(364\) 0 0
\(365\) 3.65705 0.191419
\(366\) 0 0
\(367\) −1.70395 −0.0889455 −0.0444727 0.999011i \(-0.514161\pi\)
−0.0444727 + 0.999011i \(0.514161\pi\)
\(368\) 0 0
\(369\) 15.7949 0.822250
\(370\) 0 0
\(371\) −12.8654 −0.667940
\(372\) 0 0
\(373\) 22.8920 1.18530 0.592652 0.805459i \(-0.298081\pi\)
0.592652 + 0.805459i \(0.298081\pi\)
\(374\) 0 0
\(375\) 3.17148 0.163774
\(376\) 0 0
\(377\) −1.53054 −0.0788270
\(378\) 0 0
\(379\) 9.20839 0.473003 0.236502 0.971631i \(-0.423999\pi\)
0.236502 + 0.971631i \(0.423999\pi\)
\(380\) 0 0
\(381\) 65.9744 3.37997
\(382\) 0 0
\(383\) 20.4434 1.04461 0.522303 0.852760i \(-0.325073\pi\)
0.522303 + 0.852760i \(0.325073\pi\)
\(384\) 0 0
\(385\) 1.82047 0.0927796
\(386\) 0 0
\(387\) 3.68783 0.187463
\(388\) 0 0
\(389\) 8.62764 0.437439 0.218719 0.975788i \(-0.429812\pi\)
0.218719 + 0.975788i \(0.429812\pi\)
\(390\) 0 0
\(391\) 14.1431 0.715247
\(392\) 0 0
\(393\) 49.6048 2.50223
\(394\) 0 0
\(395\) 5.69919 0.286757
\(396\) 0 0
\(397\) 11.8901 0.596746 0.298373 0.954449i \(-0.403556\pi\)
0.298373 + 0.954449i \(0.403556\pi\)
\(398\) 0 0
\(399\) −6.52771 −0.326794
\(400\) 0 0
\(401\) −27.2885 −1.36272 −0.681360 0.731948i \(-0.738611\pi\)
−0.681360 + 0.731948i \(0.738611\pi\)
\(402\) 0 0
\(403\) −2.64899 −0.131956
\(404\) 0 0
\(405\) −19.6442 −0.976130
\(406\) 0 0
\(407\) 19.2714 0.955248
\(408\) 0 0
\(409\) 34.8598 1.72371 0.861853 0.507159i \(-0.169304\pi\)
0.861853 + 0.507159i \(0.169304\pi\)
\(410\) 0 0
\(411\) 70.4921 3.47712
\(412\) 0 0
\(413\) −14.9289 −0.734604
\(414\) 0 0
\(415\) −2.34295 −0.115011
\(416\) 0 0
\(417\) −5.62482 −0.275448
\(418\) 0 0
\(419\) −7.54713 −0.368701 −0.184351 0.982861i \(-0.559018\pi\)
−0.184351 + 0.982861i \(0.559018\pi\)
\(420\) 0 0
\(421\) 29.4348 1.43457 0.717283 0.696782i \(-0.245386\pi\)
0.717283 + 0.696782i \(0.245386\pi\)
\(422\) 0 0
\(423\) −85.5215 −4.15820
\(424\) 0 0
\(425\) −3.53577 −0.171510
\(426\) 0 0
\(427\) −0.226436 −0.0109580
\(428\) 0 0
\(429\) 5.77356 0.278750
\(430\) 0 0
\(431\) −2.45947 −0.118468 −0.0592342 0.998244i \(-0.518866\pi\)
−0.0592342 + 0.998244i \(0.518866\pi\)
\(432\) 0 0
\(433\) 17.1501 0.824183 0.412091 0.911143i \(-0.364798\pi\)
0.412091 + 0.911143i \(0.364798\pi\)
\(434\) 0 0
\(435\) 4.85408 0.232735
\(436\) 0 0
\(437\) 8.23303 0.393839
\(438\) 0 0
\(439\) −2.13263 −0.101785 −0.0508925 0.998704i \(-0.516207\pi\)
−0.0508925 + 0.998704i \(0.516207\pi\)
\(440\) 0 0
\(441\) 7.05826 0.336108
\(442\) 0 0
\(443\) −31.4680 −1.49509 −0.747545 0.664211i \(-0.768767\pi\)
−0.747545 + 0.664211i \(0.768767\pi\)
\(444\) 0 0
\(445\) −0.765506 −0.0362885
\(446\) 0 0
\(447\) 4.68590 0.221636
\(448\) 0 0
\(449\) −7.61436 −0.359344 −0.179672 0.983727i \(-0.557504\pi\)
−0.179672 + 0.983727i \(0.557504\pi\)
\(450\) 0 0
\(451\) 4.07382 0.191829
\(452\) 0 0
\(453\) 33.1719 1.55855
\(454\) 0 0
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) −26.1738 −1.22436 −0.612178 0.790720i \(-0.709707\pi\)
−0.612178 + 0.790720i \(0.709707\pi\)
\(458\) 0 0
\(459\) 45.5078 2.12412
\(460\) 0 0
\(461\) −16.9394 −0.788946 −0.394473 0.918908i \(-0.629073\pi\)
−0.394473 + 0.918908i \(0.629073\pi\)
\(462\) 0 0
\(463\) −14.0687 −0.653829 −0.326914 0.945054i \(-0.606009\pi\)
−0.326914 + 0.945054i \(0.606009\pi\)
\(464\) 0 0
\(465\) 8.40121 0.389597
\(466\) 0 0
\(467\) −25.8991 −1.19847 −0.599233 0.800575i \(-0.704528\pi\)
−0.599233 + 0.800575i \(0.704528\pi\)
\(468\) 0 0
\(469\) −4.40121 −0.203229
\(470\) 0 0
\(471\) −30.5954 −1.40976
\(472\) 0 0
\(473\) 0.951166 0.0437347
\(474\) 0 0
\(475\) −2.05826 −0.0944393
\(476\) 0 0
\(477\) 90.8076 4.15779
\(478\) 0 0
\(479\) −31.3681 −1.43324 −0.716622 0.697462i \(-0.754313\pi\)
−0.716622 + 0.697462i \(0.754313\pi\)
\(480\) 0 0
\(481\) −10.5860 −0.482679
\(482\) 0 0
\(483\) −12.6859 −0.577229
\(484\) 0 0
\(485\) −7.41450 −0.336675
\(486\) 0 0
\(487\) −22.6864 −1.02802 −0.514009 0.857785i \(-0.671840\pi\)
−0.514009 + 0.857785i \(0.671840\pi\)
\(488\) 0 0
\(489\) 21.6054 0.977029
\(490\) 0 0
\(491\) −11.2819 −0.509143 −0.254572 0.967054i \(-0.581934\pi\)
−0.254572 + 0.967054i \(0.581934\pi\)
\(492\) 0 0
\(493\) −5.41165 −0.243729
\(494\) 0 0
\(495\) −12.8493 −0.577534
\(496\) 0 0
\(497\) 2.70202 0.121202
\(498\) 0 0
\(499\) −18.6656 −0.835586 −0.417793 0.908542i \(-0.637196\pi\)
−0.417793 + 0.908542i \(0.637196\pi\)
\(500\) 0 0
\(501\) −29.8578 −1.33395
\(502\) 0 0
\(503\) −9.93698 −0.443068 −0.221534 0.975153i \(-0.571106\pi\)
−0.221534 + 0.975153i \(0.571106\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) −3.17148 −0.140850
\(508\) 0 0
\(509\) −37.7891 −1.67497 −0.837487 0.546457i \(-0.815976\pi\)
−0.837487 + 0.546457i \(0.815976\pi\)
\(510\) 0 0
\(511\) 3.65705 0.161778
\(512\) 0 0
\(513\) 26.4911 1.16961
\(514\) 0 0
\(515\) −5.65229 −0.249070
\(516\) 0 0
\(517\) −22.0577 −0.970097
\(518\) 0 0
\(519\) −33.5732 −1.47370
\(520\) 0 0
\(521\) −33.3557 −1.46134 −0.730670 0.682731i \(-0.760792\pi\)
−0.730670 + 0.682731i \(0.760792\pi\)
\(522\) 0 0
\(523\) 13.0450 0.570417 0.285208 0.958466i \(-0.407937\pi\)
0.285208 + 0.958466i \(0.407937\pi\)
\(524\) 0 0
\(525\) 3.17148 0.138415
\(526\) 0 0
\(527\) −9.36623 −0.407999
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 105.372 4.57276
\(532\) 0 0
\(533\) −2.23779 −0.0969295
\(534\) 0 0
\(535\) 12.1634 0.525870
\(536\) 0 0
\(537\) 46.3912 2.00193
\(538\) 0 0
\(539\) 1.82047 0.0784130
\(540\) 0 0
\(541\) 3.20839 0.137939 0.0689697 0.997619i \(-0.478029\pi\)
0.0689697 + 0.997619i \(0.478029\pi\)
\(542\) 0 0
\(543\) 49.5537 2.12655
\(544\) 0 0
\(545\) −14.5225 −0.622075
\(546\) 0 0
\(547\) 39.9436 1.70786 0.853932 0.520385i \(-0.174211\pi\)
0.853932 + 0.520385i \(0.174211\pi\)
\(548\) 0 0
\(549\) 1.59824 0.0682113
\(550\) 0 0
\(551\) −3.15025 −0.134205
\(552\) 0 0
\(553\) 5.69919 0.242354
\(554\) 0 0
\(555\) 33.5732 1.42510
\(556\) 0 0
\(557\) −7.68873 −0.325782 −0.162891 0.986644i \(-0.552082\pi\)
−0.162891 + 0.986644i \(0.552082\pi\)
\(558\) 0 0
\(559\) −0.522485 −0.0220988
\(560\) 0 0
\(561\) 20.4140 0.861880
\(562\) 0 0
\(563\) −15.5666 −0.656052 −0.328026 0.944669i \(-0.606383\pi\)
−0.328026 + 0.944669i \(0.606383\pi\)
\(564\) 0 0
\(565\) −2.56939 −0.108095
\(566\) 0 0
\(567\) −19.6442 −0.824980
\(568\) 0 0
\(569\) −25.5305 −1.07030 −0.535148 0.844758i \(-0.679744\pi\)
−0.535148 + 0.844758i \(0.679744\pi\)
\(570\) 0 0
\(571\) 1.97776 0.0827666 0.0413833 0.999143i \(-0.486824\pi\)
0.0413833 + 0.999143i \(0.486824\pi\)
\(572\) 0 0
\(573\) −15.6831 −0.655170
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −43.1719 −1.79727 −0.898636 0.438695i \(-0.855441\pi\)
−0.898636 + 0.438695i \(0.855441\pi\)
\(578\) 0 0
\(579\) 17.3567 0.721320
\(580\) 0 0
\(581\) −2.34295 −0.0972020
\(582\) 0 0
\(583\) 23.4211 0.970003
\(584\) 0 0
\(585\) 7.05826 0.291823
\(586\) 0 0
\(587\) 11.3879 0.470030 0.235015 0.971992i \(-0.424486\pi\)
0.235015 + 0.971992i \(0.424486\pi\)
\(588\) 0 0
\(589\) −5.45231 −0.224658
\(590\) 0 0
\(591\) 38.0260 1.56418
\(592\) 0 0
\(593\) 47.1293 1.93537 0.967683 0.252169i \(-0.0811439\pi\)
0.967683 + 0.252169i \(0.0811439\pi\)
\(594\) 0 0
\(595\) −3.53577 −0.144953
\(596\) 0 0
\(597\) −43.5471 −1.78227
\(598\) 0 0
\(599\) −12.5921 −0.514499 −0.257250 0.966345i \(-0.582816\pi\)
−0.257250 + 0.966345i \(0.582816\pi\)
\(600\) 0 0
\(601\) 30.3373 1.23748 0.618742 0.785594i \(-0.287643\pi\)
0.618742 + 0.785594i \(0.287643\pi\)
\(602\) 0 0
\(603\) 31.0649 1.26506
\(604\) 0 0
\(605\) 7.68590 0.312476
\(606\) 0 0
\(607\) −27.9375 −1.13395 −0.566973 0.823736i \(-0.691886\pi\)
−0.566973 + 0.823736i \(0.691886\pi\)
\(608\) 0 0
\(609\) 4.85408 0.196697
\(610\) 0 0
\(611\) 12.1165 0.490182
\(612\) 0 0
\(613\) 8.23689 0.332685 0.166343 0.986068i \(-0.446804\pi\)
0.166343 + 0.986068i \(0.446804\pi\)
\(614\) 0 0
\(615\) 7.09710 0.286183
\(616\) 0 0
\(617\) −30.2752 −1.21883 −0.609416 0.792850i \(-0.708596\pi\)
−0.609416 + 0.792850i \(0.708596\pi\)
\(618\) 0 0
\(619\) −8.67978 −0.348870 −0.174435 0.984669i \(-0.555810\pi\)
−0.174435 + 0.984669i \(0.555810\pi\)
\(620\) 0 0
\(621\) 51.4827 2.06593
\(622\) 0 0
\(623\) −0.765506 −0.0306694
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 11.8835 0.474581
\(628\) 0 0
\(629\) −37.4296 −1.49242
\(630\) 0 0
\(631\) 17.8901 0.712193 0.356096 0.934449i \(-0.384107\pi\)
0.356096 + 0.934449i \(0.384107\pi\)
\(632\) 0 0
\(633\) 42.0104 1.66976
\(634\) 0 0
\(635\) 20.8024 0.825519
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −19.0715 −0.754459
\(640\) 0 0
\(641\) 32.0743 1.26686 0.633429 0.773801i \(-0.281647\pi\)
0.633429 + 0.773801i \(0.281647\pi\)
\(642\) 0 0
\(643\) 38.8801 1.53328 0.766641 0.642076i \(-0.221927\pi\)
0.766641 + 0.642076i \(0.221927\pi\)
\(644\) 0 0
\(645\) 1.65705 0.0652462
\(646\) 0 0
\(647\) 38.6395 1.51907 0.759537 0.650464i \(-0.225425\pi\)
0.759537 + 0.650464i \(0.225425\pi\)
\(648\) 0 0
\(649\) 27.1776 1.06681
\(650\) 0 0
\(651\) 8.40121 0.329269
\(652\) 0 0
\(653\) −50.2704 −1.96723 −0.983617 0.180273i \(-0.942302\pi\)
−0.983617 + 0.180273i \(0.942302\pi\)
\(654\) 0 0
\(655\) 15.6409 0.611142
\(656\) 0 0
\(657\) −25.8124 −1.00704
\(658\) 0 0
\(659\) 0.896772 0.0349333 0.0174666 0.999847i \(-0.494440\pi\)
0.0174666 + 0.999847i \(0.494440\pi\)
\(660\) 0 0
\(661\) −19.4566 −0.756776 −0.378388 0.925647i \(-0.623521\pi\)
−0.378388 + 0.925647i \(0.623521\pi\)
\(662\) 0 0
\(663\) −11.2136 −0.435501
\(664\) 0 0
\(665\) −2.05826 −0.0798158
\(666\) 0 0
\(667\) −6.12217 −0.237051
\(668\) 0 0
\(669\) 13.8247 0.534493
\(670\) 0 0
\(671\) 0.412218 0.0159135
\(672\) 0 0
\(673\) −23.1719 −0.893213 −0.446606 0.894731i \(-0.647368\pi\)
−0.446606 + 0.894731i \(0.647368\pi\)
\(674\) 0 0
\(675\) −12.8707 −0.495392
\(676\) 0 0
\(677\) 26.4661 1.01717 0.508587 0.861011i \(-0.330168\pi\)
0.508587 + 0.861011i \(0.330168\pi\)
\(678\) 0 0
\(679\) −7.41450 −0.284542
\(680\) 0 0
\(681\) −18.9777 −0.727229
\(682\) 0 0
\(683\) 1.89905 0.0726651 0.0363326 0.999340i \(-0.488432\pi\)
0.0363326 + 0.999340i \(0.488432\pi\)
\(684\) 0 0
\(685\) 22.2269 0.849246
\(686\) 0 0
\(687\) 77.3367 2.95058
\(688\) 0 0
\(689\) −12.8654 −0.490134
\(690\) 0 0
\(691\) 3.66704 0.139501 0.0697504 0.997564i \(-0.477780\pi\)
0.0697504 + 0.997564i \(0.477780\pi\)
\(692\) 0 0
\(693\) −12.8493 −0.488105
\(694\) 0 0
\(695\) −1.77356 −0.0672751
\(696\) 0 0
\(697\) −7.91232 −0.299701
\(698\) 0 0
\(699\) −9.02320 −0.341289
\(700\) 0 0
\(701\) 16.3851 0.618856 0.309428 0.950923i \(-0.399862\pi\)
0.309428 + 0.950923i \(0.399862\pi\)
\(702\) 0 0
\(703\) −21.7887 −0.821775
\(704\) 0 0
\(705\) −38.4272 −1.44725
\(706\) 0 0
\(707\) 2.00000 0.0752177
\(708\) 0 0
\(709\) 12.4325 0.466914 0.233457 0.972367i \(-0.424996\pi\)
0.233457 + 0.972367i \(0.424996\pi\)
\(710\) 0 0
\(711\) −40.2264 −1.50861
\(712\) 0 0
\(713\) −10.5960 −0.396822
\(714\) 0 0
\(715\) 1.82047 0.0680816
\(716\) 0 0
\(717\) −92.8876 −3.46895
\(718\) 0 0
\(719\) 35.2496 1.31459 0.657295 0.753634i \(-0.271701\pi\)
0.657295 + 0.753634i \(0.271701\pi\)
\(720\) 0 0
\(721\) −5.65229 −0.210502
\(722\) 0 0
\(723\) −6.32785 −0.235335
\(724\) 0 0
\(725\) 1.53054 0.0568429
\(726\) 0 0
\(727\) 27.2922 1.01221 0.506106 0.862471i \(-0.331085\pi\)
0.506106 + 0.862471i \(0.331085\pi\)
\(728\) 0 0
\(729\) 16.1970 0.599890
\(730\) 0 0
\(731\) −1.84739 −0.0683281
\(732\) 0 0
\(733\) 40.9123 1.51113 0.755566 0.655073i \(-0.227362\pi\)
0.755566 + 0.655073i \(0.227362\pi\)
\(734\) 0 0
\(735\) 3.17148 0.116982
\(736\) 0 0
\(737\) 8.01225 0.295135
\(738\) 0 0
\(739\) 6.26382 0.230418 0.115209 0.993341i \(-0.463246\pi\)
0.115209 + 0.993341i \(0.463246\pi\)
\(740\) 0 0
\(741\) −6.52771 −0.239802
\(742\) 0 0
\(743\) 33.4671 1.22779 0.613894 0.789388i \(-0.289602\pi\)
0.613894 + 0.789388i \(0.289602\pi\)
\(744\) 0 0
\(745\) 1.47751 0.0541320
\(746\) 0 0
\(747\) 16.5372 0.605063
\(748\) 0 0
\(749\) 12.1634 0.444442
\(750\) 0 0
\(751\) −15.7740 −0.575603 −0.287801 0.957690i \(-0.592924\pi\)
−0.287801 + 0.957690i \(0.592924\pi\)
\(752\) 0 0
\(753\) 24.9721 0.910033
\(754\) 0 0
\(755\) 10.4595 0.380659
\(756\) 0 0
\(757\) −1.95310 −0.0709866 −0.0354933 0.999370i \(-0.511300\pi\)
−0.0354933 + 0.999370i \(0.511300\pi\)
\(758\) 0 0
\(759\) 23.0943 0.838268
\(760\) 0 0
\(761\) −12.2472 −0.443962 −0.221981 0.975051i \(-0.571252\pi\)
−0.221981 + 0.975051i \(0.571252\pi\)
\(762\) 0 0
\(763\) −14.5225 −0.525749
\(764\) 0 0
\(765\) 24.9564 0.902300
\(766\) 0 0
\(767\) −14.9289 −0.539052
\(768\) 0 0
\(769\) −2.98582 −0.107671 −0.0538356 0.998550i \(-0.517145\pi\)
−0.0538356 + 0.998550i \(0.517145\pi\)
\(770\) 0 0
\(771\) 95.2297 3.42961
\(772\) 0 0
\(773\) 2.56939 0.0924144 0.0462072 0.998932i \(-0.485287\pi\)
0.0462072 + 0.998932i \(0.485287\pi\)
\(774\) 0 0
\(775\) 2.64899 0.0951546
\(776\) 0 0
\(777\) 33.5732 1.20443
\(778\) 0 0
\(779\) −4.60595 −0.165025
\(780\) 0 0
\(781\) −4.91893 −0.176013
\(782\) 0 0
\(783\) −19.6991 −0.703989
\(784\) 0 0
\(785\) −9.64706 −0.344318
\(786\) 0 0
\(787\) −19.4211 −0.692287 −0.346144 0.938182i \(-0.612509\pi\)
−0.346144 + 0.938182i \(0.612509\pi\)
\(788\) 0 0
\(789\) −43.7157 −1.55632
\(790\) 0 0
\(791\) −2.56939 −0.0913569
\(792\) 0 0
\(793\) −0.226436 −0.00804096
\(794\) 0 0
\(795\) 40.8024 1.44711
\(796\) 0 0
\(797\) −20.4960 −0.726007 −0.363003 0.931788i \(-0.618249\pi\)
−0.363003 + 0.931788i \(0.618249\pi\)
\(798\) 0 0
\(799\) 42.8412 1.51561
\(800\) 0 0
\(801\) 5.40314 0.190911
\(802\) 0 0
\(803\) −6.65753 −0.234939
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 63.0809 2.22055
\(808\) 0 0
\(809\) 28.2913 0.994669 0.497334 0.867559i \(-0.334312\pi\)
0.497334 + 0.867559i \(0.334312\pi\)
\(810\) 0 0
\(811\) 15.9062 0.558542 0.279271 0.960212i \(-0.409907\pi\)
0.279271 + 0.960212i \(0.409907\pi\)
\(812\) 0 0
\(813\) 81.1197 2.84499
\(814\) 0 0
\(815\) 6.81241 0.238628
\(816\) 0 0
\(817\) −1.07541 −0.0376238
\(818\) 0 0
\(819\) 7.05826 0.246635
\(820\) 0 0
\(821\) −49.9947 −1.74483 −0.872413 0.488769i \(-0.837446\pi\)
−0.872413 + 0.488769i \(0.837446\pi\)
\(822\) 0 0
\(823\) 2.69156 0.0938219 0.0469109 0.998899i \(-0.485062\pi\)
0.0469109 + 0.998899i \(0.485062\pi\)
\(824\) 0 0
\(825\) −5.77356 −0.201010
\(826\) 0 0
\(827\) 37.2780 1.29628 0.648142 0.761520i \(-0.275547\pi\)
0.648142 + 0.761520i \(0.275547\pi\)
\(828\) 0 0
\(829\) 38.8356 1.34882 0.674408 0.738359i \(-0.264399\pi\)
0.674408 + 0.738359i \(0.264399\pi\)
\(830\) 0 0
\(831\) −57.6048 −1.99829
\(832\) 0 0
\(833\) −3.53577 −0.122507
\(834\) 0 0
\(835\) −9.41450 −0.325802
\(836\) 0 0
\(837\) −34.0943 −1.17847
\(838\) 0 0
\(839\) 45.0397 1.55494 0.777471 0.628919i \(-0.216502\pi\)
0.777471 + 0.628919i \(0.216502\pi\)
\(840\) 0 0
\(841\) −26.6574 −0.919222
\(842\) 0 0
\(843\) −56.9512 −1.96150
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 7.68590 0.264091
\(848\) 0 0
\(849\) −20.3856 −0.699631
\(850\) 0 0
\(851\) −42.3439 −1.45153
\(852\) 0 0
\(853\) 28.6802 0.981993 0.490996 0.871162i \(-0.336633\pi\)
0.490996 + 0.871162i \(0.336633\pi\)
\(854\) 0 0
\(855\) 14.5277 0.496837
\(856\) 0 0
\(857\) −5.72619 −0.195603 −0.0978015 0.995206i \(-0.531181\pi\)
−0.0978015 + 0.995206i \(0.531181\pi\)
\(858\) 0 0
\(859\) 2.04929 0.0699208 0.0349604 0.999389i \(-0.488869\pi\)
0.0349604 + 0.999389i \(0.488869\pi\)
\(860\) 0 0
\(861\) 7.09710 0.241869
\(862\) 0 0
\(863\) −37.8052 −1.28691 −0.643453 0.765486i \(-0.722499\pi\)
−0.643453 + 0.765486i \(0.722499\pi\)
\(864\) 0 0
\(865\) −10.5860 −0.359934
\(866\) 0 0
\(867\) 14.2663 0.484508
\(868\) 0 0
\(869\) −10.3752 −0.351954
\(870\) 0 0
\(871\) −4.40121 −0.149129
\(872\) 0 0
\(873\) 52.3334 1.77122
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 23.9749 0.809575 0.404788 0.914411i \(-0.367345\pi\)
0.404788 + 0.914411i \(0.367345\pi\)
\(878\) 0 0
\(879\) 81.1842 2.73828
\(880\) 0 0
\(881\) 33.7357 1.13658 0.568292 0.822827i \(-0.307604\pi\)
0.568292 + 0.822827i \(0.307604\pi\)
\(882\) 0 0
\(883\) 3.96356 0.133384 0.0666922 0.997774i \(-0.478755\pi\)
0.0666922 + 0.997774i \(0.478755\pi\)
\(884\) 0 0
\(885\) 47.3467 1.59154
\(886\) 0 0
\(887\) −48.2965 −1.62164 −0.810819 0.585297i \(-0.800978\pi\)
−0.810819 + 0.585297i \(0.800978\pi\)
\(888\) 0 0
\(889\) 20.8024 0.697691
\(890\) 0 0
\(891\) 35.7617 1.19806
\(892\) 0 0
\(893\) 24.9389 0.834549
\(894\) 0 0
\(895\) 14.6276 0.488948
\(896\) 0 0
\(897\) −12.6859 −0.423570
\(898\) 0 0
\(899\) 4.05439 0.135222
\(900\) 0 0
\(901\) −45.4893 −1.51547
\(902\) 0 0
\(903\) 1.65705 0.0551431
\(904\) 0 0
\(905\) 15.6248 0.519386
\(906\) 0 0
\(907\) −7.56180 −0.251085 −0.125543 0.992088i \(-0.540067\pi\)
−0.125543 + 0.992088i \(0.540067\pi\)
\(908\) 0 0
\(909\) −14.1165 −0.468215
\(910\) 0 0
\(911\) −51.8118 −1.71660 −0.858301 0.513146i \(-0.828480\pi\)
−0.858301 + 0.513146i \(0.828480\pi\)
\(912\) 0 0
\(913\) 4.26526 0.141160
\(914\) 0 0
\(915\) 0.718135 0.0237408
\(916\) 0 0
\(917\) 15.6409 0.516509
\(918\) 0 0
\(919\) 10.6759 0.352166 0.176083 0.984375i \(-0.443657\pi\)
0.176083 + 0.984375i \(0.443657\pi\)
\(920\) 0 0
\(921\) 56.7048 1.86849
\(922\) 0 0
\(923\) 2.70202 0.0889380
\(924\) 0 0
\(925\) 10.5860 0.348065
\(926\) 0 0
\(927\) 39.8953 1.31033
\(928\) 0 0
\(929\) −17.4182 −0.571474 −0.285737 0.958308i \(-0.592238\pi\)
−0.285737 + 0.958308i \(0.592238\pi\)
\(930\) 0 0
\(931\) −2.05826 −0.0674567
\(932\) 0 0
\(933\) −49.6769 −1.62635
\(934\) 0 0
\(935\) 6.43676 0.210504
\(936\) 0 0
\(937\) −17.0336 −0.556464 −0.278232 0.960514i \(-0.589748\pi\)
−0.278232 + 0.960514i \(0.589748\pi\)
\(938\) 0 0
\(939\) −26.6276 −0.868960
\(940\) 0 0
\(941\) 49.0615 1.59936 0.799679 0.600428i \(-0.205003\pi\)
0.799679 + 0.600428i \(0.205003\pi\)
\(942\) 0 0
\(943\) −8.95117 −0.291490
\(944\) 0 0
\(945\) −12.8707 −0.418683
\(946\) 0 0
\(947\) 7.23972 0.235259 0.117630 0.993058i \(-0.462470\pi\)
0.117630 + 0.993058i \(0.462470\pi\)
\(948\) 0 0
\(949\) 3.65705 0.118713
\(950\) 0 0
\(951\) 73.0866 2.36999
\(952\) 0 0
\(953\) −28.5333 −0.924284 −0.462142 0.886806i \(-0.652919\pi\)
−0.462142 + 0.886806i \(0.652919\pi\)
\(954\) 0 0
\(955\) −4.94504 −0.160018
\(956\) 0 0
\(957\) −8.83669 −0.285650
\(958\) 0 0
\(959\) 22.2269 0.717744
\(960\) 0 0
\(961\) −23.9828 −0.773640
\(962\) 0 0
\(963\) −85.8525 −2.76656
\(964\) 0 0
\(965\) 5.47275 0.176174
\(966\) 0 0
\(967\) −7.92278 −0.254779 −0.127390 0.991853i \(-0.540660\pi\)
−0.127390 + 0.991853i \(0.540660\pi\)
\(968\) 0 0
\(969\) −23.0805 −0.741453
\(970\) 0 0
\(971\) −1.55279 −0.0498313 −0.0249156 0.999690i \(-0.507932\pi\)
−0.0249156 + 0.999690i \(0.507932\pi\)
\(972\) 0 0
\(973\) −1.77356 −0.0568579
\(974\) 0 0
\(975\) 3.17148 0.101569
\(976\) 0 0
\(977\) −15.0966 −0.482984 −0.241492 0.970403i \(-0.577637\pi\)
−0.241492 + 0.970403i \(0.577637\pi\)
\(978\) 0 0
\(979\) 1.39358 0.0445390
\(980\) 0 0
\(981\) 102.503 3.27268
\(982\) 0 0
\(983\) −46.9028 −1.49597 −0.747984 0.663717i \(-0.768978\pi\)
−0.747984 + 0.663717i \(0.768978\pi\)
\(984\) 0 0
\(985\) 11.9900 0.382033
\(986\) 0 0
\(987\) −38.4272 −1.22315
\(988\) 0 0
\(989\) −2.08994 −0.0664562
\(990\) 0 0
\(991\) −18.6910 −0.593739 −0.296870 0.954918i \(-0.595943\pi\)
−0.296870 + 0.954918i \(0.595943\pi\)
\(992\) 0 0
\(993\) −82.2917 −2.61145
\(994\) 0 0
\(995\) −13.7309 −0.435298
\(996\) 0 0
\(997\) −13.4405 −0.425666 −0.212833 0.977089i \(-0.568269\pi\)
−0.212833 + 0.977089i \(0.568269\pi\)
\(998\) 0 0
\(999\) −136.249 −4.31071
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 3640.2.a.r.1.1 4
4.3 odd 2 7280.2.a.bz.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3640.2.a.r.1.1 4 1.1 even 1 trivial
7280.2.a.bz.1.4 4 4.3 odd 2