Properties

Label 2496.4.a.k.1.1
Level $2496$
Weight $4$
Character 2496.1
Self dual yes
Analytic conductor $147.269$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2496,4,Mod(1,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, 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("2496.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2496.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: \(147.268767374\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2496.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -6.00000 q^{5} +20.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -6.00000 q^{5} +20.0000 q^{7} +9.00000 q^{9} -24.0000 q^{11} -13.0000 q^{13} -18.0000 q^{15} -30.0000 q^{17} +16.0000 q^{19} +60.0000 q^{21} -72.0000 q^{23} -89.0000 q^{25} +27.0000 q^{27} +282.000 q^{29} +164.000 q^{31} -72.0000 q^{33} -120.000 q^{35} -110.000 q^{37} -39.0000 q^{39} -126.000 q^{41} -164.000 q^{43} -54.0000 q^{45} -204.000 q^{47} +57.0000 q^{49} -90.0000 q^{51} +738.000 q^{53} +144.000 q^{55} +48.0000 q^{57} -120.000 q^{59} -614.000 q^{61} +180.000 q^{63} +78.0000 q^{65} -848.000 q^{67} -216.000 q^{69} +132.000 q^{71} +218.000 q^{73} -267.000 q^{75} -480.000 q^{77} -1096.00 q^{79} +81.0000 q^{81} -552.000 q^{83} +180.000 q^{85} +846.000 q^{87} +210.000 q^{89} -260.000 q^{91} +492.000 q^{93} -96.0000 q^{95} -1726.00 q^{97} -216.000 q^{99} +O(q^{100})\)

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
\(5\) −6.00000 −0.536656 −0.268328 0.963328i \(-0.586471\pi\)
−0.268328 + 0.963328i \(0.586471\pi\)
\(6\) 0 0
\(7\) 20.0000 1.07990 0.539949 0.841698i \(-0.318443\pi\)
0.539949 + 0.841698i \(0.318443\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −24.0000 −0.657843 −0.328921 0.944357i \(-0.606685\pi\)
−0.328921 + 0.944357i \(0.606685\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) −18.0000 −0.309839
\(16\) 0 0
\(17\) −30.0000 −0.428004 −0.214002 0.976833i \(-0.568650\pi\)
−0.214002 + 0.976833i \(0.568650\pi\)
\(18\) 0 0
\(19\) 16.0000 0.193192 0.0965961 0.995324i \(-0.469204\pi\)
0.0965961 + 0.995324i \(0.469204\pi\)
\(20\) 0 0
\(21\) 60.0000 0.623480
\(22\) 0 0
\(23\) −72.0000 −0.652741 −0.326370 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) 0 0
\(25\) −89.0000 −0.712000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 282.000 1.80573 0.902864 0.429927i \(-0.141461\pi\)
0.902864 + 0.429927i \(0.141461\pi\)
\(30\) 0 0
\(31\) 164.000 0.950170 0.475085 0.879940i \(-0.342417\pi\)
0.475085 + 0.879940i \(0.342417\pi\)
\(32\) 0 0
\(33\) −72.0000 −0.379806
\(34\) 0 0
\(35\) −120.000 −0.579534
\(36\) 0 0
\(37\) −110.000 −0.488754 −0.244377 0.969680i \(-0.578583\pi\)
−0.244377 + 0.969680i \(0.578583\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) −126.000 −0.479949 −0.239974 0.970779i \(-0.577139\pi\)
−0.239974 + 0.970779i \(0.577139\pi\)
\(42\) 0 0
\(43\) −164.000 −0.581622 −0.290811 0.956780i \(-0.593925\pi\)
−0.290811 + 0.956780i \(0.593925\pi\)
\(44\) 0 0
\(45\) −54.0000 −0.178885
\(46\) 0 0
\(47\) −204.000 −0.633116 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) 0 0
\(51\) −90.0000 −0.247108
\(52\) 0 0
\(53\) 738.000 1.91268 0.956341 0.292255i \(-0.0944055\pi\)
0.956341 + 0.292255i \(0.0944055\pi\)
\(54\) 0 0
\(55\) 144.000 0.353036
\(56\) 0 0
\(57\) 48.0000 0.111540
\(58\) 0 0
\(59\) −120.000 −0.264791 −0.132396 0.991197i \(-0.542267\pi\)
−0.132396 + 0.991197i \(0.542267\pi\)
\(60\) 0 0
\(61\) −614.000 −1.28876 −0.644382 0.764703i \(-0.722885\pi\)
−0.644382 + 0.764703i \(0.722885\pi\)
\(62\) 0 0
\(63\) 180.000 0.359966
\(64\) 0 0
\(65\) 78.0000 0.148842
\(66\) 0 0
\(67\) −848.000 −1.54626 −0.773132 0.634245i \(-0.781311\pi\)
−0.773132 + 0.634245i \(0.781311\pi\)
\(68\) 0 0
\(69\) −216.000 −0.376860
\(70\) 0 0
\(71\) 132.000 0.220641 0.110321 0.993896i \(-0.464812\pi\)
0.110321 + 0.993896i \(0.464812\pi\)
\(72\) 0 0
\(73\) 218.000 0.349520 0.174760 0.984611i \(-0.444085\pi\)
0.174760 + 0.984611i \(0.444085\pi\)
\(74\) 0 0
\(75\) −267.000 −0.411073
\(76\) 0 0
\(77\) −480.000 −0.710404
\(78\) 0 0
\(79\) −1096.00 −1.56088 −0.780441 0.625230i \(-0.785005\pi\)
−0.780441 + 0.625230i \(0.785005\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −552.000 −0.729998 −0.364999 0.931008i \(-0.618931\pi\)
−0.364999 + 0.931008i \(0.618931\pi\)
\(84\) 0 0
\(85\) 180.000 0.229691
\(86\) 0 0
\(87\) 846.000 1.04254
\(88\) 0 0
\(89\) 210.000 0.250112 0.125056 0.992150i \(-0.460089\pi\)
0.125056 + 0.992150i \(0.460089\pi\)
\(90\) 0 0
\(91\) −260.000 −0.299510
\(92\) 0 0
\(93\) 492.000 0.548581
\(94\) 0 0
\(95\) −96.0000 −0.103678
\(96\) 0 0
\(97\) −1726.00 −1.80669 −0.903344 0.428917i \(-0.858895\pi\)
−0.903344 + 0.428917i \(0.858895\pi\)
\(98\) 0 0
\(99\) −216.000 −0.219281
\(100\) 0 0
\(101\) −798.000 −0.786178 −0.393089 0.919500i \(-0.628594\pi\)
−0.393089 + 0.919500i \(0.628594\pi\)
\(102\) 0 0
\(103\) −520.000 −0.497448 −0.248724 0.968574i \(-0.580011\pi\)
−0.248724 + 0.968574i \(0.580011\pi\)
\(104\) 0 0
\(105\) −360.000 −0.334594
\(106\) 0 0
\(107\) −12.0000 −0.0108419 −0.00542095 0.999985i \(-0.501726\pi\)
−0.00542095 + 0.999985i \(0.501726\pi\)
\(108\) 0 0
\(109\) 1834.00 1.61161 0.805804 0.592182i \(-0.201733\pi\)
0.805804 + 0.592182i \(0.201733\pi\)
\(110\) 0 0
\(111\) −330.000 −0.282182
\(112\) 0 0
\(113\) −366.000 −0.304694 −0.152347 0.988327i \(-0.548683\pi\)
−0.152347 + 0.988327i \(0.548683\pi\)
\(114\) 0 0
\(115\) 432.000 0.350297
\(116\) 0 0
\(117\) −117.000 −0.0924500
\(118\) 0 0
\(119\) −600.000 −0.462201
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) 0 0
\(123\) −378.000 −0.277098
\(124\) 0 0
\(125\) 1284.00 0.918756
\(126\) 0 0
\(127\) 2144.00 1.49803 0.749013 0.662556i \(-0.230528\pi\)
0.749013 + 0.662556i \(0.230528\pi\)
\(128\) 0 0
\(129\) −492.000 −0.335800
\(130\) 0 0
\(131\) 2748.00 1.83278 0.916389 0.400289i \(-0.131090\pi\)
0.916389 + 0.400289i \(0.131090\pi\)
\(132\) 0 0
\(133\) 320.000 0.208628
\(134\) 0 0
\(135\) −162.000 −0.103280
\(136\) 0 0
\(137\) 2754.00 1.71745 0.858723 0.512440i \(-0.171258\pi\)
0.858723 + 0.512440i \(0.171258\pi\)
\(138\) 0 0
\(139\) −2252.00 −1.37419 −0.687094 0.726568i \(-0.741114\pi\)
−0.687094 + 0.726568i \(0.741114\pi\)
\(140\) 0 0
\(141\) −612.000 −0.365530
\(142\) 0 0
\(143\) 312.000 0.182453
\(144\) 0 0
\(145\) −1692.00 −0.969055
\(146\) 0 0
\(147\) 171.000 0.0959445
\(148\) 0 0
\(149\) 1770.00 0.973182 0.486591 0.873630i \(-0.338240\pi\)
0.486591 + 0.873630i \(0.338240\pi\)
\(150\) 0 0
\(151\) −988.000 −0.532466 −0.266233 0.963909i \(-0.585779\pi\)
−0.266233 + 0.963909i \(0.585779\pi\)
\(152\) 0 0
\(153\) −270.000 −0.142668
\(154\) 0 0
\(155\) −984.000 −0.509915
\(156\) 0 0
\(157\) −326.000 −0.165717 −0.0828587 0.996561i \(-0.526405\pi\)
−0.0828587 + 0.996561i \(0.526405\pi\)
\(158\) 0 0
\(159\) 2214.00 1.10429
\(160\) 0 0
\(161\) −1440.00 −0.704894
\(162\) 0 0
\(163\) −1496.00 −0.718870 −0.359435 0.933170i \(-0.617031\pi\)
−0.359435 + 0.933170i \(0.617031\pi\)
\(164\) 0 0
\(165\) 432.000 0.203825
\(166\) 0 0
\(167\) 1116.00 0.517118 0.258559 0.965995i \(-0.416752\pi\)
0.258559 + 0.965995i \(0.416752\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 144.000 0.0643974
\(172\) 0 0
\(173\) −4374.00 −1.92225 −0.961124 0.276116i \(-0.910953\pi\)
−0.961124 + 0.276116i \(0.910953\pi\)
\(174\) 0 0
\(175\) −1780.00 −0.768888
\(176\) 0 0
\(177\) −360.000 −0.152877
\(178\) 0 0
\(179\) −12.0000 −0.00501074 −0.00250537 0.999997i \(-0.500797\pi\)
−0.00250537 + 0.999997i \(0.500797\pi\)
\(180\) 0 0
\(181\) −4718.00 −1.93749 −0.968746 0.248053i \(-0.920209\pi\)
−0.968746 + 0.248053i \(0.920209\pi\)
\(182\) 0 0
\(183\) −1842.00 −0.744069
\(184\) 0 0
\(185\) 660.000 0.262293
\(186\) 0 0
\(187\) 720.000 0.281559
\(188\) 0 0
\(189\) 540.000 0.207827
\(190\) 0 0
\(191\) −1368.00 −0.518246 −0.259123 0.965844i \(-0.583434\pi\)
−0.259123 + 0.965844i \(0.583434\pi\)
\(192\) 0 0
\(193\) −3310.00 −1.23450 −0.617251 0.786766i \(-0.711754\pi\)
−0.617251 + 0.786766i \(0.711754\pi\)
\(194\) 0 0
\(195\) 234.000 0.0859338
\(196\) 0 0
\(197\) −3126.00 −1.13055 −0.565275 0.824903i \(-0.691230\pi\)
−0.565275 + 0.824903i \(0.691230\pi\)
\(198\) 0 0
\(199\) 4664.00 1.66142 0.830709 0.556707i \(-0.187935\pi\)
0.830709 + 0.556707i \(0.187935\pi\)
\(200\) 0 0
\(201\) −2544.00 −0.892736
\(202\) 0 0
\(203\) 5640.00 1.95000
\(204\) 0 0
\(205\) 756.000 0.257567
\(206\) 0 0
\(207\) −648.000 −0.217580
\(208\) 0 0
\(209\) −384.000 −0.127090
\(210\) 0 0
\(211\) 556.000 0.181406 0.0907029 0.995878i \(-0.471089\pi\)
0.0907029 + 0.995878i \(0.471089\pi\)
\(212\) 0 0
\(213\) 396.000 0.127387
\(214\) 0 0
\(215\) 984.000 0.312131
\(216\) 0 0
\(217\) 3280.00 1.02609
\(218\) 0 0
\(219\) 654.000 0.201796
\(220\) 0 0
\(221\) 390.000 0.118707
\(222\) 0 0
\(223\) −268.000 −0.0804781 −0.0402390 0.999190i \(-0.512812\pi\)
−0.0402390 + 0.999190i \(0.512812\pi\)
\(224\) 0 0
\(225\) −801.000 −0.237333
\(226\) 0 0
\(227\) −1800.00 −0.526300 −0.263150 0.964755i \(-0.584761\pi\)
−0.263150 + 0.964755i \(0.584761\pi\)
\(228\) 0 0
\(229\) −2990.00 −0.862816 −0.431408 0.902157i \(-0.641983\pi\)
−0.431408 + 0.902157i \(0.641983\pi\)
\(230\) 0 0
\(231\) −1440.00 −0.410152
\(232\) 0 0
\(233\) 2826.00 0.794581 0.397291 0.917693i \(-0.369951\pi\)
0.397291 + 0.917693i \(0.369951\pi\)
\(234\) 0 0
\(235\) 1224.00 0.339766
\(236\) 0 0
\(237\) −3288.00 −0.901175
\(238\) 0 0
\(239\) −1812.00 −0.490412 −0.245206 0.969471i \(-0.578856\pi\)
−0.245206 + 0.969471i \(0.578856\pi\)
\(240\) 0 0
\(241\) −1582.00 −0.422845 −0.211422 0.977395i \(-0.567810\pi\)
−0.211422 + 0.977395i \(0.567810\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −342.000 −0.0891820
\(246\) 0 0
\(247\) −208.000 −0.0535819
\(248\) 0 0
\(249\) −1656.00 −0.421465
\(250\) 0 0
\(251\) −2148.00 −0.540162 −0.270081 0.962838i \(-0.587050\pi\)
−0.270081 + 0.962838i \(0.587050\pi\)
\(252\) 0 0
\(253\) 1728.00 0.429401
\(254\) 0 0
\(255\) 540.000 0.132612
\(256\) 0 0
\(257\) −558.000 −0.135436 −0.0677181 0.997704i \(-0.521572\pi\)
−0.0677181 + 0.997704i \(0.521572\pi\)
\(258\) 0 0
\(259\) −2200.00 −0.527804
\(260\) 0 0
\(261\) 2538.00 0.601909
\(262\) 0 0
\(263\) 2112.00 0.495177 0.247588 0.968865i \(-0.420362\pi\)
0.247588 + 0.968865i \(0.420362\pi\)
\(264\) 0 0
\(265\) −4428.00 −1.02645
\(266\) 0 0
\(267\) 630.000 0.144402
\(268\) 0 0
\(269\) −5046.00 −1.14372 −0.571859 0.820352i \(-0.693777\pi\)
−0.571859 + 0.820352i \(0.693777\pi\)
\(270\) 0 0
\(271\) −3796.00 −0.850888 −0.425444 0.904985i \(-0.639882\pi\)
−0.425444 + 0.904985i \(0.639882\pi\)
\(272\) 0 0
\(273\) −780.000 −0.172922
\(274\) 0 0
\(275\) 2136.00 0.468384
\(276\) 0 0
\(277\) −5582.00 −1.21079 −0.605397 0.795924i \(-0.706986\pi\)
−0.605397 + 0.795924i \(0.706986\pi\)
\(278\) 0 0
\(279\) 1476.00 0.316723
\(280\) 0 0
\(281\) −1950.00 −0.413976 −0.206988 0.978343i \(-0.566366\pi\)
−0.206988 + 0.978343i \(0.566366\pi\)
\(282\) 0 0
\(283\) 4732.00 0.993951 0.496976 0.867765i \(-0.334444\pi\)
0.496976 + 0.867765i \(0.334444\pi\)
\(284\) 0 0
\(285\) −288.000 −0.0598584
\(286\) 0 0
\(287\) −2520.00 −0.518296
\(288\) 0 0
\(289\) −4013.00 −0.816813
\(290\) 0 0
\(291\) −5178.00 −1.04309
\(292\) 0 0
\(293\) −4998.00 −0.996540 −0.498270 0.867022i \(-0.666031\pi\)
−0.498270 + 0.867022i \(0.666031\pi\)
\(294\) 0 0
\(295\) 720.000 0.142102
\(296\) 0 0
\(297\) −648.000 −0.126602
\(298\) 0 0
\(299\) 936.000 0.181038
\(300\) 0 0
\(301\) −3280.00 −0.628093
\(302\) 0 0
\(303\) −2394.00 −0.453900
\(304\) 0 0
\(305\) 3684.00 0.691624
\(306\) 0 0
\(307\) −6824.00 −1.26862 −0.634310 0.773079i \(-0.718716\pi\)
−0.634310 + 0.773079i \(0.718716\pi\)
\(308\) 0 0
\(309\) −1560.00 −0.287202
\(310\) 0 0
\(311\) −8760.00 −1.59722 −0.798608 0.601852i \(-0.794430\pi\)
−0.798608 + 0.601852i \(0.794430\pi\)
\(312\) 0 0
\(313\) 3962.00 0.715481 0.357740 0.933821i \(-0.383547\pi\)
0.357740 + 0.933821i \(0.383547\pi\)
\(314\) 0 0
\(315\) −1080.00 −0.193178
\(316\) 0 0
\(317\) −7086.00 −1.25549 −0.627744 0.778420i \(-0.716021\pi\)
−0.627744 + 0.778420i \(0.716021\pi\)
\(318\) 0 0
\(319\) −6768.00 −1.18788
\(320\) 0 0
\(321\) −36.0000 −0.00625958
\(322\) 0 0
\(323\) −480.000 −0.0826870
\(324\) 0 0
\(325\) 1157.00 0.197473
\(326\) 0 0
\(327\) 5502.00 0.930463
\(328\) 0 0
\(329\) −4080.00 −0.683701
\(330\) 0 0
\(331\) 9016.00 1.49717 0.748586 0.663037i \(-0.230733\pi\)
0.748586 + 0.663037i \(0.230733\pi\)
\(332\) 0 0
\(333\) −990.000 −0.162918
\(334\) 0 0
\(335\) 5088.00 0.829812
\(336\) 0 0
\(337\) 2306.00 0.372747 0.186374 0.982479i \(-0.440327\pi\)
0.186374 + 0.982479i \(0.440327\pi\)
\(338\) 0 0
\(339\) −1098.00 −0.175915
\(340\) 0 0
\(341\) −3936.00 −0.625063
\(342\) 0 0
\(343\) −5720.00 −0.900440
\(344\) 0 0
\(345\) 1296.00 0.202244
\(346\) 0 0
\(347\) 11076.0 1.71352 0.856759 0.515717i \(-0.172474\pi\)
0.856759 + 0.515717i \(0.172474\pi\)
\(348\) 0 0
\(349\) −2342.00 −0.359210 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) 4650.00 0.701118 0.350559 0.936541i \(-0.385992\pi\)
0.350559 + 0.936541i \(0.385992\pi\)
\(354\) 0 0
\(355\) −792.000 −0.118408
\(356\) 0 0
\(357\) −1800.00 −0.266852
\(358\) 0 0
\(359\) −11268.0 −1.65655 −0.828276 0.560320i \(-0.810678\pi\)
−0.828276 + 0.560320i \(0.810678\pi\)
\(360\) 0 0
\(361\) −6603.00 −0.962677
\(362\) 0 0
\(363\) −2265.00 −0.327498
\(364\) 0 0
\(365\) −1308.00 −0.187572
\(366\) 0 0
\(367\) −7288.00 −1.03660 −0.518298 0.855200i \(-0.673434\pi\)
−0.518298 + 0.855200i \(0.673434\pi\)
\(368\) 0 0
\(369\) −1134.00 −0.159983
\(370\) 0 0
\(371\) 14760.0 2.06550
\(372\) 0 0
\(373\) 9970.00 1.38399 0.691993 0.721904i \(-0.256733\pi\)
0.691993 + 0.721904i \(0.256733\pi\)
\(374\) 0 0
\(375\) 3852.00 0.530444
\(376\) 0 0
\(377\) −3666.00 −0.500819
\(378\) 0 0
\(379\) −13448.0 −1.82263 −0.911316 0.411708i \(-0.864932\pi\)
−0.911316 + 0.411708i \(0.864932\pi\)
\(380\) 0 0
\(381\) 6432.00 0.864885
\(382\) 0 0
\(383\) 11820.0 1.57696 0.788478 0.615064i \(-0.210870\pi\)
0.788478 + 0.615064i \(0.210870\pi\)
\(384\) 0 0
\(385\) 2880.00 0.381243
\(386\) 0 0
\(387\) −1476.00 −0.193874
\(388\) 0 0
\(389\) −174.000 −0.0226790 −0.0113395 0.999936i \(-0.503610\pi\)
−0.0113395 + 0.999936i \(0.503610\pi\)
\(390\) 0 0
\(391\) 2160.00 0.279376
\(392\) 0 0
\(393\) 8244.00 1.05815
\(394\) 0 0
\(395\) 6576.00 0.837657
\(396\) 0 0
\(397\) 2986.00 0.377489 0.188744 0.982026i \(-0.439558\pi\)
0.188744 + 0.982026i \(0.439558\pi\)
\(398\) 0 0
\(399\) 960.000 0.120451
\(400\) 0 0
\(401\) −10566.0 −1.31581 −0.657906 0.753100i \(-0.728558\pi\)
−0.657906 + 0.753100i \(0.728558\pi\)
\(402\) 0 0
\(403\) −2132.00 −0.263530
\(404\) 0 0
\(405\) −486.000 −0.0596285
\(406\) 0 0
\(407\) 2640.00 0.321523
\(408\) 0 0
\(409\) −7270.00 −0.878920 −0.439460 0.898262i \(-0.644830\pi\)
−0.439460 + 0.898262i \(0.644830\pi\)
\(410\) 0 0
\(411\) 8262.00 0.991568
\(412\) 0 0
\(413\) −2400.00 −0.285947
\(414\) 0 0
\(415\) 3312.00 0.391758
\(416\) 0 0
\(417\) −6756.00 −0.793388
\(418\) 0 0
\(419\) 7308.00 0.852074 0.426037 0.904706i \(-0.359909\pi\)
0.426037 + 0.904706i \(0.359909\pi\)
\(420\) 0 0
\(421\) 5938.00 0.687412 0.343706 0.939077i \(-0.388318\pi\)
0.343706 + 0.939077i \(0.388318\pi\)
\(422\) 0 0
\(423\) −1836.00 −0.211039
\(424\) 0 0
\(425\) 2670.00 0.304739
\(426\) 0 0
\(427\) −12280.0 −1.39174
\(428\) 0 0
\(429\) 936.000 0.105339
\(430\) 0 0
\(431\) 11532.0 1.28881 0.644405 0.764685i \(-0.277105\pi\)
0.644405 + 0.764685i \(0.277105\pi\)
\(432\) 0 0
\(433\) −718.000 −0.0796879 −0.0398440 0.999206i \(-0.512686\pi\)
−0.0398440 + 0.999206i \(0.512686\pi\)
\(434\) 0 0
\(435\) −5076.00 −0.559484
\(436\) 0 0
\(437\) −1152.00 −0.126104
\(438\) 0 0
\(439\) 8984.00 0.976726 0.488363 0.872640i \(-0.337594\pi\)
0.488363 + 0.872640i \(0.337594\pi\)
\(440\) 0 0
\(441\) 513.000 0.0553936
\(442\) 0 0
\(443\) −2604.00 −0.279277 −0.139639 0.990203i \(-0.544594\pi\)
−0.139639 + 0.990203i \(0.544594\pi\)
\(444\) 0 0
\(445\) −1260.00 −0.134224
\(446\) 0 0
\(447\) 5310.00 0.561867
\(448\) 0 0
\(449\) −13206.0 −1.38804 −0.694020 0.719956i \(-0.744162\pi\)
−0.694020 + 0.719956i \(0.744162\pi\)
\(450\) 0 0
\(451\) 3024.00 0.315731
\(452\) 0 0
\(453\) −2964.00 −0.307419
\(454\) 0 0
\(455\) 1560.00 0.160734
\(456\) 0 0
\(457\) 8426.00 0.862476 0.431238 0.902238i \(-0.358077\pi\)
0.431238 + 0.902238i \(0.358077\pi\)
\(458\) 0 0
\(459\) −810.000 −0.0823694
\(460\) 0 0
\(461\) −16686.0 −1.68578 −0.842890 0.538086i \(-0.819148\pi\)
−0.842890 + 0.538086i \(0.819148\pi\)
\(462\) 0 0
\(463\) 15932.0 1.59919 0.799593 0.600543i \(-0.205049\pi\)
0.799593 + 0.600543i \(0.205049\pi\)
\(464\) 0 0
\(465\) −2952.00 −0.294399
\(466\) 0 0
\(467\) −18540.0 −1.83711 −0.918553 0.395297i \(-0.870642\pi\)
−0.918553 + 0.395297i \(0.870642\pi\)
\(468\) 0 0
\(469\) −16960.0 −1.66981
\(470\) 0 0
\(471\) −978.000 −0.0956770
\(472\) 0 0
\(473\) 3936.00 0.382616
\(474\) 0 0
\(475\) −1424.00 −0.137553
\(476\) 0 0
\(477\) 6642.00 0.637560
\(478\) 0 0
\(479\) 6180.00 0.589502 0.294751 0.955574i \(-0.404763\pi\)
0.294751 + 0.955574i \(0.404763\pi\)
\(480\) 0 0
\(481\) 1430.00 0.135556
\(482\) 0 0
\(483\) −4320.00 −0.406971
\(484\) 0 0
\(485\) 10356.0 0.969571
\(486\) 0 0
\(487\) 11756.0 1.09387 0.546936 0.837175i \(-0.315794\pi\)
0.546936 + 0.837175i \(0.315794\pi\)
\(488\) 0 0
\(489\) −4488.00 −0.415040
\(490\) 0 0
\(491\) −1908.00 −0.175370 −0.0876852 0.996148i \(-0.527947\pi\)
−0.0876852 + 0.996148i \(0.527947\pi\)
\(492\) 0 0
\(493\) −8460.00 −0.772858
\(494\) 0 0
\(495\) 1296.00 0.117679
\(496\) 0 0
\(497\) 2640.00 0.238270
\(498\) 0 0
\(499\) 8944.00 0.802382 0.401191 0.915995i \(-0.368596\pi\)
0.401191 + 0.915995i \(0.368596\pi\)
\(500\) 0 0
\(501\) 3348.00 0.298558
\(502\) 0 0
\(503\) −6528.00 −0.578666 −0.289333 0.957228i \(-0.593434\pi\)
−0.289333 + 0.957228i \(0.593434\pi\)
\(504\) 0 0
\(505\) 4788.00 0.421907
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) 12114.0 1.05490 0.527450 0.849586i \(-0.323148\pi\)
0.527450 + 0.849586i \(0.323148\pi\)
\(510\) 0 0
\(511\) 4360.00 0.377446
\(512\) 0 0
\(513\) 432.000 0.0371799
\(514\) 0 0
\(515\) 3120.00 0.266958
\(516\) 0 0
\(517\) 4896.00 0.416491
\(518\) 0 0
\(519\) −13122.0 −1.10981
\(520\) 0 0
\(521\) −14310.0 −1.20333 −0.601663 0.798750i \(-0.705495\pi\)
−0.601663 + 0.798750i \(0.705495\pi\)
\(522\) 0 0
\(523\) 18340.0 1.53337 0.766685 0.642024i \(-0.221905\pi\)
0.766685 + 0.642024i \(0.221905\pi\)
\(524\) 0 0
\(525\) −5340.00 −0.443918
\(526\) 0 0
\(527\) −4920.00 −0.406677
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) −1080.00 −0.0882637
\(532\) 0 0
\(533\) 1638.00 0.133114
\(534\) 0 0
\(535\) 72.0000 0.00581838
\(536\) 0 0
\(537\) −36.0000 −0.00289295
\(538\) 0 0
\(539\) −1368.00 −0.109321
\(540\) 0 0
\(541\) −9254.00 −0.735417 −0.367708 0.929941i \(-0.619858\pi\)
−0.367708 + 0.929941i \(0.619858\pi\)
\(542\) 0 0
\(543\) −14154.0 −1.11861
\(544\) 0 0
\(545\) −11004.0 −0.864880
\(546\) 0 0
\(547\) −17444.0 −1.36353 −0.681766 0.731571i \(-0.738788\pi\)
−0.681766 + 0.731571i \(0.738788\pi\)
\(548\) 0 0
\(549\) −5526.00 −0.429588
\(550\) 0 0
\(551\) 4512.00 0.348852
\(552\) 0 0
\(553\) −21920.0 −1.68559
\(554\) 0 0
\(555\) 1980.00 0.151435
\(556\) 0 0
\(557\) 3714.00 0.282526 0.141263 0.989972i \(-0.454884\pi\)
0.141263 + 0.989972i \(0.454884\pi\)
\(558\) 0 0
\(559\) 2132.00 0.161313
\(560\) 0 0
\(561\) 2160.00 0.162558
\(562\) 0 0
\(563\) 13812.0 1.03394 0.516968 0.856004i \(-0.327060\pi\)
0.516968 + 0.856004i \(0.327060\pi\)
\(564\) 0 0
\(565\) 2196.00 0.163516
\(566\) 0 0
\(567\) 1620.00 0.119989
\(568\) 0 0
\(569\) −15942.0 −1.17456 −0.587279 0.809385i \(-0.699801\pi\)
−0.587279 + 0.809385i \(0.699801\pi\)
\(570\) 0 0
\(571\) −1604.00 −0.117557 −0.0587787 0.998271i \(-0.518721\pi\)
−0.0587787 + 0.998271i \(0.518721\pi\)
\(572\) 0 0
\(573\) −4104.00 −0.299210
\(574\) 0 0
\(575\) 6408.00 0.464751
\(576\) 0 0
\(577\) −10654.0 −0.768686 −0.384343 0.923190i \(-0.625572\pi\)
−0.384343 + 0.923190i \(0.625572\pi\)
\(578\) 0 0
\(579\) −9930.00 −0.712740
\(580\) 0 0
\(581\) −11040.0 −0.788324
\(582\) 0 0
\(583\) −17712.0 −1.25824
\(584\) 0 0
\(585\) 702.000 0.0496139
\(586\) 0 0
\(587\) 9984.00 0.702017 0.351008 0.936372i \(-0.385839\pi\)
0.351008 + 0.936372i \(0.385839\pi\)
\(588\) 0 0
\(589\) 2624.00 0.183565
\(590\) 0 0
\(591\) −9378.00 −0.652723
\(592\) 0 0
\(593\) 12618.0 0.873793 0.436896 0.899512i \(-0.356078\pi\)
0.436896 + 0.899512i \(0.356078\pi\)
\(594\) 0 0
\(595\) 3600.00 0.248043
\(596\) 0 0
\(597\) 13992.0 0.959220
\(598\) 0 0
\(599\) 11184.0 0.762881 0.381441 0.924393i \(-0.375428\pi\)
0.381441 + 0.924393i \(0.375428\pi\)
\(600\) 0 0
\(601\) 2810.00 0.190719 0.0953596 0.995443i \(-0.469600\pi\)
0.0953596 + 0.995443i \(0.469600\pi\)
\(602\) 0 0
\(603\) −7632.00 −0.515421
\(604\) 0 0
\(605\) 4530.00 0.304414
\(606\) 0 0
\(607\) 1064.00 0.0711473 0.0355737 0.999367i \(-0.488674\pi\)
0.0355737 + 0.999367i \(0.488674\pi\)
\(608\) 0 0
\(609\) 16920.0 1.12583
\(610\) 0 0
\(611\) 2652.00 0.175595
\(612\) 0 0
\(613\) 20914.0 1.37799 0.688996 0.724766i \(-0.258052\pi\)
0.688996 + 0.724766i \(0.258052\pi\)
\(614\) 0 0
\(615\) 2268.00 0.148707
\(616\) 0 0
\(617\) 9714.00 0.633826 0.316913 0.948455i \(-0.397354\pi\)
0.316913 + 0.948455i \(0.397354\pi\)
\(618\) 0 0
\(619\) 14848.0 0.964122 0.482061 0.876138i \(-0.339888\pi\)
0.482061 + 0.876138i \(0.339888\pi\)
\(620\) 0 0
\(621\) −1944.00 −0.125620
\(622\) 0 0
\(623\) 4200.00 0.270095
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 0 0
\(627\) −1152.00 −0.0733755
\(628\) 0 0
\(629\) 3300.00 0.209189
\(630\) 0 0
\(631\) 19172.0 1.20955 0.604774 0.796397i \(-0.293263\pi\)
0.604774 + 0.796397i \(0.293263\pi\)
\(632\) 0 0
\(633\) 1668.00 0.104735
\(634\) 0 0
\(635\) −12864.0 −0.803925
\(636\) 0 0
\(637\) −741.000 −0.0460902
\(638\) 0 0
\(639\) 1188.00 0.0735470
\(640\) 0 0
\(641\) −11502.0 −0.708739 −0.354369 0.935105i \(-0.615304\pi\)
−0.354369 + 0.935105i \(0.615304\pi\)
\(642\) 0 0
\(643\) 15568.0 0.954809 0.477404 0.878684i \(-0.341578\pi\)
0.477404 + 0.878684i \(0.341578\pi\)
\(644\) 0 0
\(645\) 2952.00 0.180209
\(646\) 0 0
\(647\) 1128.00 0.0685414 0.0342707 0.999413i \(-0.489089\pi\)
0.0342707 + 0.999413i \(0.489089\pi\)
\(648\) 0 0
\(649\) 2880.00 0.174191
\(650\) 0 0
\(651\) 9840.00 0.592412
\(652\) 0 0
\(653\) −8118.00 −0.486496 −0.243248 0.969964i \(-0.578213\pi\)
−0.243248 + 0.969964i \(0.578213\pi\)
\(654\) 0 0
\(655\) −16488.0 −0.983572
\(656\) 0 0
\(657\) 1962.00 0.116507
\(658\) 0 0
\(659\) −13572.0 −0.802261 −0.401131 0.916021i \(-0.631383\pi\)
−0.401131 + 0.916021i \(0.631383\pi\)
\(660\) 0 0
\(661\) 13138.0 0.773085 0.386542 0.922272i \(-0.373669\pi\)
0.386542 + 0.922272i \(0.373669\pi\)
\(662\) 0 0
\(663\) 1170.00 0.0685355
\(664\) 0 0
\(665\) −1920.00 −0.111962
\(666\) 0 0
\(667\) −20304.0 −1.17867
\(668\) 0 0
\(669\) −804.000 −0.0464640
\(670\) 0 0
\(671\) 14736.0 0.847805
\(672\) 0 0
\(673\) −718.000 −0.0411246 −0.0205623 0.999789i \(-0.506546\pi\)
−0.0205623 + 0.999789i \(0.506546\pi\)
\(674\) 0 0
\(675\) −2403.00 −0.137024
\(676\) 0 0
\(677\) 2994.00 0.169969 0.0849843 0.996382i \(-0.472916\pi\)
0.0849843 + 0.996382i \(0.472916\pi\)
\(678\) 0 0
\(679\) −34520.0 −1.95104
\(680\) 0 0
\(681\) −5400.00 −0.303860
\(682\) 0 0
\(683\) −27384.0 −1.53414 −0.767071 0.641562i \(-0.778287\pi\)
−0.767071 + 0.641562i \(0.778287\pi\)
\(684\) 0 0
\(685\) −16524.0 −0.921678
\(686\) 0 0
\(687\) −8970.00 −0.498147
\(688\) 0 0
\(689\) −9594.00 −0.530482
\(690\) 0 0
\(691\) −27632.0 −1.52123 −0.760616 0.649202i \(-0.775103\pi\)
−0.760616 + 0.649202i \(0.775103\pi\)
\(692\) 0 0
\(693\) −4320.00 −0.236801
\(694\) 0 0
\(695\) 13512.0 0.737467
\(696\) 0 0
\(697\) 3780.00 0.205420
\(698\) 0 0
\(699\) 8478.00 0.458752
\(700\) 0 0
\(701\) −19062.0 −1.02705 −0.513525 0.858075i \(-0.671661\pi\)
−0.513525 + 0.858075i \(0.671661\pi\)
\(702\) 0 0
\(703\) −1760.00 −0.0944234
\(704\) 0 0
\(705\) 3672.00 0.196164
\(706\) 0 0
\(707\) −15960.0 −0.848992
\(708\) 0 0
\(709\) −3854.00 −0.204147 −0.102073 0.994777i \(-0.532548\pi\)
−0.102073 + 0.994777i \(0.532548\pi\)
\(710\) 0 0
\(711\) −9864.00 −0.520294
\(712\) 0 0
\(713\) −11808.0 −0.620215
\(714\) 0 0
\(715\) −1872.00 −0.0979144
\(716\) 0 0
\(717\) −5436.00 −0.283140
\(718\) 0 0
\(719\) 20976.0 1.08800 0.544001 0.839085i \(-0.316909\pi\)
0.544001 + 0.839085i \(0.316909\pi\)
\(720\) 0 0
\(721\) −10400.0 −0.537193
\(722\) 0 0
\(723\) −4746.00 −0.244130
\(724\) 0 0
\(725\) −25098.0 −1.28568
\(726\) 0 0
\(727\) −29464.0 −1.50311 −0.751554 0.659672i \(-0.770695\pi\)
−0.751554 + 0.659672i \(0.770695\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 4920.00 0.248937
\(732\) 0 0
\(733\) 2698.00 0.135952 0.0679761 0.997687i \(-0.478346\pi\)
0.0679761 + 0.997687i \(0.478346\pi\)
\(734\) 0 0
\(735\) −1026.00 −0.0514892
\(736\) 0 0
\(737\) 20352.0 1.01720
\(738\) 0 0
\(739\) −632.000 −0.0314594 −0.0157297 0.999876i \(-0.505007\pi\)
−0.0157297 + 0.999876i \(0.505007\pi\)
\(740\) 0 0
\(741\) −624.000 −0.0309355
\(742\) 0 0
\(743\) −20844.0 −1.02920 −0.514598 0.857432i \(-0.672059\pi\)
−0.514598 + 0.857432i \(0.672059\pi\)
\(744\) 0 0
\(745\) −10620.0 −0.522264
\(746\) 0 0
\(747\) −4968.00 −0.243333
\(748\) 0 0
\(749\) −240.000 −0.0117082
\(750\) 0 0
\(751\) 272.000 0.0132163 0.00660814 0.999978i \(-0.497897\pi\)
0.00660814 + 0.999978i \(0.497897\pi\)
\(752\) 0 0
\(753\) −6444.00 −0.311862
\(754\) 0 0
\(755\) 5928.00 0.285751
\(756\) 0 0
\(757\) −37550.0 −1.80288 −0.901439 0.432907i \(-0.857488\pi\)
−0.901439 + 0.432907i \(0.857488\pi\)
\(758\) 0 0
\(759\) 5184.00 0.247915
\(760\) 0 0
\(761\) 33330.0 1.58766 0.793832 0.608138i \(-0.208083\pi\)
0.793832 + 0.608138i \(0.208083\pi\)
\(762\) 0 0
\(763\) 36680.0 1.74037
\(764\) 0 0
\(765\) 1620.00 0.0765637
\(766\) 0 0
\(767\) 1560.00 0.0734398
\(768\) 0 0
\(769\) −15406.0 −0.722438 −0.361219 0.932481i \(-0.617639\pi\)
−0.361219 + 0.932481i \(0.617639\pi\)
\(770\) 0 0
\(771\) −1674.00 −0.0781941
\(772\) 0 0
\(773\) 29514.0 1.37328 0.686640 0.726998i \(-0.259085\pi\)
0.686640 + 0.726998i \(0.259085\pi\)
\(774\) 0 0
\(775\) −14596.0 −0.676521
\(776\) 0 0
\(777\) −6600.00 −0.304728
\(778\) 0 0
\(779\) −2016.00 −0.0927223
\(780\) 0 0
\(781\) −3168.00 −0.145147
\(782\) 0 0
\(783\) 7614.00 0.347512
\(784\) 0 0
\(785\) 1956.00 0.0889333
\(786\) 0 0
\(787\) −33176.0 −1.50266 −0.751332 0.659924i \(-0.770588\pi\)
−0.751332 + 0.659924i \(0.770588\pi\)
\(788\) 0 0
\(789\) 6336.00 0.285890
\(790\) 0 0
\(791\) −7320.00 −0.329038
\(792\) 0 0
\(793\) 7982.00 0.357439
\(794\) 0 0
\(795\) −13284.0 −0.592623
\(796\) 0 0
\(797\) 16746.0 0.744258 0.372129 0.928181i \(-0.378628\pi\)
0.372129 + 0.928181i \(0.378628\pi\)
\(798\) 0 0
\(799\) 6120.00 0.270976
\(800\) 0 0
\(801\) 1890.00 0.0833706
\(802\) 0 0
\(803\) −5232.00 −0.229929
\(804\) 0 0
\(805\) 8640.00 0.378286
\(806\) 0 0
\(807\) −15138.0 −0.660326
\(808\) 0 0
\(809\) −15846.0 −0.688647 −0.344324 0.938851i \(-0.611892\pi\)
−0.344324 + 0.938851i \(0.611892\pi\)
\(810\) 0 0
\(811\) −22952.0 −0.993778 −0.496889 0.867814i \(-0.665524\pi\)
−0.496889 + 0.867814i \(0.665524\pi\)
\(812\) 0 0
\(813\) −11388.0 −0.491260
\(814\) 0 0
\(815\) 8976.00 0.385786
\(816\) 0 0
\(817\) −2624.00 −0.112365
\(818\) 0 0
\(819\) −2340.00 −0.0998367
\(820\) 0 0
\(821\) 37146.0 1.57906 0.789528 0.613715i \(-0.210326\pi\)
0.789528 + 0.613715i \(0.210326\pi\)
\(822\) 0 0
\(823\) −9592.00 −0.406265 −0.203133 0.979151i \(-0.565112\pi\)
−0.203133 + 0.979151i \(0.565112\pi\)
\(824\) 0 0
\(825\) 6408.00 0.270422
\(826\) 0 0
\(827\) 39960.0 1.68022 0.840112 0.542413i \(-0.182489\pi\)
0.840112 + 0.542413i \(0.182489\pi\)
\(828\) 0 0
\(829\) 3706.00 0.155265 0.0776325 0.996982i \(-0.475264\pi\)
0.0776325 + 0.996982i \(0.475264\pi\)
\(830\) 0 0
\(831\) −16746.0 −0.699052
\(832\) 0 0
\(833\) −1710.00 −0.0711260
\(834\) 0 0
\(835\) −6696.00 −0.277515
\(836\) 0 0
\(837\) 4428.00 0.182860
\(838\) 0 0
\(839\) 9756.00 0.401448 0.200724 0.979648i \(-0.435671\pi\)
0.200724 + 0.979648i \(0.435671\pi\)
\(840\) 0 0
\(841\) 55135.0 2.26065
\(842\) 0 0
\(843\) −5850.00 −0.239009
\(844\) 0 0
\(845\) −1014.00 −0.0412813
\(846\) 0 0
\(847\) −15100.0 −0.612565
\(848\) 0 0
\(849\) 14196.0 0.573858
\(850\) 0 0
\(851\) 7920.00 0.319029
\(852\) 0 0
\(853\) −11342.0 −0.455267 −0.227633 0.973747i \(-0.573099\pi\)
−0.227633 + 0.973747i \(0.573099\pi\)
\(854\) 0 0
\(855\) −864.000 −0.0345593
\(856\) 0 0
\(857\) −16134.0 −0.643089 −0.321544 0.946895i \(-0.604202\pi\)
−0.321544 + 0.946895i \(0.604202\pi\)
\(858\) 0 0
\(859\) 20932.0 0.831421 0.415710 0.909497i \(-0.363533\pi\)
0.415710 + 0.909497i \(0.363533\pi\)
\(860\) 0 0
\(861\) −7560.00 −0.299238
\(862\) 0 0
\(863\) 10044.0 0.396178 0.198089 0.980184i \(-0.436526\pi\)
0.198089 + 0.980184i \(0.436526\pi\)
\(864\) 0 0
\(865\) 26244.0 1.03159
\(866\) 0 0
\(867\) −12039.0 −0.471587
\(868\) 0 0
\(869\) 26304.0 1.02681
\(870\) 0 0
\(871\) 11024.0 0.428856
\(872\) 0 0
\(873\) −15534.0 −0.602229
\(874\) 0 0
\(875\) 25680.0 0.992163
\(876\) 0 0
\(877\) 26314.0 1.01318 0.506591 0.862186i \(-0.330905\pi\)
0.506591 + 0.862186i \(0.330905\pi\)
\(878\) 0 0
\(879\) −14994.0 −0.575353
\(880\) 0 0
\(881\) 37506.0 1.43429 0.717145 0.696924i \(-0.245449\pi\)
0.717145 + 0.696924i \(0.245449\pi\)
\(882\) 0 0
\(883\) 6388.00 0.243458 0.121729 0.992563i \(-0.461156\pi\)
0.121729 + 0.992563i \(0.461156\pi\)
\(884\) 0 0
\(885\) 2160.00 0.0820425
\(886\) 0 0
\(887\) −5472.00 −0.207138 −0.103569 0.994622i \(-0.533026\pi\)
−0.103569 + 0.994622i \(0.533026\pi\)
\(888\) 0 0
\(889\) 42880.0 1.61772
\(890\) 0 0
\(891\) −1944.00 −0.0730937
\(892\) 0 0
\(893\) −3264.00 −0.122313
\(894\) 0 0
\(895\) 72.0000 0.00268904
\(896\) 0 0
\(897\) 2808.00 0.104522
\(898\) 0 0
\(899\) 46248.0 1.71575
\(900\) 0 0
\(901\) −22140.0 −0.818635
\(902\) 0 0
\(903\) −9840.00 −0.362630
\(904\) 0 0
\(905\) 28308.0 1.03977
\(906\) 0 0
\(907\) 7180.00 0.262853 0.131427 0.991326i \(-0.458044\pi\)
0.131427 + 0.991326i \(0.458044\pi\)
\(908\) 0 0
\(909\) −7182.00 −0.262059
\(910\) 0 0
\(911\) 27624.0 1.00464 0.502318 0.864683i \(-0.332481\pi\)
0.502318 + 0.864683i \(0.332481\pi\)
\(912\) 0 0
\(913\) 13248.0 0.480224
\(914\) 0 0
\(915\) 11052.0 0.399309
\(916\) 0 0
\(917\) 54960.0 1.97921
\(918\) 0 0
\(919\) −30256.0 −1.08602 −0.543011 0.839726i \(-0.682716\pi\)
−0.543011 + 0.839726i \(0.682716\pi\)
\(920\) 0 0
\(921\) −20472.0 −0.732438
\(922\) 0 0
\(923\) −1716.00 −0.0611948
\(924\) 0 0
\(925\) 9790.00 0.347993
\(926\) 0 0
\(927\) −4680.00 −0.165816
\(928\) 0 0
\(929\) −1926.00 −0.0680194 −0.0340097 0.999422i \(-0.510828\pi\)
−0.0340097 + 0.999422i \(0.510828\pi\)
\(930\) 0 0
\(931\) 912.000 0.0321048
\(932\) 0 0
\(933\) −26280.0 −0.922153
\(934\) 0 0
\(935\) −4320.00 −0.151101
\(936\) 0 0
\(937\) 3962.00 0.138135 0.0690677 0.997612i \(-0.477998\pi\)
0.0690677 + 0.997612i \(0.477998\pi\)
\(938\) 0 0
\(939\) 11886.0 0.413083
\(940\) 0 0
\(941\) 1074.00 0.0372066 0.0186033 0.999827i \(-0.494078\pi\)
0.0186033 + 0.999827i \(0.494078\pi\)
\(942\) 0 0
\(943\) 9072.00 0.313282
\(944\) 0 0
\(945\) −3240.00 −0.111531
\(946\) 0 0
\(947\) −4848.00 −0.166356 −0.0831778 0.996535i \(-0.526507\pi\)
−0.0831778 + 0.996535i \(0.526507\pi\)
\(948\) 0 0
\(949\) −2834.00 −0.0969394
\(950\) 0 0
\(951\) −21258.0 −0.724856
\(952\) 0 0
\(953\) 762.000 0.0259009 0.0129505 0.999916i \(-0.495878\pi\)
0.0129505 + 0.999916i \(0.495878\pi\)
\(954\) 0 0
\(955\) 8208.00 0.278120
\(956\) 0 0
\(957\) −20304.0 −0.685826
\(958\) 0 0
\(959\) 55080.0 1.85467
\(960\) 0 0
\(961\) −2895.00 −0.0971770
\(962\) 0 0
\(963\) −108.000 −0.00361397
\(964\) 0 0
\(965\) 19860.0 0.662504
\(966\) 0 0
\(967\) 35804.0 1.19067 0.595336 0.803477i \(-0.297019\pi\)
0.595336 + 0.803477i \(0.297019\pi\)
\(968\) 0 0
\(969\) −1440.00 −0.0477394
\(970\) 0 0
\(971\) 4260.00 0.140793 0.0703964 0.997519i \(-0.477574\pi\)
0.0703964 + 0.997519i \(0.477574\pi\)
\(972\) 0 0
\(973\) −45040.0 −1.48398
\(974\) 0 0
\(975\) 3471.00 0.114011
\(976\) 0 0
\(977\) −28710.0 −0.940137 −0.470069 0.882630i \(-0.655771\pi\)
−0.470069 + 0.882630i \(0.655771\pi\)
\(978\) 0 0
\(979\) −5040.00 −0.164534
\(980\) 0 0
\(981\) 16506.0 0.537203
\(982\) 0 0
\(983\) −49524.0 −1.60689 −0.803444 0.595381i \(-0.797001\pi\)
−0.803444 + 0.595381i \(0.797001\pi\)
\(984\) 0 0
\(985\) 18756.0 0.606717
\(986\) 0 0
\(987\) −12240.0 −0.394735
\(988\) 0 0
\(989\) 11808.0 0.379649
\(990\) 0 0
\(991\) 44408.0 1.42348 0.711739 0.702444i \(-0.247908\pi\)
0.711739 + 0.702444i \(0.247908\pi\)
\(992\) 0 0
\(993\) 27048.0 0.864393
\(994\) 0 0
\(995\) −27984.0 −0.891610
\(996\) 0 0
\(997\) −18398.0 −0.584424 −0.292212 0.956354i \(-0.594391\pi\)
−0.292212 + 0.956354i \(0.594391\pi\)
\(998\) 0 0
\(999\) −2970.00 −0.0940607
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 2496.4.a.k.1.1 1
4.3 odd 2 2496.4.a.b.1.1 1
8.3 odd 2 624.4.a.i.1.1 1
8.5 even 2 78.4.a.e.1.1 1
24.5 odd 2 234.4.a.b.1.1 1
24.11 even 2 1872.4.a.e.1.1 1
40.29 even 2 1950.4.a.c.1.1 1
104.5 odd 4 1014.4.b.c.337.1 2
104.21 odd 4 1014.4.b.c.337.2 2
104.77 even 2 1014.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.e.1.1 1 8.5 even 2
234.4.a.b.1.1 1 24.5 odd 2
624.4.a.i.1.1 1 8.3 odd 2
1014.4.a.b.1.1 1 104.77 even 2
1014.4.b.c.337.1 2 104.5 odd 4
1014.4.b.c.337.2 2 104.21 odd 4
1872.4.a.e.1.1 1 24.11 even 2
1950.4.a.c.1.1 1 40.29 even 2
2496.4.a.b.1.1 1 4.3 odd 2
2496.4.a.k.1.1 1 1.1 even 1 trivial