Properties

Label 1216.4.a.a.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 19)
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)
 
\(f(q)\) \(=\) \(q-5.00000 q^{3} +12.0000 q^{5} -11.0000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-5.00000 q^{3} +12.0000 q^{5} -11.0000 q^{7} -2.00000 q^{9} -54.0000 q^{11} -11.0000 q^{13} -60.0000 q^{15} -93.0000 q^{17} +19.0000 q^{19} +55.0000 q^{21} -183.000 q^{23} +19.0000 q^{25} +145.000 q^{27} +249.000 q^{29} -56.0000 q^{31} +270.000 q^{33} -132.000 q^{35} +250.000 q^{37} +55.0000 q^{39} +240.000 q^{41} -196.000 q^{43} -24.0000 q^{45} +168.000 q^{47} -222.000 q^{49} +465.000 q^{51} -435.000 q^{53} -648.000 q^{55} -95.0000 q^{57} +195.000 q^{59} +358.000 q^{61} +22.0000 q^{63} -132.000 q^{65} -961.000 q^{67} +915.000 q^{69} +246.000 q^{71} +353.000 q^{73} -95.0000 q^{75} +594.000 q^{77} +34.0000 q^{79} -671.000 q^{81} +234.000 q^{83} -1116.00 q^{85} -1245.00 q^{87} -168.000 q^{89} +121.000 q^{91} +280.000 q^{93} +228.000 q^{95} +758.000 q^{97} +108.000 q^{99} +O(q^{100})\)

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
\(3\) −5.00000 −0.962250 −0.481125 0.876652i \(-0.659772\pi\)
−0.481125 + 0.876652i \(0.659772\pi\)
\(4\) 0 0
\(5\) 12.0000 1.07331 0.536656 0.843801i \(-0.319687\pi\)
0.536656 + 0.843801i \(0.319687\pi\)
\(6\) 0 0
\(7\) −11.0000 −0.593944 −0.296972 0.954886i \(-0.595977\pi\)
−0.296972 + 0.954886i \(0.595977\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.0740741
\(10\) 0 0
\(11\) −54.0000 −1.48015 −0.740073 0.672526i \(-0.765209\pi\)
−0.740073 + 0.672526i \(0.765209\pi\)
\(12\) 0 0
\(13\) −11.0000 −0.234681 −0.117340 0.993092i \(-0.537437\pi\)
−0.117340 + 0.993092i \(0.537437\pi\)
\(14\) 0 0
\(15\) −60.0000 −1.03280
\(16\) 0 0
\(17\) −93.0000 −1.32681 −0.663406 0.748259i \(-0.730890\pi\)
−0.663406 + 0.748259i \(0.730890\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) 55.0000 0.571523
\(22\) 0 0
\(23\) −183.000 −1.65905 −0.829525 0.558470i \(-0.811389\pi\)
−0.829525 + 0.558470i \(0.811389\pi\)
\(24\) 0 0
\(25\) 19.0000 0.152000
\(26\) 0 0
\(27\) 145.000 1.03353
\(28\) 0 0
\(29\) 249.000 1.59442 0.797209 0.603703i \(-0.206309\pi\)
0.797209 + 0.603703i \(0.206309\pi\)
\(30\) 0 0
\(31\) −56.0000 −0.324448 −0.162224 0.986754i \(-0.551867\pi\)
−0.162224 + 0.986754i \(0.551867\pi\)
\(32\) 0 0
\(33\) 270.000 1.42427
\(34\) 0 0
\(35\) −132.000 −0.637488
\(36\) 0 0
\(37\) 250.000 1.11080 0.555402 0.831582i \(-0.312564\pi\)
0.555402 + 0.831582i \(0.312564\pi\)
\(38\) 0 0
\(39\) 55.0000 0.225822
\(40\) 0 0
\(41\) 240.000 0.914188 0.457094 0.889418i \(-0.348890\pi\)
0.457094 + 0.889418i \(0.348890\pi\)
\(42\) 0 0
\(43\) −196.000 −0.695110 −0.347555 0.937660i \(-0.612988\pi\)
−0.347555 + 0.937660i \(0.612988\pi\)
\(44\) 0 0
\(45\) −24.0000 −0.0795046
\(46\) 0 0
\(47\) 168.000 0.521390 0.260695 0.965421i \(-0.416048\pi\)
0.260695 + 0.965421i \(0.416048\pi\)
\(48\) 0 0
\(49\) −222.000 −0.647230
\(50\) 0 0
\(51\) 465.000 1.27673
\(52\) 0 0
\(53\) −435.000 −1.12739 −0.563697 0.825982i \(-0.690621\pi\)
−0.563697 + 0.825982i \(0.690621\pi\)
\(54\) 0 0
\(55\) −648.000 −1.58866
\(56\) 0 0
\(57\) −95.0000 −0.220755
\(58\) 0 0
\(59\) 195.000 0.430285 0.215143 0.976583i \(-0.430978\pi\)
0.215143 + 0.976583i \(0.430978\pi\)
\(60\) 0 0
\(61\) 358.000 0.751430 0.375715 0.926735i \(-0.377397\pi\)
0.375715 + 0.926735i \(0.377397\pi\)
\(62\) 0 0
\(63\) 22.0000 0.0439959
\(64\) 0 0
\(65\) −132.000 −0.251886
\(66\) 0 0
\(67\) −961.000 −1.75231 −0.876155 0.482029i \(-0.839900\pi\)
−0.876155 + 0.482029i \(0.839900\pi\)
\(68\) 0 0
\(69\) 915.000 1.59642
\(70\) 0 0
\(71\) 246.000 0.411195 0.205597 0.978637i \(-0.434086\pi\)
0.205597 + 0.978637i \(0.434086\pi\)
\(72\) 0 0
\(73\) 353.000 0.565966 0.282983 0.959125i \(-0.408676\pi\)
0.282983 + 0.959125i \(0.408676\pi\)
\(74\) 0 0
\(75\) −95.0000 −0.146262
\(76\) 0 0
\(77\) 594.000 0.879124
\(78\) 0 0
\(79\) 34.0000 0.0484215 0.0242108 0.999707i \(-0.492293\pi\)
0.0242108 + 0.999707i \(0.492293\pi\)
\(80\) 0 0
\(81\) −671.000 −0.920439
\(82\) 0 0
\(83\) 234.000 0.309456 0.154728 0.987957i \(-0.450550\pi\)
0.154728 + 0.987957i \(0.450550\pi\)
\(84\) 0 0
\(85\) −1116.00 −1.42408
\(86\) 0 0
\(87\) −1245.00 −1.53423
\(88\) 0 0
\(89\) −168.000 −0.200089 −0.100045 0.994983i \(-0.531899\pi\)
−0.100045 + 0.994983i \(0.531899\pi\)
\(90\) 0 0
\(91\) 121.000 0.139387
\(92\) 0 0
\(93\) 280.000 0.312201
\(94\) 0 0
\(95\) 228.000 0.246235
\(96\) 0 0
\(97\) 758.000 0.793435 0.396718 0.917941i \(-0.370149\pi\)
0.396718 + 0.917941i \(0.370149\pi\)
\(98\) 0 0
\(99\) 108.000 0.109640
\(100\) 0 0
\(101\) 726.000 0.715245 0.357622 0.933866i \(-0.383588\pi\)
0.357622 + 0.933866i \(0.383588\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.00191326 −0.000956630 1.00000i \(-0.500305\pi\)
−0.000956630 1.00000i \(0.500305\pi\)
\(104\) 0 0
\(105\) 660.000 0.613423
\(106\) 0 0
\(107\) 1413.00 1.27663 0.638317 0.769773i \(-0.279631\pi\)
0.638317 + 0.769773i \(0.279631\pi\)
\(108\) 0 0
\(109\) −389.000 −0.341830 −0.170915 0.985286i \(-0.554672\pi\)
−0.170915 + 0.985286i \(0.554672\pi\)
\(110\) 0 0
\(111\) −1250.00 −1.06887
\(112\) 0 0
\(113\) 342.000 0.284714 0.142357 0.989815i \(-0.454532\pi\)
0.142357 + 0.989815i \(0.454532\pi\)
\(114\) 0 0
\(115\) −2196.00 −1.78068
\(116\) 0 0
\(117\) 22.0000 0.0173838
\(118\) 0 0
\(119\) 1023.00 0.788053
\(120\) 0 0
\(121\) 1585.00 1.19083
\(122\) 0 0
\(123\) −1200.00 −0.879678
\(124\) 0 0
\(125\) −1272.00 −0.910169
\(126\) 0 0
\(127\) 1150.00 0.803512 0.401756 0.915747i \(-0.368400\pi\)
0.401756 + 0.915747i \(0.368400\pi\)
\(128\) 0 0
\(129\) 980.000 0.668870
\(130\) 0 0
\(131\) −1452.00 −0.968411 −0.484205 0.874954i \(-0.660891\pi\)
−0.484205 + 0.874954i \(0.660891\pi\)
\(132\) 0 0
\(133\) −209.000 −0.136260
\(134\) 0 0
\(135\) 1740.00 1.10930
\(136\) 0 0
\(137\) −1689.00 −1.05329 −0.526646 0.850085i \(-0.676551\pi\)
−0.526646 + 0.850085i \(0.676551\pi\)
\(138\) 0 0
\(139\) 2144.00 1.30829 0.654143 0.756371i \(-0.273030\pi\)
0.654143 + 0.756371i \(0.273030\pi\)
\(140\) 0 0
\(141\) −840.000 −0.501708
\(142\) 0 0
\(143\) 594.000 0.347362
\(144\) 0 0
\(145\) 2988.00 1.71131
\(146\) 0 0
\(147\) 1110.00 0.622798
\(148\) 0 0
\(149\) 3000.00 1.64946 0.824730 0.565527i \(-0.191327\pi\)
0.824730 + 0.565527i \(0.191327\pi\)
\(150\) 0 0
\(151\) 1006.00 0.542166 0.271083 0.962556i \(-0.412618\pi\)
0.271083 + 0.962556i \(0.412618\pi\)
\(152\) 0 0
\(153\) 186.000 0.0982824
\(154\) 0 0
\(155\) −672.000 −0.348234
\(156\) 0 0
\(157\) −2846.00 −1.44672 −0.723362 0.690469i \(-0.757404\pi\)
−0.723362 + 0.690469i \(0.757404\pi\)
\(158\) 0 0
\(159\) 2175.00 1.08483
\(160\) 0 0
\(161\) 2013.00 0.985383
\(162\) 0 0
\(163\) −1600.00 −0.768845 −0.384422 0.923157i \(-0.625599\pi\)
−0.384422 + 0.923157i \(0.625599\pi\)
\(164\) 0 0
\(165\) 3240.00 1.52869
\(166\) 0 0
\(167\) 2004.00 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −2076.00 −0.944925
\(170\) 0 0
\(171\) −38.0000 −0.0169938
\(172\) 0 0
\(173\) 462.000 0.203036 0.101518 0.994834i \(-0.467630\pi\)
0.101518 + 0.994834i \(0.467630\pi\)
\(174\) 0 0
\(175\) −209.000 −0.0902795
\(176\) 0 0
\(177\) −975.000 −0.414042
\(178\) 0 0
\(179\) 720.000 0.300644 0.150322 0.988637i \(-0.451969\pi\)
0.150322 + 0.988637i \(0.451969\pi\)
\(180\) 0 0
\(181\) 2338.00 0.960122 0.480061 0.877235i \(-0.340614\pi\)
0.480061 + 0.877235i \(0.340614\pi\)
\(182\) 0 0
\(183\) −1790.00 −0.723063
\(184\) 0 0
\(185\) 3000.00 1.19224
\(186\) 0 0
\(187\) 5022.00 1.96388
\(188\) 0 0
\(189\) −1595.00 −0.613858
\(190\) 0 0
\(191\) −2871.00 −1.08763 −0.543817 0.839204i \(-0.683022\pi\)
−0.543817 + 0.839204i \(0.683022\pi\)
\(192\) 0 0
\(193\) 1658.00 0.618370 0.309185 0.951002i \(-0.399944\pi\)
0.309185 + 0.951002i \(0.399944\pi\)
\(194\) 0 0
\(195\) 660.000 0.242377
\(196\) 0 0
\(197\) 4176.00 1.51029 0.755146 0.655556i \(-0.227566\pi\)
0.755146 + 0.655556i \(0.227566\pi\)
\(198\) 0 0
\(199\) 241.000 0.0858494 0.0429247 0.999078i \(-0.486332\pi\)
0.0429247 + 0.999078i \(0.486332\pi\)
\(200\) 0 0
\(201\) 4805.00 1.68616
\(202\) 0 0
\(203\) −2739.00 −0.946996
\(204\) 0 0
\(205\) 2880.00 0.981209
\(206\) 0 0
\(207\) 366.000 0.122893
\(208\) 0 0
\(209\) −1026.00 −0.339569
\(210\) 0 0
\(211\) −745.000 −0.243071 −0.121535 0.992587i \(-0.538782\pi\)
−0.121535 + 0.992587i \(0.538782\pi\)
\(212\) 0 0
\(213\) −1230.00 −0.395672
\(214\) 0 0
\(215\) −2352.00 −0.746070
\(216\) 0 0
\(217\) 616.000 0.192704
\(218\) 0 0
\(219\) −1765.00 −0.544601
\(220\) 0 0
\(221\) 1023.00 0.311377
\(222\) 0 0
\(223\) 1978.00 0.593976 0.296988 0.954881i \(-0.404018\pi\)
0.296988 + 0.954881i \(0.404018\pi\)
\(224\) 0 0
\(225\) −38.0000 −0.0112593
\(226\) 0 0
\(227\) 5355.00 1.56574 0.782872 0.622183i \(-0.213754\pi\)
0.782872 + 0.622183i \(0.213754\pi\)
\(228\) 0 0
\(229\) 6370.00 1.83817 0.919086 0.394057i \(-0.128929\pi\)
0.919086 + 0.394057i \(0.128929\pi\)
\(230\) 0 0
\(231\) −2970.00 −0.845938
\(232\) 0 0
\(233\) −2838.00 −0.797955 −0.398978 0.916961i \(-0.630635\pi\)
−0.398978 + 0.916961i \(0.630635\pi\)
\(234\) 0 0
\(235\) 2016.00 0.559614
\(236\) 0 0
\(237\) −170.000 −0.0465936
\(238\) 0 0
\(239\) 369.000 0.0998687 0.0499344 0.998753i \(-0.484099\pi\)
0.0499344 + 0.998753i \(0.484099\pi\)
\(240\) 0 0
\(241\) 6608.00 1.76622 0.883109 0.469167i \(-0.155446\pi\)
0.883109 + 0.469167i \(0.155446\pi\)
\(242\) 0 0
\(243\) −560.000 −0.147835
\(244\) 0 0
\(245\) −2664.00 −0.694680
\(246\) 0 0
\(247\) −209.000 −0.0538395
\(248\) 0 0
\(249\) −1170.00 −0.297774
\(250\) 0 0
\(251\) 4674.00 1.17538 0.587690 0.809086i \(-0.300038\pi\)
0.587690 + 0.809086i \(0.300038\pi\)
\(252\) 0 0
\(253\) 9882.00 2.45564
\(254\) 0 0
\(255\) 5580.00 1.37033
\(256\) 0 0
\(257\) 4512.00 1.09514 0.547570 0.836760i \(-0.315553\pi\)
0.547570 + 0.836760i \(0.315553\pi\)
\(258\) 0 0
\(259\) −2750.00 −0.659756
\(260\) 0 0
\(261\) −498.000 −0.118105
\(262\) 0 0
\(263\) −3768.00 −0.883440 −0.441720 0.897153i \(-0.645632\pi\)
−0.441720 + 0.897153i \(0.645632\pi\)
\(264\) 0 0
\(265\) −5220.00 −1.21005
\(266\) 0 0
\(267\) 840.000 0.192536
\(268\) 0 0
\(269\) −4758.00 −1.07844 −0.539220 0.842165i \(-0.681281\pi\)
−0.539220 + 0.842165i \(0.681281\pi\)
\(270\) 0 0
\(271\) 2041.00 0.457498 0.228749 0.973485i \(-0.426537\pi\)
0.228749 + 0.973485i \(0.426537\pi\)
\(272\) 0 0
\(273\) −605.000 −0.134126
\(274\) 0 0
\(275\) −1026.00 −0.224982
\(276\) 0 0
\(277\) −1964.00 −0.426012 −0.213006 0.977051i \(-0.568325\pi\)
−0.213006 + 0.977051i \(0.568325\pi\)
\(278\) 0 0
\(279\) 112.000 0.0240332
\(280\) 0 0
\(281\) −5496.00 −1.16678 −0.583388 0.812194i \(-0.698273\pi\)
−0.583388 + 0.812194i \(0.698273\pi\)
\(282\) 0 0
\(283\) 3098.00 0.650731 0.325366 0.945588i \(-0.394513\pi\)
0.325366 + 0.945588i \(0.394513\pi\)
\(284\) 0 0
\(285\) −1140.00 −0.236940
\(286\) 0 0
\(287\) −2640.00 −0.542977
\(288\) 0 0
\(289\) 3736.00 0.760432
\(290\) 0 0
\(291\) −3790.00 −0.763484
\(292\) 0 0
\(293\) −117.000 −0.0233284 −0.0116642 0.999932i \(-0.503713\pi\)
−0.0116642 + 0.999932i \(0.503713\pi\)
\(294\) 0 0
\(295\) 2340.00 0.461831
\(296\) 0 0
\(297\) −7830.00 −1.52977
\(298\) 0 0
\(299\) 2013.00 0.389347
\(300\) 0 0
\(301\) 2156.00 0.412856
\(302\) 0 0
\(303\) −3630.00 −0.688244
\(304\) 0 0
\(305\) 4296.00 0.806519
\(306\) 0 0
\(307\) −1420.00 −0.263986 −0.131993 0.991251i \(-0.542138\pi\)
−0.131993 + 0.991251i \(0.542138\pi\)
\(308\) 0 0
\(309\) 10.0000 0.00184104
\(310\) 0 0
\(311\) 6561.00 1.19627 0.598135 0.801395i \(-0.295909\pi\)
0.598135 + 0.801395i \(0.295909\pi\)
\(312\) 0 0
\(313\) −1483.00 −0.267809 −0.133904 0.990994i \(-0.542751\pi\)
−0.133904 + 0.990994i \(0.542751\pi\)
\(314\) 0 0
\(315\) 264.000 0.0472213
\(316\) 0 0
\(317\) 1239.00 0.219524 0.109762 0.993958i \(-0.464991\pi\)
0.109762 + 0.993958i \(0.464991\pi\)
\(318\) 0 0
\(319\) −13446.0 −2.35997
\(320\) 0 0
\(321\) −7065.00 −1.22844
\(322\) 0 0
\(323\) −1767.00 −0.304392
\(324\) 0 0
\(325\) −209.000 −0.0356715
\(326\) 0 0
\(327\) 1945.00 0.328926
\(328\) 0 0
\(329\) −1848.00 −0.309676
\(330\) 0 0
\(331\) −8899.00 −1.47774 −0.738872 0.673846i \(-0.764641\pi\)
−0.738872 + 0.673846i \(0.764641\pi\)
\(332\) 0 0
\(333\) −500.000 −0.0822818
\(334\) 0 0
\(335\) −11532.0 −1.88078
\(336\) 0 0
\(337\) 5816.00 0.940112 0.470056 0.882637i \(-0.344234\pi\)
0.470056 + 0.882637i \(0.344234\pi\)
\(338\) 0 0
\(339\) −1710.00 −0.273966
\(340\) 0 0
\(341\) 3024.00 0.480231
\(342\) 0 0
\(343\) 6215.00 0.978363
\(344\) 0 0
\(345\) 10980.0 1.71346
\(346\) 0 0
\(347\) −1578.00 −0.244125 −0.122063 0.992522i \(-0.538951\pi\)
−0.122063 + 0.992522i \(0.538951\pi\)
\(348\) 0 0
\(349\) −1658.00 −0.254300 −0.127150 0.991883i \(-0.540583\pi\)
−0.127150 + 0.991883i \(0.540583\pi\)
\(350\) 0 0
\(351\) −1595.00 −0.242549
\(352\) 0 0
\(353\) −11367.0 −1.71389 −0.856947 0.515405i \(-0.827641\pi\)
−0.856947 + 0.515405i \(0.827641\pi\)
\(354\) 0 0
\(355\) 2952.00 0.441341
\(356\) 0 0
\(357\) −5115.00 −0.758304
\(358\) 0 0
\(359\) −2553.00 −0.375326 −0.187663 0.982233i \(-0.560091\pi\)
−0.187663 + 0.982233i \(0.560091\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) −7925.00 −1.14588
\(364\) 0 0
\(365\) 4236.00 0.607459
\(366\) 0 0
\(367\) 196.000 0.0278777 0.0139389 0.999903i \(-0.495563\pi\)
0.0139389 + 0.999903i \(0.495563\pi\)
\(368\) 0 0
\(369\) −480.000 −0.0677176
\(370\) 0 0
\(371\) 4785.00 0.669609
\(372\) 0 0
\(373\) −9353.00 −1.29834 −0.649169 0.760644i \(-0.724883\pi\)
−0.649169 + 0.760644i \(0.724883\pi\)
\(374\) 0 0
\(375\) 6360.00 0.875811
\(376\) 0 0
\(377\) −2739.00 −0.374180
\(378\) 0 0
\(379\) 3827.00 0.518680 0.259340 0.965786i \(-0.416495\pi\)
0.259340 + 0.965786i \(0.416495\pi\)
\(380\) 0 0
\(381\) −5750.00 −0.773180
\(382\) 0 0
\(383\) −5694.00 −0.759660 −0.379830 0.925056i \(-0.624018\pi\)
−0.379830 + 0.925056i \(0.624018\pi\)
\(384\) 0 0
\(385\) 7128.00 0.943575
\(386\) 0 0
\(387\) 392.000 0.0514896
\(388\) 0 0
\(389\) −1290.00 −0.168138 −0.0840689 0.996460i \(-0.526792\pi\)
−0.0840689 + 0.996460i \(0.526792\pi\)
\(390\) 0 0
\(391\) 17019.0 2.20125
\(392\) 0 0
\(393\) 7260.00 0.931854
\(394\) 0 0
\(395\) 408.000 0.0519714
\(396\) 0 0
\(397\) −6536.00 −0.826278 −0.413139 0.910668i \(-0.635568\pi\)
−0.413139 + 0.910668i \(0.635568\pi\)
\(398\) 0 0
\(399\) 1045.00 0.131116
\(400\) 0 0
\(401\) 2328.00 0.289912 0.144956 0.989438i \(-0.453696\pi\)
0.144956 + 0.989438i \(0.453696\pi\)
\(402\) 0 0
\(403\) 616.000 0.0761418
\(404\) 0 0
\(405\) −8052.00 −0.987919
\(406\) 0 0
\(407\) −13500.0 −1.64415
\(408\) 0 0
\(409\) −6676.00 −0.807107 −0.403554 0.914956i \(-0.632225\pi\)
−0.403554 + 0.914956i \(0.632225\pi\)
\(410\) 0 0
\(411\) 8445.00 1.01353
\(412\) 0 0
\(413\) −2145.00 −0.255565
\(414\) 0 0
\(415\) 2808.00 0.332143
\(416\) 0 0
\(417\) −10720.0 −1.25890
\(418\) 0 0
\(419\) −8136.00 −0.948615 −0.474307 0.880359i \(-0.657301\pi\)
−0.474307 + 0.880359i \(0.657301\pi\)
\(420\) 0 0
\(421\) 8665.00 1.00310 0.501551 0.865128i \(-0.332763\pi\)
0.501551 + 0.865128i \(0.332763\pi\)
\(422\) 0 0
\(423\) −336.000 −0.0386215
\(424\) 0 0
\(425\) −1767.00 −0.201676
\(426\) 0 0
\(427\) −3938.00 −0.446307
\(428\) 0 0
\(429\) −2970.00 −0.334249
\(430\) 0 0
\(431\) −750.000 −0.0838196 −0.0419098 0.999121i \(-0.513344\pi\)
−0.0419098 + 0.999121i \(0.513344\pi\)
\(432\) 0 0
\(433\) −4858.00 −0.539170 −0.269585 0.962977i \(-0.586887\pi\)
−0.269585 + 0.962977i \(0.586887\pi\)
\(434\) 0 0
\(435\) −14940.0 −1.64671
\(436\) 0 0
\(437\) −3477.00 −0.380612
\(438\) 0 0
\(439\) −6500.00 −0.706670 −0.353335 0.935497i \(-0.614952\pi\)
−0.353335 + 0.935497i \(0.614952\pi\)
\(440\) 0 0
\(441\) 444.000 0.0479430
\(442\) 0 0
\(443\) 3486.00 0.373871 0.186936 0.982372i \(-0.440144\pi\)
0.186936 + 0.982372i \(0.440144\pi\)
\(444\) 0 0
\(445\) −2016.00 −0.214759
\(446\) 0 0
\(447\) −15000.0 −1.58719
\(448\) 0 0
\(449\) −15030.0 −1.57975 −0.789877 0.613265i \(-0.789856\pi\)
−0.789877 + 0.613265i \(0.789856\pi\)
\(450\) 0 0
\(451\) −12960.0 −1.35313
\(452\) 0 0
\(453\) −5030.00 −0.521700
\(454\) 0 0
\(455\) 1452.00 0.149606
\(456\) 0 0
\(457\) −2959.00 −0.302880 −0.151440 0.988466i \(-0.548391\pi\)
−0.151440 + 0.988466i \(0.548391\pi\)
\(458\) 0 0
\(459\) −13485.0 −1.37130
\(460\) 0 0
\(461\) 156.000 0.0157606 0.00788031 0.999969i \(-0.497492\pi\)
0.00788031 + 0.999969i \(0.497492\pi\)
\(462\) 0 0
\(463\) −4484.00 −0.450085 −0.225042 0.974349i \(-0.572252\pi\)
−0.225042 + 0.974349i \(0.572252\pi\)
\(464\) 0 0
\(465\) 3360.00 0.335089
\(466\) 0 0
\(467\) 8766.00 0.868613 0.434306 0.900765i \(-0.356994\pi\)
0.434306 + 0.900765i \(0.356994\pi\)
\(468\) 0 0
\(469\) 10571.0 1.04077
\(470\) 0 0
\(471\) 14230.0 1.39211
\(472\) 0 0
\(473\) 10584.0 1.02886
\(474\) 0 0
\(475\) 361.000 0.0348712
\(476\) 0 0
\(477\) 870.000 0.0835106
\(478\) 0 0
\(479\) 18996.0 1.81200 0.906001 0.423275i \(-0.139119\pi\)
0.906001 + 0.423275i \(0.139119\pi\)
\(480\) 0 0
\(481\) −2750.00 −0.260684
\(482\) 0 0
\(483\) −10065.0 −0.948185
\(484\) 0 0
\(485\) 9096.00 0.851604
\(486\) 0 0
\(487\) 7450.00 0.693207 0.346603 0.938012i \(-0.387335\pi\)
0.346603 + 0.938012i \(0.387335\pi\)
\(488\) 0 0
\(489\) 8000.00 0.739821
\(490\) 0 0
\(491\) 6180.00 0.568023 0.284012 0.958821i \(-0.408335\pi\)
0.284012 + 0.958821i \(0.408335\pi\)
\(492\) 0 0
\(493\) −23157.0 −2.11549
\(494\) 0 0
\(495\) 1296.00 0.117679
\(496\) 0 0
\(497\) −2706.00 −0.244227
\(498\) 0 0
\(499\) 2576.00 0.231097 0.115549 0.993302i \(-0.463137\pi\)
0.115549 + 0.993302i \(0.463137\pi\)
\(500\) 0 0
\(501\) −10020.0 −0.893534
\(502\) 0 0
\(503\) 10545.0 0.934748 0.467374 0.884060i \(-0.345200\pi\)
0.467374 + 0.884060i \(0.345200\pi\)
\(504\) 0 0
\(505\) 8712.00 0.767681
\(506\) 0 0
\(507\) 10380.0 0.909254
\(508\) 0 0
\(509\) 14694.0 1.27957 0.639784 0.768555i \(-0.279024\pi\)
0.639784 + 0.768555i \(0.279024\pi\)
\(510\) 0 0
\(511\) −3883.00 −0.336152
\(512\) 0 0
\(513\) 2755.00 0.237108
\(514\) 0 0
\(515\) −24.0000 −0.00205353
\(516\) 0 0
\(517\) −9072.00 −0.771733
\(518\) 0 0
\(519\) −2310.00 −0.195371
\(520\) 0 0
\(521\) 10332.0 0.868816 0.434408 0.900716i \(-0.356958\pi\)
0.434408 + 0.900716i \(0.356958\pi\)
\(522\) 0 0
\(523\) 10937.0 0.914420 0.457210 0.889359i \(-0.348849\pi\)
0.457210 + 0.889359i \(0.348849\pi\)
\(524\) 0 0
\(525\) 1045.00 0.0868715
\(526\) 0 0
\(527\) 5208.00 0.430482
\(528\) 0 0
\(529\) 21322.0 1.75245
\(530\) 0 0
\(531\) −390.000 −0.0318730
\(532\) 0 0
\(533\) −2640.00 −0.214542
\(534\) 0 0
\(535\) 16956.0 1.37023
\(536\) 0 0
\(537\) −3600.00 −0.289295
\(538\) 0 0
\(539\) 11988.0 0.957996
\(540\) 0 0
\(541\) −18578.0 −1.47640 −0.738198 0.674584i \(-0.764323\pi\)
−0.738198 + 0.674584i \(0.764323\pi\)
\(542\) 0 0
\(543\) −11690.0 −0.923878
\(544\) 0 0
\(545\) −4668.00 −0.366890
\(546\) 0 0
\(547\) 21404.0 1.67307 0.836535 0.547914i \(-0.184578\pi\)
0.836535 + 0.547914i \(0.184578\pi\)
\(548\) 0 0
\(549\) −716.000 −0.0556614
\(550\) 0 0
\(551\) 4731.00 0.365785
\(552\) 0 0
\(553\) −374.000 −0.0287597
\(554\) 0 0
\(555\) −15000.0 −1.14723
\(556\) 0 0
\(557\) 3948.00 0.300327 0.150163 0.988661i \(-0.452020\pi\)
0.150163 + 0.988661i \(0.452020\pi\)
\(558\) 0 0
\(559\) 2156.00 0.163129
\(560\) 0 0
\(561\) −25110.0 −1.88974
\(562\) 0 0
\(563\) 5724.00 0.428486 0.214243 0.976780i \(-0.431271\pi\)
0.214243 + 0.976780i \(0.431271\pi\)
\(564\) 0 0
\(565\) 4104.00 0.305587
\(566\) 0 0
\(567\) 7381.00 0.546689
\(568\) 0 0
\(569\) −20592.0 −1.51716 −0.758578 0.651582i \(-0.774105\pi\)
−0.758578 + 0.651582i \(0.774105\pi\)
\(570\) 0 0
\(571\) 20684.0 1.51593 0.757967 0.652293i \(-0.226193\pi\)
0.757967 + 0.652293i \(0.226193\pi\)
\(572\) 0 0
\(573\) 14355.0 1.04658
\(574\) 0 0
\(575\) −3477.00 −0.252176
\(576\) 0 0
\(577\) −19573.0 −1.41219 −0.706096 0.708116i \(-0.749545\pi\)
−0.706096 + 0.708116i \(0.749545\pi\)
\(578\) 0 0
\(579\) −8290.00 −0.595027
\(580\) 0 0
\(581\) −2574.00 −0.183800
\(582\) 0 0
\(583\) 23490.0 1.66871
\(584\) 0 0
\(585\) 264.000 0.0186582
\(586\) 0 0
\(587\) 13524.0 0.950929 0.475464 0.879735i \(-0.342280\pi\)
0.475464 + 0.879735i \(0.342280\pi\)
\(588\) 0 0
\(589\) −1064.00 −0.0744335
\(590\) 0 0
\(591\) −20880.0 −1.45328
\(592\) 0 0
\(593\) 8994.00 0.622832 0.311416 0.950274i \(-0.399197\pi\)
0.311416 + 0.950274i \(0.399197\pi\)
\(594\) 0 0
\(595\) 12276.0 0.845827
\(596\) 0 0
\(597\) −1205.00 −0.0826087
\(598\) 0 0
\(599\) −10128.0 −0.690850 −0.345425 0.938446i \(-0.612265\pi\)
−0.345425 + 0.938446i \(0.612265\pi\)
\(600\) 0 0
\(601\) −22696.0 −1.54041 −0.770207 0.637794i \(-0.779847\pi\)
−0.770207 + 0.637794i \(0.779847\pi\)
\(602\) 0 0
\(603\) 1922.00 0.129801
\(604\) 0 0
\(605\) 19020.0 1.27814
\(606\) 0 0
\(607\) 5182.00 0.346509 0.173254 0.984877i \(-0.444572\pi\)
0.173254 + 0.984877i \(0.444572\pi\)
\(608\) 0 0
\(609\) 13695.0 0.911247
\(610\) 0 0
\(611\) −1848.00 −0.122360
\(612\) 0 0
\(613\) −10082.0 −0.664287 −0.332144 0.943229i \(-0.607772\pi\)
−0.332144 + 0.943229i \(0.607772\pi\)
\(614\) 0 0
\(615\) −14400.0 −0.944169
\(616\) 0 0
\(617\) −12174.0 −0.794338 −0.397169 0.917745i \(-0.630007\pi\)
−0.397169 + 0.917745i \(0.630007\pi\)
\(618\) 0 0
\(619\) 7490.00 0.486347 0.243173 0.969983i \(-0.421812\pi\)
0.243173 + 0.969983i \(0.421812\pi\)
\(620\) 0 0
\(621\) −26535.0 −1.71467
\(622\) 0 0
\(623\) 1848.00 0.118842
\(624\) 0 0
\(625\) −17639.0 −1.12890
\(626\) 0 0
\(627\) 5130.00 0.326750
\(628\) 0 0
\(629\) −23250.0 −1.47383
\(630\) 0 0
\(631\) −11072.0 −0.698525 −0.349263 0.937025i \(-0.613568\pi\)
−0.349263 + 0.937025i \(0.613568\pi\)
\(632\) 0 0
\(633\) 3725.00 0.233895
\(634\) 0 0
\(635\) 13800.0 0.862419
\(636\) 0 0
\(637\) 2442.00 0.151893
\(638\) 0 0
\(639\) −492.000 −0.0304589
\(640\) 0 0
\(641\) −18894.0 −1.16422 −0.582112 0.813108i \(-0.697774\pi\)
−0.582112 + 0.813108i \(0.697774\pi\)
\(642\) 0 0
\(643\) −19834.0 −1.21645 −0.608224 0.793765i \(-0.708118\pi\)
−0.608224 + 0.793765i \(0.708118\pi\)
\(644\) 0 0
\(645\) 11760.0 0.717906
\(646\) 0 0
\(647\) −3375.00 −0.205077 −0.102539 0.994729i \(-0.532697\pi\)
−0.102539 + 0.994729i \(0.532697\pi\)
\(648\) 0 0
\(649\) −10530.0 −0.636885
\(650\) 0 0
\(651\) −3080.00 −0.185430
\(652\) 0 0
\(653\) 24948.0 1.49509 0.747543 0.664214i \(-0.231234\pi\)
0.747543 + 0.664214i \(0.231234\pi\)
\(654\) 0 0
\(655\) −17424.0 −1.03941
\(656\) 0 0
\(657\) −706.000 −0.0419234
\(658\) 0 0
\(659\) −9879.00 −0.583962 −0.291981 0.956424i \(-0.594314\pi\)
−0.291981 + 0.956424i \(0.594314\pi\)
\(660\) 0 0
\(661\) 14155.0 0.832928 0.416464 0.909152i \(-0.363269\pi\)
0.416464 + 0.909152i \(0.363269\pi\)
\(662\) 0 0
\(663\) −5115.00 −0.299623
\(664\) 0 0
\(665\) −2508.00 −0.146250
\(666\) 0 0
\(667\) −45567.0 −2.64522
\(668\) 0 0
\(669\) −9890.00 −0.571554
\(670\) 0 0
\(671\) −19332.0 −1.11223
\(672\) 0 0
\(673\) 8948.00 0.512511 0.256256 0.966609i \(-0.417511\pi\)
0.256256 + 0.966609i \(0.417511\pi\)
\(674\) 0 0
\(675\) 2755.00 0.157096
\(676\) 0 0
\(677\) 11511.0 0.653477 0.326738 0.945115i \(-0.394050\pi\)
0.326738 + 0.945115i \(0.394050\pi\)
\(678\) 0 0
\(679\) −8338.00 −0.471256
\(680\) 0 0
\(681\) −26775.0 −1.50664
\(682\) 0 0
\(683\) −10476.0 −0.586900 −0.293450 0.955974i \(-0.594803\pi\)
−0.293450 + 0.955974i \(0.594803\pi\)
\(684\) 0 0
\(685\) −20268.0 −1.13051
\(686\) 0 0
\(687\) −31850.0 −1.76878
\(688\) 0 0
\(689\) 4785.00 0.264578
\(690\) 0 0
\(691\) 30098.0 1.65699 0.828496 0.559995i \(-0.189197\pi\)
0.828496 + 0.559995i \(0.189197\pi\)
\(692\) 0 0
\(693\) −1188.00 −0.0651203
\(694\) 0 0
\(695\) 25728.0 1.40420
\(696\) 0 0
\(697\) −22320.0 −1.21296
\(698\) 0 0
\(699\) 14190.0 0.767833
\(700\) 0 0
\(701\) 14700.0 0.792028 0.396014 0.918245i \(-0.370393\pi\)
0.396014 + 0.918245i \(0.370393\pi\)
\(702\) 0 0
\(703\) 4750.00 0.254836
\(704\) 0 0
\(705\) −10080.0 −0.538489
\(706\) 0 0
\(707\) −7986.00 −0.424815
\(708\) 0 0
\(709\) −31178.0 −1.65150 −0.825751 0.564035i \(-0.809248\pi\)
−0.825751 + 0.564035i \(0.809248\pi\)
\(710\) 0 0
\(711\) −68.0000 −0.00358678
\(712\) 0 0
\(713\) 10248.0 0.538276
\(714\) 0 0
\(715\) 7128.00 0.372828
\(716\) 0 0
\(717\) −1845.00 −0.0960987
\(718\) 0 0
\(719\) 33285.0 1.72645 0.863227 0.504815i \(-0.168439\pi\)
0.863227 + 0.504815i \(0.168439\pi\)
\(720\) 0 0
\(721\) 22.0000 0.00113637
\(722\) 0 0
\(723\) −33040.0 −1.69954
\(724\) 0 0
\(725\) 4731.00 0.242352
\(726\) 0 0
\(727\) 34729.0 1.77170 0.885851 0.463970i \(-0.153575\pi\)
0.885851 + 0.463970i \(0.153575\pi\)
\(728\) 0 0
\(729\) 20917.0 1.06269
\(730\) 0 0
\(731\) 18228.0 0.922280
\(732\) 0 0
\(733\) −4196.00 −0.211436 −0.105718 0.994396i \(-0.533714\pi\)
−0.105718 + 0.994396i \(0.533714\pi\)
\(734\) 0 0
\(735\) 13320.0 0.668457
\(736\) 0 0
\(737\) 51894.0 2.59368
\(738\) 0 0
\(739\) −10744.0 −0.534810 −0.267405 0.963584i \(-0.586166\pi\)
−0.267405 + 0.963584i \(0.586166\pi\)
\(740\) 0 0
\(741\) 1045.00 0.0518071
\(742\) 0 0
\(743\) 2208.00 0.109022 0.0545112 0.998513i \(-0.482640\pi\)
0.0545112 + 0.998513i \(0.482640\pi\)
\(744\) 0 0
\(745\) 36000.0 1.77039
\(746\) 0 0
\(747\) −468.000 −0.0229227
\(748\) 0 0
\(749\) −15543.0 −0.758249
\(750\) 0 0
\(751\) −13160.0 −0.639434 −0.319717 0.947513i \(-0.603588\pi\)
−0.319717 + 0.947513i \(0.603588\pi\)
\(752\) 0 0
\(753\) −23370.0 −1.13101
\(754\) 0 0
\(755\) 12072.0 0.581914
\(756\) 0 0
\(757\) −758.000 −0.0363936 −0.0181968 0.999834i \(-0.505793\pi\)
−0.0181968 + 0.999834i \(0.505793\pi\)
\(758\) 0 0
\(759\) −49410.0 −2.36294
\(760\) 0 0
\(761\) 4851.00 0.231076 0.115538 0.993303i \(-0.463141\pi\)
0.115538 + 0.993303i \(0.463141\pi\)
\(762\) 0 0
\(763\) 4279.00 0.203028
\(764\) 0 0
\(765\) 2232.00 0.105488
\(766\) 0 0
\(767\) −2145.00 −0.100980
\(768\) 0 0
\(769\) −33091.0 −1.55175 −0.775873 0.630890i \(-0.782690\pi\)
−0.775873 + 0.630890i \(0.782690\pi\)
\(770\) 0 0
\(771\) −22560.0 −1.05380
\(772\) 0 0
\(773\) −42357.0 −1.97086 −0.985430 0.170079i \(-0.945598\pi\)
−0.985430 + 0.170079i \(0.945598\pi\)
\(774\) 0 0
\(775\) −1064.00 −0.0493161
\(776\) 0 0
\(777\) 13750.0 0.634850
\(778\) 0 0
\(779\) 4560.00 0.209729
\(780\) 0 0
\(781\) −13284.0 −0.608629
\(782\) 0 0
\(783\) 36105.0 1.64788
\(784\) 0 0
\(785\) −34152.0 −1.55279
\(786\) 0 0
\(787\) −39877.0 −1.80618 −0.903089 0.429454i \(-0.858706\pi\)
−0.903089 + 0.429454i \(0.858706\pi\)
\(788\) 0 0
\(789\) 18840.0 0.850091
\(790\) 0 0
\(791\) −3762.00 −0.169104
\(792\) 0 0
\(793\) −3938.00 −0.176346
\(794\) 0 0
\(795\) 26100.0 1.16437
\(796\) 0 0
\(797\) 30033.0 1.33478 0.667392 0.744706i \(-0.267410\pi\)
0.667392 + 0.744706i \(0.267410\pi\)
\(798\) 0 0
\(799\) −15624.0 −0.691786
\(800\) 0 0
\(801\) 336.000 0.0148214
\(802\) 0 0
\(803\) −19062.0 −0.837713
\(804\) 0 0
\(805\) 24156.0 1.05762
\(806\) 0 0
\(807\) 23790.0 1.03773
\(808\) 0 0
\(809\) 585.000 0.0254234 0.0127117 0.999919i \(-0.495954\pi\)
0.0127117 + 0.999919i \(0.495954\pi\)
\(810\) 0 0
\(811\) 28361.0 1.22798 0.613989 0.789315i \(-0.289564\pi\)
0.613989 + 0.789315i \(0.289564\pi\)
\(812\) 0 0
\(813\) −10205.0 −0.440228
\(814\) 0 0
\(815\) −19200.0 −0.825211
\(816\) 0 0
\(817\) −3724.00 −0.159469
\(818\) 0 0
\(819\) −242.000 −0.0103250
\(820\) 0 0
\(821\) −25068.0 −1.06563 −0.532813 0.846233i \(-0.678865\pi\)
−0.532813 + 0.846233i \(0.678865\pi\)
\(822\) 0 0
\(823\) −10901.0 −0.461707 −0.230854 0.972989i \(-0.574152\pi\)
−0.230854 + 0.972989i \(0.574152\pi\)
\(824\) 0 0
\(825\) 5130.00 0.216489
\(826\) 0 0
\(827\) 12027.0 0.505707 0.252854 0.967505i \(-0.418631\pi\)
0.252854 + 0.967505i \(0.418631\pi\)
\(828\) 0 0
\(829\) 19339.0 0.810219 0.405109 0.914268i \(-0.367233\pi\)
0.405109 + 0.914268i \(0.367233\pi\)
\(830\) 0 0
\(831\) 9820.00 0.409930
\(832\) 0 0
\(833\) 20646.0 0.858753
\(834\) 0 0
\(835\) 24048.0 0.996665
\(836\) 0 0
\(837\) −8120.00 −0.335326
\(838\) 0 0
\(839\) 13188.0 0.542670 0.271335 0.962485i \(-0.412535\pi\)
0.271335 + 0.962485i \(0.412535\pi\)
\(840\) 0 0
\(841\) 37612.0 1.54217
\(842\) 0 0
\(843\) 27480.0 1.12273
\(844\) 0 0
\(845\) −24912.0 −1.01420
\(846\) 0 0
\(847\) −17435.0 −0.707289
\(848\) 0 0
\(849\) −15490.0 −0.626167
\(850\) 0 0
\(851\) −45750.0 −1.84288
\(852\) 0 0
\(853\) 4678.00 0.187775 0.0938873 0.995583i \(-0.470071\pi\)
0.0938873 + 0.995583i \(0.470071\pi\)
\(854\) 0 0
\(855\) −456.000 −0.0182396
\(856\) 0 0
\(857\) 15252.0 0.607933 0.303966 0.952683i \(-0.401689\pi\)
0.303966 + 0.952683i \(0.401689\pi\)
\(858\) 0 0
\(859\) −610.000 −0.0242293 −0.0121146 0.999927i \(-0.503856\pi\)
−0.0121146 + 0.999927i \(0.503856\pi\)
\(860\) 0 0
\(861\) 13200.0 0.522479
\(862\) 0 0
\(863\) −774.000 −0.0305299 −0.0152649 0.999883i \(-0.504859\pi\)
−0.0152649 + 0.999883i \(0.504859\pi\)
\(864\) 0 0
\(865\) 5544.00 0.217921
\(866\) 0 0
\(867\) −18680.0 −0.731726
\(868\) 0 0
\(869\) −1836.00 −0.0716709
\(870\) 0 0
\(871\) 10571.0 0.411234
\(872\) 0 0
\(873\) −1516.00 −0.0587730
\(874\) 0 0
\(875\) 13992.0 0.540590
\(876\) 0 0
\(877\) 31039.0 1.19511 0.597556 0.801827i \(-0.296139\pi\)
0.597556 + 0.801827i \(0.296139\pi\)
\(878\) 0 0
\(879\) 585.000 0.0224477
\(880\) 0 0
\(881\) 33678.0 1.28790 0.643950 0.765067i \(-0.277294\pi\)
0.643950 + 0.765067i \(0.277294\pi\)
\(882\) 0 0
\(883\) −42982.0 −1.63812 −0.819060 0.573708i \(-0.805504\pi\)
−0.819060 + 0.573708i \(0.805504\pi\)
\(884\) 0 0
\(885\) −11700.0 −0.444397
\(886\) 0 0
\(887\) −4494.00 −0.170117 −0.0850585 0.996376i \(-0.527108\pi\)
−0.0850585 + 0.996376i \(0.527108\pi\)
\(888\) 0 0
\(889\) −12650.0 −0.477241
\(890\) 0 0
\(891\) 36234.0 1.36238
\(892\) 0 0
\(893\) 3192.00 0.119615
\(894\) 0 0
\(895\) 8640.00 0.322685
\(896\) 0 0
\(897\) −10065.0 −0.374649
\(898\) 0 0
\(899\) −13944.0 −0.517306
\(900\) 0 0
\(901\) 40455.0 1.49584
\(902\) 0 0
\(903\) −10780.0 −0.397271
\(904\) 0 0
\(905\) 28056.0 1.03051
\(906\) 0 0
\(907\) −23839.0 −0.872724 −0.436362 0.899771i \(-0.643733\pi\)
−0.436362 + 0.899771i \(0.643733\pi\)
\(908\) 0 0
\(909\) −1452.00 −0.0529811
\(910\) 0 0
\(911\) 10332.0 0.375757 0.187878 0.982192i \(-0.439839\pi\)
0.187878 + 0.982192i \(0.439839\pi\)
\(912\) 0 0
\(913\) −12636.0 −0.458040
\(914\) 0 0
\(915\) −21480.0 −0.776073
\(916\) 0 0
\(917\) 15972.0 0.575182
\(918\) 0 0
\(919\) 14371.0 0.515838 0.257919 0.966166i \(-0.416963\pi\)
0.257919 + 0.966166i \(0.416963\pi\)
\(920\) 0 0
\(921\) 7100.00 0.254021
\(922\) 0 0
\(923\) −2706.00 −0.0964995
\(924\) 0 0
\(925\) 4750.00 0.168842
\(926\) 0 0
\(927\) 4.00000 0.000141723 0
\(928\) 0 0
\(929\) 26889.0 0.949623 0.474811 0.880088i \(-0.342516\pi\)
0.474811 + 0.880088i \(0.342516\pi\)
\(930\) 0 0
\(931\) −4218.00 −0.148485
\(932\) 0 0
\(933\) −32805.0 −1.15111
\(934\) 0 0
\(935\) 60264.0 2.10785
\(936\) 0 0
\(937\) 785.000 0.0273691 0.0136845 0.999906i \(-0.495644\pi\)
0.0136845 + 0.999906i \(0.495644\pi\)
\(938\) 0 0
\(939\) 7415.00 0.257699
\(940\) 0 0
\(941\) 18141.0 0.628459 0.314229 0.949347i \(-0.398254\pi\)
0.314229 + 0.949347i \(0.398254\pi\)
\(942\) 0 0
\(943\) −43920.0 −1.51668
\(944\) 0 0
\(945\) −19140.0 −0.658862
\(946\) 0 0
\(947\) 23100.0 0.792660 0.396330 0.918108i \(-0.370284\pi\)
0.396330 + 0.918108i \(0.370284\pi\)
\(948\) 0 0
\(949\) −3883.00 −0.132821
\(950\) 0 0
\(951\) −6195.00 −0.211237
\(952\) 0 0
\(953\) 45690.0 1.55304 0.776519 0.630094i \(-0.216984\pi\)
0.776519 + 0.630094i \(0.216984\pi\)
\(954\) 0 0
\(955\) −34452.0 −1.16737
\(956\) 0 0
\(957\) 67230.0 2.27089
\(958\) 0 0
\(959\) 18579.0 0.625597
\(960\) 0 0
\(961\) −26655.0 −0.894733
\(962\) 0 0
\(963\) −2826.00 −0.0945655
\(964\) 0 0
\(965\) 19896.0 0.663705
\(966\) 0 0
\(967\) −21584.0 −0.717781 −0.358891 0.933380i \(-0.616845\pi\)
−0.358891 + 0.933380i \(0.616845\pi\)
\(968\) 0 0
\(969\) 8835.00 0.292901
\(970\) 0 0
\(971\) −50556.0 −1.67087 −0.835437 0.549586i \(-0.814786\pi\)
−0.835437 + 0.549586i \(0.814786\pi\)
\(972\) 0 0
\(973\) −23584.0 −0.777049
\(974\) 0 0
\(975\) 1045.00 0.0343249
\(976\) 0 0
\(977\) 8568.00 0.280568 0.140284 0.990111i \(-0.455198\pi\)
0.140284 + 0.990111i \(0.455198\pi\)
\(978\) 0 0
\(979\) 9072.00 0.296162
\(980\) 0 0
\(981\) 778.000 0.0253207
\(982\) 0 0
\(983\) −29706.0 −0.963860 −0.481930 0.876210i \(-0.660064\pi\)
−0.481930 + 0.876210i \(0.660064\pi\)
\(984\) 0 0
\(985\) 50112.0 1.62102
\(986\) 0 0
\(987\) 9240.00 0.297986
\(988\) 0 0
\(989\) 35868.0 1.15322
\(990\) 0 0
\(991\) −30512.0 −0.978048 −0.489024 0.872270i \(-0.662647\pi\)
−0.489024 + 0.872270i \(0.662647\pi\)
\(992\) 0 0
\(993\) 44495.0 1.42196
\(994\) 0 0
\(995\) 2892.00 0.0921433
\(996\) 0 0
\(997\) −47756.0 −1.51700 −0.758499 0.651674i \(-0.774067\pi\)
−0.758499 + 0.651674i \(0.774067\pi\)
\(998\) 0 0
\(999\) 36250.0 1.14805
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.a.1.1 1
4.3 odd 2 1216.4.a.f.1.1 1
8.3 odd 2 19.4.a.a.1.1 1
8.5 even 2 304.4.a.b.1.1 1
24.11 even 2 171.4.a.d.1.1 1
40.3 even 4 475.4.b.c.324.2 2
40.19 odd 2 475.4.a.e.1.1 1
40.27 even 4 475.4.b.c.324.1 2
56.27 even 2 931.4.a.a.1.1 1
88.43 even 2 2299.4.a.b.1.1 1
152.75 even 2 361.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
19.4.a.a.1.1 1 8.3 odd 2
171.4.a.d.1.1 1 24.11 even 2
304.4.a.b.1.1 1 8.5 even 2
361.4.a.b.1.1 1 152.75 even 2
475.4.a.e.1.1 1 40.19 odd 2
475.4.b.c.324.1 2 40.27 even 4
475.4.b.c.324.2 2 40.3 even 4
931.4.a.a.1.1 1 56.27 even 2
1216.4.a.a.1.1 1 1.1 even 1 trivial
1216.4.a.f.1.1 1 4.3 odd 2
2299.4.a.b.1.1 1 88.43 even 2