Properties

Label 960.4.a.h.1.1
Level $960$
Weight $4$
Character 960.1
Self dual yes
Analytic conductor $56.642$
Analytic rank $1$
Dimension $1$
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] = [960,4,Mod(1,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.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: \(56.6418336055\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -5.00000 q^{5} +16.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -5.00000 q^{5} +16.0000 q^{7} +9.00000 q^{9} -28.0000 q^{11} +26.0000 q^{13} +15.0000 q^{15} -62.0000 q^{17} -68.0000 q^{19} -48.0000 q^{21} +208.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +58.0000 q^{29} -160.000 q^{31} +84.0000 q^{33} -80.0000 q^{35} -270.000 q^{37} -78.0000 q^{39} +282.000 q^{41} +76.0000 q^{43} -45.0000 q^{45} +280.000 q^{47} -87.0000 q^{49} +186.000 q^{51} +210.000 q^{53} +140.000 q^{55} +204.000 q^{57} +196.000 q^{59} -742.000 q^{61} +144.000 q^{63} -130.000 q^{65} +836.000 q^{67} -624.000 q^{69} +504.000 q^{71} -1062.00 q^{73} -75.0000 q^{75} -448.000 q^{77} -768.000 q^{79} +81.0000 q^{81} -1052.00 q^{83} +310.000 q^{85} -174.000 q^{87} -726.000 q^{89} +416.000 q^{91} +480.000 q^{93} +340.000 q^{95} -1406.00 q^{97} -252.000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 16.0000 0.863919 0.431959 0.901893i \(-0.357822\pi\)
0.431959 + 0.901893i \(0.357822\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −28.0000 −0.767483 −0.383742 0.923440i \(-0.625365\pi\)
−0.383742 + 0.923440i \(0.625365\pi\)
\(12\) 0 0
\(13\) 26.0000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) −62.0000 −0.884542 −0.442271 0.896882i \(-0.645827\pi\)
−0.442271 + 0.896882i \(0.645827\pi\)
\(18\) 0 0
\(19\) −68.0000 −0.821067 −0.410533 0.911846i \(-0.634657\pi\)
−0.410533 + 0.911846i \(0.634657\pi\)
\(20\) 0 0
\(21\) −48.0000 −0.498784
\(22\) 0 0
\(23\) 208.000 1.88570 0.942848 0.333224i \(-0.108136\pi\)
0.942848 + 0.333224i \(0.108136\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 58.0000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −160.000 −0.926995 −0.463498 0.886098i \(-0.653406\pi\)
−0.463498 + 0.886098i \(0.653406\pi\)
\(32\) 0 0
\(33\) 84.0000 0.443107
\(34\) 0 0
\(35\) −80.0000 −0.386356
\(36\) 0 0
\(37\) −270.000 −1.19967 −0.599834 0.800124i \(-0.704767\pi\)
−0.599834 + 0.800124i \(0.704767\pi\)
\(38\) 0 0
\(39\) −78.0000 −0.320256
\(40\) 0 0
\(41\) 282.000 1.07417 0.537085 0.843528i \(-0.319525\pi\)
0.537085 + 0.843528i \(0.319525\pi\)
\(42\) 0 0
\(43\) 76.0000 0.269532 0.134766 0.990877i \(-0.456972\pi\)
0.134766 + 0.990877i \(0.456972\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) 280.000 0.868983 0.434491 0.900676i \(-0.356928\pi\)
0.434491 + 0.900676i \(0.356928\pi\)
\(48\) 0 0
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) 186.000 0.510690
\(52\) 0 0
\(53\) 210.000 0.544259 0.272129 0.962261i \(-0.412272\pi\)
0.272129 + 0.962261i \(0.412272\pi\)
\(54\) 0 0
\(55\) 140.000 0.343229
\(56\) 0 0
\(57\) 204.000 0.474043
\(58\) 0 0
\(59\) 196.000 0.432492 0.216246 0.976339i \(-0.430619\pi\)
0.216246 + 0.976339i \(0.430619\pi\)
\(60\) 0 0
\(61\) −742.000 −1.55743 −0.778716 0.627376i \(-0.784129\pi\)
−0.778716 + 0.627376i \(0.784129\pi\)
\(62\) 0 0
\(63\) 144.000 0.287973
\(64\) 0 0
\(65\) −130.000 −0.248069
\(66\) 0 0
\(67\) 836.000 1.52438 0.762191 0.647352i \(-0.224123\pi\)
0.762191 + 0.647352i \(0.224123\pi\)
\(68\) 0 0
\(69\) −624.000 −1.08871
\(70\) 0 0
\(71\) 504.000 0.842448 0.421224 0.906957i \(-0.361601\pi\)
0.421224 + 0.906957i \(0.361601\pi\)
\(72\) 0 0
\(73\) −1062.00 −1.70271 −0.851354 0.524591i \(-0.824218\pi\)
−0.851354 + 0.524591i \(0.824218\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) −448.000 −0.663043
\(78\) 0 0
\(79\) −768.000 −1.09376 −0.546878 0.837212i \(-0.684184\pi\)
−0.546878 + 0.837212i \(0.684184\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1052.00 −1.39123 −0.695614 0.718415i \(-0.744868\pi\)
−0.695614 + 0.718415i \(0.744868\pi\)
\(84\) 0 0
\(85\) 310.000 0.395579
\(86\) 0 0
\(87\) −174.000 −0.214423
\(88\) 0 0
\(89\) −726.000 −0.864672 −0.432336 0.901712i \(-0.642311\pi\)
−0.432336 + 0.901712i \(0.642311\pi\)
\(90\) 0 0
\(91\) 416.000 0.479216
\(92\) 0 0
\(93\) 480.000 0.535201
\(94\) 0 0
\(95\) 340.000 0.367192
\(96\) 0 0
\(97\) −1406.00 −1.47173 −0.735864 0.677129i \(-0.763224\pi\)
−0.735864 + 0.677129i \(0.763224\pi\)
\(98\) 0 0
\(99\) −252.000 −0.255828
\(100\) 0 0
\(101\) −990.000 −0.975333 −0.487667 0.873030i \(-0.662152\pi\)
−0.487667 + 0.873030i \(0.662152\pi\)
\(102\) 0 0
\(103\) −736.000 −0.704080 −0.352040 0.935985i \(-0.614512\pi\)
−0.352040 + 0.935985i \(0.614512\pi\)
\(104\) 0 0
\(105\) 240.000 0.223063
\(106\) 0 0
\(107\) 1212.00 1.09503 0.547516 0.836795i \(-0.315573\pi\)
0.547516 + 0.836795i \(0.315573\pi\)
\(108\) 0 0
\(109\) 1834.00 1.61161 0.805804 0.592182i \(-0.201733\pi\)
0.805804 + 0.592182i \(0.201733\pi\)
\(110\) 0 0
\(111\) 810.000 0.692629
\(112\) 0 0
\(113\) −2046.00 −1.70329 −0.851644 0.524121i \(-0.824394\pi\)
−0.851644 + 0.524121i \(0.824394\pi\)
\(114\) 0 0
\(115\) −1040.00 −0.843309
\(116\) 0 0
\(117\) 234.000 0.184900
\(118\) 0 0
\(119\) −992.000 −0.764172
\(120\) 0 0
\(121\) −547.000 −0.410969
\(122\) 0 0
\(123\) −846.000 −0.620173
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1176.00 −0.821678 −0.410839 0.911708i \(-0.634764\pi\)
−0.410839 + 0.911708i \(0.634764\pi\)
\(128\) 0 0
\(129\) −228.000 −0.155615
\(130\) 0 0
\(131\) 12.0000 0.00800340 0.00400170 0.999992i \(-0.498726\pi\)
0.00400170 + 0.999992i \(0.498726\pi\)
\(132\) 0 0
\(133\) −1088.00 −0.709335
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) −790.000 −0.492659 −0.246329 0.969186i \(-0.579225\pi\)
−0.246329 + 0.969186i \(0.579225\pi\)
\(138\) 0 0
\(139\) −924.000 −0.563832 −0.281916 0.959439i \(-0.590970\pi\)
−0.281916 + 0.959439i \(0.590970\pi\)
\(140\) 0 0
\(141\) −840.000 −0.501708
\(142\) 0 0
\(143\) −728.000 −0.425723
\(144\) 0 0
\(145\) −290.000 −0.166091
\(146\) 0 0
\(147\) 261.000 0.146442
\(148\) 0 0
\(149\) −3022.00 −1.66156 −0.830778 0.556604i \(-0.812104\pi\)
−0.830778 + 0.556604i \(0.812104\pi\)
\(150\) 0 0
\(151\) −1736.00 −0.935587 −0.467794 0.883838i \(-0.654951\pi\)
−0.467794 + 0.883838i \(0.654951\pi\)
\(152\) 0 0
\(153\) −558.000 −0.294847
\(154\) 0 0
\(155\) 800.000 0.414565
\(156\) 0 0
\(157\) 1322.00 0.672020 0.336010 0.941858i \(-0.390922\pi\)
0.336010 + 0.941858i \(0.390922\pi\)
\(158\) 0 0
\(159\) −630.000 −0.314228
\(160\) 0 0
\(161\) 3328.00 1.62909
\(162\) 0 0
\(163\) −908.000 −0.436319 −0.218160 0.975913i \(-0.570005\pi\)
−0.218160 + 0.975913i \(0.570005\pi\)
\(164\) 0 0
\(165\) −420.000 −0.198163
\(166\) 0 0
\(167\) −1296.00 −0.600524 −0.300262 0.953857i \(-0.597074\pi\)
−0.300262 + 0.953857i \(0.597074\pi\)
\(168\) 0 0
\(169\) −1521.00 −0.692308
\(170\) 0 0
\(171\) −612.000 −0.273689
\(172\) 0 0
\(173\) −2134.00 −0.937832 −0.468916 0.883243i \(-0.655355\pi\)
−0.468916 + 0.883243i \(0.655355\pi\)
\(174\) 0 0
\(175\) 400.000 0.172784
\(176\) 0 0
\(177\) −588.000 −0.249699
\(178\) 0 0
\(179\) 1612.00 0.673109 0.336555 0.941664i \(-0.390738\pi\)
0.336555 + 0.941664i \(0.390738\pi\)
\(180\) 0 0
\(181\) −3086.00 −1.26730 −0.633648 0.773621i \(-0.718443\pi\)
−0.633648 + 0.773621i \(0.718443\pi\)
\(182\) 0 0
\(183\) 2226.00 0.899184
\(184\) 0 0
\(185\) 1350.00 0.536508
\(186\) 0 0
\(187\) 1736.00 0.678871
\(188\) 0 0
\(189\) −432.000 −0.166261
\(190\) 0 0
\(191\) 4208.00 1.59414 0.797069 0.603889i \(-0.206383\pi\)
0.797069 + 0.603889i \(0.206383\pi\)
\(192\) 0 0
\(193\) 2818.00 1.05101 0.525503 0.850792i \(-0.323877\pi\)
0.525503 + 0.850792i \(0.323877\pi\)
\(194\) 0 0
\(195\) 390.000 0.143223
\(196\) 0 0
\(197\) 418.000 0.151174 0.0755870 0.997139i \(-0.475917\pi\)
0.0755870 + 0.997139i \(0.475917\pi\)
\(198\) 0 0
\(199\) 3352.00 1.19406 0.597028 0.802221i \(-0.296348\pi\)
0.597028 + 0.802221i \(0.296348\pi\)
\(200\) 0 0
\(201\) −2508.00 −0.880103
\(202\) 0 0
\(203\) 928.000 0.320851
\(204\) 0 0
\(205\) −1410.00 −0.480384
\(206\) 0 0
\(207\) 1872.00 0.628565
\(208\) 0 0
\(209\) 1904.00 0.630155
\(210\) 0 0
\(211\) −4276.00 −1.39513 −0.697564 0.716523i \(-0.745733\pi\)
−0.697564 + 0.716523i \(0.745733\pi\)
\(212\) 0 0
\(213\) −1512.00 −0.486387
\(214\) 0 0
\(215\) −380.000 −0.120539
\(216\) 0 0
\(217\) −2560.00 −0.800848
\(218\) 0 0
\(219\) 3186.00 0.983059
\(220\) 0 0
\(221\) −1612.00 −0.490655
\(222\) 0 0
\(223\) −4712.00 −1.41497 −0.707486 0.706727i \(-0.750171\pi\)
−0.707486 + 0.706727i \(0.750171\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −732.000 −0.214029 −0.107014 0.994257i \(-0.534129\pi\)
−0.107014 + 0.994257i \(0.534129\pi\)
\(228\) 0 0
\(229\) 5186.00 1.49651 0.748254 0.663412i \(-0.230892\pi\)
0.748254 + 0.663412i \(0.230892\pi\)
\(230\) 0 0
\(231\) 1344.00 0.382808
\(232\) 0 0
\(233\) −3798.00 −1.06788 −0.533938 0.845523i \(-0.679289\pi\)
−0.533938 + 0.845523i \(0.679289\pi\)
\(234\) 0 0
\(235\) −1400.00 −0.388621
\(236\) 0 0
\(237\) 2304.00 0.631481
\(238\) 0 0
\(239\) 3120.00 0.844419 0.422209 0.906498i \(-0.361255\pi\)
0.422209 + 0.906498i \(0.361255\pi\)
\(240\) 0 0
\(241\) 1490.00 0.398255 0.199127 0.979974i \(-0.436189\pi\)
0.199127 + 0.979974i \(0.436189\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 435.000 0.113433
\(246\) 0 0
\(247\) −1768.00 −0.455446
\(248\) 0 0
\(249\) 3156.00 0.803226
\(250\) 0 0
\(251\) −5292.00 −1.33079 −0.665395 0.746492i \(-0.731737\pi\)
−0.665395 + 0.746492i \(0.731737\pi\)
\(252\) 0 0
\(253\) −5824.00 −1.44724
\(254\) 0 0
\(255\) −930.000 −0.228388
\(256\) 0 0
\(257\) −3918.00 −0.950965 −0.475483 0.879725i \(-0.657727\pi\)
−0.475483 + 0.879725i \(0.657727\pi\)
\(258\) 0 0
\(259\) −4320.00 −1.03642
\(260\) 0 0
\(261\) 522.000 0.123797
\(262\) 0 0
\(263\) −6624.00 −1.55305 −0.776527 0.630084i \(-0.783021\pi\)
−0.776527 + 0.630084i \(0.783021\pi\)
\(264\) 0 0
\(265\) −1050.00 −0.243400
\(266\) 0 0
\(267\) 2178.00 0.499219
\(268\) 0 0
\(269\) 2954.00 0.669549 0.334774 0.942298i \(-0.391340\pi\)
0.334774 + 0.942298i \(0.391340\pi\)
\(270\) 0 0
\(271\) 6576.00 1.47404 0.737018 0.675874i \(-0.236233\pi\)
0.737018 + 0.675874i \(0.236233\pi\)
\(272\) 0 0
\(273\) −1248.00 −0.276675
\(274\) 0 0
\(275\) −700.000 −0.153497
\(276\) 0 0
\(277\) −4478.00 −0.971325 −0.485662 0.874146i \(-0.661422\pi\)
−0.485662 + 0.874146i \(0.661422\pi\)
\(278\) 0 0
\(279\) −1440.00 −0.308998
\(280\) 0 0
\(281\) −6358.00 −1.34977 −0.674887 0.737921i \(-0.735808\pi\)
−0.674887 + 0.737921i \(0.735808\pi\)
\(282\) 0 0
\(283\) 860.000 0.180642 0.0903210 0.995913i \(-0.471211\pi\)
0.0903210 + 0.995913i \(0.471211\pi\)
\(284\) 0 0
\(285\) −1020.00 −0.211999
\(286\) 0 0
\(287\) 4512.00 0.927996
\(288\) 0 0
\(289\) −1069.00 −0.217586
\(290\) 0 0
\(291\) 4218.00 0.849703
\(292\) 0 0
\(293\) 5794.00 1.15525 0.577626 0.816301i \(-0.303979\pi\)
0.577626 + 0.816301i \(0.303979\pi\)
\(294\) 0 0
\(295\) −980.000 −0.193416
\(296\) 0 0
\(297\) 756.000 0.147702
\(298\) 0 0
\(299\) 5408.00 1.04600
\(300\) 0 0
\(301\) 1216.00 0.232854
\(302\) 0 0
\(303\) 2970.00 0.563109
\(304\) 0 0
\(305\) 3710.00 0.696505
\(306\) 0 0
\(307\) −6860.00 −1.27531 −0.637656 0.770321i \(-0.720096\pi\)
−0.637656 + 0.770321i \(0.720096\pi\)
\(308\) 0 0
\(309\) 2208.00 0.406501
\(310\) 0 0
\(311\) 6248.00 1.13920 0.569601 0.821922i \(-0.307098\pi\)
0.569601 + 0.821922i \(0.307098\pi\)
\(312\) 0 0
\(313\) 11018.0 1.98969 0.994847 0.101388i \(-0.0323284\pi\)
0.994847 + 0.101388i \(0.0323284\pi\)
\(314\) 0 0
\(315\) −720.000 −0.128785
\(316\) 0 0
\(317\) 954.000 0.169028 0.0845142 0.996422i \(-0.473066\pi\)
0.0845142 + 0.996422i \(0.473066\pi\)
\(318\) 0 0
\(319\) −1624.00 −0.285036
\(320\) 0 0
\(321\) −3636.00 −0.632217
\(322\) 0 0
\(323\) 4216.00 0.726268
\(324\) 0 0
\(325\) 650.000 0.110940
\(326\) 0 0
\(327\) −5502.00 −0.930463
\(328\) 0 0
\(329\) 4480.00 0.750731
\(330\) 0 0
\(331\) 9396.00 1.56027 0.780137 0.625608i \(-0.215149\pi\)
0.780137 + 0.625608i \(0.215149\pi\)
\(332\) 0 0
\(333\) −2430.00 −0.399889
\(334\) 0 0
\(335\) −4180.00 −0.681725
\(336\) 0 0
\(337\) 5074.00 0.820173 0.410087 0.912047i \(-0.365498\pi\)
0.410087 + 0.912047i \(0.365498\pi\)
\(338\) 0 0
\(339\) 6138.00 0.983394
\(340\) 0 0
\(341\) 4480.00 0.711453
\(342\) 0 0
\(343\) −6880.00 −1.08305
\(344\) 0 0
\(345\) 3120.00 0.486885
\(346\) 0 0
\(347\) 3916.00 0.605827 0.302913 0.953018i \(-0.402041\pi\)
0.302913 + 0.953018i \(0.402041\pi\)
\(348\) 0 0
\(349\) 1818.00 0.278840 0.139420 0.990233i \(-0.455476\pi\)
0.139420 + 0.990233i \(0.455476\pi\)
\(350\) 0 0
\(351\) −702.000 −0.106752
\(352\) 0 0
\(353\) −7118.00 −1.07324 −0.536619 0.843825i \(-0.680299\pi\)
−0.536619 + 0.843825i \(0.680299\pi\)
\(354\) 0 0
\(355\) −2520.00 −0.376754
\(356\) 0 0
\(357\) 2976.00 0.441195
\(358\) 0 0
\(359\) −5304.00 −0.779762 −0.389881 0.920865i \(-0.627484\pi\)
−0.389881 + 0.920865i \(0.627484\pi\)
\(360\) 0 0
\(361\) −2235.00 −0.325849
\(362\) 0 0
\(363\) 1641.00 0.237273
\(364\) 0 0
\(365\) 5310.00 0.761474
\(366\) 0 0
\(367\) −5672.00 −0.806747 −0.403373 0.915036i \(-0.632162\pi\)
−0.403373 + 0.915036i \(0.632162\pi\)
\(368\) 0 0
\(369\) 2538.00 0.358057
\(370\) 0 0
\(371\) 3360.00 0.470195
\(372\) 0 0
\(373\) −7774.00 −1.07915 −0.539574 0.841938i \(-0.681415\pi\)
−0.539574 + 0.841938i \(0.681415\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) 1508.00 0.206010
\(378\) 0 0
\(379\) −5516.00 −0.747593 −0.373797 0.927511i \(-0.621944\pi\)
−0.373797 + 0.927511i \(0.621944\pi\)
\(380\) 0 0
\(381\) 3528.00 0.474396
\(382\) 0 0
\(383\) 7128.00 0.950976 0.475488 0.879722i \(-0.342272\pi\)
0.475488 + 0.879722i \(0.342272\pi\)
\(384\) 0 0
\(385\) 2240.00 0.296522
\(386\) 0 0
\(387\) 684.000 0.0898441
\(388\) 0 0
\(389\) 10722.0 1.39750 0.698749 0.715367i \(-0.253740\pi\)
0.698749 + 0.715367i \(0.253740\pi\)
\(390\) 0 0
\(391\) −12896.0 −1.66798
\(392\) 0 0
\(393\) −36.0000 −0.00462076
\(394\) 0 0
\(395\) 3840.00 0.489143
\(396\) 0 0
\(397\) 12122.0 1.53246 0.766229 0.642568i \(-0.222131\pi\)
0.766229 + 0.642568i \(0.222131\pi\)
\(398\) 0 0
\(399\) 3264.00 0.409535
\(400\) 0 0
\(401\) 10482.0 1.30535 0.652676 0.757637i \(-0.273646\pi\)
0.652676 + 0.757637i \(0.273646\pi\)
\(402\) 0 0
\(403\) −4160.00 −0.514204
\(404\) 0 0
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) 7560.00 0.920726
\(408\) 0 0
\(409\) 3850.00 0.465453 0.232726 0.972542i \(-0.425235\pi\)
0.232726 + 0.972542i \(0.425235\pi\)
\(410\) 0 0
\(411\) 2370.00 0.284437
\(412\) 0 0
\(413\) 3136.00 0.373638
\(414\) 0 0
\(415\) 5260.00 0.622176
\(416\) 0 0
\(417\) 2772.00 0.325529
\(418\) 0 0
\(419\) −5796.00 −0.675783 −0.337892 0.941185i \(-0.609714\pi\)
−0.337892 + 0.941185i \(0.609714\pi\)
\(420\) 0 0
\(421\) −3294.00 −0.381330 −0.190665 0.981655i \(-0.561064\pi\)
−0.190665 + 0.981655i \(0.561064\pi\)
\(422\) 0 0
\(423\) 2520.00 0.289661
\(424\) 0 0
\(425\) −1550.00 −0.176908
\(426\) 0 0
\(427\) −11872.0 −1.34549
\(428\) 0 0
\(429\) 2184.00 0.245791
\(430\) 0 0
\(431\) −1696.00 −0.189544 −0.0947720 0.995499i \(-0.530212\pi\)
−0.0947720 + 0.995499i \(0.530212\pi\)
\(432\) 0 0
\(433\) −12334.0 −1.36890 −0.684451 0.729059i \(-0.739958\pi\)
−0.684451 + 0.729059i \(0.739958\pi\)
\(434\) 0 0
\(435\) 870.000 0.0958927
\(436\) 0 0
\(437\) −14144.0 −1.54828
\(438\) 0 0
\(439\) −376.000 −0.0408781 −0.0204391 0.999791i \(-0.506506\pi\)
−0.0204391 + 0.999791i \(0.506506\pi\)
\(440\) 0 0
\(441\) −783.000 −0.0845481
\(442\) 0 0
\(443\) 8028.00 0.860997 0.430499 0.902591i \(-0.358338\pi\)
0.430499 + 0.902591i \(0.358338\pi\)
\(444\) 0 0
\(445\) 3630.00 0.386693
\(446\) 0 0
\(447\) 9066.00 0.959300
\(448\) 0 0
\(449\) 8898.00 0.935240 0.467620 0.883930i \(-0.345112\pi\)
0.467620 + 0.883930i \(0.345112\pi\)
\(450\) 0 0
\(451\) −7896.00 −0.824408
\(452\) 0 0
\(453\) 5208.00 0.540162
\(454\) 0 0
\(455\) −2080.00 −0.214312
\(456\) 0 0
\(457\) 10330.0 1.05737 0.528684 0.848819i \(-0.322686\pi\)
0.528684 + 0.848819i \(0.322686\pi\)
\(458\) 0 0
\(459\) 1674.00 0.170230
\(460\) 0 0
\(461\) −1878.00 −0.189734 −0.0948668 0.995490i \(-0.530243\pi\)
−0.0948668 + 0.995490i \(0.530243\pi\)
\(462\) 0 0
\(463\) 13224.0 1.32737 0.663684 0.748013i \(-0.268992\pi\)
0.663684 + 0.748013i \(0.268992\pi\)
\(464\) 0 0
\(465\) −2400.00 −0.239349
\(466\) 0 0
\(467\) −8012.00 −0.793900 −0.396950 0.917840i \(-0.629931\pi\)
−0.396950 + 0.917840i \(0.629931\pi\)
\(468\) 0 0
\(469\) 13376.0 1.31694
\(470\) 0 0
\(471\) −3966.00 −0.387991
\(472\) 0 0
\(473\) −2128.00 −0.206862
\(474\) 0 0
\(475\) −1700.00 −0.164213
\(476\) 0 0
\(477\) 1890.00 0.181420
\(478\) 0 0
\(479\) 1792.00 0.170936 0.0854682 0.996341i \(-0.472761\pi\)
0.0854682 + 0.996341i \(0.472761\pi\)
\(480\) 0 0
\(481\) −7020.00 −0.665456
\(482\) 0 0
\(483\) −9984.00 −0.940554
\(484\) 0 0
\(485\) 7030.00 0.658177
\(486\) 0 0
\(487\) −8272.00 −0.769692 −0.384846 0.922981i \(-0.625745\pi\)
−0.384846 + 0.922981i \(0.625745\pi\)
\(488\) 0 0
\(489\) 2724.00 0.251909
\(490\) 0 0
\(491\) 516.000 0.0474272 0.0237136 0.999719i \(-0.492451\pi\)
0.0237136 + 0.999719i \(0.492451\pi\)
\(492\) 0 0
\(493\) −3596.00 −0.328511
\(494\) 0 0
\(495\) 1260.00 0.114410
\(496\) 0 0
\(497\) 8064.00 0.727807
\(498\) 0 0
\(499\) −14020.0 −1.25776 −0.628879 0.777503i \(-0.716486\pi\)
−0.628879 + 0.777503i \(0.716486\pi\)
\(500\) 0 0
\(501\) 3888.00 0.346713
\(502\) 0 0
\(503\) −1872.00 −0.165941 −0.0829705 0.996552i \(-0.526441\pi\)
−0.0829705 + 0.996552i \(0.526441\pi\)
\(504\) 0 0
\(505\) 4950.00 0.436182
\(506\) 0 0
\(507\) 4563.00 0.399704
\(508\) 0 0
\(509\) −8678.00 −0.755689 −0.377844 0.925869i \(-0.623335\pi\)
−0.377844 + 0.925869i \(0.623335\pi\)
\(510\) 0 0
\(511\) −16992.0 −1.47100
\(512\) 0 0
\(513\) 1836.00 0.158014
\(514\) 0 0
\(515\) 3680.00 0.314874
\(516\) 0 0
\(517\) −7840.00 −0.666930
\(518\) 0 0
\(519\) 6402.00 0.541458
\(520\) 0 0
\(521\) 18074.0 1.51984 0.759920 0.650017i \(-0.225238\pi\)
0.759920 + 0.650017i \(0.225238\pi\)
\(522\) 0 0
\(523\) −20852.0 −1.74339 −0.871696 0.490047i \(-0.836980\pi\)
−0.871696 + 0.490047i \(0.836980\pi\)
\(524\) 0 0
\(525\) −1200.00 −0.0997567
\(526\) 0 0
\(527\) 9920.00 0.819966
\(528\) 0 0
\(529\) 31097.0 2.55585
\(530\) 0 0
\(531\) 1764.00 0.144164
\(532\) 0 0
\(533\) 7332.00 0.595843
\(534\) 0 0
\(535\) −6060.00 −0.489713
\(536\) 0 0
\(537\) −4836.00 −0.388620
\(538\) 0 0
\(539\) 2436.00 0.194668
\(540\) 0 0
\(541\) 12410.0 0.986225 0.493112 0.869966i \(-0.335859\pi\)
0.493112 + 0.869966i \(0.335859\pi\)
\(542\) 0 0
\(543\) 9258.00 0.731674
\(544\) 0 0
\(545\) −9170.00 −0.720733
\(546\) 0 0
\(547\) 3620.00 0.282962 0.141481 0.989941i \(-0.454814\pi\)
0.141481 + 0.989941i \(0.454814\pi\)
\(548\) 0 0
\(549\) −6678.00 −0.519144
\(550\) 0 0
\(551\) −3944.00 −0.304937
\(552\) 0 0
\(553\) −12288.0 −0.944917
\(554\) 0 0
\(555\) −4050.00 −0.309753
\(556\) 0 0
\(557\) −11734.0 −0.892613 −0.446307 0.894880i \(-0.647261\pi\)
−0.446307 + 0.894880i \(0.647261\pi\)
\(558\) 0 0
\(559\) 1976.00 0.149510
\(560\) 0 0
\(561\) −5208.00 −0.391946
\(562\) 0 0
\(563\) −1372.00 −0.102705 −0.0513525 0.998681i \(-0.516353\pi\)
−0.0513525 + 0.998681i \(0.516353\pi\)
\(564\) 0 0
\(565\) 10230.0 0.761733
\(566\) 0 0
\(567\) 1296.00 0.0959910
\(568\) 0 0
\(569\) 18922.0 1.39412 0.697058 0.717015i \(-0.254492\pi\)
0.697058 + 0.717015i \(0.254492\pi\)
\(570\) 0 0
\(571\) 14596.0 1.06974 0.534872 0.844933i \(-0.320360\pi\)
0.534872 + 0.844933i \(0.320360\pi\)
\(572\) 0 0
\(573\) −12624.0 −0.920376
\(574\) 0 0
\(575\) 5200.00 0.377139
\(576\) 0 0
\(577\) −2302.00 −0.166089 −0.0830446 0.996546i \(-0.526464\pi\)
−0.0830446 + 0.996546i \(0.526464\pi\)
\(578\) 0 0
\(579\) −8454.00 −0.606798
\(580\) 0 0
\(581\) −16832.0 −1.20191
\(582\) 0 0
\(583\) −5880.00 −0.417710
\(584\) 0 0
\(585\) −1170.00 −0.0826898
\(586\) 0 0
\(587\) 23292.0 1.63776 0.818879 0.573966i \(-0.194596\pi\)
0.818879 + 0.573966i \(0.194596\pi\)
\(588\) 0 0
\(589\) 10880.0 0.761125
\(590\) 0 0
\(591\) −1254.00 −0.0872803
\(592\) 0 0
\(593\) −16542.0 −1.14553 −0.572764 0.819720i \(-0.694129\pi\)
−0.572764 + 0.819720i \(0.694129\pi\)
\(594\) 0 0
\(595\) 4960.00 0.341748
\(596\) 0 0
\(597\) −10056.0 −0.689388
\(598\) 0 0
\(599\) −7464.00 −0.509133 −0.254567 0.967055i \(-0.581933\pi\)
−0.254567 + 0.967055i \(0.581933\pi\)
\(600\) 0 0
\(601\) −17270.0 −1.17214 −0.586072 0.810259i \(-0.699326\pi\)
−0.586072 + 0.810259i \(0.699326\pi\)
\(602\) 0 0
\(603\) 7524.00 0.508128
\(604\) 0 0
\(605\) 2735.00 0.183791
\(606\) 0 0
\(607\) −984.000 −0.0657979 −0.0328990 0.999459i \(-0.510474\pi\)
−0.0328990 + 0.999459i \(0.510474\pi\)
\(608\) 0 0
\(609\) −2784.00 −0.185244
\(610\) 0 0
\(611\) 7280.00 0.482025
\(612\) 0 0
\(613\) −7278.00 −0.479536 −0.239768 0.970830i \(-0.577071\pi\)
−0.239768 + 0.970830i \(0.577071\pi\)
\(614\) 0 0
\(615\) 4230.00 0.277350
\(616\) 0 0
\(617\) 18090.0 1.18035 0.590175 0.807275i \(-0.299059\pi\)
0.590175 + 0.807275i \(0.299059\pi\)
\(618\) 0 0
\(619\) 24740.0 1.60644 0.803219 0.595684i \(-0.203119\pi\)
0.803219 + 0.595684i \(0.203119\pi\)
\(620\) 0 0
\(621\) −5616.00 −0.362902
\(622\) 0 0
\(623\) −11616.0 −0.747007
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −5712.00 −0.363820
\(628\) 0 0
\(629\) 16740.0 1.06116
\(630\) 0 0
\(631\) −19720.0 −1.24412 −0.622061 0.782969i \(-0.713704\pi\)
−0.622061 + 0.782969i \(0.713704\pi\)
\(632\) 0 0
\(633\) 12828.0 0.805477
\(634\) 0 0
\(635\) 5880.00 0.367466
\(636\) 0 0
\(637\) −2262.00 −0.140697
\(638\) 0 0
\(639\) 4536.00 0.280816
\(640\) 0 0
\(641\) −16542.0 −1.01930 −0.509649 0.860383i \(-0.670225\pi\)
−0.509649 + 0.860383i \(0.670225\pi\)
\(642\) 0 0
\(643\) −10092.0 −0.618957 −0.309479 0.950906i \(-0.600155\pi\)
−0.309479 + 0.950906i \(0.600155\pi\)
\(644\) 0 0
\(645\) 1140.00 0.0695930
\(646\) 0 0
\(647\) 14544.0 0.883746 0.441873 0.897078i \(-0.354314\pi\)
0.441873 + 0.897078i \(0.354314\pi\)
\(648\) 0 0
\(649\) −5488.00 −0.331930
\(650\) 0 0
\(651\) 7680.00 0.462370
\(652\) 0 0
\(653\) −23062.0 −1.38206 −0.691030 0.722826i \(-0.742843\pi\)
−0.691030 + 0.722826i \(0.742843\pi\)
\(654\) 0 0
\(655\) −60.0000 −0.00357923
\(656\) 0 0
\(657\) −9558.00 −0.567569
\(658\) 0 0
\(659\) −28020.0 −1.65630 −0.828152 0.560504i \(-0.810608\pi\)
−0.828152 + 0.560504i \(0.810608\pi\)
\(660\) 0 0
\(661\) 6738.00 0.396487 0.198243 0.980153i \(-0.436476\pi\)
0.198243 + 0.980153i \(0.436476\pi\)
\(662\) 0 0
\(663\) 4836.00 0.283280
\(664\) 0 0
\(665\) 5440.00 0.317224
\(666\) 0 0
\(667\) 12064.0 0.700330
\(668\) 0 0
\(669\) 14136.0 0.816935
\(670\) 0 0
\(671\) 20776.0 1.19530
\(672\) 0 0
\(673\) −14430.0 −0.826502 −0.413251 0.910617i \(-0.635607\pi\)
−0.413251 + 0.910617i \(0.635607\pi\)
\(674\) 0 0
\(675\) −675.000 −0.0384900
\(676\) 0 0
\(677\) 17890.0 1.01561 0.507805 0.861472i \(-0.330457\pi\)
0.507805 + 0.861472i \(0.330457\pi\)
\(678\) 0 0
\(679\) −22496.0 −1.27145
\(680\) 0 0
\(681\) 2196.00 0.123570
\(682\) 0 0
\(683\) 10860.0 0.608413 0.304207 0.952606i \(-0.401609\pi\)
0.304207 + 0.952606i \(0.401609\pi\)
\(684\) 0 0
\(685\) 3950.00 0.220324
\(686\) 0 0
\(687\) −15558.0 −0.864010
\(688\) 0 0
\(689\) 5460.00 0.301900
\(690\) 0 0
\(691\) −8692.00 −0.478523 −0.239261 0.970955i \(-0.576905\pi\)
−0.239261 + 0.970955i \(0.576905\pi\)
\(692\) 0 0
\(693\) −4032.00 −0.221014
\(694\) 0 0
\(695\) 4620.00 0.252153
\(696\) 0 0
\(697\) −17484.0 −0.950149
\(698\) 0 0
\(699\) 11394.0 0.616539
\(700\) 0 0
\(701\) 698.000 0.0376078 0.0188039 0.999823i \(-0.494014\pi\)
0.0188039 + 0.999823i \(0.494014\pi\)
\(702\) 0 0
\(703\) 18360.0 0.985008
\(704\) 0 0
\(705\) 4200.00 0.224370
\(706\) 0 0
\(707\) −15840.0 −0.842609
\(708\) 0 0
\(709\) −2654.00 −0.140583 −0.0702913 0.997527i \(-0.522393\pi\)
−0.0702913 + 0.997527i \(0.522393\pi\)
\(710\) 0 0
\(711\) −6912.00 −0.364585
\(712\) 0 0
\(713\) −33280.0 −1.74803
\(714\) 0 0
\(715\) 3640.00 0.190389
\(716\) 0 0
\(717\) −9360.00 −0.487525
\(718\) 0 0
\(719\) 28240.0 1.46478 0.732388 0.680887i \(-0.238406\pi\)
0.732388 + 0.680887i \(0.238406\pi\)
\(720\) 0 0
\(721\) −11776.0 −0.608268
\(722\) 0 0
\(723\) −4470.00 −0.229932
\(724\) 0 0
\(725\) 1450.00 0.0742781
\(726\) 0 0
\(727\) 8320.00 0.424445 0.212223 0.977221i \(-0.431930\pi\)
0.212223 + 0.977221i \(0.431930\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −4712.00 −0.238413
\(732\) 0 0
\(733\) 2154.00 0.108540 0.0542700 0.998526i \(-0.482717\pi\)
0.0542700 + 0.998526i \(0.482717\pi\)
\(734\) 0 0
\(735\) −1305.00 −0.0654907
\(736\) 0 0
\(737\) −23408.0 −1.16994
\(738\) 0 0
\(739\) 22380.0 1.11402 0.557011 0.830505i \(-0.311948\pi\)
0.557011 + 0.830505i \(0.311948\pi\)
\(740\) 0 0
\(741\) 5304.00 0.262952
\(742\) 0 0
\(743\) 5760.00 0.284406 0.142203 0.989837i \(-0.454581\pi\)
0.142203 + 0.989837i \(0.454581\pi\)
\(744\) 0 0
\(745\) 15110.0 0.743071
\(746\) 0 0
\(747\) −9468.00 −0.463743
\(748\) 0 0
\(749\) 19392.0 0.946019
\(750\) 0 0
\(751\) 6192.00 0.300865 0.150432 0.988620i \(-0.451933\pi\)
0.150432 + 0.988620i \(0.451933\pi\)
\(752\) 0 0
\(753\) 15876.0 0.768331
\(754\) 0 0
\(755\) 8680.00 0.418407
\(756\) 0 0
\(757\) 13666.0 0.656142 0.328071 0.944653i \(-0.393602\pi\)
0.328071 + 0.944653i \(0.393602\pi\)
\(758\) 0 0
\(759\) 17472.0 0.835564
\(760\) 0 0
\(761\) −32022.0 −1.52536 −0.762678 0.646778i \(-0.776116\pi\)
−0.762678 + 0.646778i \(0.776116\pi\)
\(762\) 0 0
\(763\) 29344.0 1.39230
\(764\) 0 0
\(765\) 2790.00 0.131860
\(766\) 0 0
\(767\) 5096.00 0.239903
\(768\) 0 0
\(769\) 22786.0 1.06851 0.534255 0.845323i \(-0.320592\pi\)
0.534255 + 0.845323i \(0.320592\pi\)
\(770\) 0 0
\(771\) 11754.0 0.549040
\(772\) 0 0
\(773\) −8286.00 −0.385546 −0.192773 0.981243i \(-0.561748\pi\)
−0.192773 + 0.981243i \(0.561748\pi\)
\(774\) 0 0
\(775\) −4000.00 −0.185399
\(776\) 0 0
\(777\) 12960.0 0.598375
\(778\) 0 0
\(779\) −19176.0 −0.881966
\(780\) 0 0
\(781\) −14112.0 −0.646565
\(782\) 0 0
\(783\) −1566.00 −0.0714742
\(784\) 0 0
\(785\) −6610.00 −0.300536
\(786\) 0 0
\(787\) −25804.0 −1.16876 −0.584379 0.811481i \(-0.698662\pi\)
−0.584379 + 0.811481i \(0.698662\pi\)
\(788\) 0 0
\(789\) 19872.0 0.896656
\(790\) 0 0
\(791\) −32736.0 −1.47150
\(792\) 0 0
\(793\) −19292.0 −0.863908
\(794\) 0 0
\(795\) 3150.00 0.140527
\(796\) 0 0
\(797\) −17670.0 −0.785324 −0.392662 0.919683i \(-0.628446\pi\)
−0.392662 + 0.919683i \(0.628446\pi\)
\(798\) 0 0
\(799\) −17360.0 −0.768652
\(800\) 0 0
\(801\) −6534.00 −0.288224
\(802\) 0 0
\(803\) 29736.0 1.30680
\(804\) 0 0
\(805\) −16640.0 −0.728550
\(806\) 0 0
\(807\) −8862.00 −0.386564
\(808\) 0 0
\(809\) −7398.00 −0.321508 −0.160754 0.986995i \(-0.551393\pi\)
−0.160754 + 0.986995i \(0.551393\pi\)
\(810\) 0 0
\(811\) −28108.0 −1.21702 −0.608511 0.793545i \(-0.708233\pi\)
−0.608511 + 0.793545i \(0.708233\pi\)
\(812\) 0 0
\(813\) −19728.0 −0.851035
\(814\) 0 0
\(815\) 4540.00 0.195128
\(816\) 0 0
\(817\) −5168.00 −0.221304
\(818\) 0 0
\(819\) 3744.00 0.159739
\(820\) 0 0
\(821\) −30830.0 −1.31057 −0.655283 0.755384i \(-0.727451\pi\)
−0.655283 + 0.755384i \(0.727451\pi\)
\(822\) 0 0
\(823\) −5872.00 −0.248706 −0.124353 0.992238i \(-0.539686\pi\)
−0.124353 + 0.992238i \(0.539686\pi\)
\(824\) 0 0
\(825\) 2100.00 0.0886214
\(826\) 0 0
\(827\) −16308.0 −0.685713 −0.342857 0.939388i \(-0.611394\pi\)
−0.342857 + 0.939388i \(0.611394\pi\)
\(828\) 0 0
\(829\) −28294.0 −1.18539 −0.592697 0.805426i \(-0.701937\pi\)
−0.592697 + 0.805426i \(0.701937\pi\)
\(830\) 0 0
\(831\) 13434.0 0.560795
\(832\) 0 0
\(833\) 5394.00 0.224359
\(834\) 0 0
\(835\) 6480.00 0.268562
\(836\) 0 0
\(837\) 4320.00 0.178400
\(838\) 0 0
\(839\) −20536.0 −0.845032 −0.422516 0.906356i \(-0.638853\pi\)
−0.422516 + 0.906356i \(0.638853\pi\)
\(840\) 0 0
\(841\) −21025.0 −0.862069
\(842\) 0 0
\(843\) 19074.0 0.779292
\(844\) 0 0
\(845\) 7605.00 0.309609
\(846\) 0 0
\(847\) −8752.00 −0.355044
\(848\) 0 0
\(849\) −2580.00 −0.104294
\(850\) 0 0
\(851\) −56160.0 −2.26221
\(852\) 0 0
\(853\) −27710.0 −1.11228 −0.556139 0.831090i \(-0.687718\pi\)
−0.556139 + 0.831090i \(0.687718\pi\)
\(854\) 0 0
\(855\) 3060.00 0.122397
\(856\) 0 0
\(857\) 12858.0 0.512510 0.256255 0.966609i \(-0.417511\pi\)
0.256255 + 0.966609i \(0.417511\pi\)
\(858\) 0 0
\(859\) −3148.00 −0.125039 −0.0625194 0.998044i \(-0.519914\pi\)
−0.0625194 + 0.998044i \(0.519914\pi\)
\(860\) 0 0
\(861\) −13536.0 −0.535779
\(862\) 0 0
\(863\) −48456.0 −1.91131 −0.955656 0.294487i \(-0.904851\pi\)
−0.955656 + 0.294487i \(0.904851\pi\)
\(864\) 0 0
\(865\) 10670.0 0.419411
\(866\) 0 0
\(867\) 3207.00 0.125623
\(868\) 0 0
\(869\) 21504.0 0.839440
\(870\) 0 0
\(871\) 21736.0 0.845576
\(872\) 0 0
\(873\) −12654.0 −0.490576
\(874\) 0 0
\(875\) −2000.00 −0.0772712
\(876\) 0 0
\(877\) −9478.00 −0.364937 −0.182468 0.983212i \(-0.558409\pi\)
−0.182468 + 0.983212i \(0.558409\pi\)
\(878\) 0 0
\(879\) −17382.0 −0.666986
\(880\) 0 0
\(881\) 8178.00 0.312740 0.156370 0.987699i \(-0.450021\pi\)
0.156370 + 0.987699i \(0.450021\pi\)
\(882\) 0 0
\(883\) −316.000 −0.0120433 −0.00602166 0.999982i \(-0.501917\pi\)
−0.00602166 + 0.999982i \(0.501917\pi\)
\(884\) 0 0
\(885\) 2940.00 0.111669
\(886\) 0 0
\(887\) 6304.00 0.238633 0.119317 0.992856i \(-0.461930\pi\)
0.119317 + 0.992856i \(0.461930\pi\)
\(888\) 0 0
\(889\) −18816.0 −0.709863
\(890\) 0 0
\(891\) −2268.00 −0.0852759
\(892\) 0 0
\(893\) −19040.0 −0.713493
\(894\) 0 0
\(895\) −8060.00 −0.301024
\(896\) 0 0
\(897\) −16224.0 −0.603906
\(898\) 0 0
\(899\) −9280.00 −0.344277
\(900\) 0 0
\(901\) −13020.0 −0.481420
\(902\) 0 0
\(903\) −3648.00 −0.134438
\(904\) 0 0
\(905\) 15430.0 0.566752
\(906\) 0 0
\(907\) 1596.00 0.0584281 0.0292141 0.999573i \(-0.490700\pi\)
0.0292141 + 0.999573i \(0.490700\pi\)
\(908\) 0 0
\(909\) −8910.00 −0.325111
\(910\) 0 0
\(911\) 25792.0 0.938010 0.469005 0.883196i \(-0.344613\pi\)
0.469005 + 0.883196i \(0.344613\pi\)
\(912\) 0 0
\(913\) 29456.0 1.06775
\(914\) 0 0
\(915\) −11130.0 −0.402127
\(916\) 0 0
\(917\) 192.000 0.00691428
\(918\) 0 0
\(919\) 9736.00 0.349468 0.174734 0.984616i \(-0.444093\pi\)
0.174734 + 0.984616i \(0.444093\pi\)
\(920\) 0 0
\(921\) 20580.0 0.736302
\(922\) 0 0
\(923\) 13104.0 0.467306
\(924\) 0 0
\(925\) −6750.00 −0.239934
\(926\) 0 0
\(927\) −6624.00 −0.234693
\(928\) 0 0
\(929\) −94.0000 −0.00331974 −0.00165987 0.999999i \(-0.500528\pi\)
−0.00165987 + 0.999999i \(0.500528\pi\)
\(930\) 0 0
\(931\) 5916.00 0.208259
\(932\) 0 0
\(933\) −18744.0 −0.657718
\(934\) 0 0
\(935\) −8680.00 −0.303600
\(936\) 0 0
\(937\) −8678.00 −0.302559 −0.151280 0.988491i \(-0.548339\pi\)
−0.151280 + 0.988491i \(0.548339\pi\)
\(938\) 0 0
\(939\) −33054.0 −1.14875
\(940\) 0 0
\(941\) −28406.0 −0.984069 −0.492035 0.870576i \(-0.663747\pi\)
−0.492035 + 0.870576i \(0.663747\pi\)
\(942\) 0 0
\(943\) 58656.0 2.02556
\(944\) 0 0
\(945\) 2160.00 0.0743543
\(946\) 0 0
\(947\) 31988.0 1.09765 0.548823 0.835939i \(-0.315076\pi\)
0.548823 + 0.835939i \(0.315076\pi\)
\(948\) 0 0
\(949\) −27612.0 −0.944493
\(950\) 0 0
\(951\) −2862.00 −0.0975885
\(952\) 0 0
\(953\) 6714.00 0.228214 0.114107 0.993468i \(-0.463599\pi\)
0.114107 + 0.993468i \(0.463599\pi\)
\(954\) 0 0
\(955\) −21040.0 −0.712920
\(956\) 0 0
\(957\) 4872.00 0.164566
\(958\) 0 0
\(959\) −12640.0 −0.425617
\(960\) 0 0
\(961\) −4191.00 −0.140680
\(962\) 0 0
\(963\) 10908.0 0.365011
\(964\) 0 0
\(965\) −14090.0 −0.470024
\(966\) 0 0
\(967\) −15312.0 −0.509204 −0.254602 0.967046i \(-0.581945\pi\)
−0.254602 + 0.967046i \(0.581945\pi\)
\(968\) 0 0
\(969\) −12648.0 −0.419311
\(970\) 0 0
\(971\) −8540.00 −0.282247 −0.141123 0.989992i \(-0.545071\pi\)
−0.141123 + 0.989992i \(0.545071\pi\)
\(972\) 0 0
\(973\) −14784.0 −0.487105
\(974\) 0 0
\(975\) −1950.00 −0.0640513
\(976\) 0 0
\(977\) −8126.00 −0.266094 −0.133047 0.991110i \(-0.542476\pi\)
−0.133047 + 0.991110i \(0.542476\pi\)
\(978\) 0 0
\(979\) 20328.0 0.663622
\(980\) 0 0
\(981\) 16506.0 0.537203
\(982\) 0 0
\(983\) −1392.00 −0.0451657 −0.0225829 0.999745i \(-0.507189\pi\)
−0.0225829 + 0.999745i \(0.507189\pi\)
\(984\) 0 0
\(985\) −2090.00 −0.0676070
\(986\) 0 0
\(987\) −13440.0 −0.433435
\(988\) 0 0
\(989\) 15808.0 0.508256
\(990\) 0 0
\(991\) 48832.0 1.56529 0.782644 0.622470i \(-0.213871\pi\)
0.782644 + 0.622470i \(0.213871\pi\)
\(992\) 0 0
\(993\) −28188.0 −0.900825
\(994\) 0 0
\(995\) −16760.0 −0.533998
\(996\) 0 0
\(997\) −46926.0 −1.49063 −0.745317 0.666711i \(-0.767702\pi\)
−0.745317 + 0.666711i \(0.767702\pi\)
\(998\) 0 0
\(999\) 7290.00 0.230876
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 960.4.a.h.1.1 1
4.3 odd 2 960.4.a.u.1.1 1
8.3 odd 2 120.4.a.c.1.1 1
8.5 even 2 240.4.a.l.1.1 1
24.5 odd 2 720.4.a.l.1.1 1
24.11 even 2 360.4.a.b.1.1 1
40.3 even 4 600.4.f.c.49.1 2
40.13 odd 4 1200.4.f.o.49.2 2
40.19 odd 2 600.4.a.q.1.1 1
40.27 even 4 600.4.f.c.49.2 2
40.29 even 2 1200.4.a.c.1.1 1
40.37 odd 4 1200.4.f.o.49.1 2
120.59 even 2 1800.4.a.bb.1.1 1
120.83 odd 4 1800.4.f.r.649.2 2
120.107 odd 4 1800.4.f.r.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.c.1.1 1 8.3 odd 2
240.4.a.l.1.1 1 8.5 even 2
360.4.a.b.1.1 1 24.11 even 2
600.4.a.q.1.1 1 40.19 odd 2
600.4.f.c.49.1 2 40.3 even 4
600.4.f.c.49.2 2 40.27 even 4
720.4.a.l.1.1 1 24.5 odd 2
960.4.a.h.1.1 1 1.1 even 1 trivial
960.4.a.u.1.1 1 4.3 odd 2
1200.4.a.c.1.1 1 40.29 even 2
1200.4.f.o.49.1 2 40.37 odd 4
1200.4.f.o.49.2 2 40.13 odd 4
1800.4.a.bb.1.1 1 120.59 even 2
1800.4.f.r.649.1 2 120.107 odd 4
1800.4.f.r.649.2 2 120.83 odd 4