Properties

Label 1856.4.a.f.1.1
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
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] = [1856,4,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, 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("1856.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.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: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1856.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.00000 q^{3} +15.0000 q^{5} -18.0000 q^{7} +22.0000 q^{9} +O(q^{10})\) \(q+7.00000 q^{3} +15.0000 q^{5} -18.0000 q^{7} +22.0000 q^{9} -27.0000 q^{11} +57.0000 q^{13} +105.000 q^{15} -44.0000 q^{17} -152.000 q^{19} -126.000 q^{21} -152.000 q^{23} +100.000 q^{25} -35.0000 q^{27} +29.0000 q^{29} -173.000 q^{31} -189.000 q^{33} -270.000 q^{35} +120.000 q^{37} +399.000 q^{39} -314.000 q^{41} -339.000 q^{43} +330.000 q^{45} -357.000 q^{47} -19.0000 q^{49} -308.000 q^{51} +59.0000 q^{53} -405.000 q^{55} -1064.00 q^{57} +572.000 q^{59} +420.000 q^{61} -396.000 q^{63} +855.000 q^{65} -660.000 q^{67} -1064.00 q^{69} +726.000 q^{71} +1004.00 q^{73} +700.000 q^{75} +486.000 q^{77} +361.000 q^{79} -839.000 q^{81} +168.000 q^{83} -660.000 q^{85} +203.000 q^{87} +58.0000 q^{89} -1026.00 q^{91} -1211.00 q^{93} -2280.00 q^{95} -1206.00 q^{97} -594.000 q^{99} +O(q^{100})\)

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
\(3\) 7.00000 1.34715 0.673575 0.739119i \(-0.264758\pi\)
0.673575 + 0.739119i \(0.264758\pi\)
\(4\) 0 0
\(5\) 15.0000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −18.0000 −0.971909 −0.485954 0.873984i \(-0.661528\pi\)
−0.485954 + 0.873984i \(0.661528\pi\)
\(8\) 0 0
\(9\) 22.0000 0.814815
\(10\) 0 0
\(11\) −27.0000 −0.740073 −0.370037 0.929017i \(-0.620655\pi\)
−0.370037 + 0.929017i \(0.620655\pi\)
\(12\) 0 0
\(13\) 57.0000 1.21607 0.608037 0.793909i \(-0.291957\pi\)
0.608037 + 0.793909i \(0.291957\pi\)
\(14\) 0 0
\(15\) 105.000 1.80739
\(16\) 0 0
\(17\) −44.0000 −0.627739 −0.313870 0.949466i \(-0.601625\pi\)
−0.313870 + 0.949466i \(0.601625\pi\)
\(18\) 0 0
\(19\) −152.000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −126.000 −1.30931
\(22\) 0 0
\(23\) −152.000 −1.37801 −0.689004 0.724757i \(-0.741952\pi\)
−0.689004 + 0.724757i \(0.741952\pi\)
\(24\) 0 0
\(25\) 100.000 0.800000
\(26\) 0 0
\(27\) −35.0000 −0.249472
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) −173.000 −1.00231 −0.501157 0.865357i \(-0.667092\pi\)
−0.501157 + 0.865357i \(0.667092\pi\)
\(32\) 0 0
\(33\) −189.000 −0.996990
\(34\) 0 0
\(35\) −270.000 −1.30395
\(36\) 0 0
\(37\) 120.000 0.533186 0.266593 0.963809i \(-0.414102\pi\)
0.266593 + 0.963809i \(0.414102\pi\)
\(38\) 0 0
\(39\) 399.000 1.63823
\(40\) 0 0
\(41\) −314.000 −1.19606 −0.598031 0.801473i \(-0.704050\pi\)
−0.598031 + 0.801473i \(0.704050\pi\)
\(42\) 0 0
\(43\) −339.000 −1.20226 −0.601128 0.799153i \(-0.705282\pi\)
−0.601128 + 0.799153i \(0.705282\pi\)
\(44\) 0 0
\(45\) 330.000 1.09319
\(46\) 0 0
\(47\) −357.000 −1.10795 −0.553977 0.832532i \(-0.686890\pi\)
−0.553977 + 0.832532i \(0.686890\pi\)
\(48\) 0 0
\(49\) −19.0000 −0.0553936
\(50\) 0 0
\(51\) −308.000 −0.845659
\(52\) 0 0
\(53\) 59.0000 0.152911 0.0764554 0.997073i \(-0.475640\pi\)
0.0764554 + 0.997073i \(0.475640\pi\)
\(54\) 0 0
\(55\) −405.000 −0.992913
\(56\) 0 0
\(57\) −1064.00 −2.47246
\(58\) 0 0
\(59\) 572.000 1.26217 0.631085 0.775713i \(-0.282610\pi\)
0.631085 + 0.775713i \(0.282610\pi\)
\(60\) 0 0
\(61\) 420.000 0.881565 0.440783 0.897614i \(-0.354701\pi\)
0.440783 + 0.897614i \(0.354701\pi\)
\(62\) 0 0
\(63\) −396.000 −0.791926
\(64\) 0 0
\(65\) 855.000 1.63153
\(66\) 0 0
\(67\) −660.000 −1.20346 −0.601730 0.798699i \(-0.705522\pi\)
−0.601730 + 0.798699i \(0.705522\pi\)
\(68\) 0 0
\(69\) −1064.00 −1.85638
\(70\) 0 0
\(71\) 726.000 1.21353 0.606763 0.794883i \(-0.292468\pi\)
0.606763 + 0.794883i \(0.292468\pi\)
\(72\) 0 0
\(73\) 1004.00 1.60972 0.804858 0.593467i \(-0.202241\pi\)
0.804858 + 0.593467i \(0.202241\pi\)
\(74\) 0 0
\(75\) 700.000 1.07772
\(76\) 0 0
\(77\) 486.000 0.719284
\(78\) 0 0
\(79\) 361.000 0.514122 0.257061 0.966395i \(-0.417246\pi\)
0.257061 + 0.966395i \(0.417246\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) 0 0
\(83\) 168.000 0.222173 0.111087 0.993811i \(-0.464567\pi\)
0.111087 + 0.993811i \(0.464567\pi\)
\(84\) 0 0
\(85\) −660.000 −0.842201
\(86\) 0 0
\(87\) 203.000 0.250160
\(88\) 0 0
\(89\) 58.0000 0.0690785 0.0345393 0.999403i \(-0.489004\pi\)
0.0345393 + 0.999403i \(0.489004\pi\)
\(90\) 0 0
\(91\) −1026.00 −1.18191
\(92\) 0 0
\(93\) −1211.00 −1.35027
\(94\) 0 0
\(95\) −2280.00 −2.46235
\(96\) 0 0
\(97\) −1206.00 −1.26238 −0.631189 0.775629i \(-0.717433\pi\)
−0.631189 + 0.775629i \(0.717433\pi\)
\(98\) 0 0
\(99\) −594.000 −0.603023
\(100\) 0 0
\(101\) −1440.00 −1.41867 −0.709333 0.704873i \(-0.751004\pi\)
−0.709333 + 0.704873i \(0.751004\pi\)
\(102\) 0 0
\(103\) 1858.00 1.77742 0.888710 0.458471i \(-0.151603\pi\)
0.888710 + 0.458471i \(0.151603\pi\)
\(104\) 0 0
\(105\) −1890.00 −1.75662
\(106\) 0 0
\(107\) 1914.00 1.72928 0.864642 0.502389i \(-0.167545\pi\)
0.864642 + 0.502389i \(0.167545\pi\)
\(108\) 0 0
\(109\) −989.000 −0.869074 −0.434537 0.900654i \(-0.643088\pi\)
−0.434537 + 0.900654i \(0.643088\pi\)
\(110\) 0 0
\(111\) 840.000 0.718282
\(112\) 0 0
\(113\) −278.000 −0.231434 −0.115717 0.993282i \(-0.536917\pi\)
−0.115717 + 0.993282i \(0.536917\pi\)
\(114\) 0 0
\(115\) −2280.00 −1.84879
\(116\) 0 0
\(117\) 1254.00 0.990875
\(118\) 0 0
\(119\) 792.000 0.610105
\(120\) 0 0
\(121\) −602.000 −0.452292
\(122\) 0 0
\(123\) −2198.00 −1.61128
\(124\) 0 0
\(125\) −375.000 −0.268328
\(126\) 0 0
\(127\) −600.000 −0.419224 −0.209612 0.977785i \(-0.567220\pi\)
−0.209612 + 0.977785i \(0.567220\pi\)
\(128\) 0 0
\(129\) −2373.00 −1.61962
\(130\) 0 0
\(131\) 1652.00 1.10180 0.550900 0.834571i \(-0.314284\pi\)
0.550900 + 0.834571i \(0.314284\pi\)
\(132\) 0 0
\(133\) 2736.00 1.78377
\(134\) 0 0
\(135\) −525.000 −0.334702
\(136\) 0 0
\(137\) −1212.00 −0.755826 −0.377913 0.925841i \(-0.623358\pi\)
−0.377913 + 0.925841i \(0.623358\pi\)
\(138\) 0 0
\(139\) −1328.00 −0.810356 −0.405178 0.914238i \(-0.632790\pi\)
−0.405178 + 0.914238i \(0.632790\pi\)
\(140\) 0 0
\(141\) −2499.00 −1.49258
\(142\) 0 0
\(143\) −1539.00 −0.899984
\(144\) 0 0
\(145\) 435.000 0.249136
\(146\) 0 0
\(147\) −133.000 −0.0746235
\(148\) 0 0
\(149\) 1777.00 0.977030 0.488515 0.872555i \(-0.337539\pi\)
0.488515 + 0.872555i \(0.337539\pi\)
\(150\) 0 0
\(151\) −1934.00 −1.04230 −0.521148 0.853466i \(-0.674496\pi\)
−0.521148 + 0.853466i \(0.674496\pi\)
\(152\) 0 0
\(153\) −968.000 −0.511491
\(154\) 0 0
\(155\) −2595.00 −1.34474
\(156\) 0 0
\(157\) −734.000 −0.373118 −0.186559 0.982444i \(-0.559734\pi\)
−0.186559 + 0.982444i \(0.559734\pi\)
\(158\) 0 0
\(159\) 413.000 0.205994
\(160\) 0 0
\(161\) 2736.00 1.33930
\(162\) 0 0
\(163\) 3337.00 1.60352 0.801761 0.597645i \(-0.203897\pi\)
0.801761 + 0.597645i \(0.203897\pi\)
\(164\) 0 0
\(165\) −2835.00 −1.33760
\(166\) 0 0
\(167\) −198.000 −0.0917467 −0.0458734 0.998947i \(-0.514607\pi\)
−0.0458734 + 0.998947i \(0.514607\pi\)
\(168\) 0 0
\(169\) 1052.00 0.478835
\(170\) 0 0
\(171\) −3344.00 −1.49545
\(172\) 0 0
\(173\) 2598.00 1.14175 0.570874 0.821038i \(-0.306605\pi\)
0.570874 + 0.821038i \(0.306605\pi\)
\(174\) 0 0
\(175\) −1800.00 −0.777527
\(176\) 0 0
\(177\) 4004.00 1.70033
\(178\) 0 0
\(179\) 1510.00 0.630518 0.315259 0.949006i \(-0.397909\pi\)
0.315259 + 0.949006i \(0.397909\pi\)
\(180\) 0 0
\(181\) 2865.00 1.17654 0.588270 0.808665i \(-0.299809\pi\)
0.588270 + 0.808665i \(0.299809\pi\)
\(182\) 0 0
\(183\) 2940.00 1.18760
\(184\) 0 0
\(185\) 1800.00 0.715344
\(186\) 0 0
\(187\) 1188.00 0.464573
\(188\) 0 0
\(189\) 630.000 0.242464
\(190\) 0 0
\(191\) −1936.00 −0.733424 −0.366712 0.930334i \(-0.619517\pi\)
−0.366712 + 0.930334i \(0.619517\pi\)
\(192\) 0 0
\(193\) 3850.00 1.43590 0.717951 0.696094i \(-0.245080\pi\)
0.717951 + 0.696094i \(0.245080\pi\)
\(194\) 0 0
\(195\) 5985.00 2.19792
\(196\) 0 0
\(197\) −3034.00 −1.09728 −0.548638 0.836060i \(-0.684854\pi\)
−0.548638 + 0.836060i \(0.684854\pi\)
\(198\) 0 0
\(199\) 246.000 0.0876305 0.0438153 0.999040i \(-0.486049\pi\)
0.0438153 + 0.999040i \(0.486049\pi\)
\(200\) 0 0
\(201\) −4620.00 −1.62124
\(202\) 0 0
\(203\) −522.000 −0.180479
\(204\) 0 0
\(205\) −4710.00 −1.60469
\(206\) 0 0
\(207\) −3344.00 −1.12282
\(208\) 0 0
\(209\) 4104.00 1.35828
\(210\) 0 0
\(211\) 1493.00 0.487120 0.243560 0.969886i \(-0.421685\pi\)
0.243560 + 0.969886i \(0.421685\pi\)
\(212\) 0 0
\(213\) 5082.00 1.63480
\(214\) 0 0
\(215\) −5085.00 −1.61300
\(216\) 0 0
\(217\) 3114.00 0.974157
\(218\) 0 0
\(219\) 7028.00 2.16853
\(220\) 0 0
\(221\) −2508.00 −0.763377
\(222\) 0 0
\(223\) 1402.00 0.421008 0.210504 0.977593i \(-0.432489\pi\)
0.210504 + 0.977593i \(0.432489\pi\)
\(224\) 0 0
\(225\) 2200.00 0.651852
\(226\) 0 0
\(227\) −1166.00 −0.340926 −0.170463 0.985364i \(-0.554526\pi\)
−0.170463 + 0.985364i \(0.554526\pi\)
\(228\) 0 0
\(229\) 1466.00 0.423039 0.211520 0.977374i \(-0.432159\pi\)
0.211520 + 0.977374i \(0.432159\pi\)
\(230\) 0 0
\(231\) 3402.00 0.968983
\(232\) 0 0
\(233\) 847.000 0.238149 0.119075 0.992885i \(-0.462007\pi\)
0.119075 + 0.992885i \(0.462007\pi\)
\(234\) 0 0
\(235\) −5355.00 −1.48648
\(236\) 0 0
\(237\) 2527.00 0.692600
\(238\) 0 0
\(239\) −444.000 −0.120167 −0.0600836 0.998193i \(-0.519137\pi\)
−0.0600836 + 0.998193i \(0.519137\pi\)
\(240\) 0 0
\(241\) 5297.00 1.41581 0.707904 0.706309i \(-0.249641\pi\)
0.707904 + 0.706309i \(0.249641\pi\)
\(242\) 0 0
\(243\) −4928.00 −1.30095
\(244\) 0 0
\(245\) −285.000 −0.0743183
\(246\) 0 0
\(247\) −8664.00 −2.23189
\(248\) 0 0
\(249\) 1176.00 0.299301
\(250\) 0 0
\(251\) −4061.00 −1.02123 −0.510614 0.859810i \(-0.670582\pi\)
−0.510614 + 0.859810i \(0.670582\pi\)
\(252\) 0 0
\(253\) 4104.00 1.01983
\(254\) 0 0
\(255\) −4620.00 −1.13457
\(256\) 0 0
\(257\) −6843.00 −1.66091 −0.830456 0.557084i \(-0.811920\pi\)
−0.830456 + 0.557084i \(0.811920\pi\)
\(258\) 0 0
\(259\) −2160.00 −0.518208
\(260\) 0 0
\(261\) 638.000 0.151307
\(262\) 0 0
\(263\) 3191.00 0.748158 0.374079 0.927397i \(-0.377959\pi\)
0.374079 + 0.927397i \(0.377959\pi\)
\(264\) 0 0
\(265\) 885.000 0.205151
\(266\) 0 0
\(267\) 406.000 0.0930592
\(268\) 0 0
\(269\) −7436.00 −1.68543 −0.842715 0.538359i \(-0.819044\pi\)
−0.842715 + 0.538359i \(0.819044\pi\)
\(270\) 0 0
\(271\) −2755.00 −0.617544 −0.308772 0.951136i \(-0.599918\pi\)
−0.308772 + 0.951136i \(0.599918\pi\)
\(272\) 0 0
\(273\) −7182.00 −1.59221
\(274\) 0 0
\(275\) −2700.00 −0.592059
\(276\) 0 0
\(277\) 2410.00 0.522754 0.261377 0.965237i \(-0.415823\pi\)
0.261377 + 0.965237i \(0.415823\pi\)
\(278\) 0 0
\(279\) −3806.00 −0.816700
\(280\) 0 0
\(281\) 2235.00 0.474480 0.237240 0.971451i \(-0.423757\pi\)
0.237240 + 0.971451i \(0.423757\pi\)
\(282\) 0 0
\(283\) 1648.00 0.346161 0.173080 0.984908i \(-0.444628\pi\)
0.173080 + 0.984908i \(0.444628\pi\)
\(284\) 0 0
\(285\) −15960.0 −3.31715
\(286\) 0 0
\(287\) 5652.00 1.16246
\(288\) 0 0
\(289\) −2977.00 −0.605943
\(290\) 0 0
\(291\) −8442.00 −1.70061
\(292\) 0 0
\(293\) −2878.00 −0.573838 −0.286919 0.957955i \(-0.592631\pi\)
−0.286919 + 0.957955i \(0.592631\pi\)
\(294\) 0 0
\(295\) 8580.00 1.69338
\(296\) 0 0
\(297\) 945.000 0.184628
\(298\) 0 0
\(299\) −8664.00 −1.67576
\(300\) 0 0
\(301\) 6102.00 1.16848
\(302\) 0 0
\(303\) −10080.0 −1.91116
\(304\) 0 0
\(305\) 6300.00 1.18274
\(306\) 0 0
\(307\) −4903.00 −0.911495 −0.455748 0.890109i \(-0.650628\pi\)
−0.455748 + 0.890109i \(0.650628\pi\)
\(308\) 0 0
\(309\) 13006.0 2.39445
\(310\) 0 0
\(311\) −32.0000 −0.00583458 −0.00291729 0.999996i \(-0.500929\pi\)
−0.00291729 + 0.999996i \(0.500929\pi\)
\(312\) 0 0
\(313\) −5455.00 −0.985095 −0.492548 0.870285i \(-0.663934\pi\)
−0.492548 + 0.870285i \(0.663934\pi\)
\(314\) 0 0
\(315\) −5940.00 −1.06248
\(316\) 0 0
\(317\) 9552.00 1.69241 0.846205 0.532858i \(-0.178882\pi\)
0.846205 + 0.532858i \(0.178882\pi\)
\(318\) 0 0
\(319\) −783.000 −0.137428
\(320\) 0 0
\(321\) 13398.0 2.32961
\(322\) 0 0
\(323\) 6688.00 1.15211
\(324\) 0 0
\(325\) 5700.00 0.972859
\(326\) 0 0
\(327\) −6923.00 −1.17077
\(328\) 0 0
\(329\) 6426.00 1.07683
\(330\) 0 0
\(331\) 2433.00 0.404017 0.202009 0.979384i \(-0.435253\pi\)
0.202009 + 0.979384i \(0.435253\pi\)
\(332\) 0 0
\(333\) 2640.00 0.434448
\(334\) 0 0
\(335\) −9900.00 −1.61461
\(336\) 0 0
\(337\) −7460.00 −1.20585 −0.602926 0.797797i \(-0.705999\pi\)
−0.602926 + 0.797797i \(0.705999\pi\)
\(338\) 0 0
\(339\) −1946.00 −0.311776
\(340\) 0 0
\(341\) 4671.00 0.741785
\(342\) 0 0
\(343\) 6516.00 1.02575
\(344\) 0 0
\(345\) −15960.0 −2.49060
\(346\) 0 0
\(347\) −4002.00 −0.619131 −0.309566 0.950878i \(-0.600184\pi\)
−0.309566 + 0.950878i \(0.600184\pi\)
\(348\) 0 0
\(349\) 2167.00 0.332369 0.166185 0.986095i \(-0.446855\pi\)
0.166185 + 0.986095i \(0.446855\pi\)
\(350\) 0 0
\(351\) −1995.00 −0.303377
\(352\) 0 0
\(353\) −2354.00 −0.354931 −0.177466 0.984127i \(-0.556790\pi\)
−0.177466 + 0.984127i \(0.556790\pi\)
\(354\) 0 0
\(355\) 10890.0 1.62812
\(356\) 0 0
\(357\) 5544.00 0.821904
\(358\) 0 0
\(359\) −3471.00 −0.510285 −0.255143 0.966903i \(-0.582122\pi\)
−0.255143 + 0.966903i \(0.582122\pi\)
\(360\) 0 0
\(361\) 16245.0 2.36842
\(362\) 0 0
\(363\) −4214.00 −0.609305
\(364\) 0 0
\(365\) 15060.0 2.15966
\(366\) 0 0
\(367\) −10880.0 −1.54750 −0.773748 0.633493i \(-0.781621\pi\)
−0.773748 + 0.633493i \(0.781621\pi\)
\(368\) 0 0
\(369\) −6908.00 −0.974569
\(370\) 0 0
\(371\) −1062.00 −0.148615
\(372\) 0 0
\(373\) −8827.00 −1.22532 −0.612661 0.790346i \(-0.709901\pi\)
−0.612661 + 0.790346i \(0.709901\pi\)
\(374\) 0 0
\(375\) −2625.00 −0.361478
\(376\) 0 0
\(377\) 1653.00 0.225819
\(378\) 0 0
\(379\) 876.000 0.118726 0.0593629 0.998236i \(-0.481093\pi\)
0.0593629 + 0.998236i \(0.481093\pi\)
\(380\) 0 0
\(381\) −4200.00 −0.564757
\(382\) 0 0
\(383\) 5446.00 0.726573 0.363287 0.931677i \(-0.381655\pi\)
0.363287 + 0.931677i \(0.381655\pi\)
\(384\) 0 0
\(385\) 7290.00 0.965020
\(386\) 0 0
\(387\) −7458.00 −0.979616
\(388\) 0 0
\(389\) 2216.00 0.288832 0.144416 0.989517i \(-0.453870\pi\)
0.144416 + 0.989517i \(0.453870\pi\)
\(390\) 0 0
\(391\) 6688.00 0.865030
\(392\) 0 0
\(393\) 11564.0 1.48429
\(394\) 0 0
\(395\) 5415.00 0.689768
\(396\) 0 0
\(397\) −6823.00 −0.862561 −0.431280 0.902218i \(-0.641938\pi\)
−0.431280 + 0.902218i \(0.641938\pi\)
\(398\) 0 0
\(399\) 19152.0 2.40301
\(400\) 0 0
\(401\) −5589.00 −0.696013 −0.348007 0.937492i \(-0.613141\pi\)
−0.348007 + 0.937492i \(0.613141\pi\)
\(402\) 0 0
\(403\) −9861.00 −1.21889
\(404\) 0 0
\(405\) −12585.0 −1.54408
\(406\) 0 0
\(407\) −3240.00 −0.394597
\(408\) 0 0
\(409\) −8258.00 −0.998366 −0.499183 0.866496i \(-0.666367\pi\)
−0.499183 + 0.866496i \(0.666367\pi\)
\(410\) 0 0
\(411\) −8484.00 −1.01821
\(412\) 0 0
\(413\) −10296.0 −1.22671
\(414\) 0 0
\(415\) 2520.00 0.298077
\(416\) 0 0
\(417\) −9296.00 −1.09167
\(418\) 0 0
\(419\) −3946.00 −0.460083 −0.230041 0.973181i \(-0.573886\pi\)
−0.230041 + 0.973181i \(0.573886\pi\)
\(420\) 0 0
\(421\) −1380.00 −0.159756 −0.0798778 0.996805i \(-0.525453\pi\)
−0.0798778 + 0.996805i \(0.525453\pi\)
\(422\) 0 0
\(423\) −7854.00 −0.902777
\(424\) 0 0
\(425\) −4400.00 −0.502191
\(426\) 0 0
\(427\) −7560.00 −0.856801
\(428\) 0 0
\(429\) −10773.0 −1.21241
\(430\) 0 0
\(431\) −5820.00 −0.650440 −0.325220 0.945638i \(-0.605438\pi\)
−0.325220 + 0.945638i \(0.605438\pi\)
\(432\) 0 0
\(433\) −14504.0 −1.60974 −0.804870 0.593451i \(-0.797765\pi\)
−0.804870 + 0.593451i \(0.797765\pi\)
\(434\) 0 0
\(435\) 3045.00 0.335624
\(436\) 0 0
\(437\) 23104.0 2.52909
\(438\) 0 0
\(439\) 14912.0 1.62121 0.810605 0.585594i \(-0.199139\pi\)
0.810605 + 0.585594i \(0.199139\pi\)
\(440\) 0 0
\(441\) −418.000 −0.0451355
\(442\) 0 0
\(443\) −7180.00 −0.770050 −0.385025 0.922906i \(-0.625807\pi\)
−0.385025 + 0.922906i \(0.625807\pi\)
\(444\) 0 0
\(445\) 870.000 0.0926786
\(446\) 0 0
\(447\) 12439.0 1.31621
\(448\) 0 0
\(449\) 1398.00 0.146939 0.0734696 0.997297i \(-0.476593\pi\)
0.0734696 + 0.997297i \(0.476593\pi\)
\(450\) 0 0
\(451\) 8478.00 0.885174
\(452\) 0 0
\(453\) −13538.0 −1.40413
\(454\) 0 0
\(455\) −15390.0 −1.58570
\(456\) 0 0
\(457\) −5658.00 −0.579147 −0.289573 0.957156i \(-0.593513\pi\)
−0.289573 + 0.957156i \(0.593513\pi\)
\(458\) 0 0
\(459\) 1540.00 0.156604
\(460\) 0 0
\(461\) −11410.0 −1.15275 −0.576374 0.817186i \(-0.695533\pi\)
−0.576374 + 0.817186i \(0.695533\pi\)
\(462\) 0 0
\(463\) −2560.00 −0.256962 −0.128481 0.991712i \(-0.541010\pi\)
−0.128481 + 0.991712i \(0.541010\pi\)
\(464\) 0 0
\(465\) −18165.0 −1.81157
\(466\) 0 0
\(467\) −10395.0 −1.03003 −0.515014 0.857182i \(-0.672213\pi\)
−0.515014 + 0.857182i \(0.672213\pi\)
\(468\) 0 0
\(469\) 11880.0 1.16965
\(470\) 0 0
\(471\) −5138.00 −0.502647
\(472\) 0 0
\(473\) 9153.00 0.889758
\(474\) 0 0
\(475\) −15200.0 −1.46826
\(476\) 0 0
\(477\) 1298.00 0.124594
\(478\) 0 0
\(479\) −5627.00 −0.536752 −0.268376 0.963314i \(-0.586487\pi\)
−0.268376 + 0.963314i \(0.586487\pi\)
\(480\) 0 0
\(481\) 6840.00 0.648393
\(482\) 0 0
\(483\) 19152.0 1.80424
\(484\) 0 0
\(485\) −18090.0 −1.69366
\(486\) 0 0
\(487\) 8738.00 0.813053 0.406526 0.913639i \(-0.366740\pi\)
0.406526 + 0.913639i \(0.366740\pi\)
\(488\) 0 0
\(489\) 23359.0 2.16019
\(490\) 0 0
\(491\) −3177.00 −0.292008 −0.146004 0.989284i \(-0.546641\pi\)
−0.146004 + 0.989284i \(0.546641\pi\)
\(492\) 0 0
\(493\) −1276.00 −0.116568
\(494\) 0 0
\(495\) −8910.00 −0.809040
\(496\) 0 0
\(497\) −13068.0 −1.17944
\(498\) 0 0
\(499\) 5816.00 0.521763 0.260882 0.965371i \(-0.415987\pi\)
0.260882 + 0.965371i \(0.415987\pi\)
\(500\) 0 0
\(501\) −1386.00 −0.123597
\(502\) 0 0
\(503\) −2267.00 −0.200955 −0.100478 0.994939i \(-0.532037\pi\)
−0.100478 + 0.994939i \(0.532037\pi\)
\(504\) 0 0
\(505\) −21600.0 −1.90334
\(506\) 0 0
\(507\) 7364.00 0.645063
\(508\) 0 0
\(509\) −6481.00 −0.564372 −0.282186 0.959360i \(-0.591060\pi\)
−0.282186 + 0.959360i \(0.591060\pi\)
\(510\) 0 0
\(511\) −18072.0 −1.56450
\(512\) 0 0
\(513\) 5320.00 0.457863
\(514\) 0 0
\(515\) 27870.0 2.38466
\(516\) 0 0
\(517\) 9639.00 0.819967
\(518\) 0 0
\(519\) 18186.0 1.53811
\(520\) 0 0
\(521\) −21357.0 −1.79591 −0.897953 0.440091i \(-0.854946\pi\)
−0.897953 + 0.440091i \(0.854946\pi\)
\(522\) 0 0
\(523\) 6192.00 0.517700 0.258850 0.965917i \(-0.416656\pi\)
0.258850 + 0.965917i \(0.416656\pi\)
\(524\) 0 0
\(525\) −12600.0 −1.04745
\(526\) 0 0
\(527\) 7612.00 0.629192
\(528\) 0 0
\(529\) 10937.0 0.898907
\(530\) 0 0
\(531\) 12584.0 1.02844
\(532\) 0 0
\(533\) −17898.0 −1.45450
\(534\) 0 0
\(535\) 28710.0 2.32008
\(536\) 0 0
\(537\) 10570.0 0.849403
\(538\) 0 0
\(539\) 513.000 0.0409953
\(540\) 0 0
\(541\) 19800.0 1.57351 0.786755 0.617266i \(-0.211760\pi\)
0.786755 + 0.617266i \(0.211760\pi\)
\(542\) 0 0
\(543\) 20055.0 1.58498
\(544\) 0 0
\(545\) −14835.0 −1.16598
\(546\) 0 0
\(547\) −10526.0 −0.822777 −0.411389 0.911460i \(-0.634956\pi\)
−0.411389 + 0.911460i \(0.634956\pi\)
\(548\) 0 0
\(549\) 9240.00 0.718313
\(550\) 0 0
\(551\) −4408.00 −0.340811
\(552\) 0 0
\(553\) −6498.00 −0.499680
\(554\) 0 0
\(555\) 12600.0 0.963676
\(556\) 0 0
\(557\) 11834.0 0.900220 0.450110 0.892973i \(-0.351385\pi\)
0.450110 + 0.892973i \(0.351385\pi\)
\(558\) 0 0
\(559\) −19323.0 −1.46203
\(560\) 0 0
\(561\) 8316.00 0.625850
\(562\) 0 0
\(563\) −21771.0 −1.62973 −0.814865 0.579650i \(-0.803189\pi\)
−0.814865 + 0.579650i \(0.803189\pi\)
\(564\) 0 0
\(565\) −4170.00 −0.310501
\(566\) 0 0
\(567\) 15102.0 1.11856
\(568\) 0 0
\(569\) 20778.0 1.53086 0.765430 0.643519i \(-0.222526\pi\)
0.765430 + 0.643519i \(0.222526\pi\)
\(570\) 0 0
\(571\) 5596.00 0.410132 0.205066 0.978748i \(-0.434259\pi\)
0.205066 + 0.978748i \(0.434259\pi\)
\(572\) 0 0
\(573\) −13552.0 −0.988033
\(574\) 0 0
\(575\) −15200.0 −1.10241
\(576\) 0 0
\(577\) −14908.0 −1.07561 −0.537806 0.843069i \(-0.680747\pi\)
−0.537806 + 0.843069i \(0.680747\pi\)
\(578\) 0 0
\(579\) 26950.0 1.93438
\(580\) 0 0
\(581\) −3024.00 −0.215932
\(582\) 0 0
\(583\) −1593.00 −0.113165
\(584\) 0 0
\(585\) 18810.0 1.32940
\(586\) 0 0
\(587\) −8136.00 −0.572076 −0.286038 0.958218i \(-0.592338\pi\)
−0.286038 + 0.958218i \(0.592338\pi\)
\(588\) 0 0
\(589\) 26296.0 1.83957
\(590\) 0 0
\(591\) −21238.0 −1.47820
\(592\) 0 0
\(593\) −3521.00 −0.243828 −0.121914 0.992541i \(-0.538903\pi\)
−0.121914 + 0.992541i \(0.538903\pi\)
\(594\) 0 0
\(595\) 11880.0 0.818542
\(596\) 0 0
\(597\) 1722.00 0.118052
\(598\) 0 0
\(599\) 7749.00 0.528574 0.264287 0.964444i \(-0.414863\pi\)
0.264287 + 0.964444i \(0.414863\pi\)
\(600\) 0 0
\(601\) 5054.00 0.343023 0.171512 0.985182i \(-0.445135\pi\)
0.171512 + 0.985182i \(0.445135\pi\)
\(602\) 0 0
\(603\) −14520.0 −0.980597
\(604\) 0 0
\(605\) −9030.00 −0.606813
\(606\) 0 0
\(607\) 27389.0 1.83144 0.915721 0.401815i \(-0.131620\pi\)
0.915721 + 0.401815i \(0.131620\pi\)
\(608\) 0 0
\(609\) −3654.00 −0.243132
\(610\) 0 0
\(611\) −20349.0 −1.34735
\(612\) 0 0
\(613\) 2375.00 0.156485 0.0782425 0.996934i \(-0.475069\pi\)
0.0782425 + 0.996934i \(0.475069\pi\)
\(614\) 0 0
\(615\) −32970.0 −2.16175
\(616\) 0 0
\(617\) 3412.00 0.222629 0.111314 0.993785i \(-0.464494\pi\)
0.111314 + 0.993785i \(0.464494\pi\)
\(618\) 0 0
\(619\) −17659.0 −1.14665 −0.573324 0.819329i \(-0.694346\pi\)
−0.573324 + 0.819329i \(0.694346\pi\)
\(620\) 0 0
\(621\) 5320.00 0.343775
\(622\) 0 0
\(623\) −1044.00 −0.0671380
\(624\) 0 0
\(625\) −18125.0 −1.16000
\(626\) 0 0
\(627\) 28728.0 1.82980
\(628\) 0 0
\(629\) −5280.00 −0.334702
\(630\) 0 0
\(631\) 12742.0 0.803884 0.401942 0.915665i \(-0.368335\pi\)
0.401942 + 0.915665i \(0.368335\pi\)
\(632\) 0 0
\(633\) 10451.0 0.656224
\(634\) 0 0
\(635\) −9000.00 −0.562447
\(636\) 0 0
\(637\) −1083.00 −0.0673627
\(638\) 0 0
\(639\) 15972.0 0.988799
\(640\) 0 0
\(641\) −10988.0 −0.677067 −0.338533 0.940954i \(-0.609931\pi\)
−0.338533 + 0.940954i \(0.609931\pi\)
\(642\) 0 0
\(643\) 13190.0 0.808962 0.404481 0.914546i \(-0.367452\pi\)
0.404481 + 0.914546i \(0.367452\pi\)
\(644\) 0 0
\(645\) −35595.0 −2.17295
\(646\) 0 0
\(647\) 22924.0 1.39295 0.696473 0.717583i \(-0.254752\pi\)
0.696473 + 0.717583i \(0.254752\pi\)
\(648\) 0 0
\(649\) −15444.0 −0.934099
\(650\) 0 0
\(651\) 21798.0 1.31234
\(652\) 0 0
\(653\) 10278.0 0.615941 0.307970 0.951396i \(-0.400350\pi\)
0.307970 + 0.951396i \(0.400350\pi\)
\(654\) 0 0
\(655\) 24780.0 1.47822
\(656\) 0 0
\(657\) 22088.0 1.31162
\(658\) 0 0
\(659\) 10607.0 0.626996 0.313498 0.949589i \(-0.398499\pi\)
0.313498 + 0.949589i \(0.398499\pi\)
\(660\) 0 0
\(661\) 24698.0 1.45331 0.726657 0.687000i \(-0.241073\pi\)
0.726657 + 0.687000i \(0.241073\pi\)
\(662\) 0 0
\(663\) −17556.0 −1.02838
\(664\) 0 0
\(665\) 41040.0 2.39318
\(666\) 0 0
\(667\) −4408.00 −0.255890
\(668\) 0 0
\(669\) 9814.00 0.567162
\(670\) 0 0
\(671\) −11340.0 −0.652423
\(672\) 0 0
\(673\) 16801.0 0.962305 0.481152 0.876637i \(-0.340218\pi\)
0.481152 + 0.876637i \(0.340218\pi\)
\(674\) 0 0
\(675\) −3500.00 −0.199578
\(676\) 0 0
\(677\) 1258.00 0.0714163 0.0357082 0.999362i \(-0.488631\pi\)
0.0357082 + 0.999362i \(0.488631\pi\)
\(678\) 0 0
\(679\) 21708.0 1.22692
\(680\) 0 0
\(681\) −8162.00 −0.459278
\(682\) 0 0
\(683\) 1256.00 0.0703653 0.0351827 0.999381i \(-0.488799\pi\)
0.0351827 + 0.999381i \(0.488799\pi\)
\(684\) 0 0
\(685\) −18180.0 −1.01405
\(686\) 0 0
\(687\) 10262.0 0.569898
\(688\) 0 0
\(689\) 3363.00 0.185951
\(690\) 0 0
\(691\) 15996.0 0.880632 0.440316 0.897843i \(-0.354866\pi\)
0.440316 + 0.897843i \(0.354866\pi\)
\(692\) 0 0
\(693\) 10692.0 0.586083
\(694\) 0 0
\(695\) −19920.0 −1.08721
\(696\) 0 0
\(697\) 13816.0 0.750815
\(698\) 0 0
\(699\) 5929.00 0.320823
\(700\) 0 0
\(701\) 189.000 0.0101832 0.00509161 0.999987i \(-0.498379\pi\)
0.00509161 + 0.999987i \(0.498379\pi\)
\(702\) 0 0
\(703\) −18240.0 −0.978570
\(704\) 0 0
\(705\) −37485.0 −2.00251
\(706\) 0 0
\(707\) 25920.0 1.37881
\(708\) 0 0
\(709\) −2431.00 −0.128770 −0.0643851 0.997925i \(-0.520509\pi\)
−0.0643851 + 0.997925i \(0.520509\pi\)
\(710\) 0 0
\(711\) 7942.00 0.418915
\(712\) 0 0
\(713\) 26296.0 1.38120
\(714\) 0 0
\(715\) −23085.0 −1.20745
\(716\) 0 0
\(717\) −3108.00 −0.161883
\(718\) 0 0
\(719\) −3630.00 −0.188284 −0.0941420 0.995559i \(-0.530011\pi\)
−0.0941420 + 0.995559i \(0.530011\pi\)
\(720\) 0 0
\(721\) −33444.0 −1.72749
\(722\) 0 0
\(723\) 37079.0 1.90731
\(724\) 0 0
\(725\) 2900.00 0.148556
\(726\) 0 0
\(727\) −1712.00 −0.0873378 −0.0436689 0.999046i \(-0.513905\pi\)
−0.0436689 + 0.999046i \(0.513905\pi\)
\(728\) 0 0
\(729\) −11843.0 −0.601687
\(730\) 0 0
\(731\) 14916.0 0.754703
\(732\) 0 0
\(733\) −26584.0 −1.33957 −0.669783 0.742557i \(-0.733613\pi\)
−0.669783 + 0.742557i \(0.733613\pi\)
\(734\) 0 0
\(735\) −1995.00 −0.100118
\(736\) 0 0
\(737\) 17820.0 0.890649
\(738\) 0 0
\(739\) −8125.00 −0.404442 −0.202221 0.979340i \(-0.564816\pi\)
−0.202221 + 0.979340i \(0.564816\pi\)
\(740\) 0 0
\(741\) −60648.0 −3.00669
\(742\) 0 0
\(743\) −31804.0 −1.57036 −0.785179 0.619269i \(-0.787429\pi\)
−0.785179 + 0.619269i \(0.787429\pi\)
\(744\) 0 0
\(745\) 26655.0 1.31082
\(746\) 0 0
\(747\) 3696.00 0.181030
\(748\) 0 0
\(749\) −34452.0 −1.68071
\(750\) 0 0
\(751\) −37504.0 −1.82229 −0.911145 0.412085i \(-0.864801\pi\)
−0.911145 + 0.412085i \(0.864801\pi\)
\(752\) 0 0
\(753\) −28427.0 −1.37575
\(754\) 0 0
\(755\) −29010.0 −1.39839
\(756\) 0 0
\(757\) 76.0000 0.00364897 0.00182448 0.999998i \(-0.499419\pi\)
0.00182448 + 0.999998i \(0.499419\pi\)
\(758\) 0 0
\(759\) 28728.0 1.37386
\(760\) 0 0
\(761\) −23374.0 −1.11341 −0.556706 0.830709i \(-0.687935\pi\)
−0.556706 + 0.830709i \(0.687935\pi\)
\(762\) 0 0
\(763\) 17802.0 0.844660
\(764\) 0 0
\(765\) −14520.0 −0.686238
\(766\) 0 0
\(767\) 32604.0 1.53489
\(768\) 0 0
\(769\) −5448.00 −0.255475 −0.127737 0.991808i \(-0.540771\pi\)
−0.127737 + 0.991808i \(0.540771\pi\)
\(770\) 0 0
\(771\) −47901.0 −2.23750
\(772\) 0 0
\(773\) 16062.0 0.747361 0.373680 0.927558i \(-0.378096\pi\)
0.373680 + 0.927558i \(0.378096\pi\)
\(774\) 0 0
\(775\) −17300.0 −0.801851
\(776\) 0 0
\(777\) −15120.0 −0.698104
\(778\) 0 0
\(779\) 47728.0 2.19516
\(780\) 0 0
\(781\) −19602.0 −0.898098
\(782\) 0 0
\(783\) −1015.00 −0.0463259
\(784\) 0 0
\(785\) −11010.0 −0.500591
\(786\) 0 0
\(787\) −19182.0 −0.868824 −0.434412 0.900714i \(-0.643044\pi\)
−0.434412 + 0.900714i \(0.643044\pi\)
\(788\) 0 0
\(789\) 22337.0 1.00788
\(790\) 0 0
\(791\) 5004.00 0.224933
\(792\) 0 0
\(793\) 23940.0 1.07205
\(794\) 0 0
\(795\) 6195.00 0.276370
\(796\) 0 0
\(797\) −5556.00 −0.246931 −0.123465 0.992349i \(-0.539401\pi\)
−0.123465 + 0.992349i \(0.539401\pi\)
\(798\) 0 0
\(799\) 15708.0 0.695506
\(800\) 0 0
\(801\) 1276.00 0.0562862
\(802\) 0 0
\(803\) −27108.0 −1.19131
\(804\) 0 0
\(805\) 41040.0 1.79686
\(806\) 0 0
\(807\) −52052.0 −2.27053
\(808\) 0 0
\(809\) 2616.00 0.113688 0.0568440 0.998383i \(-0.481896\pi\)
0.0568440 + 0.998383i \(0.481896\pi\)
\(810\) 0 0
\(811\) 9626.00 0.416787 0.208394 0.978045i \(-0.433176\pi\)
0.208394 + 0.978045i \(0.433176\pi\)
\(812\) 0 0
\(813\) −19285.0 −0.831924
\(814\) 0 0
\(815\) 50055.0 2.15135
\(816\) 0 0
\(817\) 51528.0 2.20653
\(818\) 0 0
\(819\) −22572.0 −0.963040
\(820\) 0 0
\(821\) −16007.0 −0.680448 −0.340224 0.940344i \(-0.610503\pi\)
−0.340224 + 0.940344i \(0.610503\pi\)
\(822\) 0 0
\(823\) 23216.0 0.983304 0.491652 0.870792i \(-0.336393\pi\)
0.491652 + 0.870792i \(0.336393\pi\)
\(824\) 0 0
\(825\) −18900.0 −0.797592
\(826\) 0 0
\(827\) 46765.0 1.96636 0.983179 0.182644i \(-0.0584654\pi\)
0.983179 + 0.182644i \(0.0584654\pi\)
\(828\) 0 0
\(829\) 37820.0 1.58449 0.792245 0.610203i \(-0.208912\pi\)
0.792245 + 0.610203i \(0.208912\pi\)
\(830\) 0 0
\(831\) 16870.0 0.704228
\(832\) 0 0
\(833\) 836.000 0.0347727
\(834\) 0 0
\(835\) −2970.00 −0.123091
\(836\) 0 0
\(837\) 6055.00 0.250049
\(838\) 0 0
\(839\) 6747.00 0.277631 0.138815 0.990318i \(-0.455671\pi\)
0.138815 + 0.990318i \(0.455671\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 15645.0 0.639196
\(844\) 0 0
\(845\) 15780.0 0.642424
\(846\) 0 0
\(847\) 10836.0 0.439586
\(848\) 0 0
\(849\) 11536.0 0.466330
\(850\) 0 0
\(851\) −18240.0 −0.734735
\(852\) 0 0
\(853\) 13274.0 0.532817 0.266409 0.963860i \(-0.414163\pi\)
0.266409 + 0.963860i \(0.414163\pi\)
\(854\) 0 0
\(855\) −50160.0 −2.00636
\(856\) 0 0
\(857\) 39069.0 1.55726 0.778630 0.627483i \(-0.215915\pi\)
0.778630 + 0.627483i \(0.215915\pi\)
\(858\) 0 0
\(859\) −35329.0 −1.40327 −0.701636 0.712536i \(-0.747547\pi\)
−0.701636 + 0.712536i \(0.747547\pi\)
\(860\) 0 0
\(861\) 39564.0 1.56601
\(862\) 0 0
\(863\) −36378.0 −1.43490 −0.717452 0.696608i \(-0.754692\pi\)
−0.717452 + 0.696608i \(0.754692\pi\)
\(864\) 0 0
\(865\) 38970.0 1.53181
\(866\) 0 0
\(867\) −20839.0 −0.816297
\(868\) 0 0
\(869\) −9747.00 −0.380488
\(870\) 0 0
\(871\) −37620.0 −1.46350
\(872\) 0 0
\(873\) −26532.0 −1.02860
\(874\) 0 0
\(875\) 6750.00 0.260790
\(876\) 0 0
\(877\) 219.000 0.00843227 0.00421614 0.999991i \(-0.498658\pi\)
0.00421614 + 0.999991i \(0.498658\pi\)
\(878\) 0 0
\(879\) −20146.0 −0.773046
\(880\) 0 0
\(881\) −5598.00 −0.214077 −0.107038 0.994255i \(-0.534137\pi\)
−0.107038 + 0.994255i \(0.534137\pi\)
\(882\) 0 0
\(883\) −5886.00 −0.224326 −0.112163 0.993690i \(-0.535778\pi\)
−0.112163 + 0.993690i \(0.535778\pi\)
\(884\) 0 0
\(885\) 60060.0 2.28124
\(886\) 0 0
\(887\) −29673.0 −1.12325 −0.561624 0.827392i \(-0.689823\pi\)
−0.561624 + 0.827392i \(0.689823\pi\)
\(888\) 0 0
\(889\) 10800.0 0.407447
\(890\) 0 0
\(891\) 22653.0 0.851744
\(892\) 0 0
\(893\) 54264.0 2.03346
\(894\) 0 0
\(895\) 22650.0 0.845928
\(896\) 0 0
\(897\) −60648.0 −2.25750
\(898\) 0 0
\(899\) −5017.00 −0.186125
\(900\) 0 0
\(901\) −2596.00 −0.0959881
\(902\) 0 0
\(903\) 42714.0 1.57412
\(904\) 0 0
\(905\) 42975.0 1.57849
\(906\) 0 0
\(907\) −13668.0 −0.500373 −0.250187 0.968198i \(-0.580492\pi\)
−0.250187 + 0.968198i \(0.580492\pi\)
\(908\) 0 0
\(909\) −31680.0 −1.15595
\(910\) 0 0
\(911\) −4963.00 −0.180496 −0.0902478 0.995919i \(-0.528766\pi\)
−0.0902478 + 0.995919i \(0.528766\pi\)
\(912\) 0 0
\(913\) −4536.00 −0.164425
\(914\) 0 0
\(915\) 44100.0 1.59333
\(916\) 0 0
\(917\) −29736.0 −1.07085
\(918\) 0 0
\(919\) 9954.00 0.357293 0.178646 0.983913i \(-0.442828\pi\)
0.178646 + 0.983913i \(0.442828\pi\)
\(920\) 0 0
\(921\) −34321.0 −1.22792
\(922\) 0 0
\(923\) 41382.0 1.47574
\(924\) 0 0
\(925\) 12000.0 0.426549
\(926\) 0 0
\(927\) 40876.0 1.44827
\(928\) 0 0
\(929\) 30930.0 1.09234 0.546168 0.837676i \(-0.316086\pi\)
0.546168 + 0.837676i \(0.316086\pi\)
\(930\) 0 0
\(931\) 2888.00 0.101665
\(932\) 0 0
\(933\) −224.000 −0.00786005
\(934\) 0 0
\(935\) 17820.0 0.623290
\(936\) 0 0
\(937\) −38770.0 −1.35172 −0.675859 0.737030i \(-0.736227\pi\)
−0.675859 + 0.737030i \(0.736227\pi\)
\(938\) 0 0
\(939\) −38185.0 −1.32707
\(940\) 0 0
\(941\) 30115.0 1.04327 0.521637 0.853167i \(-0.325322\pi\)
0.521637 + 0.853167i \(0.325322\pi\)
\(942\) 0 0
\(943\) 47728.0 1.64818
\(944\) 0 0
\(945\) 9450.00 0.325300
\(946\) 0 0
\(947\) −31319.0 −1.07469 −0.537345 0.843363i \(-0.680573\pi\)
−0.537345 + 0.843363i \(0.680573\pi\)
\(948\) 0 0
\(949\) 57228.0 1.95753
\(950\) 0 0
\(951\) 66864.0 2.27993
\(952\) 0 0
\(953\) 7623.00 0.259111 0.129556 0.991572i \(-0.458645\pi\)
0.129556 + 0.991572i \(0.458645\pi\)
\(954\) 0 0
\(955\) −29040.0 −0.983992
\(956\) 0 0
\(957\) −5481.00 −0.185136
\(958\) 0 0
\(959\) 21816.0 0.734594
\(960\) 0 0
\(961\) 138.000 0.00463227
\(962\) 0 0
\(963\) 42108.0 1.40905
\(964\) 0 0
\(965\) 57750.0 1.92646
\(966\) 0 0
\(967\) −6269.00 −0.208477 −0.104239 0.994552i \(-0.533241\pi\)
−0.104239 + 0.994552i \(0.533241\pi\)
\(968\) 0 0
\(969\) 46816.0 1.55206
\(970\) 0 0
\(971\) 34020.0 1.12436 0.562180 0.827015i \(-0.309963\pi\)
0.562180 + 0.827015i \(0.309963\pi\)
\(972\) 0 0
\(973\) 23904.0 0.787592
\(974\) 0 0
\(975\) 39900.0 1.31059
\(976\) 0 0
\(977\) −51291.0 −1.67957 −0.839787 0.542915i \(-0.817320\pi\)
−0.839787 + 0.542915i \(0.817320\pi\)
\(978\) 0 0
\(979\) −1566.00 −0.0511232
\(980\) 0 0
\(981\) −21758.0 −0.708134
\(982\) 0 0
\(983\) 38687.0 1.25526 0.627632 0.778511i \(-0.284024\pi\)
0.627632 + 0.778511i \(0.284024\pi\)
\(984\) 0 0
\(985\) −45510.0 −1.47215
\(986\) 0 0
\(987\) 44982.0 1.45065
\(988\) 0 0
\(989\) 51528.0 1.65672
\(990\) 0 0
\(991\) −3158.00 −0.101228 −0.0506141 0.998718i \(-0.516118\pi\)
−0.0506141 + 0.998718i \(0.516118\pi\)
\(992\) 0 0
\(993\) 17031.0 0.544272
\(994\) 0 0
\(995\) 3690.00 0.117569
\(996\) 0 0
\(997\) 21944.0 0.697065 0.348532 0.937297i \(-0.386680\pi\)
0.348532 + 0.937297i \(0.386680\pi\)
\(998\) 0 0
\(999\) −4200.00 −0.133015
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 1856.4.a.f.1.1 1
4.3 odd 2 1856.4.a.c.1.1 1
8.3 odd 2 464.4.a.b.1.1 1
8.5 even 2 58.4.a.b.1.1 1
24.5 odd 2 522.4.a.b.1.1 1
40.29 even 2 1450.4.a.d.1.1 1
232.173 even 2 1682.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.4.a.b.1.1 1 8.5 even 2
464.4.a.b.1.1 1 8.3 odd 2
522.4.a.b.1.1 1 24.5 odd 2
1450.4.a.d.1.1 1 40.29 even 2
1682.4.a.a.1.1 1 232.173 even 2
1856.4.a.c.1.1 1 4.3 odd 2
1856.4.a.f.1.1 1 1.1 even 1 trivial