Properties

Label 6032.2.a.z.1.8
Level $6032$
Weight $2$
Character 6032.1
Self dual yes
Analytic conductor $48.166$
Analytic rank $1$
Dimension $10$
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] = [6032,2,Mod(1,6032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6032.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: \(48.1657624992\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} - 17x^{8} + 47x^{7} + 104x^{6} - 235x^{5} - 283x^{4} + 364x^{3} + 330x^{2} + 12x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3016)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-1.58994\) of defining polynomial
Character \(\chi\) \(=\) 6032.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.58994 q^{3} +3.10645 q^{5} -2.36044 q^{7} -0.472088 q^{9} +O(q^{10})\) \(q+1.58994 q^{3} +3.10645 q^{5} -2.36044 q^{7} -0.472088 q^{9} +0.816398 q^{11} -1.00000 q^{13} +4.93907 q^{15} -3.91558 q^{17} -3.83032 q^{19} -3.75296 q^{21} +0.187045 q^{23} +4.65003 q^{25} -5.52041 q^{27} -1.00000 q^{29} -5.86942 q^{31} +1.29802 q^{33} -7.33259 q^{35} -3.70681 q^{37} -1.58994 q^{39} +6.03084 q^{41} -7.79773 q^{43} -1.46652 q^{45} -12.8759 q^{47} -1.42832 q^{49} -6.22554 q^{51} +0.553149 q^{53} +2.53610 q^{55} -6.08998 q^{57} -1.22849 q^{59} +0.251767 q^{61} +1.11434 q^{63} -3.10645 q^{65} +12.9180 q^{67} +0.297391 q^{69} -4.57668 q^{71} +3.62147 q^{73} +7.39327 q^{75} -1.92706 q^{77} +7.07472 q^{79} -7.36087 q^{81} -12.6738 q^{83} -12.1635 q^{85} -1.58994 q^{87} +7.27356 q^{89} +2.36044 q^{91} -9.33202 q^{93} -11.8987 q^{95} -7.35022 q^{97} -0.385412 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9} - 14 q^{11} - 10 q^{13} - 7 q^{15} + 5 q^{17} - 11 q^{19} - 7 q^{23} + 10 q^{25} - 21 q^{27} - 10 q^{29} - 5 q^{31} + 5 q^{33} - 11 q^{35} + 8 q^{37} + 3 q^{39} + 14 q^{41} - 35 q^{43} + 7 q^{45} - 7 q^{49} - 20 q^{51} - 11 q^{53} - 8 q^{55} + 4 q^{57} - 23 q^{59} - 8 q^{61} - 43 q^{63} - 4 q^{65} - 27 q^{67} + 10 q^{69} - 3 q^{71} + 7 q^{73} - 23 q^{75} + 2 q^{77} - 9 q^{79} - 6 q^{81} - 48 q^{83} - 6 q^{85} + 3 q^{87} + 20 q^{89} - 3 q^{91} - 11 q^{93} - 11 q^{95} + q^{97} - 54 q^{99}+O(q^{100}) \) 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
\(3\) 1.58994 0.917953 0.458976 0.888448i \(-0.348216\pi\)
0.458976 + 0.888448i \(0.348216\pi\)
\(4\) 0 0
\(5\) 3.10645 1.38925 0.694623 0.719374i \(-0.255571\pi\)
0.694623 + 0.719374i \(0.255571\pi\)
\(6\) 0 0
\(7\) −2.36044 −0.892163 −0.446081 0.894992i \(-0.647181\pi\)
−0.446081 + 0.894992i \(0.647181\pi\)
\(8\) 0 0
\(9\) −0.472088 −0.157363
\(10\) 0 0
\(11\) 0.816398 0.246153 0.123077 0.992397i \(-0.460724\pi\)
0.123077 + 0.992397i \(0.460724\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 4.93907 1.27526
\(16\) 0 0
\(17\) −3.91558 −0.949667 −0.474834 0.880076i \(-0.657492\pi\)
−0.474834 + 0.880076i \(0.657492\pi\)
\(18\) 0 0
\(19\) −3.83032 −0.878736 −0.439368 0.898307i \(-0.644798\pi\)
−0.439368 + 0.898307i \(0.644798\pi\)
\(20\) 0 0
\(21\) −3.75296 −0.818963
\(22\) 0 0
\(23\) 0.187045 0.0390016 0.0195008 0.999810i \(-0.493792\pi\)
0.0195008 + 0.999810i \(0.493792\pi\)
\(24\) 0 0
\(25\) 4.65003 0.930006
\(26\) 0 0
\(27\) −5.52041 −1.06240
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −5.86942 −1.05418 −0.527089 0.849810i \(-0.676717\pi\)
−0.527089 + 0.849810i \(0.676717\pi\)
\(32\) 0 0
\(33\) 1.29802 0.225957
\(34\) 0 0
\(35\) −7.33259 −1.23943
\(36\) 0 0
\(37\) −3.70681 −0.609395 −0.304698 0.952449i \(-0.598555\pi\)
−0.304698 + 0.952449i \(0.598555\pi\)
\(38\) 0 0
\(39\) −1.58994 −0.254594
\(40\) 0 0
\(41\) 6.03084 0.941859 0.470929 0.882171i \(-0.343919\pi\)
0.470929 + 0.882171i \(0.343919\pi\)
\(42\) 0 0
\(43\) −7.79773 −1.18914 −0.594571 0.804043i \(-0.702678\pi\)
−0.594571 + 0.804043i \(0.702678\pi\)
\(44\) 0 0
\(45\) −1.46652 −0.218616
\(46\) 0 0
\(47\) −12.8759 −1.87815 −0.939075 0.343714i \(-0.888315\pi\)
−0.939075 + 0.343714i \(0.888315\pi\)
\(48\) 0 0
\(49\) −1.42832 −0.204046
\(50\) 0 0
\(51\) −6.22554 −0.871750
\(52\) 0 0
\(53\) 0.553149 0.0759808 0.0379904 0.999278i \(-0.487904\pi\)
0.0379904 + 0.999278i \(0.487904\pi\)
\(54\) 0 0
\(55\) 2.53610 0.341968
\(56\) 0 0
\(57\) −6.08998 −0.806638
\(58\) 0 0
\(59\) −1.22849 −0.159936 −0.0799679 0.996797i \(-0.525482\pi\)
−0.0799679 + 0.996797i \(0.525482\pi\)
\(60\) 0 0
\(61\) 0.251767 0.0322354 0.0161177 0.999870i \(-0.494869\pi\)
0.0161177 + 0.999870i \(0.494869\pi\)
\(62\) 0 0
\(63\) 1.11434 0.140393
\(64\) 0 0
\(65\) −3.10645 −0.385308
\(66\) 0 0
\(67\) 12.9180 1.57818 0.789091 0.614276i \(-0.210552\pi\)
0.789091 + 0.614276i \(0.210552\pi\)
\(68\) 0 0
\(69\) 0.297391 0.0358016
\(70\) 0 0
\(71\) −4.57668 −0.543151 −0.271576 0.962417i \(-0.587545\pi\)
−0.271576 + 0.962417i \(0.587545\pi\)
\(72\) 0 0
\(73\) 3.62147 0.423861 0.211931 0.977285i \(-0.432025\pi\)
0.211931 + 0.977285i \(0.432025\pi\)
\(74\) 0 0
\(75\) 7.39327 0.853701
\(76\) 0 0
\(77\) −1.92706 −0.219609
\(78\) 0 0
\(79\) 7.07472 0.795968 0.397984 0.917392i \(-0.369710\pi\)
0.397984 + 0.917392i \(0.369710\pi\)
\(80\) 0 0
\(81\) −7.36087 −0.817874
\(82\) 0 0
\(83\) −12.6738 −1.39113 −0.695564 0.718464i \(-0.744845\pi\)
−0.695564 + 0.718464i \(0.744845\pi\)
\(84\) 0 0
\(85\) −12.1635 −1.31932
\(86\) 0 0
\(87\) −1.58994 −0.170460
\(88\) 0 0
\(89\) 7.27356 0.770996 0.385498 0.922709i \(-0.374030\pi\)
0.385498 + 0.922709i \(0.374030\pi\)
\(90\) 0 0
\(91\) 2.36044 0.247441
\(92\) 0 0
\(93\) −9.33202 −0.967686
\(94\) 0 0
\(95\) −11.8987 −1.22078
\(96\) 0 0
\(97\) −7.35022 −0.746302 −0.373151 0.927771i \(-0.621723\pi\)
−0.373151 + 0.927771i \(0.621723\pi\)
\(98\) 0 0
\(99\) −0.385412 −0.0387353
\(100\) 0 0
\(101\) 10.5242 1.04720 0.523598 0.851965i \(-0.324589\pi\)
0.523598 + 0.851965i \(0.324589\pi\)
\(102\) 0 0
\(103\) −0.249989 −0.0246322 −0.0123161 0.999924i \(-0.503920\pi\)
−0.0123161 + 0.999924i \(0.503920\pi\)
\(104\) 0 0
\(105\) −11.6584 −1.13774
\(106\) 0 0
\(107\) 8.88422 0.858870 0.429435 0.903098i \(-0.358713\pi\)
0.429435 + 0.903098i \(0.358713\pi\)
\(108\) 0 0
\(109\) 6.40079 0.613085 0.306542 0.951857i \(-0.400828\pi\)
0.306542 + 0.951857i \(0.400828\pi\)
\(110\) 0 0
\(111\) −5.89360 −0.559396
\(112\) 0 0
\(113\) 3.52842 0.331925 0.165963 0.986132i \(-0.446927\pi\)
0.165963 + 0.986132i \(0.446927\pi\)
\(114\) 0 0
\(115\) 0.581046 0.0541828
\(116\) 0 0
\(117\) 0.472088 0.0436446
\(118\) 0 0
\(119\) 9.24249 0.847258
\(120\) 0 0
\(121\) −10.3335 −0.939409
\(122\) 0 0
\(123\) 9.58867 0.864582
\(124\) 0 0
\(125\) −1.08717 −0.0972391
\(126\) 0 0
\(127\) −13.4823 −1.19636 −0.598178 0.801363i \(-0.704108\pi\)
−0.598178 + 0.801363i \(0.704108\pi\)
\(128\) 0 0
\(129\) −12.3979 −1.09158
\(130\) 0 0
\(131\) 10.4241 0.910760 0.455380 0.890297i \(-0.349503\pi\)
0.455380 + 0.890297i \(0.349503\pi\)
\(132\) 0 0
\(133\) 9.04125 0.783975
\(134\) 0 0
\(135\) −17.1489 −1.47594
\(136\) 0 0
\(137\) −11.3018 −0.965575 −0.482788 0.875737i \(-0.660376\pi\)
−0.482788 + 0.875737i \(0.660376\pi\)
\(138\) 0 0
\(139\) −22.4425 −1.90355 −0.951774 0.306799i \(-0.900742\pi\)
−0.951774 + 0.306799i \(0.900742\pi\)
\(140\) 0 0
\(141\) −20.4720 −1.72405
\(142\) 0 0
\(143\) −0.816398 −0.0682706
\(144\) 0 0
\(145\) −3.10645 −0.257977
\(146\) 0 0
\(147\) −2.27095 −0.187304
\(148\) 0 0
\(149\) 17.7352 1.45293 0.726463 0.687206i \(-0.241163\pi\)
0.726463 + 0.687206i \(0.241163\pi\)
\(150\) 0 0
\(151\) 5.73675 0.466850 0.233425 0.972375i \(-0.425007\pi\)
0.233425 + 0.972375i \(0.425007\pi\)
\(152\) 0 0
\(153\) 1.84850 0.149442
\(154\) 0 0
\(155\) −18.2330 −1.46451
\(156\) 0 0
\(157\) 11.6950 0.933364 0.466682 0.884425i \(-0.345449\pi\)
0.466682 + 0.884425i \(0.345449\pi\)
\(158\) 0 0
\(159\) 0.879474 0.0697468
\(160\) 0 0
\(161\) −0.441509 −0.0347958
\(162\) 0 0
\(163\) 10.7803 0.844376 0.422188 0.906508i \(-0.361262\pi\)
0.422188 + 0.906508i \(0.361262\pi\)
\(164\) 0 0
\(165\) 4.03225 0.313910
\(166\) 0 0
\(167\) −22.0165 −1.70369 −0.851845 0.523794i \(-0.824516\pi\)
−0.851845 + 0.523794i \(0.824516\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 1.80825 0.138280
\(172\) 0 0
\(173\) −11.0552 −0.840510 −0.420255 0.907406i \(-0.638059\pi\)
−0.420255 + 0.907406i \(0.638059\pi\)
\(174\) 0 0
\(175\) −10.9761 −0.829716
\(176\) 0 0
\(177\) −1.95323 −0.146813
\(178\) 0 0
\(179\) −8.83680 −0.660493 −0.330247 0.943895i \(-0.607132\pi\)
−0.330247 + 0.943895i \(0.607132\pi\)
\(180\) 0 0
\(181\) 22.6553 1.68395 0.841976 0.539515i \(-0.181392\pi\)
0.841976 + 0.539515i \(0.181392\pi\)
\(182\) 0 0
\(183\) 0.400294 0.0295906
\(184\) 0 0
\(185\) −11.5150 −0.846600
\(186\) 0 0
\(187\) −3.19667 −0.233764
\(188\) 0 0
\(189\) 13.0306 0.947837
\(190\) 0 0
\(191\) 10.8841 0.787546 0.393773 0.919208i \(-0.371170\pi\)
0.393773 + 0.919208i \(0.371170\pi\)
\(192\) 0 0
\(193\) 17.1059 1.23131 0.615654 0.788017i \(-0.288892\pi\)
0.615654 + 0.788017i \(0.288892\pi\)
\(194\) 0 0
\(195\) −4.93907 −0.353694
\(196\) 0 0
\(197\) −6.58584 −0.469222 −0.234611 0.972089i \(-0.575382\pi\)
−0.234611 + 0.972089i \(0.575382\pi\)
\(198\) 0 0
\(199\) −10.2648 −0.727653 −0.363826 0.931467i \(-0.618530\pi\)
−0.363826 + 0.931467i \(0.618530\pi\)
\(200\) 0 0
\(201\) 20.5388 1.44870
\(202\) 0 0
\(203\) 2.36044 0.165670
\(204\) 0 0
\(205\) 18.7345 1.30847
\(206\) 0 0
\(207\) −0.0883018 −0.00613740
\(208\) 0 0
\(209\) −3.12707 −0.216304
\(210\) 0 0
\(211\) 3.19865 0.220204 0.110102 0.993920i \(-0.464882\pi\)
0.110102 + 0.993920i \(0.464882\pi\)
\(212\) 0 0
\(213\) −7.27664 −0.498587
\(214\) 0 0
\(215\) −24.2232 −1.65201
\(216\) 0 0
\(217\) 13.8544 0.940498
\(218\) 0 0
\(219\) 5.75793 0.389085
\(220\) 0 0
\(221\) 3.91558 0.263390
\(222\) 0 0
\(223\) 23.7923 1.59325 0.796624 0.604476i \(-0.206617\pi\)
0.796624 + 0.604476i \(0.206617\pi\)
\(224\) 0 0
\(225\) −2.19522 −0.146348
\(226\) 0 0
\(227\) −5.44160 −0.361171 −0.180586 0.983559i \(-0.557799\pi\)
−0.180586 + 0.983559i \(0.557799\pi\)
\(228\) 0 0
\(229\) 2.31309 0.152853 0.0764265 0.997075i \(-0.475649\pi\)
0.0764265 + 0.997075i \(0.475649\pi\)
\(230\) 0 0
\(231\) −3.06391 −0.201590
\(232\) 0 0
\(233\) −3.59167 −0.235298 −0.117649 0.993055i \(-0.537536\pi\)
−0.117649 + 0.993055i \(0.537536\pi\)
\(234\) 0 0
\(235\) −39.9985 −2.60921
\(236\) 0 0
\(237\) 11.2484 0.730661
\(238\) 0 0
\(239\) −4.14058 −0.267832 −0.133916 0.990993i \(-0.542755\pi\)
−0.133916 + 0.990993i \(0.542755\pi\)
\(240\) 0 0
\(241\) −23.4205 −1.50865 −0.754324 0.656502i \(-0.772035\pi\)
−0.754324 + 0.656502i \(0.772035\pi\)
\(242\) 0 0
\(243\) 4.85790 0.311634
\(244\) 0 0
\(245\) −4.43701 −0.283470
\(246\) 0 0
\(247\) 3.83032 0.243718
\(248\) 0 0
\(249\) −20.1505 −1.27699
\(250\) 0 0
\(251\) 10.9924 0.693835 0.346917 0.937896i \(-0.387228\pi\)
0.346917 + 0.937896i \(0.387228\pi\)
\(252\) 0 0
\(253\) 0.152703 0.00960037
\(254\) 0 0
\(255\) −19.3393 −1.21108
\(256\) 0 0
\(257\) −9.69758 −0.604918 −0.302459 0.953162i \(-0.597808\pi\)
−0.302459 + 0.953162i \(0.597808\pi\)
\(258\) 0 0
\(259\) 8.74969 0.543680
\(260\) 0 0
\(261\) 0.472088 0.0292215
\(262\) 0 0
\(263\) −28.5062 −1.75777 −0.878883 0.477037i \(-0.841711\pi\)
−0.878883 + 0.477037i \(0.841711\pi\)
\(264\) 0 0
\(265\) 1.71833 0.105556
\(266\) 0 0
\(267\) 11.5645 0.707738
\(268\) 0 0
\(269\) 27.1593 1.65593 0.827967 0.560777i \(-0.189498\pi\)
0.827967 + 0.560777i \(0.189498\pi\)
\(270\) 0 0
\(271\) −1.29120 −0.0784349 −0.0392174 0.999231i \(-0.512486\pi\)
−0.0392174 + 0.999231i \(0.512486\pi\)
\(272\) 0 0
\(273\) 3.75296 0.227140
\(274\) 0 0
\(275\) 3.79628 0.228924
\(276\) 0 0
\(277\) 28.1612 1.69204 0.846020 0.533151i \(-0.178992\pi\)
0.846020 + 0.533151i \(0.178992\pi\)
\(278\) 0 0
\(279\) 2.77088 0.165888
\(280\) 0 0
\(281\) −23.4881 −1.40118 −0.700591 0.713563i \(-0.747080\pi\)
−0.700591 + 0.713563i \(0.747080\pi\)
\(282\) 0 0
\(283\) 2.88422 0.171449 0.0857244 0.996319i \(-0.472680\pi\)
0.0857244 + 0.996319i \(0.472680\pi\)
\(284\) 0 0
\(285\) −18.9182 −1.12062
\(286\) 0 0
\(287\) −14.2354 −0.840291
\(288\) 0 0
\(289\) −1.66824 −0.0981320
\(290\) 0 0
\(291\) −11.6864 −0.685070
\(292\) 0 0
\(293\) −17.7155 −1.03495 −0.517475 0.855698i \(-0.673128\pi\)
−0.517475 + 0.855698i \(0.673128\pi\)
\(294\) 0 0
\(295\) −3.81624 −0.222190
\(296\) 0 0
\(297\) −4.50686 −0.261514
\(298\) 0 0
\(299\) −0.187045 −0.0108171
\(300\) 0 0
\(301\) 18.4061 1.06091
\(302\) 0 0
\(303\) 16.7328 0.961277
\(304\) 0 0
\(305\) 0.782100 0.0447829
\(306\) 0 0
\(307\) −21.5927 −1.23236 −0.616181 0.787604i \(-0.711321\pi\)
−0.616181 + 0.787604i \(0.711321\pi\)
\(308\) 0 0
\(309\) −0.397468 −0.0226112
\(310\) 0 0
\(311\) −29.4332 −1.66901 −0.834503 0.551004i \(-0.814245\pi\)
−0.834503 + 0.551004i \(0.814245\pi\)
\(312\) 0 0
\(313\) 9.18393 0.519106 0.259553 0.965729i \(-0.416425\pi\)
0.259553 + 0.965729i \(0.416425\pi\)
\(314\) 0 0
\(315\) 3.46163 0.195041
\(316\) 0 0
\(317\) 15.2241 0.855068 0.427534 0.903999i \(-0.359382\pi\)
0.427534 + 0.903999i \(0.359382\pi\)
\(318\) 0 0
\(319\) −0.816398 −0.0457095
\(320\) 0 0
\(321\) 14.1254 0.788402
\(322\) 0 0
\(323\) 14.9979 0.834507
\(324\) 0 0
\(325\) −4.65003 −0.257937
\(326\) 0 0
\(327\) 10.1769 0.562783
\(328\) 0 0
\(329\) 30.3929 1.67561
\(330\) 0 0
\(331\) 8.50024 0.467215 0.233608 0.972331i \(-0.424947\pi\)
0.233608 + 0.972331i \(0.424947\pi\)
\(332\) 0 0
\(333\) 1.74994 0.0958961
\(334\) 0 0
\(335\) 40.1290 2.19248
\(336\) 0 0
\(337\) −11.4735 −0.625000 −0.312500 0.949918i \(-0.601166\pi\)
−0.312500 + 0.949918i \(0.601166\pi\)
\(338\) 0 0
\(339\) 5.60997 0.304692
\(340\) 0 0
\(341\) −4.79178 −0.259489
\(342\) 0 0
\(343\) 19.8945 1.07420
\(344\) 0 0
\(345\) 0.923829 0.0497373
\(346\) 0 0
\(347\) −25.5955 −1.37404 −0.687020 0.726639i \(-0.741081\pi\)
−0.687020 + 0.726639i \(0.741081\pi\)
\(348\) 0 0
\(349\) 19.4461 1.04093 0.520464 0.853884i \(-0.325759\pi\)
0.520464 + 0.853884i \(0.325759\pi\)
\(350\) 0 0
\(351\) 5.52041 0.294658
\(352\) 0 0
\(353\) 3.02630 0.161074 0.0805368 0.996752i \(-0.474337\pi\)
0.0805368 + 0.996752i \(0.474337\pi\)
\(354\) 0 0
\(355\) −14.2172 −0.754571
\(356\) 0 0
\(357\) 14.6950 0.777743
\(358\) 0 0
\(359\) 25.2742 1.33392 0.666962 0.745092i \(-0.267594\pi\)
0.666962 + 0.745092i \(0.267594\pi\)
\(360\) 0 0
\(361\) −4.32864 −0.227823
\(362\) 0 0
\(363\) −16.4296 −0.862333
\(364\) 0 0
\(365\) 11.2499 0.588848
\(366\) 0 0
\(367\) 24.7174 1.29024 0.645120 0.764082i \(-0.276808\pi\)
0.645120 + 0.764082i \(0.276808\pi\)
\(368\) 0 0
\(369\) −2.84709 −0.148213
\(370\) 0 0
\(371\) −1.30567 −0.0677873
\(372\) 0 0
\(373\) −27.7618 −1.43745 −0.718726 0.695293i \(-0.755274\pi\)
−0.718726 + 0.695293i \(0.755274\pi\)
\(374\) 0 0
\(375\) −1.72853 −0.0892609
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) −11.8263 −0.607474 −0.303737 0.952756i \(-0.598234\pi\)
−0.303737 + 0.952756i \(0.598234\pi\)
\(380\) 0 0
\(381\) −21.4360 −1.09820
\(382\) 0 0
\(383\) 8.17709 0.417830 0.208915 0.977934i \(-0.433007\pi\)
0.208915 + 0.977934i \(0.433007\pi\)
\(384\) 0 0
\(385\) −5.98631 −0.305091
\(386\) 0 0
\(387\) 3.68121 0.187127
\(388\) 0 0
\(389\) 21.6722 1.09882 0.549412 0.835552i \(-0.314852\pi\)
0.549412 + 0.835552i \(0.314852\pi\)
\(390\) 0 0
\(391\) −0.732390 −0.0370386
\(392\) 0 0
\(393\) 16.5738 0.836035
\(394\) 0 0
\(395\) 21.9773 1.10580
\(396\) 0 0
\(397\) −9.24134 −0.463809 −0.231905 0.972739i \(-0.574496\pi\)
−0.231905 + 0.972739i \(0.574496\pi\)
\(398\) 0 0
\(399\) 14.3750 0.719652
\(400\) 0 0
\(401\) −36.5336 −1.82440 −0.912200 0.409746i \(-0.865617\pi\)
−0.912200 + 0.409746i \(0.865617\pi\)
\(402\) 0 0
\(403\) 5.86942 0.292376
\(404\) 0 0
\(405\) −22.8662 −1.13623
\(406\) 0 0
\(407\) −3.02623 −0.150005
\(408\) 0 0
\(409\) 2.39512 0.118431 0.0592154 0.998245i \(-0.481140\pi\)
0.0592154 + 0.998245i \(0.481140\pi\)
\(410\) 0 0
\(411\) −17.9691 −0.886353
\(412\) 0 0
\(413\) 2.89978 0.142689
\(414\) 0 0
\(415\) −39.3704 −1.93262
\(416\) 0 0
\(417\) −35.6823 −1.74737
\(418\) 0 0
\(419\) 10.4109 0.508605 0.254302 0.967125i \(-0.418154\pi\)
0.254302 + 0.967125i \(0.418154\pi\)
\(420\) 0 0
\(421\) −6.38045 −0.310964 −0.155482 0.987839i \(-0.549693\pi\)
−0.155482 + 0.987839i \(0.549693\pi\)
\(422\) 0 0
\(423\) 6.07858 0.295551
\(424\) 0 0
\(425\) −18.2076 −0.883196
\(426\) 0 0
\(427\) −0.594280 −0.0287592
\(428\) 0 0
\(429\) −1.29802 −0.0626692
\(430\) 0 0
\(431\) 2.42218 0.116673 0.0583363 0.998297i \(-0.481420\pi\)
0.0583363 + 0.998297i \(0.481420\pi\)
\(432\) 0 0
\(433\) 15.8771 0.763006 0.381503 0.924368i \(-0.375407\pi\)
0.381503 + 0.924368i \(0.375407\pi\)
\(434\) 0 0
\(435\) −4.93907 −0.236810
\(436\) 0 0
\(437\) −0.716443 −0.0342721
\(438\) 0 0
\(439\) 22.9516 1.09542 0.547709 0.836669i \(-0.315500\pi\)
0.547709 + 0.836669i \(0.315500\pi\)
\(440\) 0 0
\(441\) 0.674293 0.0321092
\(442\) 0 0
\(443\) −2.76948 −0.131582 −0.0657911 0.997833i \(-0.520957\pi\)
−0.0657911 + 0.997833i \(0.520957\pi\)
\(444\) 0 0
\(445\) 22.5950 1.07110
\(446\) 0 0
\(447\) 28.1979 1.33372
\(448\) 0 0
\(449\) 27.1512 1.28135 0.640673 0.767814i \(-0.278656\pi\)
0.640673 + 0.767814i \(0.278656\pi\)
\(450\) 0 0
\(451\) 4.92356 0.231842
\(452\) 0 0
\(453\) 9.12109 0.428546
\(454\) 0 0
\(455\) 7.33259 0.343757
\(456\) 0 0
\(457\) −12.4511 −0.582440 −0.291220 0.956656i \(-0.594061\pi\)
−0.291220 + 0.956656i \(0.594061\pi\)
\(458\) 0 0
\(459\) 21.6156 1.00893
\(460\) 0 0
\(461\) 37.5424 1.74853 0.874263 0.485453i \(-0.161345\pi\)
0.874263 + 0.485453i \(0.161345\pi\)
\(462\) 0 0
\(463\) 21.0931 0.980279 0.490140 0.871644i \(-0.336946\pi\)
0.490140 + 0.871644i \(0.336946\pi\)
\(464\) 0 0
\(465\) −28.9895 −1.34435
\(466\) 0 0
\(467\) 7.35843 0.340508 0.170254 0.985400i \(-0.445541\pi\)
0.170254 + 0.985400i \(0.445541\pi\)
\(468\) 0 0
\(469\) −30.4921 −1.40800
\(470\) 0 0
\(471\) 18.5944 0.856784
\(472\) 0 0
\(473\) −6.36605 −0.292711
\(474\) 0 0
\(475\) −17.8111 −0.817230
\(476\) 0 0
\(477\) −0.261135 −0.0119565
\(478\) 0 0
\(479\) 12.9261 0.590607 0.295304 0.955403i \(-0.404579\pi\)
0.295304 + 0.955403i \(0.404579\pi\)
\(480\) 0 0
\(481\) 3.70681 0.169016
\(482\) 0 0
\(483\) −0.701973 −0.0319409
\(484\) 0 0
\(485\) −22.8331 −1.03680
\(486\) 0 0
\(487\) 4.88214 0.221231 0.110615 0.993863i \(-0.464718\pi\)
0.110615 + 0.993863i \(0.464718\pi\)
\(488\) 0 0
\(489\) 17.1400 0.775097
\(490\) 0 0
\(491\) 4.86864 0.219719 0.109859 0.993947i \(-0.464960\pi\)
0.109859 + 0.993947i \(0.464960\pi\)
\(492\) 0 0
\(493\) 3.91558 0.176349
\(494\) 0 0
\(495\) −1.19726 −0.0538129
\(496\) 0 0
\(497\) 10.8030 0.484579
\(498\) 0 0
\(499\) −3.00575 −0.134556 −0.0672780 0.997734i \(-0.521431\pi\)
−0.0672780 + 0.997734i \(0.521431\pi\)
\(500\) 0 0
\(501\) −35.0050 −1.56391
\(502\) 0 0
\(503\) −11.1121 −0.495463 −0.247732 0.968829i \(-0.579685\pi\)
−0.247732 + 0.968829i \(0.579685\pi\)
\(504\) 0 0
\(505\) 32.6929 1.45481
\(506\) 0 0
\(507\) 1.58994 0.0706118
\(508\) 0 0
\(509\) −6.51993 −0.288991 −0.144495 0.989505i \(-0.546156\pi\)
−0.144495 + 0.989505i \(0.546156\pi\)
\(510\) 0 0
\(511\) −8.54827 −0.378153
\(512\) 0 0
\(513\) 21.1450 0.933573
\(514\) 0 0
\(515\) −0.776580 −0.0342202
\(516\) 0 0
\(517\) −10.5119 −0.462313
\(518\) 0 0
\(519\) −17.5771 −0.771549
\(520\) 0 0
\(521\) −23.3042 −1.02098 −0.510488 0.859885i \(-0.670535\pi\)
−0.510488 + 0.859885i \(0.670535\pi\)
\(522\) 0 0
\(523\) 14.0635 0.614955 0.307477 0.951555i \(-0.400515\pi\)
0.307477 + 0.951555i \(0.400515\pi\)
\(524\) 0 0
\(525\) −17.4514 −0.761641
\(526\) 0 0
\(527\) 22.9822 1.00112
\(528\) 0 0
\(529\) −22.9650 −0.998479
\(530\) 0 0
\(531\) 0.579955 0.0251679
\(532\) 0 0
\(533\) −6.03084 −0.261225
\(534\) 0 0
\(535\) 27.5984 1.19318
\(536\) 0 0
\(537\) −14.0500 −0.606301
\(538\) 0 0
\(539\) −1.16608 −0.0502266
\(540\) 0 0
\(541\) −28.7461 −1.23589 −0.617945 0.786222i \(-0.712034\pi\)
−0.617945 + 0.786222i \(0.712034\pi\)
\(542\) 0 0
\(543\) 36.0205 1.54579
\(544\) 0 0
\(545\) 19.8837 0.851726
\(546\) 0 0
\(547\) 5.19168 0.221980 0.110990 0.993822i \(-0.464598\pi\)
0.110990 + 0.993822i \(0.464598\pi\)
\(548\) 0 0
\(549\) −0.118856 −0.00507265
\(550\) 0 0
\(551\) 3.83032 0.163177
\(552\) 0 0
\(553\) −16.6994 −0.710133
\(554\) 0 0
\(555\) −18.3082 −0.777139
\(556\) 0 0
\(557\) −23.8076 −1.00876 −0.504380 0.863482i \(-0.668279\pi\)
−0.504380 + 0.863482i \(0.668279\pi\)
\(558\) 0 0
\(559\) 7.79773 0.329809
\(560\) 0 0
\(561\) −5.08252 −0.214584
\(562\) 0 0
\(563\) 13.9175 0.586552 0.293276 0.956028i \(-0.405254\pi\)
0.293276 + 0.956028i \(0.405254\pi\)
\(564\) 0 0
\(565\) 10.9608 0.461126
\(566\) 0 0
\(567\) 17.3749 0.729677
\(568\) 0 0
\(569\) −8.89659 −0.372964 −0.186482 0.982458i \(-0.559709\pi\)
−0.186482 + 0.982458i \(0.559709\pi\)
\(570\) 0 0
\(571\) −20.6817 −0.865502 −0.432751 0.901513i \(-0.642457\pi\)
−0.432751 + 0.901513i \(0.642457\pi\)
\(572\) 0 0
\(573\) 17.3051 0.722930
\(574\) 0 0
\(575\) 0.869765 0.0362717
\(576\) 0 0
\(577\) −3.79935 −0.158169 −0.0790844 0.996868i \(-0.525200\pi\)
−0.0790844 + 0.996868i \(0.525200\pi\)
\(578\) 0 0
\(579\) 27.1973 1.13028
\(580\) 0 0
\(581\) 29.9157 1.24111
\(582\) 0 0
\(583\) 0.451590 0.0187029
\(584\) 0 0
\(585\) 1.46652 0.0606331
\(586\) 0 0
\(587\) −6.91359 −0.285354 −0.142677 0.989769i \(-0.545571\pi\)
−0.142677 + 0.989769i \(0.545571\pi\)
\(588\) 0 0
\(589\) 22.4818 0.926344
\(590\) 0 0
\(591\) −10.4711 −0.430724
\(592\) 0 0
\(593\) −21.6513 −0.889114 −0.444557 0.895751i \(-0.646639\pi\)
−0.444557 + 0.895751i \(0.646639\pi\)
\(594\) 0 0
\(595\) 28.7113 1.17705
\(596\) 0 0
\(597\) −16.3204 −0.667951
\(598\) 0 0
\(599\) −46.8921 −1.91596 −0.957979 0.286838i \(-0.907396\pi\)
−0.957979 + 0.286838i \(0.907396\pi\)
\(600\) 0 0
\(601\) −32.4501 −1.32367 −0.661833 0.749651i \(-0.730221\pi\)
−0.661833 + 0.749651i \(0.730221\pi\)
\(602\) 0 0
\(603\) −6.09842 −0.248347
\(604\) 0 0
\(605\) −32.1005 −1.30507
\(606\) 0 0
\(607\) −0.745772 −0.0302700 −0.0151350 0.999885i \(-0.504818\pi\)
−0.0151350 + 0.999885i \(0.504818\pi\)
\(608\) 0 0
\(609\) 3.75296 0.152078
\(610\) 0 0
\(611\) 12.8759 0.520905
\(612\) 0 0
\(613\) −15.6677 −0.632814 −0.316407 0.948623i \(-0.602477\pi\)
−0.316407 + 0.948623i \(0.602477\pi\)
\(614\) 0 0
\(615\) 29.7867 1.20112
\(616\) 0 0
\(617\) −30.7252 −1.23695 −0.618474 0.785805i \(-0.712249\pi\)
−0.618474 + 0.785805i \(0.712249\pi\)
\(618\) 0 0
\(619\) 4.49750 0.180770 0.0903849 0.995907i \(-0.471190\pi\)
0.0903849 + 0.995907i \(0.471190\pi\)
\(620\) 0 0
\(621\) −1.03257 −0.0414355
\(622\) 0 0
\(623\) −17.1688 −0.687854
\(624\) 0 0
\(625\) −26.6274 −1.06509
\(626\) 0 0
\(627\) −4.97185 −0.198557
\(628\) 0 0
\(629\) 14.5143 0.578723
\(630\) 0 0
\(631\) 19.5314 0.777534 0.388767 0.921336i \(-0.372901\pi\)
0.388767 + 0.921336i \(0.372901\pi\)
\(632\) 0 0
\(633\) 5.08567 0.202137
\(634\) 0 0
\(635\) −41.8819 −1.66203
\(636\) 0 0
\(637\) 1.42832 0.0565921
\(638\) 0 0
\(639\) 2.16059 0.0854718
\(640\) 0 0
\(641\) 40.6112 1.60405 0.802024 0.597292i \(-0.203757\pi\)
0.802024 + 0.597292i \(0.203757\pi\)
\(642\) 0 0
\(643\) 34.9483 1.37823 0.689113 0.724654i \(-0.258000\pi\)
0.689113 + 0.724654i \(0.258000\pi\)
\(644\) 0 0
\(645\) −38.5135 −1.51647
\(646\) 0 0
\(647\) −23.8896 −0.939198 −0.469599 0.882880i \(-0.655602\pi\)
−0.469599 + 0.882880i \(0.655602\pi\)
\(648\) 0 0
\(649\) −1.00294 −0.0393687
\(650\) 0 0
\(651\) 22.0277 0.863333
\(652\) 0 0
\(653\) −14.8680 −0.581830 −0.290915 0.956749i \(-0.593960\pi\)
−0.290915 + 0.956749i \(0.593960\pi\)
\(654\) 0 0
\(655\) 32.3820 1.26527
\(656\) 0 0
\(657\) −1.70965 −0.0666999
\(658\) 0 0
\(659\) −21.4462 −0.835427 −0.417714 0.908579i \(-0.637168\pi\)
−0.417714 + 0.908579i \(0.637168\pi\)
\(660\) 0 0
\(661\) 2.63100 0.102334 0.0511669 0.998690i \(-0.483706\pi\)
0.0511669 + 0.998690i \(0.483706\pi\)
\(662\) 0 0
\(663\) 6.22554 0.241780
\(664\) 0 0
\(665\) 28.0862 1.08914
\(666\) 0 0
\(667\) −0.187045 −0.00724242
\(668\) 0 0
\(669\) 37.8283 1.46253
\(670\) 0 0
\(671\) 0.205542 0.00793485
\(672\) 0 0
\(673\) −20.1979 −0.778573 −0.389287 0.921117i \(-0.627278\pi\)
−0.389287 + 0.921117i \(0.627278\pi\)
\(674\) 0 0
\(675\) −25.6701 −0.988042
\(676\) 0 0
\(677\) −1.10452 −0.0424502 −0.0212251 0.999775i \(-0.506757\pi\)
−0.0212251 + 0.999775i \(0.506757\pi\)
\(678\) 0 0
\(679\) 17.3498 0.665823
\(680\) 0 0
\(681\) −8.65182 −0.331538
\(682\) 0 0
\(683\) −17.5799 −0.672678 −0.336339 0.941741i \(-0.609189\pi\)
−0.336339 + 0.941741i \(0.609189\pi\)
\(684\) 0 0
\(685\) −35.1084 −1.34142
\(686\) 0 0
\(687\) 3.67767 0.140312
\(688\) 0 0
\(689\) −0.553149 −0.0210733
\(690\) 0 0
\(691\) 15.7339 0.598547 0.299273 0.954167i \(-0.403256\pi\)
0.299273 + 0.954167i \(0.403256\pi\)
\(692\) 0 0
\(693\) 0.909742 0.0345582
\(694\) 0 0
\(695\) −69.7165 −2.64450
\(696\) 0 0
\(697\) −23.6142 −0.894452
\(698\) 0 0
\(699\) −5.71054 −0.215992
\(700\) 0 0
\(701\) −48.7119 −1.83982 −0.919912 0.392126i \(-0.871740\pi\)
−0.919912 + 0.392126i \(0.871740\pi\)
\(702\) 0 0
\(703\) 14.1983 0.535497
\(704\) 0 0
\(705\) −63.5952 −2.39513
\(706\) 0 0
\(707\) −24.8417 −0.934269
\(708\) 0 0
\(709\) −12.0483 −0.452483 −0.226241 0.974071i \(-0.572644\pi\)
−0.226241 + 0.974071i \(0.572644\pi\)
\(710\) 0 0
\(711\) −3.33989 −0.125256
\(712\) 0 0
\(713\) −1.09785 −0.0411146
\(714\) 0 0
\(715\) −2.53610 −0.0948447
\(716\) 0 0
\(717\) −6.58328 −0.245857
\(718\) 0 0
\(719\) −39.8103 −1.48467 −0.742337 0.670027i \(-0.766283\pi\)
−0.742337 + 0.670027i \(0.766283\pi\)
\(720\) 0 0
\(721\) 0.590085 0.0219759
\(722\) 0 0
\(723\) −37.2372 −1.38487
\(724\) 0 0
\(725\) −4.65003 −0.172698
\(726\) 0 0
\(727\) −4.08086 −0.151351 −0.0756754 0.997133i \(-0.524111\pi\)
−0.0756754 + 0.997133i \(0.524111\pi\)
\(728\) 0 0
\(729\) 29.8064 1.10394
\(730\) 0 0
\(731\) 30.5326 1.12929
\(732\) 0 0
\(733\) 25.6048 0.945735 0.472867 0.881134i \(-0.343219\pi\)
0.472867 + 0.881134i \(0.343219\pi\)
\(734\) 0 0
\(735\) −7.05458 −0.260212
\(736\) 0 0
\(737\) 10.5462 0.388475
\(738\) 0 0
\(739\) −10.9915 −0.404328 −0.202164 0.979352i \(-0.564797\pi\)
−0.202164 + 0.979352i \(0.564797\pi\)
\(740\) 0 0
\(741\) 6.08998 0.223721
\(742\) 0 0
\(743\) 8.42606 0.309122 0.154561 0.987983i \(-0.450604\pi\)
0.154561 + 0.987983i \(0.450604\pi\)
\(744\) 0 0
\(745\) 55.0935 2.01847
\(746\) 0 0
\(747\) 5.98314 0.218912
\(748\) 0 0
\(749\) −20.9707 −0.766252
\(750\) 0 0
\(751\) 15.8856 0.579674 0.289837 0.957076i \(-0.406399\pi\)
0.289837 + 0.957076i \(0.406399\pi\)
\(752\) 0 0
\(753\) 17.4773 0.636907
\(754\) 0 0
\(755\) 17.8209 0.648570
\(756\) 0 0
\(757\) −32.8803 −1.19506 −0.597528 0.801848i \(-0.703850\pi\)
−0.597528 + 0.801848i \(0.703850\pi\)
\(758\) 0 0
\(759\) 0.242789 0.00881269
\(760\) 0 0
\(761\) 19.4632 0.705541 0.352770 0.935710i \(-0.385240\pi\)
0.352770 + 0.935710i \(0.385240\pi\)
\(762\) 0 0
\(763\) −15.1087 −0.546971
\(764\) 0 0
\(765\) 5.74227 0.207612
\(766\) 0 0
\(767\) 1.22849 0.0443582
\(768\) 0 0
\(769\) 6.22888 0.224619 0.112310 0.993673i \(-0.464175\pi\)
0.112310 + 0.993673i \(0.464175\pi\)
\(770\) 0 0
\(771\) −15.4186 −0.555286
\(772\) 0 0
\(773\) −13.4007 −0.481990 −0.240995 0.970526i \(-0.577474\pi\)
−0.240995 + 0.970526i \(0.577474\pi\)
\(774\) 0 0
\(775\) −27.2930 −0.980392
\(776\) 0 0
\(777\) 13.9115 0.499072
\(778\) 0 0
\(779\) −23.1000 −0.827645
\(780\) 0 0
\(781\) −3.73639 −0.133699
\(782\) 0 0
\(783\) 5.52041 0.197284
\(784\) 0 0
\(785\) 36.3300 1.29667
\(786\) 0 0
\(787\) −6.05665 −0.215896 −0.107948 0.994157i \(-0.534428\pi\)
−0.107948 + 0.994157i \(0.534428\pi\)
\(788\) 0 0
\(789\) −45.3231 −1.61355
\(790\) 0 0
\(791\) −8.32862 −0.296131
\(792\) 0 0
\(793\) −0.251767 −0.00894050
\(794\) 0 0
\(795\) 2.73204 0.0968955
\(796\) 0 0
\(797\) 29.0664 1.02959 0.514793 0.857314i \(-0.327869\pi\)
0.514793 + 0.857314i \(0.327869\pi\)
\(798\) 0 0
\(799\) 50.4168 1.78362
\(800\) 0 0
\(801\) −3.43376 −0.121326
\(802\) 0 0
\(803\) 2.95656 0.104335
\(804\) 0 0
\(805\) −1.37153 −0.0483399
\(806\) 0 0
\(807\) 43.1817 1.52007
\(808\) 0 0
\(809\) −40.9128 −1.43842 −0.719209 0.694794i \(-0.755495\pi\)
−0.719209 + 0.694794i \(0.755495\pi\)
\(810\) 0 0
\(811\) −19.8842 −0.698228 −0.349114 0.937080i \(-0.613517\pi\)
−0.349114 + 0.937080i \(0.613517\pi\)
\(812\) 0 0
\(813\) −2.05293 −0.0719995
\(814\) 0 0
\(815\) 33.4884 1.17305
\(816\) 0 0
\(817\) 29.8678 1.04494
\(818\) 0 0
\(819\) −1.11434 −0.0389380
\(820\) 0 0
\(821\) −47.7971 −1.66813 −0.834064 0.551667i \(-0.813992\pi\)
−0.834064 + 0.551667i \(0.813992\pi\)
\(822\) 0 0
\(823\) −6.16596 −0.214932 −0.107466 0.994209i \(-0.534274\pi\)
−0.107466 + 0.994209i \(0.534274\pi\)
\(824\) 0 0
\(825\) 6.03585 0.210141
\(826\) 0 0
\(827\) −1.87980 −0.0653671 −0.0326836 0.999466i \(-0.510405\pi\)
−0.0326836 + 0.999466i \(0.510405\pi\)
\(828\) 0 0
\(829\) 28.0363 0.973742 0.486871 0.873474i \(-0.338138\pi\)
0.486871 + 0.873474i \(0.338138\pi\)
\(830\) 0 0
\(831\) 44.7746 1.55321
\(832\) 0 0
\(833\) 5.59270 0.193776
\(834\) 0 0
\(835\) −68.3932 −2.36684
\(836\) 0 0
\(837\) 32.4016 1.11996
\(838\) 0 0
\(839\) −40.4393 −1.39612 −0.698059 0.716040i \(-0.745953\pi\)
−0.698059 + 0.716040i \(0.745953\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −37.3447 −1.28622
\(844\) 0 0
\(845\) 3.10645 0.106865
\(846\) 0 0
\(847\) 24.3916 0.838105
\(848\) 0 0
\(849\) 4.58573 0.157382
\(850\) 0 0
\(851\) −0.693340 −0.0237674
\(852\) 0 0
\(853\) −6.55069 −0.224291 −0.112146 0.993692i \(-0.535772\pi\)
−0.112146 + 0.993692i \(0.535772\pi\)
\(854\) 0 0
\(855\) 5.61723 0.192105
\(856\) 0 0
\(857\) −19.0739 −0.651553 −0.325777 0.945447i \(-0.605626\pi\)
−0.325777 + 0.945447i \(0.605626\pi\)
\(858\) 0 0
\(859\) −23.0570 −0.786695 −0.393348 0.919390i \(-0.628683\pi\)
−0.393348 + 0.919390i \(0.628683\pi\)
\(860\) 0 0
\(861\) −22.6335 −0.771347
\(862\) 0 0
\(863\) 49.2810 1.67754 0.838772 0.544484i \(-0.183274\pi\)
0.838772 + 0.544484i \(0.183274\pi\)
\(864\) 0 0
\(865\) −34.3424 −1.16768
\(866\) 0 0
\(867\) −2.65241 −0.0900805
\(868\) 0 0
\(869\) 5.77579 0.195930
\(870\) 0 0
\(871\) −12.9180 −0.437709
\(872\) 0 0
\(873\) 3.46995 0.117440
\(874\) 0 0
\(875\) 2.56619 0.0867531
\(876\) 0 0
\(877\) 36.6539 1.23771 0.618856 0.785504i \(-0.287596\pi\)
0.618856 + 0.785504i \(0.287596\pi\)
\(878\) 0 0
\(879\) −28.1666 −0.950036
\(880\) 0 0
\(881\) 30.6245 1.03177 0.515883 0.856659i \(-0.327464\pi\)
0.515883 + 0.856659i \(0.327464\pi\)
\(882\) 0 0
\(883\) −51.5865 −1.73603 −0.868013 0.496542i \(-0.834603\pi\)
−0.868013 + 0.496542i \(0.834603\pi\)
\(884\) 0 0
\(885\) −6.06760 −0.203960
\(886\) 0 0
\(887\) 9.52716 0.319891 0.159945 0.987126i \(-0.448868\pi\)
0.159945 + 0.987126i \(0.448868\pi\)
\(888\) 0 0
\(889\) 31.8240 1.06734
\(890\) 0 0
\(891\) −6.00940 −0.201322
\(892\) 0 0
\(893\) 49.3190 1.65040
\(894\) 0 0
\(895\) −27.4511 −0.917588
\(896\) 0 0
\(897\) −0.297391 −0.00992959
\(898\) 0 0
\(899\) 5.86942 0.195756
\(900\) 0 0
\(901\) −2.16590 −0.0721565
\(902\) 0 0
\(903\) 29.2646 0.973864
\(904\) 0 0
\(905\) 70.3774 2.33942
\(906\) 0 0
\(907\) 9.17351 0.304601 0.152301 0.988334i \(-0.451332\pi\)
0.152301 + 0.988334i \(0.451332\pi\)
\(908\) 0 0
\(909\) −4.96834 −0.164790
\(910\) 0 0
\(911\) 31.1004 1.03040 0.515201 0.857069i \(-0.327717\pi\)
0.515201 + 0.857069i \(0.327717\pi\)
\(912\) 0 0
\(913\) −10.3468 −0.342431
\(914\) 0 0
\(915\) 1.24349 0.0411086
\(916\) 0 0
\(917\) −24.6055 −0.812546
\(918\) 0 0
\(919\) 9.60301 0.316774 0.158387 0.987377i \(-0.449371\pi\)
0.158387 + 0.987377i \(0.449371\pi\)
\(920\) 0 0
\(921\) −34.3312 −1.13125
\(922\) 0 0
\(923\) 4.57668 0.150643
\(924\) 0 0
\(925\) −17.2368 −0.566741
\(926\) 0 0
\(927\) 0.118017 0.00387619
\(928\) 0 0
\(929\) −0.204374 −0.00670530 −0.00335265 0.999994i \(-0.501067\pi\)
−0.00335265 + 0.999994i \(0.501067\pi\)
\(930\) 0 0
\(931\) 5.47093 0.179302
\(932\) 0 0
\(933\) −46.7971 −1.53207
\(934\) 0 0
\(935\) −9.93030 −0.324755
\(936\) 0 0
\(937\) −4.79992 −0.156807 −0.0784033 0.996922i \(-0.524982\pi\)
−0.0784033 + 0.996922i \(0.524982\pi\)
\(938\) 0 0
\(939\) 14.6019 0.476515
\(940\) 0 0
\(941\) 60.2934 1.96551 0.982754 0.184916i \(-0.0592014\pi\)
0.982754 + 0.184916i \(0.0592014\pi\)
\(942\) 0 0
\(943\) 1.12804 0.0367340
\(944\) 0 0
\(945\) 40.4789 1.31678
\(946\) 0 0
\(947\) 36.5730 1.18846 0.594232 0.804294i \(-0.297456\pi\)
0.594232 + 0.804294i \(0.297456\pi\)
\(948\) 0 0
\(949\) −3.62147 −0.117558
\(950\) 0 0
\(951\) 24.2053 0.784912
\(952\) 0 0
\(953\) 5.90927 0.191420 0.0957100 0.995409i \(-0.469488\pi\)
0.0957100 + 0.995409i \(0.469488\pi\)
\(954\) 0 0
\(955\) 33.8109 1.09409
\(956\) 0 0
\(957\) −1.29802 −0.0419592
\(958\) 0 0
\(959\) 26.6772 0.861450
\(960\) 0 0
\(961\) 3.45005 0.111292
\(962\) 0 0
\(963\) −4.19414 −0.135154
\(964\) 0 0
\(965\) 53.1386 1.71059
\(966\) 0 0
\(967\) −4.57062 −0.146981 −0.0734906 0.997296i \(-0.523414\pi\)
−0.0734906 + 0.997296i \(0.523414\pi\)
\(968\) 0 0
\(969\) 23.8458 0.766038
\(970\) 0 0
\(971\) 29.0080 0.930912 0.465456 0.885071i \(-0.345890\pi\)
0.465456 + 0.885071i \(0.345890\pi\)
\(972\) 0 0
\(973\) 52.9742 1.69827
\(974\) 0 0
\(975\) −7.39327 −0.236774
\(976\) 0 0
\(977\) 14.8889 0.476338 0.238169 0.971224i \(-0.423453\pi\)
0.238169 + 0.971224i \(0.423453\pi\)
\(978\) 0 0
\(979\) 5.93812 0.189783
\(980\) 0 0
\(981\) −3.02174 −0.0964767
\(982\) 0 0
\(983\) 15.9148 0.507602 0.253801 0.967256i \(-0.418319\pi\)
0.253801 + 0.967256i \(0.418319\pi\)
\(984\) 0 0
\(985\) −20.4586 −0.651865
\(986\) 0 0
\(987\) 48.3229 1.53813
\(988\) 0 0
\(989\) −1.45853 −0.0463785
\(990\) 0 0
\(991\) −10.4040 −0.330495 −0.165247 0.986252i \(-0.552842\pi\)
−0.165247 + 0.986252i \(0.552842\pi\)
\(992\) 0 0
\(993\) 13.5149 0.428882
\(994\) 0 0
\(995\) −31.8871 −1.01089
\(996\) 0 0
\(997\) −24.4556 −0.774517 −0.387259 0.921971i \(-0.626578\pi\)
−0.387259 + 0.921971i \(0.626578\pi\)
\(998\) 0 0
\(999\) 20.4631 0.647424
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 6032.2.a.z.1.8 10
4.3 odd 2 3016.2.a.h.1.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3016.2.a.h.1.3 10 4.3 odd 2
6032.2.a.z.1.8 10 1.1 even 1 trivial