Properties

Label 3720.2.a.r.1.2
Level $3720$
Weight $2$
Character 3720.1
Self dual yes
Analytic conductor $29.704$
Analytic rank $0$
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] = [3720,2,Mod(1,3720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3720, 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("3720.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3720.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: \(29.7043495519\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.78292.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.09502\) of defining polynomial
Character \(\chi\) \(=\) 3720.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.09502 q^{7} +1.00000 q^{9} +5.25711 q^{11} +1.24976 q^{13} +1.00000 q^{15} -2.64620 q^{17} +3.25711 q^{19} +1.09502 q^{21} -6.99097 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.89596 q^{29} +1.00000 q^{31} -5.25711 q^{33} +1.09502 q^{35} -3.24976 q^{37} -1.24976 q^{39} +4.00000 q^{41} -4.34477 q^{43} -1.00000 q^{45} +7.14571 q^{47} -5.80094 q^{49} +2.64620 q^{51} +6.80094 q^{53} -5.25711 q^{55} -3.25711 q^{57} -8.80829 q^{59} +2.91234 q^{61} -1.09502 q^{63} -1.24976 q^{65} +1.43979 q^{67} +6.99097 q^{69} -4.24073 q^{71} -8.69689 q^{73} -1.00000 q^{75} -5.75662 q^{77} +12.0933 q^{79} +1.00000 q^{81} +0.0433434 q^{83} +2.64620 q^{85} -5.89596 q^{87} +4.63885 q^{89} -1.36850 q^{91} -1.00000 q^{93} -3.25711 q^{95} +0.309478 q^{97} +5.25711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} + q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 7 q^{19} - q^{21} - q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} + 2 q^{39}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.09502 −0.413877 −0.206939 0.978354i \(-0.566350\pi\)
−0.206939 + 0.978354i \(0.566350\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.25711 1.58508 0.792539 0.609822i \(-0.208759\pi\)
0.792539 + 0.609822i \(0.208759\pi\)
\(12\) 0 0
\(13\) 1.24976 0.346620 0.173310 0.984867i \(-0.444554\pi\)
0.173310 + 0.984867i \(0.444554\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −2.64620 −0.641798 −0.320899 0.947113i \(-0.603985\pi\)
−0.320899 + 0.947113i \(0.603985\pi\)
\(18\) 0 0
\(19\) 3.25711 0.747232 0.373616 0.927584i \(-0.378118\pi\)
0.373616 + 0.927584i \(0.378118\pi\)
\(20\) 0 0
\(21\) 1.09502 0.238952
\(22\) 0 0
\(23\) −6.99097 −1.45772 −0.728859 0.684664i \(-0.759949\pi\)
−0.728859 + 0.684664i \(0.759949\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.89596 1.09485 0.547426 0.836854i \(-0.315608\pi\)
0.547426 + 0.836854i \(0.315608\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) −5.25711 −0.915145
\(34\) 0 0
\(35\) 1.09502 0.185092
\(36\) 0 0
\(37\) −3.24976 −0.534257 −0.267128 0.963661i \(-0.586075\pi\)
−0.267128 + 0.963661i \(0.586075\pi\)
\(38\) 0 0
\(39\) −1.24976 −0.200121
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −4.34477 −0.662571 −0.331286 0.943530i \(-0.607482\pi\)
−0.331286 + 0.943530i \(0.607482\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 7.14571 1.04231 0.521155 0.853462i \(-0.325502\pi\)
0.521155 + 0.853462i \(0.325502\pi\)
\(48\) 0 0
\(49\) −5.80094 −0.828706
\(50\) 0 0
\(51\) 2.64620 0.370542
\(52\) 0 0
\(53\) 6.80094 0.934181 0.467090 0.884210i \(-0.345302\pi\)
0.467090 + 0.884210i \(0.345302\pi\)
\(54\) 0 0
\(55\) −5.25711 −0.708868
\(56\) 0 0
\(57\) −3.25711 −0.431414
\(58\) 0 0
\(59\) −8.80829 −1.14674 −0.573371 0.819296i \(-0.694365\pi\)
−0.573371 + 0.819296i \(0.694365\pi\)
\(60\) 0 0
\(61\) 2.91234 0.372886 0.186443 0.982466i \(-0.440304\pi\)
0.186443 + 0.982466i \(0.440304\pi\)
\(62\) 0 0
\(63\) −1.09502 −0.137959
\(64\) 0 0
\(65\) −1.24976 −0.155013
\(66\) 0 0
\(67\) 1.43979 0.175898 0.0879491 0.996125i \(-0.471969\pi\)
0.0879491 + 0.996125i \(0.471969\pi\)
\(68\) 0 0
\(69\) 6.99097 0.841614
\(70\) 0 0
\(71\) −4.24073 −0.503282 −0.251641 0.967821i \(-0.580970\pi\)
−0.251641 + 0.967821i \(0.580970\pi\)
\(72\) 0 0
\(73\) −8.69689 −1.01789 −0.508947 0.860798i \(-0.669965\pi\)
−0.508947 + 0.860798i \(0.669965\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −5.75662 −0.656027
\(78\) 0 0
\(79\) 12.0933 1.36061 0.680304 0.732931i \(-0.261848\pi\)
0.680304 + 0.732931i \(0.261848\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.0433434 0.00475755 0.00237878 0.999997i \(-0.499243\pi\)
0.00237878 + 0.999997i \(0.499243\pi\)
\(84\) 0 0
\(85\) 2.64620 0.287021
\(86\) 0 0
\(87\) −5.89596 −0.632113
\(88\) 0 0
\(89\) 4.63885 0.491717 0.245858 0.969306i \(-0.420930\pi\)
0.245858 + 0.969306i \(0.420930\pi\)
\(90\) 0 0
\(91\) −1.36850 −0.143458
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −3.25711 −0.334172
\(96\) 0 0
\(97\) 0.309478 0.0314228 0.0157114 0.999877i \(-0.494999\pi\)
0.0157114 + 0.999877i \(0.494999\pi\)
\(98\) 0 0
\(99\) 5.25711 0.528359
\(100\) 0 0
\(101\) 16.4789 1.63971 0.819857 0.572568i \(-0.194053\pi\)
0.819857 + 0.572568i \(0.194053\pi\)
\(102\) 0 0
\(103\) −15.0417 −1.48210 −0.741050 0.671450i \(-0.765672\pi\)
−0.741050 + 0.671450i \(0.765672\pi\)
\(104\) 0 0
\(105\) −1.09502 −0.106863
\(106\) 0 0
\(107\) −10.0786 −0.974339 −0.487169 0.873308i \(-0.661971\pi\)
−0.487169 + 0.873308i \(0.661971\pi\)
\(108\) 0 0
\(109\) 10.2481 0.981588 0.490794 0.871276i \(-0.336707\pi\)
0.490794 + 0.871276i \(0.336707\pi\)
\(110\) 0 0
\(111\) 3.24976 0.308453
\(112\) 0 0
\(113\) 10.1547 0.955278 0.477639 0.878556i \(-0.341493\pi\)
0.477639 + 0.878556i \(0.341493\pi\)
\(114\) 0 0
\(115\) 6.99097 0.651911
\(116\) 0 0
\(117\) 1.24976 0.115540
\(118\) 0 0
\(119\) 2.89763 0.265626
\(120\) 0 0
\(121\) 16.6372 1.51247
\(122\) 0 0
\(123\) −4.00000 −0.360668
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 21.7033 1.92585 0.962927 0.269763i \(-0.0869453\pi\)
0.962927 + 0.269763i \(0.0869453\pi\)
\(128\) 0 0
\(129\) 4.34477 0.382536
\(130\) 0 0
\(131\) −19.0997 −1.66875 −0.834375 0.551197i \(-0.814171\pi\)
−0.834375 + 0.551197i \(0.814171\pi\)
\(132\) 0 0
\(133\) −3.56659 −0.309262
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 10.3367 0.883126 0.441563 0.897230i \(-0.354424\pi\)
0.441563 + 0.897230i \(0.354424\pi\)
\(138\) 0 0
\(139\) −1.80997 −0.153519 −0.0767597 0.997050i \(-0.524457\pi\)
−0.0767597 + 0.997050i \(0.524457\pi\)
\(140\) 0 0
\(141\) −7.14571 −0.601777
\(142\) 0 0
\(143\) 6.57010 0.549419
\(144\) 0 0
\(145\) −5.89596 −0.489632
\(146\) 0 0
\(147\) 5.80094 0.478453
\(148\) 0 0
\(149\) 15.2571 1.24991 0.624955 0.780660i \(-0.285117\pi\)
0.624955 + 0.780660i \(0.285117\pi\)
\(150\) 0 0
\(151\) 6.26613 0.509931 0.254965 0.966950i \(-0.417936\pi\)
0.254965 + 0.966950i \(0.417936\pi\)
\(152\) 0 0
\(153\) −2.64620 −0.213933
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) 21.2391 1.69506 0.847530 0.530747i \(-0.178089\pi\)
0.847530 + 0.530747i \(0.178089\pi\)
\(158\) 0 0
\(159\) −6.80094 −0.539350
\(160\) 0 0
\(161\) 7.65523 0.603317
\(162\) 0 0
\(163\) 15.3511 1.20239 0.601197 0.799101i \(-0.294691\pi\)
0.601197 + 0.799101i \(0.294691\pi\)
\(164\) 0 0
\(165\) 5.25711 0.409265
\(166\) 0 0
\(167\) 18.6690 1.44465 0.722323 0.691555i \(-0.243074\pi\)
0.722323 + 0.691555i \(0.243074\pi\)
\(168\) 0 0
\(169\) −11.4381 −0.879855
\(170\) 0 0
\(171\) 3.25711 0.249077
\(172\) 0 0
\(173\) 13.4118 1.01968 0.509842 0.860268i \(-0.329704\pi\)
0.509842 + 0.860268i \(0.329704\pi\)
\(174\) 0 0
\(175\) −1.09502 −0.0827755
\(176\) 0 0
\(177\) 8.80829 0.662072
\(178\) 0 0
\(179\) −24.4962 −1.83093 −0.915464 0.402399i \(-0.868176\pi\)
−0.915464 + 0.402399i \(0.868176\pi\)
\(180\) 0 0
\(181\) 17.0343 1.26615 0.633075 0.774090i \(-0.281792\pi\)
0.633075 + 0.774090i \(0.281792\pi\)
\(182\) 0 0
\(183\) −2.91234 −0.215286
\(184\) 0 0
\(185\) 3.24976 0.238927
\(186\) 0 0
\(187\) −13.9114 −1.01730
\(188\) 0 0
\(189\) 1.09502 0.0796507
\(190\) 0 0
\(191\) 12.4988 0.904382 0.452191 0.891921i \(-0.350642\pi\)
0.452191 + 0.891921i \(0.350642\pi\)
\(192\) 0 0
\(193\) −5.60188 −0.403232 −0.201616 0.979465i \(-0.564619\pi\)
−0.201616 + 0.979465i \(0.564619\pi\)
\(194\) 0 0
\(195\) 1.24976 0.0894968
\(196\) 0 0
\(197\) 24.2481 1.72760 0.863802 0.503831i \(-0.168077\pi\)
0.863802 + 0.503831i \(0.168077\pi\)
\(198\) 0 0
\(199\) −4.61091 −0.326859 −0.163429 0.986555i \(-0.552256\pi\)
−0.163429 + 0.986555i \(0.552256\pi\)
\(200\) 0 0
\(201\) −1.43979 −0.101555
\(202\) 0 0
\(203\) −6.45617 −0.453134
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) −6.99097 −0.485906
\(208\) 0 0
\(209\) 17.1230 1.18442
\(210\) 0 0
\(211\) −2.35947 −0.162433 −0.0812165 0.996696i \(-0.525881\pi\)
−0.0812165 + 0.996696i \(0.525881\pi\)
\(212\) 0 0
\(213\) 4.24073 0.290570
\(214\) 0 0
\(215\) 4.34477 0.296311
\(216\) 0 0
\(217\) −1.09502 −0.0743346
\(218\) 0 0
\(219\) 8.69689 0.587682
\(220\) 0 0
\(221\) −3.30710 −0.222460
\(222\) 0 0
\(223\) 0.792889 0.0530958 0.0265479 0.999648i \(-0.491549\pi\)
0.0265479 + 0.999648i \(0.491549\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 22.7976 1.51313 0.756564 0.653919i \(-0.226876\pi\)
0.756564 + 0.653919i \(0.226876\pi\)
\(228\) 0 0
\(229\) −6.60539 −0.436497 −0.218248 0.975893i \(-0.570034\pi\)
−0.218248 + 0.975893i \(0.570034\pi\)
\(230\) 0 0
\(231\) 5.75662 0.378758
\(232\) 0 0
\(233\) 28.7417 1.88293 0.941466 0.337109i \(-0.109449\pi\)
0.941466 + 0.337109i \(0.109449\pi\)
\(234\) 0 0
\(235\) −7.14571 −0.466135
\(236\) 0 0
\(237\) −12.0933 −0.785547
\(238\) 0 0
\(239\) −2.10139 −0.135928 −0.0679638 0.997688i \(-0.521650\pi\)
−0.0679638 + 0.997688i \(0.521650\pi\)
\(240\) 0 0
\(241\) 4.11945 0.265357 0.132678 0.991159i \(-0.457642\pi\)
0.132678 + 0.991159i \(0.457642\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 5.80094 0.370608
\(246\) 0 0
\(247\) 4.07059 0.259005
\(248\) 0 0
\(249\) −0.0433434 −0.00274677
\(250\) 0 0
\(251\) 24.1161 1.52219 0.761097 0.648638i \(-0.224661\pi\)
0.761097 + 0.648638i \(0.224661\pi\)
\(252\) 0 0
\(253\) −36.7523 −2.31060
\(254\) 0 0
\(255\) −2.64620 −0.165711
\(256\) 0 0
\(257\) −4.09334 −0.255336 −0.127668 0.991817i \(-0.540749\pi\)
−0.127668 + 0.991817i \(0.540749\pi\)
\(258\) 0 0
\(259\) 3.55854 0.221117
\(260\) 0 0
\(261\) 5.89596 0.364950
\(262\) 0 0
\(263\) 10.5289 0.649241 0.324620 0.945844i \(-0.394763\pi\)
0.324620 + 0.945844i \(0.394763\pi\)
\(264\) 0 0
\(265\) −6.80094 −0.417778
\(266\) 0 0
\(267\) −4.63885 −0.283893
\(268\) 0 0
\(269\) −5.61728 −0.342492 −0.171246 0.985228i \(-0.554779\pi\)
−0.171246 + 0.985228i \(0.554779\pi\)
\(270\) 0 0
\(271\) −26.8556 −1.63136 −0.815682 0.578501i \(-0.803638\pi\)
−0.815682 + 0.578501i \(0.803638\pi\)
\(272\) 0 0
\(273\) 1.36850 0.0828255
\(274\) 0 0
\(275\) 5.25711 0.317015
\(276\) 0 0
\(277\) −31.1578 −1.87209 −0.936044 0.351883i \(-0.885542\pi\)
−0.936044 + 0.351883i \(0.885542\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 22.3948 1.33596 0.667980 0.744179i \(-0.267159\pi\)
0.667980 + 0.744179i \(0.267159\pi\)
\(282\) 0 0
\(283\) −0.616096 −0.0366231 −0.0183116 0.999832i \(-0.505829\pi\)
−0.0183116 + 0.999832i \(0.505829\pi\)
\(284\) 0 0
\(285\) 3.25711 0.192934
\(286\) 0 0
\(287\) −4.38007 −0.258547
\(288\) 0 0
\(289\) −9.99762 −0.588096
\(290\) 0 0
\(291\) −0.309478 −0.0181419
\(292\) 0 0
\(293\) −28.0400 −1.63811 −0.819057 0.573712i \(-0.805503\pi\)
−0.819057 + 0.573712i \(0.805503\pi\)
\(294\) 0 0
\(295\) 8.80829 0.512839
\(296\) 0 0
\(297\) −5.25711 −0.305048
\(298\) 0 0
\(299\) −8.73701 −0.505274
\(300\) 0 0
\(301\) 4.75760 0.274223
\(302\) 0 0
\(303\) −16.4789 −0.946689
\(304\) 0 0
\(305\) −2.91234 −0.166760
\(306\) 0 0
\(307\) −19.6118 −1.11930 −0.559651 0.828728i \(-0.689065\pi\)
−0.559651 + 0.828728i \(0.689065\pi\)
\(308\) 0 0
\(309\) 15.0417 0.855691
\(310\) 0 0
\(311\) 31.2750 1.77344 0.886722 0.462302i \(-0.152976\pi\)
0.886722 + 0.462302i \(0.152976\pi\)
\(312\) 0 0
\(313\) −0.630799 −0.0356549 −0.0178274 0.999841i \(-0.505675\pi\)
−0.0178274 + 0.999841i \(0.505675\pi\)
\(314\) 0 0
\(315\) 1.09502 0.0616972
\(316\) 0 0
\(317\) −3.88272 −0.218075 −0.109038 0.994038i \(-0.534777\pi\)
−0.109038 + 0.994038i \(0.534777\pi\)
\(318\) 0 0
\(319\) 30.9957 1.73542
\(320\) 0 0
\(321\) 10.0786 0.562535
\(322\) 0 0
\(323\) −8.61896 −0.479572
\(324\) 0 0
\(325\) 1.24976 0.0693240
\(326\) 0 0
\(327\) −10.2481 −0.566720
\(328\) 0 0
\(329\) −7.82467 −0.431388
\(330\) 0 0
\(331\) −11.3504 −0.623877 −0.311938 0.950102i \(-0.600978\pi\)
−0.311938 + 0.950102i \(0.600978\pi\)
\(332\) 0 0
\(333\) −3.24976 −0.178086
\(334\) 0 0
\(335\) −1.43979 −0.0786640
\(336\) 0 0
\(337\) 20.6435 1.12453 0.562263 0.826959i \(-0.309931\pi\)
0.562263 + 0.826959i \(0.309931\pi\)
\(338\) 0 0
\(339\) −10.1547 −0.551530
\(340\) 0 0
\(341\) 5.25711 0.284688
\(342\) 0 0
\(343\) 14.0172 0.756860
\(344\) 0 0
\(345\) −6.99097 −0.376381
\(346\) 0 0
\(347\) −21.6166 −1.16044 −0.580219 0.814460i \(-0.697033\pi\)
−0.580219 + 0.814460i \(0.697033\pi\)
\(348\) 0 0
\(349\) −4.42341 −0.236780 −0.118390 0.992967i \(-0.537773\pi\)
−0.118390 + 0.992967i \(0.537773\pi\)
\(350\) 0 0
\(351\) −1.24976 −0.0667070
\(352\) 0 0
\(353\) 15.8680 0.844569 0.422285 0.906463i \(-0.361228\pi\)
0.422285 + 0.906463i \(0.361228\pi\)
\(354\) 0 0
\(355\) 4.24073 0.225074
\(356\) 0 0
\(357\) −2.89763 −0.153359
\(358\) 0 0
\(359\) 8.59767 0.453768 0.226884 0.973922i \(-0.427146\pi\)
0.226884 + 0.973922i \(0.427146\pi\)
\(360\) 0 0
\(361\) −8.39126 −0.441645
\(362\) 0 0
\(363\) −16.6372 −0.873225
\(364\) 0 0
\(365\) 8.69689 0.455216
\(366\) 0 0
\(367\) −12.3389 −0.644084 −0.322042 0.946725i \(-0.604369\pi\)
−0.322042 + 0.946725i \(0.604369\pi\)
\(368\) 0 0
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) −7.44714 −0.386636
\(372\) 0 0
\(373\) 28.2528 1.46287 0.731437 0.681909i \(-0.238850\pi\)
0.731437 + 0.681909i \(0.238850\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 7.36850 0.379497
\(378\) 0 0
\(379\) −32.0172 −1.64462 −0.822308 0.569043i \(-0.807314\pi\)
−0.822308 + 0.569043i \(0.807314\pi\)
\(380\) 0 0
\(381\) −21.7033 −1.11189
\(382\) 0 0
\(383\) 11.0082 0.562493 0.281247 0.959636i \(-0.409252\pi\)
0.281247 + 0.959636i \(0.409252\pi\)
\(384\) 0 0
\(385\) 5.75662 0.293384
\(386\) 0 0
\(387\) −4.34477 −0.220857
\(388\) 0 0
\(389\) −4.60020 −0.233239 −0.116620 0.993177i \(-0.537206\pi\)
−0.116620 + 0.993177i \(0.537206\pi\)
\(390\) 0 0
\(391\) 18.4995 0.935560
\(392\) 0 0
\(393\) 19.0997 0.963453
\(394\) 0 0
\(395\) −12.0933 −0.608482
\(396\) 0 0
\(397\) −6.47892 −0.325168 −0.162584 0.986695i \(-0.551983\pi\)
−0.162584 + 0.986695i \(0.551983\pi\)
\(398\) 0 0
\(399\) 3.56659 0.178553
\(400\) 0 0
\(401\) 0.0506947 0.00253157 0.00126579 0.999999i \(-0.499597\pi\)
0.00126579 + 0.999999i \(0.499597\pi\)
\(402\) 0 0
\(403\) 1.24976 0.0622548
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −17.0843 −0.846838
\(408\) 0 0
\(409\) 18.1867 0.899273 0.449637 0.893212i \(-0.351553\pi\)
0.449637 + 0.893212i \(0.351553\pi\)
\(410\) 0 0
\(411\) −10.3367 −0.509873
\(412\) 0 0
\(413\) 9.64522 0.474610
\(414\) 0 0
\(415\) −0.0433434 −0.00212764
\(416\) 0 0
\(417\) 1.80997 0.0886345
\(418\) 0 0
\(419\) 13.4798 0.658530 0.329265 0.944238i \(-0.393199\pi\)
0.329265 + 0.944238i \(0.393199\pi\)
\(420\) 0 0
\(421\) 7.14571 0.348261 0.174130 0.984723i \(-0.444289\pi\)
0.174130 + 0.984723i \(0.444289\pi\)
\(422\) 0 0
\(423\) 7.14571 0.347436
\(424\) 0 0
\(425\) −2.64620 −0.128360
\(426\) 0 0
\(427\) −3.18905 −0.154329
\(428\) 0 0
\(429\) −6.57010 −0.317207
\(430\) 0 0
\(431\) 39.6551 1.91012 0.955060 0.296414i \(-0.0957907\pi\)
0.955060 + 0.296414i \(0.0957907\pi\)
\(432\) 0 0
\(433\) 13.4011 0.644018 0.322009 0.946737i \(-0.395642\pi\)
0.322009 + 0.946737i \(0.395642\pi\)
\(434\) 0 0
\(435\) 5.89596 0.282689
\(436\) 0 0
\(437\) −22.7703 −1.08925
\(438\) 0 0
\(439\) −26.8943 −1.28359 −0.641797 0.766875i \(-0.721811\pi\)
−0.641797 + 0.766875i \(0.721811\pi\)
\(440\) 0 0
\(441\) −5.80094 −0.276235
\(442\) 0 0
\(443\) −17.7692 −0.844238 −0.422119 0.906540i \(-0.638714\pi\)
−0.422119 + 0.906540i \(0.638714\pi\)
\(444\) 0 0
\(445\) −4.63885 −0.219903
\(446\) 0 0
\(447\) −15.2571 −0.721636
\(448\) 0 0
\(449\) 16.8642 0.795870 0.397935 0.917414i \(-0.369727\pi\)
0.397935 + 0.917414i \(0.369727\pi\)
\(450\) 0 0
\(451\) 21.0284 0.990190
\(452\) 0 0
\(453\) −6.26613 −0.294409
\(454\) 0 0
\(455\) 1.36850 0.0641564
\(456\) 0 0
\(457\) 9.67728 0.452684 0.226342 0.974048i \(-0.427323\pi\)
0.226342 + 0.974048i \(0.427323\pi\)
\(458\) 0 0
\(459\) 2.64620 0.123514
\(460\) 0 0
\(461\) 15.2442 0.709995 0.354998 0.934867i \(-0.384482\pi\)
0.354998 + 0.934867i \(0.384482\pi\)
\(462\) 0 0
\(463\) −42.3028 −1.96598 −0.982988 0.183668i \(-0.941203\pi\)
−0.982988 + 0.183668i \(0.941203\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) 13.1277 0.607475 0.303738 0.952756i \(-0.401765\pi\)
0.303738 + 0.952756i \(0.401765\pi\)
\(468\) 0 0
\(469\) −1.57659 −0.0728002
\(470\) 0 0
\(471\) −21.2391 −0.978644
\(472\) 0 0
\(473\) −22.8409 −1.05023
\(474\) 0 0
\(475\) 3.25711 0.149446
\(476\) 0 0
\(477\) 6.80094 0.311394
\(478\) 0 0
\(479\) 19.0791 0.871747 0.435874 0.900008i \(-0.356439\pi\)
0.435874 + 0.900008i \(0.356439\pi\)
\(480\) 0 0
\(481\) −4.06140 −0.185184
\(482\) 0 0
\(483\) −7.65523 −0.348325
\(484\) 0 0
\(485\) −0.309478 −0.0140527
\(486\) 0 0
\(487\) 21.2365 0.962318 0.481159 0.876633i \(-0.340216\pi\)
0.481159 + 0.876633i \(0.340216\pi\)
\(488\) 0 0
\(489\) −15.3511 −0.694203
\(490\) 0 0
\(491\) −7.66797 −0.346051 −0.173025 0.984917i \(-0.555354\pi\)
−0.173025 + 0.984917i \(0.555354\pi\)
\(492\) 0 0
\(493\) −15.6019 −0.702673
\(494\) 0 0
\(495\) −5.25711 −0.236289
\(496\) 0 0
\(497\) 4.64367 0.208297
\(498\) 0 0
\(499\) −38.7582 −1.73505 −0.867527 0.497390i \(-0.834292\pi\)
−0.867527 + 0.497390i \(0.834292\pi\)
\(500\) 0 0
\(501\) −18.6690 −0.834067
\(502\) 0 0
\(503\) −12.9249 −0.576292 −0.288146 0.957587i \(-0.593039\pi\)
−0.288146 + 0.957587i \(0.593039\pi\)
\(504\) 0 0
\(505\) −16.4789 −0.733302
\(506\) 0 0
\(507\) 11.4381 0.507984
\(508\) 0 0
\(509\) −3.05421 −0.135375 −0.0676877 0.997707i \(-0.521562\pi\)
−0.0676877 + 0.997707i \(0.521562\pi\)
\(510\) 0 0
\(511\) 9.52324 0.421283
\(512\) 0 0
\(513\) −3.25711 −0.143805
\(514\) 0 0
\(515\) 15.0417 0.662815
\(516\) 0 0
\(517\) 37.5658 1.65214
\(518\) 0 0
\(519\) −13.4118 −0.588715
\(520\) 0 0
\(521\) −6.13415 −0.268742 −0.134371 0.990931i \(-0.542901\pi\)
−0.134371 + 0.990931i \(0.542901\pi\)
\(522\) 0 0
\(523\) 14.6984 0.642714 0.321357 0.946958i \(-0.395861\pi\)
0.321357 + 0.946958i \(0.395861\pi\)
\(524\) 0 0
\(525\) 1.09502 0.0477904
\(526\) 0 0
\(527\) −2.64620 −0.115270
\(528\) 0 0
\(529\) 25.8737 1.12494
\(530\) 0 0
\(531\) −8.80829 −0.382247
\(532\) 0 0
\(533\) 4.99902 0.216532
\(534\) 0 0
\(535\) 10.0786 0.435737
\(536\) 0 0
\(537\) 24.4962 1.05709
\(538\) 0 0
\(539\) −30.4962 −1.31356
\(540\) 0 0
\(541\) 20.5542 0.883694 0.441847 0.897090i \(-0.354323\pi\)
0.441847 + 0.897090i \(0.354323\pi\)
\(542\) 0 0
\(543\) −17.0343 −0.731012
\(544\) 0 0
\(545\) −10.2481 −0.438979
\(546\) 0 0
\(547\) 7.08285 0.302841 0.151420 0.988469i \(-0.451615\pi\)
0.151420 + 0.988469i \(0.451615\pi\)
\(548\) 0 0
\(549\) 2.91234 0.124295
\(550\) 0 0
\(551\) 19.2038 0.818108
\(552\) 0 0
\(553\) −13.2424 −0.563124
\(554\) 0 0
\(555\) −3.24976 −0.137944
\(556\) 0 0
\(557\) 2.18338 0.0925128 0.0462564 0.998930i \(-0.485271\pi\)
0.0462564 + 0.998930i \(0.485271\pi\)
\(558\) 0 0
\(559\) −5.42990 −0.229660
\(560\) 0 0
\(561\) 13.9114 0.587338
\(562\) 0 0
\(563\) −34.5164 −1.45469 −0.727346 0.686271i \(-0.759246\pi\)
−0.727346 + 0.686271i \(0.759246\pi\)
\(564\) 0 0
\(565\) −10.1547 −0.427213
\(566\) 0 0
\(567\) −1.09502 −0.0459864
\(568\) 0 0
\(569\) 5.65453 0.237050 0.118525 0.992951i \(-0.462183\pi\)
0.118525 + 0.992951i \(0.462183\pi\)
\(570\) 0 0
\(571\) 10.1900 0.426440 0.213220 0.977004i \(-0.431605\pi\)
0.213220 + 0.977004i \(0.431605\pi\)
\(572\) 0 0
\(573\) −12.4988 −0.522145
\(574\) 0 0
\(575\) −6.99097 −0.291544
\(576\) 0 0
\(577\) 1.70327 0.0709080 0.0354540 0.999371i \(-0.488712\pi\)
0.0354540 + 0.999371i \(0.488712\pi\)
\(578\) 0 0
\(579\) 5.60188 0.232806
\(580\) 0 0
\(581\) −0.0474617 −0.00196904
\(582\) 0 0
\(583\) 35.7533 1.48075
\(584\) 0 0
\(585\) −1.24976 −0.0516710
\(586\) 0 0
\(587\) 12.7042 0.524360 0.262180 0.965019i \(-0.415559\pi\)
0.262180 + 0.965019i \(0.415559\pi\)
\(588\) 0 0
\(589\) 3.25711 0.134207
\(590\) 0 0
\(591\) −24.2481 −0.997433
\(592\) 0 0
\(593\) −28.7023 −1.17866 −0.589331 0.807892i \(-0.700609\pi\)
−0.589331 + 0.807892i \(0.700609\pi\)
\(594\) 0 0
\(595\) −2.89763 −0.118791
\(596\) 0 0
\(597\) 4.61091 0.188712
\(598\) 0 0
\(599\) 12.8597 0.525432 0.262716 0.964873i \(-0.415382\pi\)
0.262716 + 0.964873i \(0.415382\pi\)
\(600\) 0 0
\(601\) 12.5289 0.511065 0.255533 0.966800i \(-0.417749\pi\)
0.255533 + 0.966800i \(0.417749\pi\)
\(602\) 0 0
\(603\) 1.43979 0.0586327
\(604\) 0 0
\(605\) −16.6372 −0.676397
\(606\) 0 0
\(607\) −5.22916 −0.212245 −0.106123 0.994353i \(-0.533844\pi\)
−0.106123 + 0.994353i \(0.533844\pi\)
\(608\) 0 0
\(609\) 6.45617 0.261617
\(610\) 0 0
\(611\) 8.93039 0.361285
\(612\) 0 0
\(613\) −16.8697 −0.681360 −0.340680 0.940179i \(-0.610657\pi\)
−0.340680 + 0.940179i \(0.610657\pi\)
\(614\) 0 0
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 43.3551 1.74541 0.872706 0.488246i \(-0.162363\pi\)
0.872706 + 0.488246i \(0.162363\pi\)
\(618\) 0 0
\(619\) −14.5448 −0.584605 −0.292303 0.956326i \(-0.594421\pi\)
−0.292303 + 0.956326i \(0.594421\pi\)
\(620\) 0 0
\(621\) 6.99097 0.280538
\(622\) 0 0
\(623\) −5.07962 −0.203510
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −17.1230 −0.683825
\(628\) 0 0
\(629\) 8.59950 0.342885
\(630\) 0 0
\(631\) 10.4970 0.417878 0.208939 0.977929i \(-0.432999\pi\)
0.208939 + 0.977929i \(0.432999\pi\)
\(632\) 0 0
\(633\) 2.35947 0.0937807
\(634\) 0 0
\(635\) −21.7033 −0.861268
\(636\) 0 0
\(637\) −7.24976 −0.287246
\(638\) 0 0
\(639\) −4.24073 −0.167761
\(640\) 0 0
\(641\) −33.2864 −1.31473 −0.657367 0.753571i \(-0.728330\pi\)
−0.657367 + 0.753571i \(0.728330\pi\)
\(642\) 0 0
\(643\) 19.3585 0.763425 0.381712 0.924281i \(-0.375335\pi\)
0.381712 + 0.924281i \(0.375335\pi\)
\(644\) 0 0
\(645\) −4.34477 −0.171075
\(646\) 0 0
\(647\) −19.3732 −0.761639 −0.380820 0.924649i \(-0.624358\pi\)
−0.380820 + 0.924649i \(0.624358\pi\)
\(648\) 0 0
\(649\) −46.3061 −1.81767
\(650\) 0 0
\(651\) 1.09502 0.0429171
\(652\) 0 0
\(653\) 0.456168 0.0178512 0.00892561 0.999960i \(-0.497159\pi\)
0.00892561 + 0.999960i \(0.497159\pi\)
\(654\) 0 0
\(655\) 19.0997 0.746288
\(656\) 0 0
\(657\) −8.69689 −0.339298
\(658\) 0 0
\(659\) 49.2158 1.91718 0.958588 0.284797i \(-0.0919263\pi\)
0.958588 + 0.284797i \(0.0919263\pi\)
\(660\) 0 0
\(661\) −45.1984 −1.75802 −0.879008 0.476807i \(-0.841794\pi\)
−0.879008 + 0.476807i \(0.841794\pi\)
\(662\) 0 0
\(663\) 3.30710 0.128437
\(664\) 0 0
\(665\) 3.56659 0.138306
\(666\) 0 0
\(667\) −41.2185 −1.59599
\(668\) 0 0
\(669\) −0.792889 −0.0306549
\(670\) 0 0
\(671\) 15.3105 0.591054
\(672\) 0 0
\(673\) 15.1611 0.584418 0.292209 0.956354i \(-0.405610\pi\)
0.292209 + 0.956354i \(0.405610\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −3.78386 −0.145426 −0.0727128 0.997353i \(-0.523166\pi\)
−0.0727128 + 0.997353i \(0.523166\pi\)
\(678\) 0 0
\(679\) −0.338884 −0.0130052
\(680\) 0 0
\(681\) −22.7976 −0.873605
\(682\) 0 0
\(683\) −25.3479 −0.969911 −0.484955 0.874539i \(-0.661164\pi\)
−0.484955 + 0.874539i \(0.661164\pi\)
\(684\) 0 0
\(685\) −10.3367 −0.394946
\(686\) 0 0
\(687\) 6.60539 0.252011
\(688\) 0 0
\(689\) 8.49951 0.323806
\(690\) 0 0
\(691\) 22.0500 0.838821 0.419411 0.907797i \(-0.362237\pi\)
0.419411 + 0.907797i \(0.362237\pi\)
\(692\) 0 0
\(693\) −5.75662 −0.218676
\(694\) 0 0
\(695\) 1.80997 0.0686560
\(696\) 0 0
\(697\) −10.5848 −0.400928
\(698\) 0 0
\(699\) −28.7417 −1.08711
\(700\) 0 0
\(701\) −4.72819 −0.178581 −0.0892906 0.996006i \(-0.528460\pi\)
−0.0892906 + 0.996006i \(0.528460\pi\)
\(702\) 0 0
\(703\) −10.5848 −0.399213
\(704\) 0 0
\(705\) 7.14571 0.269123
\(706\) 0 0
\(707\) −18.0447 −0.678640
\(708\) 0 0
\(709\) −41.5305 −1.55971 −0.779855 0.625960i \(-0.784707\pi\)
−0.779855 + 0.625960i \(0.784707\pi\)
\(710\) 0 0
\(711\) 12.0933 0.453536
\(712\) 0 0
\(713\) −6.99097 −0.261814
\(714\) 0 0
\(715\) −6.57010 −0.245708
\(716\) 0 0
\(717\) 2.10139 0.0784778
\(718\) 0 0
\(719\) −18.2028 −0.678849 −0.339425 0.940633i \(-0.610232\pi\)
−0.339425 + 0.940633i \(0.610232\pi\)
\(720\) 0 0
\(721\) 16.4709 0.613407
\(722\) 0 0
\(723\) −4.11945 −0.153204
\(724\) 0 0
\(725\) 5.89596 0.218970
\(726\) 0 0
\(727\) 2.21209 0.0820418 0.0410209 0.999158i \(-0.486939\pi\)
0.0410209 + 0.999158i \(0.486939\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 11.4971 0.425237
\(732\) 0 0
\(733\) −39.0578 −1.44263 −0.721317 0.692605i \(-0.756463\pi\)
−0.721317 + 0.692605i \(0.756463\pi\)
\(734\) 0 0
\(735\) −5.80094 −0.213971
\(736\) 0 0
\(737\) 7.56912 0.278812
\(738\) 0 0
\(739\) −1.11491 −0.0410126 −0.0205063 0.999790i \(-0.506528\pi\)
−0.0205063 + 0.999790i \(0.506528\pi\)
\(740\) 0 0
\(741\) −4.07059 −0.149537
\(742\) 0 0
\(743\) −26.7556 −0.981569 −0.490785 0.871281i \(-0.663290\pi\)
−0.490785 + 0.871281i \(0.663290\pi\)
\(744\) 0 0
\(745\) −15.2571 −0.558977
\(746\) 0 0
\(747\) 0.0433434 0.00158585
\(748\) 0 0
\(749\) 11.0363 0.403257
\(750\) 0 0
\(751\) −14.0783 −0.513723 −0.256862 0.966448i \(-0.582688\pi\)
−0.256862 + 0.966448i \(0.582688\pi\)
\(752\) 0 0
\(753\) −24.1161 −0.878839
\(754\) 0 0
\(755\) −6.26613 −0.228048
\(756\) 0 0
\(757\) 2.16209 0.0785825 0.0392913 0.999228i \(-0.487490\pi\)
0.0392913 + 0.999228i \(0.487490\pi\)
\(758\) 0 0
\(759\) 36.7523 1.33402
\(760\) 0 0
\(761\) −3.46450 −0.125588 −0.0627940 0.998027i \(-0.520001\pi\)
−0.0627940 + 0.998027i \(0.520001\pi\)
\(762\) 0 0
\(763\) −11.2218 −0.406257
\(764\) 0 0
\(765\) 2.64620 0.0956736
\(766\) 0 0
\(767\) −11.0082 −0.397483
\(768\) 0 0
\(769\) −14.0786 −0.507689 −0.253844 0.967245i \(-0.581695\pi\)
−0.253844 + 0.967245i \(0.581695\pi\)
\(770\) 0 0
\(771\) 4.09334 0.147418
\(772\) 0 0
\(773\) −45.8327 −1.64849 −0.824244 0.566235i \(-0.808400\pi\)
−0.824244 + 0.566235i \(0.808400\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 0 0
\(777\) −3.55854 −0.127662
\(778\) 0 0
\(779\) 13.0284 0.466792
\(780\) 0 0
\(781\) −22.2940 −0.797740
\(782\) 0 0
\(783\) −5.89596 −0.210704
\(784\) 0 0
\(785\) −21.2391 −0.758054
\(786\) 0 0
\(787\) 12.6234 0.449977 0.224989 0.974361i \(-0.427765\pi\)
0.224989 + 0.974361i \(0.427765\pi\)
\(788\) 0 0
\(789\) −10.5289 −0.374839
\(790\) 0 0
\(791\) −11.1196 −0.395368
\(792\) 0 0
\(793\) 3.63971 0.129250
\(794\) 0 0
\(795\) 6.80094 0.241204
\(796\) 0 0
\(797\) 11.9841 0.424499 0.212249 0.977216i \(-0.431921\pi\)
0.212249 + 0.977216i \(0.431921\pi\)
\(798\) 0 0
\(799\) −18.9090 −0.668952
\(800\) 0 0
\(801\) 4.63885 0.163906
\(802\) 0 0
\(803\) −45.7205 −1.61344
\(804\) 0 0
\(805\) −7.65523 −0.269811
\(806\) 0 0
\(807\) 5.61728 0.197738
\(808\) 0 0
\(809\) 14.9611 0.526003 0.263002 0.964795i \(-0.415288\pi\)
0.263002 + 0.964795i \(0.415288\pi\)
\(810\) 0 0
\(811\) 16.9604 0.595559 0.297780 0.954635i \(-0.403754\pi\)
0.297780 + 0.954635i \(0.403754\pi\)
\(812\) 0 0
\(813\) 26.8556 0.941868
\(814\) 0 0
\(815\) −15.3511 −0.537727
\(816\) 0 0
\(817\) −14.1514 −0.495094
\(818\) 0 0
\(819\) −1.36850 −0.0478194
\(820\) 0 0
\(821\) −10.8167 −0.377506 −0.188753 0.982025i \(-0.560445\pi\)
−0.188753 + 0.982025i \(0.560445\pi\)
\(822\) 0 0
\(823\) −11.4085 −0.397675 −0.198838 0.980032i \(-0.563717\pi\)
−0.198838 + 0.980032i \(0.563717\pi\)
\(824\) 0 0
\(825\) −5.25711 −0.183029
\(826\) 0 0
\(827\) 40.6556 1.41373 0.706867 0.707347i \(-0.250108\pi\)
0.706867 + 0.707347i \(0.250108\pi\)
\(828\) 0 0
\(829\) 13.0078 0.451781 0.225890 0.974153i \(-0.427471\pi\)
0.225890 + 0.974153i \(0.427471\pi\)
\(830\) 0 0
\(831\) 31.1578 1.08085
\(832\) 0 0
\(833\) 15.3504 0.531861
\(834\) 0 0
\(835\) −18.6690 −0.646066
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) −54.8677 −1.89424 −0.947121 0.320876i \(-0.896023\pi\)
−0.947121 + 0.320876i \(0.896023\pi\)
\(840\) 0 0
\(841\) 5.76229 0.198700
\(842\) 0 0
\(843\) −22.3948 −0.771317
\(844\) 0 0
\(845\) 11.4381 0.393483
\(846\) 0 0
\(847\) −18.2180 −0.625977
\(848\) 0 0
\(849\) 0.616096 0.0211444
\(850\) 0 0
\(851\) 22.7189 0.778796
\(852\) 0 0
\(853\) −57.6338 −1.97334 −0.986672 0.162721i \(-0.947973\pi\)
−0.986672 + 0.162721i \(0.947973\pi\)
\(854\) 0 0
\(855\) −3.25711 −0.111391
\(856\) 0 0
\(857\) −8.34731 −0.285139 −0.142569 0.989785i \(-0.545536\pi\)
−0.142569 + 0.989785i \(0.545536\pi\)
\(858\) 0 0
\(859\) −25.7886 −0.879894 −0.439947 0.898024i \(-0.645003\pi\)
−0.439947 + 0.898024i \(0.645003\pi\)
\(860\) 0 0
\(861\) 4.38007 0.149272
\(862\) 0 0
\(863\) 8.32672 0.283445 0.141722 0.989906i \(-0.454736\pi\)
0.141722 + 0.989906i \(0.454736\pi\)
\(864\) 0 0
\(865\) −13.4118 −0.456016
\(866\) 0 0
\(867\) 9.99762 0.339537
\(868\) 0 0
\(869\) 63.5760 2.15667
\(870\) 0 0
\(871\) 1.79938 0.0609698
\(872\) 0 0
\(873\) 0.309478 0.0104743
\(874\) 0 0
\(875\) 1.09502 0.0370183
\(876\) 0 0
\(877\) 57.7377 1.94967 0.974833 0.222938i \(-0.0715649\pi\)
0.974833 + 0.222938i \(0.0715649\pi\)
\(878\) 0 0
\(879\) 28.0400 0.945765
\(880\) 0 0
\(881\) −32.6002 −1.09833 −0.549164 0.835714i \(-0.685054\pi\)
−0.549164 + 0.835714i \(0.685054\pi\)
\(882\) 0 0
\(883\) −28.0955 −0.945489 −0.472744 0.881200i \(-0.656737\pi\)
−0.472744 + 0.881200i \(0.656737\pi\)
\(884\) 0 0
\(885\) −8.80829 −0.296087
\(886\) 0 0
\(887\) −36.1741 −1.21461 −0.607304 0.794469i \(-0.707749\pi\)
−0.607304 + 0.794469i \(0.707749\pi\)
\(888\) 0 0
\(889\) −23.7654 −0.797067
\(890\) 0 0
\(891\) 5.25711 0.176120
\(892\) 0 0
\(893\) 23.2743 0.778846
\(894\) 0 0
\(895\) 24.4962 0.818816
\(896\) 0 0
\(897\) 8.73701 0.291720
\(898\) 0 0
\(899\) 5.89596 0.196641
\(900\) 0 0
\(901\) −17.9966 −0.599555
\(902\) 0 0
\(903\) −4.75760 −0.158323
\(904\) 0 0
\(905\) −17.0343 −0.566240
\(906\) 0 0
\(907\) −14.3488 −0.476443 −0.238222 0.971211i \(-0.576564\pi\)
−0.238222 + 0.971211i \(0.576564\pi\)
\(908\) 0 0
\(909\) 16.4789 0.546571
\(910\) 0 0
\(911\) 7.48579 0.248015 0.124008 0.992281i \(-0.460425\pi\)
0.124008 + 0.992281i \(0.460425\pi\)
\(912\) 0 0
\(913\) 0.227861 0.00754109
\(914\) 0 0
\(915\) 2.91234 0.0962788
\(916\) 0 0
\(917\) 20.9145 0.690658
\(918\) 0 0
\(919\) −9.84077 −0.324617 −0.162309 0.986740i \(-0.551894\pi\)
−0.162309 + 0.986740i \(0.551894\pi\)
\(920\) 0 0
\(921\) 19.6118 0.646230
\(922\) 0 0
\(923\) −5.29987 −0.174447
\(924\) 0 0
\(925\) −3.24976 −0.106851
\(926\) 0 0
\(927\) −15.0417 −0.494033
\(928\) 0 0
\(929\) 34.2227 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(930\) 0 0
\(931\) −18.8943 −0.619235
\(932\) 0 0
\(933\) −31.2750 −1.02390
\(934\) 0 0
\(935\) 13.9114 0.454950
\(936\) 0 0
\(937\) −51.5838 −1.68517 −0.842585 0.538563i \(-0.818967\pi\)
−0.842585 + 0.538563i \(0.818967\pi\)
\(938\) 0 0
\(939\) 0.630799 0.0205853
\(940\) 0 0
\(941\) 6.44628 0.210143 0.105071 0.994465i \(-0.466493\pi\)
0.105071 + 0.994465i \(0.466493\pi\)
\(942\) 0 0
\(943\) −27.9639 −0.910629
\(944\) 0 0
\(945\) −1.09502 −0.0356209
\(946\) 0 0
\(947\) 29.2584 0.950772 0.475386 0.879777i \(-0.342308\pi\)
0.475386 + 0.879777i \(0.342308\pi\)
\(948\) 0 0
\(949\) −10.8690 −0.352822
\(950\) 0 0
\(951\) 3.88272 0.125906
\(952\) 0 0
\(953\) 49.3471 1.59851 0.799255 0.600993i \(-0.205228\pi\)
0.799255 + 0.600993i \(0.205228\pi\)
\(954\) 0 0
\(955\) −12.4988 −0.404452
\(956\) 0 0
\(957\) −30.9957 −1.00195
\(958\) 0 0
\(959\) −11.3189 −0.365506
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −10.0786 −0.324780
\(964\) 0 0
\(965\) 5.60188 0.180331
\(966\) 0 0
\(967\) 50.6842 1.62990 0.814948 0.579535i \(-0.196766\pi\)
0.814948 + 0.579535i \(0.196766\pi\)
\(968\) 0 0
\(969\) 8.61896 0.276881
\(970\) 0 0
\(971\) 6.76711 0.217167 0.108583 0.994087i \(-0.465369\pi\)
0.108583 + 0.994087i \(0.465369\pi\)
\(972\) 0 0
\(973\) 1.98194 0.0635382
\(974\) 0 0
\(975\) −1.24976 −0.0400242
\(976\) 0 0
\(977\) 16.5848 0.530595 0.265297 0.964167i \(-0.414530\pi\)
0.265297 + 0.964167i \(0.414530\pi\)
\(978\) 0 0
\(979\) 24.3869 0.779409
\(980\) 0 0
\(981\) 10.2481 0.327196
\(982\) 0 0
\(983\) 15.1171 0.482160 0.241080 0.970505i \(-0.422498\pi\)
0.241080 + 0.970505i \(0.422498\pi\)
\(984\) 0 0
\(985\) −24.2481 −0.772608
\(986\) 0 0
\(987\) 7.82467 0.249062
\(988\) 0 0
\(989\) 30.3742 0.965843
\(990\) 0 0
\(991\) 60.9948 1.93757 0.968783 0.247912i \(-0.0797443\pi\)
0.968783 + 0.247912i \(0.0797443\pi\)
\(992\) 0 0
\(993\) 11.3504 0.360196
\(994\) 0 0
\(995\) 4.61091 0.146176
\(996\) 0 0
\(997\) 40.2355 1.27427 0.637136 0.770751i \(-0.280119\pi\)
0.637136 + 0.770751i \(0.280119\pi\)
\(998\) 0 0
\(999\) 3.24976 0.102818
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 3720.2.a.r.1.2 4
4.3 odd 2 7440.2.a.bx.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.r.1.2 4 1.1 even 1 trivial
7440.2.a.bx.1.3 4 4.3 odd 2