Properties

Label 1216.4.a.d.1.1
Level $1216$
Weight $4$
Character 1216.1
Self dual yes
Analytic conductor $71.746$
Analytic rank $0$
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] = [1216,4,Mod(1,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1216.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(71.7463225670\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +8.00000 q^{5} -17.0000 q^{7} -26.0000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +8.00000 q^{5} -17.0000 q^{7} -26.0000 q^{9} +70.0000 q^{11} +61.0000 q^{13} +8.00000 q^{15} +83.0000 q^{17} -19.0000 q^{19} -17.0000 q^{21} +115.000 q^{23} -61.0000 q^{25} -53.0000 q^{27} -279.000 q^{29} -72.0000 q^{31} +70.0000 q^{33} -136.000 q^{35} +34.0000 q^{37} +61.0000 q^{39} +108.000 q^{41} +192.000 q^{43} -208.000 q^{45} -392.000 q^{47} -54.0000 q^{49} +83.0000 q^{51} -131.000 q^{53} +560.000 q^{55} -19.0000 q^{57} +609.000 q^{59} -338.000 q^{61} +442.000 q^{63} +488.000 q^{65} +461.000 q^{67} +115.000 q^{69} +750.000 q^{71} +1177.00 q^{73} -61.0000 q^{75} -1190.00 q^{77} -22.0000 q^{79} +649.000 q^{81} +810.000 q^{83} +664.000 q^{85} -279.000 q^{87} -476.000 q^{89} -1037.00 q^{91} -72.0000 q^{93} -152.000 q^{95} +1426.00 q^{97} -1820.00 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.192450 0.0962250 0.995360i \(-0.469323\pi\)
0.0962250 + 0.995360i \(0.469323\pi\)
\(4\) 0 0
\(5\) 8.00000 0.715542 0.357771 0.933809i \(-0.383537\pi\)
0.357771 + 0.933809i \(0.383537\pi\)
\(6\) 0 0
\(7\) −17.0000 −0.917914 −0.458957 0.888459i \(-0.651777\pi\)
−0.458957 + 0.888459i \(0.651777\pi\)
\(8\) 0 0
\(9\) −26.0000 −0.962963
\(10\) 0 0
\(11\) 70.0000 1.91871 0.959354 0.282204i \(-0.0910657\pi\)
0.959354 + 0.282204i \(0.0910657\pi\)
\(12\) 0 0
\(13\) 61.0000 1.30141 0.650706 0.759330i \(-0.274473\pi\)
0.650706 + 0.759330i \(0.274473\pi\)
\(14\) 0 0
\(15\) 8.00000 0.137706
\(16\) 0 0
\(17\) 83.0000 1.18414 0.592072 0.805885i \(-0.298310\pi\)
0.592072 + 0.805885i \(0.298310\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416
\(20\) 0 0
\(21\) −17.0000 −0.176653
\(22\) 0 0
\(23\) 115.000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 0 0
\(25\) −61.0000 −0.488000
\(26\) 0 0
\(27\) −53.0000 −0.377772
\(28\) 0 0
\(29\) −279.000 −1.78652 −0.893259 0.449543i \(-0.851587\pi\)
−0.893259 + 0.449543i \(0.851587\pi\)
\(30\) 0 0
\(31\) −72.0000 −0.417148 −0.208574 0.978007i \(-0.566882\pi\)
−0.208574 + 0.978007i \(0.566882\pi\)
\(32\) 0 0
\(33\) 70.0000 0.369256
\(34\) 0 0
\(35\) −136.000 −0.656806
\(36\) 0 0
\(37\) 34.0000 0.151069 0.0755347 0.997143i \(-0.475934\pi\)
0.0755347 + 0.997143i \(0.475934\pi\)
\(38\) 0 0
\(39\) 61.0000 0.250457
\(40\) 0 0
\(41\) 108.000 0.411385 0.205692 0.978617i \(-0.434055\pi\)
0.205692 + 0.978617i \(0.434055\pi\)
\(42\) 0 0
\(43\) 192.000 0.680924 0.340462 0.940258i \(-0.389417\pi\)
0.340462 + 0.940258i \(0.389417\pi\)
\(44\) 0 0
\(45\) −208.000 −0.689040
\(46\) 0 0
\(47\) −392.000 −1.21658 −0.608288 0.793716i \(-0.708143\pi\)
−0.608288 + 0.793716i \(0.708143\pi\)
\(48\) 0 0
\(49\) −54.0000 −0.157434
\(50\) 0 0
\(51\) 83.0000 0.227889
\(52\) 0 0
\(53\) −131.000 −0.339514 −0.169757 0.985486i \(-0.554298\pi\)
−0.169757 + 0.985486i \(0.554298\pi\)
\(54\) 0 0
\(55\) 560.000 1.37292
\(56\) 0 0
\(57\) −19.0000 −0.0441511
\(58\) 0 0
\(59\) 609.000 1.34381 0.671907 0.740635i \(-0.265475\pi\)
0.671907 + 0.740635i \(0.265475\pi\)
\(60\) 0 0
\(61\) −338.000 −0.709450 −0.354725 0.934971i \(-0.615426\pi\)
−0.354725 + 0.934971i \(0.615426\pi\)
\(62\) 0 0
\(63\) 442.000 0.883917
\(64\) 0 0
\(65\) 488.000 0.931215
\(66\) 0 0
\(67\) 461.000 0.840599 0.420299 0.907386i \(-0.361925\pi\)
0.420299 + 0.907386i \(0.361925\pi\)
\(68\) 0 0
\(69\) 115.000 0.200643
\(70\) 0 0
\(71\) 750.000 1.25364 0.626821 0.779163i \(-0.284356\pi\)
0.626821 + 0.779163i \(0.284356\pi\)
\(72\) 0 0
\(73\) 1177.00 1.88709 0.943544 0.331247i \(-0.107469\pi\)
0.943544 + 0.331247i \(0.107469\pi\)
\(74\) 0 0
\(75\) −61.0000 −0.0939156
\(76\) 0 0
\(77\) −1190.00 −1.76121
\(78\) 0 0
\(79\) −22.0000 −0.0313316 −0.0156658 0.999877i \(-0.504987\pi\)
−0.0156658 + 0.999877i \(0.504987\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) 0 0
\(83\) 810.000 1.07119 0.535597 0.844474i \(-0.320087\pi\)
0.535597 + 0.844474i \(0.320087\pi\)
\(84\) 0 0
\(85\) 664.000 0.847305
\(86\) 0 0
\(87\) −279.000 −0.343815
\(88\) 0 0
\(89\) −476.000 −0.566920 −0.283460 0.958984i \(-0.591482\pi\)
−0.283460 + 0.958984i \(0.591482\pi\)
\(90\) 0 0
\(91\) −1037.00 −1.19458
\(92\) 0 0
\(93\) −72.0000 −0.0802801
\(94\) 0 0
\(95\) −152.000 −0.164157
\(96\) 0 0
\(97\) 1426.00 1.49266 0.746332 0.665574i \(-0.231813\pi\)
0.746332 + 0.665574i \(0.231813\pi\)
\(98\) 0 0
\(99\) −1820.00 −1.84765
\(100\) 0 0
\(101\) 1230.00 1.21178 0.605889 0.795549i \(-0.292818\pi\)
0.605889 + 0.795549i \(0.292818\pi\)
\(102\) 0 0
\(103\) 310.000 0.296555 0.148278 0.988946i \(-0.452627\pi\)
0.148278 + 0.988946i \(0.452627\pi\)
\(104\) 0 0
\(105\) −136.000 −0.126402
\(106\) 0 0
\(107\) 1791.00 1.61815 0.809077 0.587702i \(-0.199967\pi\)
0.809077 + 0.587702i \(0.199967\pi\)
\(108\) 0 0
\(109\) 779.000 0.684538 0.342269 0.939602i \(-0.388805\pi\)
0.342269 + 0.939602i \(0.388805\pi\)
\(110\) 0 0
\(111\) 34.0000 0.0290733
\(112\) 0 0
\(113\) −1430.00 −1.19047 −0.595235 0.803552i \(-0.702941\pi\)
−0.595235 + 0.803552i \(0.702941\pi\)
\(114\) 0 0
\(115\) 920.000 0.746004
\(116\) 0 0
\(117\) −1586.00 −1.25321
\(118\) 0 0
\(119\) −1411.00 −1.08694
\(120\) 0 0
\(121\) 3569.00 2.68144
\(122\) 0 0
\(123\) 108.000 0.0791710
\(124\) 0 0
\(125\) −1488.00 −1.06473
\(126\) 0 0
\(127\) 466.000 0.325597 0.162798 0.986659i \(-0.447948\pi\)
0.162798 + 0.986659i \(0.447948\pi\)
\(128\) 0 0
\(129\) 192.000 0.131044
\(130\) 0 0
\(131\) −2240.00 −1.49397 −0.746984 0.664842i \(-0.768499\pi\)
−0.746984 + 0.664842i \(0.768499\pi\)
\(132\) 0 0
\(133\) 323.000 0.210584
\(134\) 0 0
\(135\) −424.000 −0.270312
\(136\) 0 0
\(137\) −1145.00 −0.714043 −0.357022 0.934096i \(-0.616208\pi\)
−0.357022 + 0.934096i \(0.616208\pi\)
\(138\) 0 0
\(139\) 2336.00 1.42545 0.712723 0.701446i \(-0.247462\pi\)
0.712723 + 0.701446i \(0.247462\pi\)
\(140\) 0 0
\(141\) −392.000 −0.234130
\(142\) 0 0
\(143\) 4270.00 2.49703
\(144\) 0 0
\(145\) −2232.00 −1.27833
\(146\) 0 0
\(147\) −54.0000 −0.0302983
\(148\) 0 0
\(149\) −3516.00 −1.93317 −0.966584 0.256351i \(-0.917480\pi\)
−0.966584 + 0.256351i \(0.917480\pi\)
\(150\) 0 0
\(151\) −390.000 −0.210184 −0.105092 0.994463i \(-0.533514\pi\)
−0.105092 + 0.994463i \(0.533514\pi\)
\(152\) 0 0
\(153\) −2158.00 −1.14029
\(154\) 0 0
\(155\) −576.000 −0.298487
\(156\) 0 0
\(157\) −2306.00 −1.17222 −0.586111 0.810231i \(-0.699342\pi\)
−0.586111 + 0.810231i \(0.699342\pi\)
\(158\) 0 0
\(159\) −131.000 −0.0653395
\(160\) 0 0
\(161\) −1955.00 −0.956991
\(162\) 0 0
\(163\) 1272.00 0.611231 0.305616 0.952155i \(-0.401138\pi\)
0.305616 + 0.952155i \(0.401138\pi\)
\(164\) 0 0
\(165\) 560.000 0.264218
\(166\) 0 0
\(167\) 1768.00 0.819233 0.409617 0.912258i \(-0.365662\pi\)
0.409617 + 0.912258i \(0.365662\pi\)
\(168\) 0 0
\(169\) 1524.00 0.693673
\(170\) 0 0
\(171\) 494.000 0.220919
\(172\) 0 0
\(173\) 3726.00 1.63747 0.818736 0.574171i \(-0.194675\pi\)
0.818736 + 0.574171i \(0.194675\pi\)
\(174\) 0 0
\(175\) 1037.00 0.447942
\(176\) 0 0
\(177\) 609.000 0.258617
\(178\) 0 0
\(179\) 1048.00 0.437604 0.218802 0.975769i \(-0.429785\pi\)
0.218802 + 0.975769i \(0.429785\pi\)
\(180\) 0 0
\(181\) −2678.00 −1.09975 −0.549873 0.835248i \(-0.685324\pi\)
−0.549873 + 0.835248i \(0.685324\pi\)
\(182\) 0 0
\(183\) −338.000 −0.136534
\(184\) 0 0
\(185\) 272.000 0.108096
\(186\) 0 0
\(187\) 5810.00 2.27203
\(188\) 0 0
\(189\) 901.000 0.346762
\(190\) 0 0
\(191\) 1443.00 0.546659 0.273329 0.961921i \(-0.411875\pi\)
0.273329 + 0.961921i \(0.411875\pi\)
\(192\) 0 0
\(193\) 890.000 0.331936 0.165968 0.986131i \(-0.446925\pi\)
0.165968 + 0.986131i \(0.446925\pi\)
\(194\) 0 0
\(195\) 488.000 0.179212
\(196\) 0 0
\(197\) 300.000 0.108498 0.0542490 0.998527i \(-0.482724\pi\)
0.0542490 + 0.998527i \(0.482724\pi\)
\(198\) 0 0
\(199\) 3339.00 1.18942 0.594712 0.803939i \(-0.297266\pi\)
0.594712 + 0.803939i \(0.297266\pi\)
\(200\) 0 0
\(201\) 461.000 0.161773
\(202\) 0 0
\(203\) 4743.00 1.63987
\(204\) 0 0
\(205\) 864.000 0.294363
\(206\) 0 0
\(207\) −2990.00 −1.00396
\(208\) 0 0
\(209\) −1330.00 −0.440182
\(210\) 0 0
\(211\) 2149.00 0.701153 0.350576 0.936534i \(-0.385986\pi\)
0.350576 + 0.936534i \(0.385986\pi\)
\(212\) 0 0
\(213\) 750.000 0.241264
\(214\) 0 0
\(215\) 1536.00 0.487229
\(216\) 0 0
\(217\) 1224.00 0.382906
\(218\) 0 0
\(219\) 1177.00 0.363170
\(220\) 0 0
\(221\) 5063.00 1.54106
\(222\) 0 0
\(223\) 3834.00 1.15132 0.575658 0.817690i \(-0.304746\pi\)
0.575658 + 0.817690i \(0.304746\pi\)
\(224\) 0 0
\(225\) 1586.00 0.469926
\(226\) 0 0
\(227\) 329.000 0.0961960 0.0480980 0.998843i \(-0.484684\pi\)
0.0480980 + 0.998843i \(0.484684\pi\)
\(228\) 0 0
\(229\) −430.000 −0.124084 −0.0620419 0.998074i \(-0.519761\pi\)
−0.0620419 + 0.998074i \(0.519761\pi\)
\(230\) 0 0
\(231\) −1190.00 −0.338945
\(232\) 0 0
\(233\) −438.000 −0.123152 −0.0615758 0.998102i \(-0.519613\pi\)
−0.0615758 + 0.998102i \(0.519613\pi\)
\(234\) 0 0
\(235\) −3136.00 −0.870511
\(236\) 0 0
\(237\) −22.0000 −0.00602976
\(238\) 0 0
\(239\) 2099.00 0.568088 0.284044 0.958811i \(-0.408324\pi\)
0.284044 + 0.958811i \(0.408324\pi\)
\(240\) 0 0
\(241\) −2996.00 −0.800786 −0.400393 0.916344i \(-0.631126\pi\)
−0.400393 + 0.916344i \(0.631126\pi\)
\(242\) 0 0
\(243\) 2080.00 0.549103
\(244\) 0 0
\(245\) −432.000 −0.112651
\(246\) 0 0
\(247\) −1159.00 −0.298564
\(248\) 0 0
\(249\) 810.000 0.206151
\(250\) 0 0
\(251\) −5518.00 −1.38762 −0.693811 0.720157i \(-0.744070\pi\)
−0.693811 + 0.720157i \(0.744070\pi\)
\(252\) 0 0
\(253\) 8050.00 2.00039
\(254\) 0 0
\(255\) 664.000 0.163064
\(256\) 0 0
\(257\) −4068.00 −0.987373 −0.493687 0.869640i \(-0.664351\pi\)
−0.493687 + 0.869640i \(0.664351\pi\)
\(258\) 0 0
\(259\) −578.000 −0.138669
\(260\) 0 0
\(261\) 7254.00 1.72035
\(262\) 0 0
\(263\) 4992.00 1.17042 0.585209 0.810883i \(-0.301012\pi\)
0.585209 + 0.810883i \(0.301012\pi\)
\(264\) 0 0
\(265\) −1048.00 −0.242936
\(266\) 0 0
\(267\) −476.000 −0.109104
\(268\) 0 0
\(269\) 1970.00 0.446517 0.223258 0.974759i \(-0.428331\pi\)
0.223258 + 0.974759i \(0.428331\pi\)
\(270\) 0 0
\(271\) −1861.00 −0.417150 −0.208575 0.978006i \(-0.566883\pi\)
−0.208575 + 0.978006i \(0.566883\pi\)
\(272\) 0 0
\(273\) −1037.00 −0.229898
\(274\) 0 0
\(275\) −4270.00 −0.936330
\(276\) 0 0
\(277\) 1268.00 0.275042 0.137521 0.990499i \(-0.456086\pi\)
0.137521 + 0.990499i \(0.456086\pi\)
\(278\) 0 0
\(279\) 1872.00 0.401698
\(280\) 0 0
\(281\) −8912.00 −1.89198 −0.945988 0.324201i \(-0.894905\pi\)
−0.945988 + 0.324201i \(0.894905\pi\)
\(282\) 0 0
\(283\) −3302.00 −0.693581 −0.346791 0.937943i \(-0.612729\pi\)
−0.346791 + 0.937943i \(0.612729\pi\)
\(284\) 0 0
\(285\) −152.000 −0.0315919
\(286\) 0 0
\(287\) −1836.00 −0.377616
\(288\) 0 0
\(289\) 1976.00 0.402198
\(290\) 0 0
\(291\) 1426.00 0.287263
\(292\) 0 0
\(293\) 5435.00 1.08367 0.541836 0.840484i \(-0.317729\pi\)
0.541836 + 0.840484i \(0.317729\pi\)
\(294\) 0 0
\(295\) 4872.00 0.961555
\(296\) 0 0
\(297\) −3710.00 −0.724835
\(298\) 0 0
\(299\) 7015.00 1.35682
\(300\) 0 0
\(301\) −3264.00 −0.625029
\(302\) 0 0
\(303\) 1230.00 0.233207
\(304\) 0 0
\(305\) −2704.00 −0.507641
\(306\) 0 0
\(307\) −1740.00 −0.323476 −0.161738 0.986834i \(-0.551710\pi\)
−0.161738 + 0.986834i \(0.551710\pi\)
\(308\) 0 0
\(309\) 310.000 0.0570721
\(310\) 0 0
\(311\) −2837.00 −0.517272 −0.258636 0.965975i \(-0.583273\pi\)
−0.258636 + 0.965975i \(0.583273\pi\)
\(312\) 0 0
\(313\) −1579.00 −0.285145 −0.142572 0.989784i \(-0.545537\pi\)
−0.142572 + 0.989784i \(0.545537\pi\)
\(314\) 0 0
\(315\) 3536.00 0.632479
\(316\) 0 0
\(317\) −10065.0 −1.78330 −0.891651 0.452723i \(-0.850452\pi\)
−0.891651 + 0.452723i \(0.850452\pi\)
\(318\) 0 0
\(319\) −19530.0 −3.42781
\(320\) 0 0
\(321\) 1791.00 0.311414
\(322\) 0 0
\(323\) −1577.00 −0.271661
\(324\) 0 0
\(325\) −3721.00 −0.635089
\(326\) 0 0
\(327\) 779.000 0.131739
\(328\) 0 0
\(329\) 6664.00 1.11671
\(330\) 0 0
\(331\) −2953.00 −0.490367 −0.245184 0.969477i \(-0.578848\pi\)
−0.245184 + 0.969477i \(0.578848\pi\)
\(332\) 0 0
\(333\) −884.000 −0.145474
\(334\) 0 0
\(335\) 3688.00 0.601483
\(336\) 0 0
\(337\) −988.000 −0.159703 −0.0798513 0.996807i \(-0.525445\pi\)
−0.0798513 + 0.996807i \(0.525445\pi\)
\(338\) 0 0
\(339\) −1430.00 −0.229106
\(340\) 0 0
\(341\) −5040.00 −0.800385
\(342\) 0 0
\(343\) 6749.00 1.06242
\(344\) 0 0
\(345\) 920.000 0.143569
\(346\) 0 0
\(347\) 4458.00 0.689677 0.344839 0.938662i \(-0.387934\pi\)
0.344839 + 0.938662i \(0.387934\pi\)
\(348\) 0 0
\(349\) 3522.00 0.540196 0.270098 0.962833i \(-0.412944\pi\)
0.270098 + 0.962833i \(0.412944\pi\)
\(350\) 0 0
\(351\) −3233.00 −0.491638
\(352\) 0 0
\(353\) 8809.00 1.32820 0.664102 0.747642i \(-0.268814\pi\)
0.664102 + 0.747642i \(0.268814\pi\)
\(354\) 0 0
\(355\) 6000.00 0.897034
\(356\) 0 0
\(357\) −1411.00 −0.209182
\(358\) 0 0
\(359\) −12611.0 −1.85399 −0.926996 0.375071i \(-0.877618\pi\)
−0.926996 + 0.375071i \(0.877618\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 3569.00 0.516044
\(364\) 0 0
\(365\) 9416.00 1.35029
\(366\) 0 0
\(367\) 9412.00 1.33870 0.669349 0.742948i \(-0.266573\pi\)
0.669349 + 0.742948i \(0.266573\pi\)
\(368\) 0 0
\(369\) −2808.00 −0.396148
\(370\) 0 0
\(371\) 2227.00 0.311644
\(372\) 0 0
\(373\) −6553.00 −0.909655 −0.454828 0.890579i \(-0.650299\pi\)
−0.454828 + 0.890579i \(0.650299\pi\)
\(374\) 0 0
\(375\) −1488.00 −0.204907
\(376\) 0 0
\(377\) −17019.0 −2.32499
\(378\) 0 0
\(379\) −14111.0 −1.91249 −0.956245 0.292568i \(-0.905490\pi\)
−0.956245 + 0.292568i \(0.905490\pi\)
\(380\) 0 0
\(381\) 466.000 0.0626612
\(382\) 0 0
\(383\) −9902.00 −1.32107 −0.660533 0.750797i \(-0.729670\pi\)
−0.660533 + 0.750797i \(0.729670\pi\)
\(384\) 0 0
\(385\) −9520.00 −1.26022
\(386\) 0 0
\(387\) −4992.00 −0.655704
\(388\) 0 0
\(389\) −258.000 −0.0336276 −0.0168138 0.999859i \(-0.505352\pi\)
−0.0168138 + 0.999859i \(0.505352\pi\)
\(390\) 0 0
\(391\) 9545.00 1.23456
\(392\) 0 0
\(393\) −2240.00 −0.287514
\(394\) 0 0
\(395\) −176.000 −0.0224190
\(396\) 0 0
\(397\) −1224.00 −0.154738 −0.0773688 0.997003i \(-0.524652\pi\)
−0.0773688 + 0.997003i \(0.524652\pi\)
\(398\) 0 0
\(399\) 323.000 0.0405269
\(400\) 0 0
\(401\) 12104.0 1.50734 0.753672 0.657251i \(-0.228281\pi\)
0.753672 + 0.657251i \(0.228281\pi\)
\(402\) 0 0
\(403\) −4392.00 −0.542881
\(404\) 0 0
\(405\) 5192.00 0.637019
\(406\) 0 0
\(407\) 2380.00 0.289858
\(408\) 0 0
\(409\) 6076.00 0.734569 0.367285 0.930109i \(-0.380287\pi\)
0.367285 + 0.930109i \(0.380287\pi\)
\(410\) 0 0
\(411\) −1145.00 −0.137418
\(412\) 0 0
\(413\) −10353.0 −1.23351
\(414\) 0 0
\(415\) 6480.00 0.766484
\(416\) 0 0
\(417\) 2336.00 0.274327
\(418\) 0 0
\(419\) −8148.00 −0.950014 −0.475007 0.879982i \(-0.657554\pi\)
−0.475007 + 0.879982i \(0.657554\pi\)
\(420\) 0 0
\(421\) 13849.0 1.60323 0.801614 0.597842i \(-0.203975\pi\)
0.801614 + 0.597842i \(0.203975\pi\)
\(422\) 0 0
\(423\) 10192.0 1.17152
\(424\) 0 0
\(425\) −5063.00 −0.577863
\(426\) 0 0
\(427\) 5746.00 0.651214
\(428\) 0 0
\(429\) 4270.00 0.480554
\(430\) 0 0
\(431\) −9322.00 −1.04182 −0.520911 0.853611i \(-0.674408\pi\)
−0.520911 + 0.853611i \(0.674408\pi\)
\(432\) 0 0
\(433\) 7090.00 0.786891 0.393445 0.919348i \(-0.371283\pi\)
0.393445 + 0.919348i \(0.371283\pi\)
\(434\) 0 0
\(435\) −2232.00 −0.246014
\(436\) 0 0
\(437\) −2185.00 −0.239182
\(438\) 0 0
\(439\) 9740.00 1.05892 0.529459 0.848336i \(-0.322395\pi\)
0.529459 + 0.848336i \(0.322395\pi\)
\(440\) 0 0
\(441\) 1404.00 0.151603
\(442\) 0 0
\(443\) 4658.00 0.499567 0.249784 0.968302i \(-0.419641\pi\)
0.249784 + 0.968302i \(0.419641\pi\)
\(444\) 0 0
\(445\) −3808.00 −0.405655
\(446\) 0 0
\(447\) −3516.00 −0.372038
\(448\) 0 0
\(449\) 14314.0 1.50450 0.752249 0.658879i \(-0.228969\pi\)
0.752249 + 0.658879i \(0.228969\pi\)
\(450\) 0 0
\(451\) 7560.00 0.789327
\(452\) 0 0
\(453\) −390.000 −0.0404499
\(454\) 0 0
\(455\) −8296.00 −0.854775
\(456\) 0 0
\(457\) 3641.00 0.372689 0.186344 0.982484i \(-0.440336\pi\)
0.186344 + 0.982484i \(0.440336\pi\)
\(458\) 0 0
\(459\) −4399.00 −0.447337
\(460\) 0 0
\(461\) 11540.0 1.16588 0.582941 0.812515i \(-0.301902\pi\)
0.582941 + 0.812515i \(0.301902\pi\)
\(462\) 0 0
\(463\) −13732.0 −1.37836 −0.689179 0.724591i \(-0.742029\pi\)
−0.689179 + 0.724591i \(0.742029\pi\)
\(464\) 0 0
\(465\) −576.000 −0.0574438
\(466\) 0 0
\(467\) −15978.0 −1.58324 −0.791621 0.611013i \(-0.790762\pi\)
−0.791621 + 0.611013i \(0.790762\pi\)
\(468\) 0 0
\(469\) −7837.00 −0.771597
\(470\) 0 0
\(471\) −2306.00 −0.225594
\(472\) 0 0
\(473\) 13440.0 1.30649
\(474\) 0 0
\(475\) 1159.00 0.111955
\(476\) 0 0
\(477\) 3406.00 0.326939
\(478\) 0 0
\(479\) −4300.00 −0.410171 −0.205086 0.978744i \(-0.565747\pi\)
−0.205086 + 0.978744i \(0.565747\pi\)
\(480\) 0 0
\(481\) 2074.00 0.196603
\(482\) 0 0
\(483\) −1955.00 −0.184173
\(484\) 0 0
\(485\) 11408.0 1.06806
\(486\) 0 0
\(487\) −12326.0 −1.14691 −0.573454 0.819238i \(-0.694397\pi\)
−0.573454 + 0.819238i \(0.694397\pi\)
\(488\) 0 0
\(489\) 1272.00 0.117632
\(490\) 0 0
\(491\) 7236.00 0.665084 0.332542 0.943088i \(-0.392094\pi\)
0.332542 + 0.943088i \(0.392094\pi\)
\(492\) 0 0
\(493\) −23157.0 −2.11549
\(494\) 0 0
\(495\) −14560.0 −1.32207
\(496\) 0 0
\(497\) −12750.0 −1.15074
\(498\) 0 0
\(499\) 1148.00 0.102989 0.0514945 0.998673i \(-0.483602\pi\)
0.0514945 + 0.998673i \(0.483602\pi\)
\(500\) 0 0
\(501\) 1768.00 0.157662
\(502\) 0 0
\(503\) 5739.00 0.508726 0.254363 0.967109i \(-0.418134\pi\)
0.254363 + 0.967109i \(0.418134\pi\)
\(504\) 0 0
\(505\) 9840.00 0.867078
\(506\) 0 0
\(507\) 1524.00 0.133497
\(508\) 0 0
\(509\) −8442.00 −0.735138 −0.367569 0.929996i \(-0.619810\pi\)
−0.367569 + 0.929996i \(0.619810\pi\)
\(510\) 0 0
\(511\) −20009.0 −1.73218
\(512\) 0 0
\(513\) 1007.00 0.0866669
\(514\) 0 0
\(515\) 2480.00 0.212198
\(516\) 0 0
\(517\) −27440.0 −2.33425
\(518\) 0 0
\(519\) 3726.00 0.315131
\(520\) 0 0
\(521\) 744.000 0.0625628 0.0312814 0.999511i \(-0.490041\pi\)
0.0312814 + 0.999511i \(0.490041\pi\)
\(522\) 0 0
\(523\) −10013.0 −0.837166 −0.418583 0.908179i \(-0.637473\pi\)
−0.418583 + 0.908179i \(0.637473\pi\)
\(524\) 0 0
\(525\) 1037.00 0.0862065
\(526\) 0 0
\(527\) −5976.00 −0.493963
\(528\) 0 0
\(529\) 1058.00 0.0869565
\(530\) 0 0
\(531\) −15834.0 −1.29404
\(532\) 0 0
\(533\) 6588.00 0.535381
\(534\) 0 0
\(535\) 14328.0 1.15786
\(536\) 0 0
\(537\) 1048.00 0.0842170
\(538\) 0 0
\(539\) −3780.00 −0.302071
\(540\) 0 0
\(541\) −14582.0 −1.15883 −0.579417 0.815031i \(-0.696720\pi\)
−0.579417 + 0.815031i \(0.696720\pi\)
\(542\) 0 0
\(543\) −2678.00 −0.211646
\(544\) 0 0
\(545\) 6232.00 0.489816
\(546\) 0 0
\(547\) −17924.0 −1.40105 −0.700526 0.713627i \(-0.747051\pi\)
−0.700526 + 0.713627i \(0.747051\pi\)
\(548\) 0 0
\(549\) 8788.00 0.683174
\(550\) 0 0
\(551\) 5301.00 0.409855
\(552\) 0 0
\(553\) 374.000 0.0287597
\(554\) 0 0
\(555\) 272.000 0.0208032
\(556\) 0 0
\(557\) −11960.0 −0.909805 −0.454903 0.890541i \(-0.650326\pi\)
−0.454903 + 0.890541i \(0.650326\pi\)
\(558\) 0 0
\(559\) 11712.0 0.886162
\(560\) 0 0
\(561\) 5810.00 0.437252
\(562\) 0 0
\(563\) −1348.00 −0.100908 −0.0504542 0.998726i \(-0.516067\pi\)
−0.0504542 + 0.998726i \(0.516067\pi\)
\(564\) 0 0
\(565\) −11440.0 −0.851831
\(566\) 0 0
\(567\) −11033.0 −0.817182
\(568\) 0 0
\(569\) −9820.00 −0.723508 −0.361754 0.932274i \(-0.617822\pi\)
−0.361754 + 0.932274i \(0.617822\pi\)
\(570\) 0 0
\(571\) 20904.0 1.53206 0.766029 0.642806i \(-0.222230\pi\)
0.766029 + 0.642806i \(0.222230\pi\)
\(572\) 0 0
\(573\) 1443.00 0.105205
\(574\) 0 0
\(575\) −7015.00 −0.508775
\(576\) 0 0
\(577\) −15093.0 −1.08896 −0.544480 0.838774i \(-0.683273\pi\)
−0.544480 + 0.838774i \(0.683273\pi\)
\(578\) 0 0
\(579\) 890.000 0.0638811
\(580\) 0 0
\(581\) −13770.0 −0.983263
\(582\) 0 0
\(583\) −9170.00 −0.651428
\(584\) 0 0
\(585\) −12688.0 −0.896725
\(586\) 0 0
\(587\) 16740.0 1.17706 0.588530 0.808476i \(-0.299707\pi\)
0.588530 + 0.808476i \(0.299707\pi\)
\(588\) 0 0
\(589\) 1368.00 0.0957003
\(590\) 0 0
\(591\) 300.000 0.0208805
\(592\) 0 0
\(593\) 3042.00 0.210658 0.105329 0.994437i \(-0.466410\pi\)
0.105329 + 0.994437i \(0.466410\pi\)
\(594\) 0 0
\(595\) −11288.0 −0.777753
\(596\) 0 0
\(597\) 3339.00 0.228905
\(598\) 0 0
\(599\) 27816.0 1.89738 0.948690 0.316207i \(-0.102409\pi\)
0.948690 + 0.316207i \(0.102409\pi\)
\(600\) 0 0
\(601\) 19320.0 1.31128 0.655640 0.755074i \(-0.272399\pi\)
0.655640 + 0.755074i \(0.272399\pi\)
\(602\) 0 0
\(603\) −11986.0 −0.809465
\(604\) 0 0
\(605\) 28552.0 1.91868
\(606\) 0 0
\(607\) 17926.0 1.19867 0.599336 0.800498i \(-0.295431\pi\)
0.599336 + 0.800498i \(0.295431\pi\)
\(608\) 0 0
\(609\) 4743.00 0.315593
\(610\) 0 0
\(611\) −23912.0 −1.58327
\(612\) 0 0
\(613\) −18294.0 −1.20536 −0.602682 0.797982i \(-0.705901\pi\)
−0.602682 + 0.797982i \(0.705901\pi\)
\(614\) 0 0
\(615\) 864.000 0.0566502
\(616\) 0 0
\(617\) 22538.0 1.47058 0.735288 0.677755i \(-0.237047\pi\)
0.735288 + 0.677755i \(0.237047\pi\)
\(618\) 0 0
\(619\) 17558.0 1.14009 0.570045 0.821614i \(-0.306926\pi\)
0.570045 + 0.821614i \(0.306926\pi\)
\(620\) 0 0
\(621\) −6095.00 −0.393855
\(622\) 0 0
\(623\) 8092.00 0.520384
\(624\) 0 0
\(625\) −4279.00 −0.273856
\(626\) 0 0
\(627\) −1330.00 −0.0847131
\(628\) 0 0
\(629\) 2822.00 0.178888
\(630\) 0 0
\(631\) −12440.0 −0.784831 −0.392416 0.919788i \(-0.628361\pi\)
−0.392416 + 0.919788i \(0.628361\pi\)
\(632\) 0 0
\(633\) 2149.00 0.134937
\(634\) 0 0
\(635\) 3728.00 0.232978
\(636\) 0 0
\(637\) −3294.00 −0.204887
\(638\) 0 0
\(639\) −19500.0 −1.20721
\(640\) 0 0
\(641\) −10842.0 −0.668071 −0.334035 0.942561i \(-0.608410\pi\)
−0.334035 + 0.942561i \(0.608410\pi\)
\(642\) 0 0
\(643\) −322.000 −0.0197487 −0.00987437 0.999951i \(-0.503143\pi\)
−0.00987437 + 0.999951i \(0.503143\pi\)
\(644\) 0 0
\(645\) 1536.00 0.0937674
\(646\) 0 0
\(647\) 2995.00 0.181987 0.0909935 0.995851i \(-0.470996\pi\)
0.0909935 + 0.995851i \(0.470996\pi\)
\(648\) 0 0
\(649\) 42630.0 2.57839
\(650\) 0 0
\(651\) 1224.00 0.0736902
\(652\) 0 0
\(653\) −14652.0 −0.878066 −0.439033 0.898471i \(-0.644679\pi\)
−0.439033 + 0.898471i \(0.644679\pi\)
\(654\) 0 0
\(655\) −17920.0 −1.06900
\(656\) 0 0
\(657\) −30602.0 −1.81720
\(658\) 0 0
\(659\) −20613.0 −1.21847 −0.609233 0.792992i \(-0.708522\pi\)
−0.609233 + 0.792992i \(0.708522\pi\)
\(660\) 0 0
\(661\) 7875.00 0.463392 0.231696 0.972788i \(-0.425573\pi\)
0.231696 + 0.972788i \(0.425573\pi\)
\(662\) 0 0
\(663\) 5063.00 0.296577
\(664\) 0 0
\(665\) 2584.00 0.150682
\(666\) 0 0
\(667\) −32085.0 −1.86257
\(668\) 0 0
\(669\) 3834.00 0.221571
\(670\) 0 0
\(671\) −23660.0 −1.36123
\(672\) 0 0
\(673\) 6216.00 0.356031 0.178016 0.984028i \(-0.443032\pi\)
0.178016 + 0.984028i \(0.443032\pi\)
\(674\) 0 0
\(675\) 3233.00 0.184353
\(676\) 0 0
\(677\) −13961.0 −0.792563 −0.396281 0.918129i \(-0.629699\pi\)
−0.396281 + 0.918129i \(0.629699\pi\)
\(678\) 0 0
\(679\) −24242.0 −1.37014
\(680\) 0 0
\(681\) 329.000 0.0185129
\(682\) 0 0
\(683\) 7788.00 0.436310 0.218155 0.975914i \(-0.429996\pi\)
0.218155 + 0.975914i \(0.429996\pi\)
\(684\) 0 0
\(685\) −9160.00 −0.510928
\(686\) 0 0
\(687\) −430.000 −0.0238799
\(688\) 0 0
\(689\) −7991.00 −0.441847
\(690\) 0 0
\(691\) 11782.0 0.648637 0.324319 0.945948i \(-0.394865\pi\)
0.324319 + 0.945948i \(0.394865\pi\)
\(692\) 0 0
\(693\) 30940.0 1.69598
\(694\) 0 0
\(695\) 18688.0 1.01997
\(696\) 0 0
\(697\) 8964.00 0.487139
\(698\) 0 0
\(699\) −438.000 −0.0237005
\(700\) 0 0
\(701\) 22700.0 1.22306 0.611532 0.791220i \(-0.290554\pi\)
0.611532 + 0.791220i \(0.290554\pi\)
\(702\) 0 0
\(703\) −646.000 −0.0346577
\(704\) 0 0
\(705\) −3136.00 −0.167530
\(706\) 0 0
\(707\) −20910.0 −1.11231
\(708\) 0 0
\(709\) −12162.0 −0.644222 −0.322111 0.946702i \(-0.604392\pi\)
−0.322111 + 0.946702i \(0.604392\pi\)
\(710\) 0 0
\(711\) 572.000 0.0301711
\(712\) 0 0
\(713\) −8280.00 −0.434907
\(714\) 0 0
\(715\) 34160.0 1.78673
\(716\) 0 0
\(717\) 2099.00 0.109329
\(718\) 0 0
\(719\) 2199.00 0.114060 0.0570298 0.998372i \(-0.481837\pi\)
0.0570298 + 0.998372i \(0.481837\pi\)
\(720\) 0 0
\(721\) −5270.00 −0.272212
\(722\) 0 0
\(723\) −2996.00 −0.154111
\(724\) 0 0
\(725\) 17019.0 0.871820
\(726\) 0 0
\(727\) −18965.0 −0.967501 −0.483750 0.875206i \(-0.660726\pi\)
−0.483750 + 0.875206i \(0.660726\pi\)
\(728\) 0 0
\(729\) −15443.0 −0.784586
\(730\) 0 0
\(731\) 15936.0 0.806312
\(732\) 0 0
\(733\) 5552.00 0.279765 0.139883 0.990168i \(-0.455328\pi\)
0.139883 + 0.990168i \(0.455328\pi\)
\(734\) 0 0
\(735\) −432.000 −0.0216797
\(736\) 0 0
\(737\) 32270.0 1.61286
\(738\) 0 0
\(739\) −13136.0 −0.653878 −0.326939 0.945046i \(-0.606017\pi\)
−0.326939 + 0.945046i \(0.606017\pi\)
\(740\) 0 0
\(741\) −1159.00 −0.0574587
\(742\) 0 0
\(743\) −13416.0 −0.662430 −0.331215 0.943555i \(-0.607459\pi\)
−0.331215 + 0.943555i \(0.607459\pi\)
\(744\) 0 0
\(745\) −28128.0 −1.38326
\(746\) 0 0
\(747\) −21060.0 −1.03152
\(748\) 0 0
\(749\) −30447.0 −1.48533
\(750\) 0 0
\(751\) −33348.0 −1.62035 −0.810177 0.586185i \(-0.800629\pi\)
−0.810177 + 0.586185i \(0.800629\pi\)
\(752\) 0 0
\(753\) −5518.00 −0.267048
\(754\) 0 0
\(755\) −3120.00 −0.150395
\(756\) 0 0
\(757\) −10606.0 −0.509223 −0.254611 0.967043i \(-0.581948\pi\)
−0.254611 + 0.967043i \(0.581948\pi\)
\(758\) 0 0
\(759\) 8050.00 0.384976
\(760\) 0 0
\(761\) −29061.0 −1.38431 −0.692155 0.721749i \(-0.743339\pi\)
−0.692155 + 0.721749i \(0.743339\pi\)
\(762\) 0 0
\(763\) −13243.0 −0.628347
\(764\) 0 0
\(765\) −17264.0 −0.815923
\(766\) 0 0
\(767\) 37149.0 1.74886
\(768\) 0 0
\(769\) −15955.0 −0.748182 −0.374091 0.927392i \(-0.622045\pi\)
−0.374091 + 0.927392i \(0.622045\pi\)
\(770\) 0 0
\(771\) −4068.00 −0.190020
\(772\) 0 0
\(773\) 36763.0 1.71057 0.855287 0.518155i \(-0.173381\pi\)
0.855287 + 0.518155i \(0.173381\pi\)
\(774\) 0 0
\(775\) 4392.00 0.203568
\(776\) 0 0
\(777\) −578.000 −0.0266868
\(778\) 0 0
\(779\) −2052.00 −0.0943781
\(780\) 0 0
\(781\) 52500.0 2.40537
\(782\) 0 0
\(783\) 14787.0 0.674897
\(784\) 0 0
\(785\) −18448.0 −0.838774
\(786\) 0 0
\(787\) −31655.0 −1.43377 −0.716886 0.697190i \(-0.754433\pi\)
−0.716886 + 0.697190i \(0.754433\pi\)
\(788\) 0 0
\(789\) 4992.00 0.225247
\(790\) 0 0
\(791\) 24310.0 1.09275
\(792\) 0 0
\(793\) −20618.0 −0.923287
\(794\) 0 0
\(795\) −1048.00 −0.0467531
\(796\) 0 0
\(797\) 12945.0 0.575327 0.287663 0.957732i \(-0.407122\pi\)
0.287663 + 0.957732i \(0.407122\pi\)
\(798\) 0 0
\(799\) −32536.0 −1.44060
\(800\) 0 0
\(801\) 12376.0 0.545923
\(802\) 0 0
\(803\) 82390.0 3.62077
\(804\) 0 0
\(805\) −15640.0 −0.684767
\(806\) 0 0
\(807\) 1970.00 0.0859322
\(808\) 0 0
\(809\) −20263.0 −0.880605 −0.440302 0.897850i \(-0.645129\pi\)
−0.440302 + 0.897850i \(0.645129\pi\)
\(810\) 0 0
\(811\) −29477.0 −1.27630 −0.638149 0.769913i \(-0.720300\pi\)
−0.638149 + 0.769913i \(0.720300\pi\)
\(812\) 0 0
\(813\) −1861.00 −0.0802806
\(814\) 0 0
\(815\) 10176.0 0.437362
\(816\) 0 0
\(817\) −3648.00 −0.156215
\(818\) 0 0
\(819\) 26962.0 1.15034
\(820\) 0 0
\(821\) 1756.00 0.0746466 0.0373233 0.999303i \(-0.488117\pi\)
0.0373233 + 0.999303i \(0.488117\pi\)
\(822\) 0 0
\(823\) 1721.00 0.0728922 0.0364461 0.999336i \(-0.488396\pi\)
0.0364461 + 0.999336i \(0.488396\pi\)
\(824\) 0 0
\(825\) −4270.00 −0.180197
\(826\) 0 0
\(827\) −3063.00 −0.128792 −0.0643960 0.997924i \(-0.520512\pi\)
−0.0643960 + 0.997924i \(0.520512\pi\)
\(828\) 0 0
\(829\) 25467.0 1.06695 0.533477 0.845814i \(-0.320885\pi\)
0.533477 + 0.845814i \(0.320885\pi\)
\(830\) 0 0
\(831\) 1268.00 0.0529319
\(832\) 0 0
\(833\) −4482.00 −0.186425
\(834\) 0 0
\(835\) 14144.0 0.586196
\(836\) 0 0
\(837\) 3816.00 0.157587
\(838\) 0 0
\(839\) 2976.00 0.122459 0.0612294 0.998124i \(-0.480498\pi\)
0.0612294 + 0.998124i \(0.480498\pi\)
\(840\) 0 0
\(841\) 53452.0 2.19164
\(842\) 0 0
\(843\) −8912.00 −0.364111
\(844\) 0 0
\(845\) 12192.0 0.496352
\(846\) 0 0
\(847\) −60673.0 −2.46133
\(848\) 0 0
\(849\) −3302.00 −0.133480
\(850\) 0 0
\(851\) 3910.00 0.157501
\(852\) 0 0
\(853\) −2126.00 −0.0853375 −0.0426687 0.999089i \(-0.513586\pi\)
−0.0426687 + 0.999089i \(0.513586\pi\)
\(854\) 0 0
\(855\) 3952.00 0.158077
\(856\) 0 0
\(857\) −4684.00 −0.186701 −0.0933503 0.995633i \(-0.529758\pi\)
−0.0933503 + 0.995633i \(0.529758\pi\)
\(858\) 0 0
\(859\) −1370.00 −0.0544165 −0.0272083 0.999630i \(-0.508662\pi\)
−0.0272083 + 0.999630i \(0.508662\pi\)
\(860\) 0 0
\(861\) −1836.00 −0.0726721
\(862\) 0 0
\(863\) 30630.0 1.20818 0.604089 0.796917i \(-0.293537\pi\)
0.604089 + 0.796917i \(0.293537\pi\)
\(864\) 0 0
\(865\) 29808.0 1.17168
\(866\) 0 0
\(867\) 1976.00 0.0774031
\(868\) 0 0
\(869\) −1540.00 −0.0601161
\(870\) 0 0
\(871\) 28121.0 1.09397
\(872\) 0 0
\(873\) −37076.0 −1.43738
\(874\) 0 0
\(875\) 25296.0 0.977327
\(876\) 0 0
\(877\) −19617.0 −0.755324 −0.377662 0.925944i \(-0.623272\pi\)
−0.377662 + 0.925944i \(0.623272\pi\)
\(878\) 0 0
\(879\) 5435.00 0.208553
\(880\) 0 0
\(881\) 1614.00 0.0617220 0.0308610 0.999524i \(-0.490175\pi\)
0.0308610 + 0.999524i \(0.490175\pi\)
\(882\) 0 0
\(883\) −35258.0 −1.34374 −0.671872 0.740667i \(-0.734510\pi\)
−0.671872 + 0.740667i \(0.734510\pi\)
\(884\) 0 0
\(885\) 4872.00 0.185051
\(886\) 0 0
\(887\) 2166.00 0.0819923 0.0409961 0.999159i \(-0.486947\pi\)
0.0409961 + 0.999159i \(0.486947\pi\)
\(888\) 0 0
\(889\) −7922.00 −0.298870
\(890\) 0 0
\(891\) 45430.0 1.70815
\(892\) 0 0
\(893\) 7448.00 0.279102
\(894\) 0 0
\(895\) 8384.00 0.313124
\(896\) 0 0
\(897\) 7015.00 0.261119
\(898\) 0 0
\(899\) 20088.0 0.745242
\(900\) 0 0
\(901\) −10873.0 −0.402033
\(902\) 0 0
\(903\) −3264.00 −0.120287
\(904\) 0 0
\(905\) −21424.0 −0.786915
\(906\) 0 0
\(907\) 30419.0 1.11361 0.556806 0.830642i \(-0.312027\pi\)
0.556806 + 0.830642i \(0.312027\pi\)
\(908\) 0 0
\(909\) −31980.0 −1.16690
\(910\) 0 0
\(911\) 32864.0 1.19521 0.597603 0.801792i \(-0.296120\pi\)
0.597603 + 0.801792i \(0.296120\pi\)
\(912\) 0 0
\(913\) 56700.0 2.05531
\(914\) 0 0
\(915\) −2704.00 −0.0976956
\(916\) 0 0
\(917\) 38080.0 1.37133
\(918\) 0 0
\(919\) 20225.0 0.725964 0.362982 0.931796i \(-0.381759\pi\)
0.362982 + 0.931796i \(0.381759\pi\)
\(920\) 0 0
\(921\) −1740.00 −0.0622529
\(922\) 0 0
\(923\) 45750.0 1.63151
\(924\) 0 0
\(925\) −2074.00 −0.0737218
\(926\) 0 0
\(927\) −8060.00 −0.285572
\(928\) 0 0
\(929\) −29415.0 −1.03883 −0.519416 0.854522i \(-0.673850\pi\)
−0.519416 + 0.854522i \(0.673850\pi\)
\(930\) 0 0
\(931\) 1026.00 0.0361179
\(932\) 0 0
\(933\) −2837.00 −0.0995490
\(934\) 0 0
\(935\) 46480.0 1.62573
\(936\) 0 0
\(937\) 3697.00 0.128896 0.0644481 0.997921i \(-0.479471\pi\)
0.0644481 + 0.997921i \(0.479471\pi\)
\(938\) 0 0
\(939\) −1579.00 −0.0548762
\(940\) 0 0
\(941\) 55245.0 1.91385 0.956926 0.290331i \(-0.0937653\pi\)
0.956926 + 0.290331i \(0.0937653\pi\)
\(942\) 0 0
\(943\) 12420.0 0.428898
\(944\) 0 0
\(945\) 7208.00 0.248123
\(946\) 0 0
\(947\) −54296.0 −1.86313 −0.931564 0.363576i \(-0.881556\pi\)
−0.931564 + 0.363576i \(0.881556\pi\)
\(948\) 0 0
\(949\) 71797.0 2.45588
\(950\) 0 0
\(951\) −10065.0 −0.343197
\(952\) 0 0
\(953\) −25770.0 −0.875941 −0.437971 0.898989i \(-0.644303\pi\)
−0.437971 + 0.898989i \(0.644303\pi\)
\(954\) 0 0
\(955\) 11544.0 0.391157
\(956\) 0 0
\(957\) −19530.0 −0.659682
\(958\) 0 0
\(959\) 19465.0 0.655430
\(960\) 0 0
\(961\) −24607.0 −0.825988
\(962\) 0 0
\(963\) −46566.0 −1.55822
\(964\) 0 0
\(965\) 7120.00 0.237514
\(966\) 0 0
\(967\) 20296.0 0.674949 0.337474 0.941335i \(-0.390427\pi\)
0.337474 + 0.941335i \(0.390427\pi\)
\(968\) 0 0
\(969\) −1577.00 −0.0522813
\(970\) 0 0
\(971\) −34476.0 −1.13943 −0.569715 0.821842i \(-0.692947\pi\)
−0.569715 + 0.821842i \(0.692947\pi\)
\(972\) 0 0
\(973\) −39712.0 −1.30844
\(974\) 0 0
\(975\) −3721.00 −0.122223
\(976\) 0 0
\(977\) 39952.0 1.30827 0.654134 0.756379i \(-0.273033\pi\)
0.654134 + 0.756379i \(0.273033\pi\)
\(978\) 0 0
\(979\) −33320.0 −1.08775
\(980\) 0 0
\(981\) −20254.0 −0.659185
\(982\) 0 0
\(983\) −56942.0 −1.84758 −0.923788 0.382904i \(-0.874924\pi\)
−0.923788 + 0.382904i \(0.874924\pi\)
\(984\) 0 0
\(985\) 2400.00 0.0776349
\(986\) 0 0
\(987\) 6664.00 0.214911
\(988\) 0 0
\(989\) 22080.0 0.709912
\(990\) 0 0
\(991\) 45772.0 1.46720 0.733600 0.679581i \(-0.237839\pi\)
0.733600 + 0.679581i \(0.237839\pi\)
\(992\) 0 0
\(993\) −2953.00 −0.0943712
\(994\) 0 0
\(995\) 26712.0 0.851083
\(996\) 0 0
\(997\) 24916.0 0.791472 0.395736 0.918364i \(-0.370490\pi\)
0.395736 + 0.918364i \(0.370490\pi\)
\(998\) 0 0
\(999\) −1802.00 −0.0570698
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 1216.4.a.d.1.1 1
4.3 odd 2 1216.4.a.c.1.1 1
8.3 odd 2 608.4.a.b.1.1 yes 1
8.5 even 2 608.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.a.1.1 1 8.5 even 2
608.4.a.b.1.1 yes 1 8.3 odd 2
1216.4.a.c.1.1 1 4.3 odd 2
1216.4.a.d.1.1 1 1.1 even 1 trivial