Properties

Label 624.4.a.i.1.1
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
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] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, 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("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
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 \(\chi\) \(=\) 624.1

$q$-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 \(-0.413529\pi\)
0.268328 + 0.963328i \(0.413529\pi\)
\(6\) 0 0
\(7\) −20.0000 −1.07990 −0.539949 0.841698i \(-0.681557\pi\)
−0.539949 + 0.841698i \(0.681557\pi\)
\(8\) 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 \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 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 \(-0.858539\pi\)
−0.902864 + 0.429927i \(0.858539\pi\)
\(30\) 0 0
\(31\) −164.000 −0.950170 −0.475085 0.879940i \(-0.657583\pi\)
−0.475085 + 0.879940i \(0.657583\pi\)
\(32\) 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 \(-0.421417\pi\)
0.244377 + 0.969680i \(0.421417\pi\)
\(38\) 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 \(-0.397473\pi\)
0.316558 + 0.948573i \(0.397473\pi\)
\(48\) 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 \(-0.905595\pi\)
−0.956341 + 0.292255i \(0.905595\pi\)
\(54\) 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 \(-0.277115\pi\)
0.644382 + 0.764703i \(0.277115\pi\)
\(62\) 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 \(-0.535188\pi\)
−0.110321 + 0.993896i \(0.535188\pi\)
\(72\) 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 \(-0.214995\pi\)
0.780441 + 0.625230i \(0.214995\pi\)
\(80\) 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 \(-0.371406\pi\)
0.393089 + 0.919500i \(0.371406\pi\)
\(102\) 0 0
\(103\) 520.000 0.497448 0.248724 0.968574i \(-0.419989\pi\)
0.248724 + 0.968574i \(0.419989\pi\)
\(104\) 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 \(-0.798267\pi\)
−0.805804 + 0.592182i \(0.798267\pi\)
\(110\) 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 \(-0.769472\pi\)
−0.749013 + 0.662556i \(0.769472\pi\)
\(128\) 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 \(-0.661760\pi\)
−0.486591 + 0.873630i \(0.661760\pi\)
\(150\) 0 0
\(151\) 988.000 0.532466 0.266233 0.963909i \(-0.414221\pi\)
0.266233 + 0.963909i \(0.414221\pi\)
\(152\) 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 \(-0.473595\pi\)
0.0828587 + 0.996561i \(0.473595\pi\)
\(158\) 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 \(-0.583248\pi\)
−0.258559 + 0.965995i \(0.583248\pi\)
\(168\) 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 \(-0.0890472\pi\)
0.961124 + 0.276116i \(0.0890472\pi\)
\(174\) 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 \(-0.0797909\pi\)
0.968746 + 0.248053i \(0.0797909\pi\)
\(182\) 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 \(-0.416566\pi\)
0.259123 + 0.965844i \(0.416566\pi\)
\(192\) 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 \(-0.308770\pi\)
0.565275 + 0.824903i \(0.308770\pi\)
\(198\) 0 0
\(199\) −4664.00 −1.66142 −0.830709 0.556707i \(-0.812065\pi\)
−0.830709 + 0.556707i \(0.812065\pi\)
\(200\) 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 \(-0.487188\pi\)
0.0402390 + 0.999190i \(0.487188\pi\)
\(224\) 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 \(-0.358017\pi\)
0.431408 + 0.902157i \(0.358017\pi\)
\(230\) 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 \(-0.421144\pi\)
0.245206 + 0.969471i \(0.421144\pi\)
\(240\) 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 \(-0.579638\pi\)
−0.247588 + 0.968865i \(0.579638\pi\)
\(264\) 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 \(-0.306223\pi\)
0.571859 + 0.820352i \(0.306223\pi\)
\(270\) 0 0
\(271\) 3796.00 0.850888 0.425444 0.904985i \(-0.360118\pi\)
0.425444 + 0.904985i \(0.360118\pi\)
\(272\) 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 \(-0.293014\pi\)
0.605397 + 0.795924i \(0.293014\pi\)
\(278\) 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 \(-0.333969\pi\)
0.498270 + 0.867022i \(0.333969\pi\)
\(294\) 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 \(-0.205570\pi\)
0.798608 + 0.601852i \(0.205570\pi\)
\(312\) 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 \(-0.283979\pi\)
0.627744 + 0.778420i \(0.283979\pi\)
\(318\) 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 \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(350\) 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 \(-0.189322\pi\)
0.828276 + 0.560320i \(0.189322\pi\)
\(360\) 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 \(-0.326566\pi\)
0.518298 + 0.855200i \(0.326566\pi\)
\(368\) 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 \(-0.743267\pi\)
−0.691993 + 0.721904i \(0.743267\pi\)
\(374\) 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 \(-0.789130\pi\)
−0.788478 + 0.615064i \(0.789130\pi\)
\(384\) 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 \(-0.496390\pi\)
0.0113395 + 0.999936i \(0.496390\pi\)
\(390\) 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 \(-0.560442\pi\)
−0.188744 + 0.982026i \(0.560442\pi\)
\(398\) 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 \(-0.611682\pi\)
−0.343706 + 0.939077i \(0.611682\pi\)
\(422\) 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 \(-0.722895\pi\)
−0.644405 + 0.764685i \(0.722895\pi\)
\(432\) 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 \(-0.662406\pi\)
−0.488363 + 0.872640i \(0.662406\pi\)
\(440\) 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 \(-0.180852\pi\)
0.842890 + 0.538086i \(0.180852\pi\)
\(462\) 0 0
\(463\) −15932.0 −1.59919 −0.799593 0.600543i \(-0.794951\pi\)
−0.799593 + 0.600543i \(0.794951\pi\)
\(464\) 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 \(-0.595237\pi\)
−0.294751 + 0.955574i \(0.595237\pi\)
\(480\) 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 \(-0.684206\pi\)
−0.546936 + 0.837175i \(0.684206\pi\)
\(488\) 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 \(-0.406566\pi\)
0.289333 + 0.957228i \(0.406566\pi\)
\(504\) 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 \(-0.676852\pi\)
−0.527450 + 0.849586i \(0.676852\pi\)
\(510\) 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 \(-0.380142\pi\)
0.367708 + 0.929941i \(0.380142\pi\)
\(542\) 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 \(-0.545116\pi\)
−0.141263 + 0.989972i \(0.545116\pi\)
\(558\) 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 \(-0.624572\pi\)
−0.381441 + 0.924393i \(0.624572\pi\)
\(600\) 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 \(-0.511326\pi\)
−0.0355737 + 0.999367i \(0.511326\pi\)
\(608\) 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 \(-0.741948\pi\)
−0.688996 + 0.724766i \(0.741948\pi\)
\(614\) 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 \(-0.706737\pi\)
−0.604774 + 0.796397i \(0.706737\pi\)
\(632\) 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 \(-0.510911\pi\)
−0.0342707 + 0.999413i \(0.510911\pi\)
\(648\) 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 \(-0.421787\pi\)
0.243248 + 0.969964i \(0.421787\pi\)
\(654\) 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 \(-0.626331\pi\)
−0.386542 + 0.922272i \(0.626331\pi\)
\(662\) 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 \(-0.527084\pi\)
−0.0849843 + 0.996382i \(0.527084\pi\)
\(678\) 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 \(-0.328339\pi\)
0.513525 + 0.858075i \(0.328339\pi\)
\(702\) 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 \(-0.467452\pi\)
0.102073 + 0.994777i \(0.467452\pi\)
\(710\) 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 \(-0.683091\pi\)
−0.544001 + 0.839085i \(0.683091\pi\)
\(720\) 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 \(-0.229305\pi\)
0.751554 + 0.659672i \(0.229305\pi\)
\(728\) 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 \(-0.521654\pi\)
−0.0679761 + 0.997687i \(0.521654\pi\)
\(734\) 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 \(-0.327941\pi\)
0.514598 + 0.857432i \(0.327941\pi\)
\(744\) 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 \(-0.502103\pi\)
−0.00660814 + 0.999978i \(0.502103\pi\)
\(752\) 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 \(-0.142512\pi\)
0.901439 + 0.432907i \(0.142512\pi\)
\(758\) 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 \(-0.740915\pi\)
−0.686640 + 0.726998i \(0.740915\pi\)
\(774\) 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 \(-0.621372\pi\)
−0.372129 + 0.928181i \(0.621372\pi\)
\(798\) 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 \(-0.789674\pi\)
−0.789528 + 0.613715i \(0.789674\pi\)
\(822\) 0 0
\(823\) 9592.00 0.406265 0.203133 0.979151i \(-0.434888\pi\)
0.203133 + 0.979151i \(0.434888\pi\)
\(824\) 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 \(-0.524736\pi\)
−0.0776325 + 0.996982i \(0.524736\pi\)
\(830\) 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 \(-0.564329\pi\)
−0.200724 + 0.979648i \(0.564329\pi\)
\(840\) 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 \(-0.426901\pi\)
0.227633 + 0.973747i \(0.426901\pi\)
\(854\) 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 \(-0.563474\pi\)
−0.198089 + 0.980184i \(0.563474\pi\)
\(864\) 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 \(-0.669095\pi\)
−0.506591 + 0.862186i \(0.669095\pi\)
\(878\) 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 \(-0.466974\pi\)
0.103569 + 0.994622i \(0.466974\pi\)
\(888\) 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 \(-0.667519\pi\)
−0.502318 + 0.864683i \(0.667519\pi\)
\(912\) 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 \(-0.317284\pi\)
0.543011 + 0.839726i \(0.317284\pi\)
\(920\) 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 \(-0.505922\pi\)
−0.0186033 + 0.999827i \(0.505922\pi\)
\(942\) 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 \(-0.702981\pi\)
−0.595336 + 0.803477i \(0.702981\pi\)
\(968\) 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 \(-0.202999\pi\)
0.803444 + 0.595381i \(0.202999\pi\)
\(984\) 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 \(-0.752092\pi\)
−0.711739 + 0.702444i \(0.752092\pi\)
\(992\) 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 \(-0.405609\pi\)
0.292212 + 0.956354i \(0.405609\pi\)
\(998\) 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 624.4.a.i.1.1 1
3.2 odd 2 1872.4.a.e.1.1 1
4.3 odd 2 78.4.a.e.1.1 1
8.3 odd 2 2496.4.a.k.1.1 1
8.5 even 2 2496.4.a.b.1.1 1
12.11 even 2 234.4.a.b.1.1 1
20.19 odd 2 1950.4.a.c.1.1 1
52.31 even 4 1014.4.b.c.337.1 2
52.47 even 4 1014.4.b.c.337.2 2
52.51 odd 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 4.3 odd 2
234.4.a.b.1.1 1 12.11 even 2
624.4.a.i.1.1 1 1.1 even 1 trivial
1014.4.a.b.1.1 1 52.51 odd 2
1014.4.b.c.337.1 2 52.31 even 4
1014.4.b.c.337.2 2 52.47 even 4
1872.4.a.e.1.1 1 3.2 odd 2
1950.4.a.c.1.1 1 20.19 odd 2
2496.4.a.b.1.1 1 8.5 even 2
2496.4.a.k.1.1 1 8.3 odd 2