Properties

Label 1350.4.a.q.1.1
Level $1350$
Weight $4$
Character 1350.1
Self dual yes
Analytic conductor $79.653$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1350,4,Mod(1,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1350.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.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: \(79.6525785077\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1350.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} -19.0000 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} -19.0000 q^{7} +8.00000 q^{8} +12.0000 q^{11} +50.0000 q^{13} -38.0000 q^{14} +16.0000 q^{16} -126.000 q^{17} +29.0000 q^{19} +24.0000 q^{22} +18.0000 q^{23} +100.000 q^{26} -76.0000 q^{28} +102.000 q^{29} -265.000 q^{31} +32.0000 q^{32} -252.000 q^{34} +65.0000 q^{37} +58.0000 q^{38} +240.000 q^{41} -367.000 q^{43} +48.0000 q^{44} +36.0000 q^{46} -72.0000 q^{47} +18.0000 q^{49} +200.000 q^{52} -636.000 q^{53} -152.000 q^{56} +204.000 q^{58} +102.000 q^{59} -103.000 q^{61} -530.000 q^{62} +64.0000 q^{64} -52.0000 q^{67} -504.000 q^{68} -582.000 q^{71} +65.0000 q^{73} +130.000 q^{74} +116.000 q^{76} -228.000 q^{77} +173.000 q^{79} +480.000 q^{82} +498.000 q^{83} -734.000 q^{86} +96.0000 q^{88} -822.000 q^{89} -950.000 q^{91} +72.0000 q^{92} -144.000 q^{94} +821.000 q^{97} +36.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −19.0000 −1.02590 −0.512952 0.858417i \(-0.671448\pi\)
−0.512952 + 0.858417i \(0.671448\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 12.0000 0.328921 0.164461 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(12\) 0 0
\(13\) 50.0000 1.06673 0.533366 0.845885i \(-0.320927\pi\)
0.533366 + 0.845885i \(0.320927\pi\)
\(14\) −38.0000 −0.725423
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −126.000 −1.79762 −0.898808 0.438342i \(-0.855566\pi\)
−0.898808 + 0.438342i \(0.855566\pi\)
\(18\) 0 0
\(19\) 29.0000 0.350161 0.175080 0.984554i \(-0.443981\pi\)
0.175080 + 0.984554i \(0.443981\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 24.0000 0.232583
\(23\) 18.0000 0.163185 0.0815926 0.996666i \(-0.473999\pi\)
0.0815926 + 0.996666i \(0.473999\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 100.000 0.754293
\(27\) 0 0
\(28\) −76.0000 −0.512952
\(29\) 102.000 0.653135 0.326568 0.945174i \(-0.394108\pi\)
0.326568 + 0.945174i \(0.394108\pi\)
\(30\) 0 0
\(31\) −265.000 −1.53534 −0.767668 0.640848i \(-0.778583\pi\)
−0.767668 + 0.640848i \(0.778583\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) −252.000 −1.27111
\(35\) 0 0
\(36\) 0 0
\(37\) 65.0000 0.288809 0.144405 0.989519i \(-0.453873\pi\)
0.144405 + 0.989519i \(0.453873\pi\)
\(38\) 58.0000 0.247601
\(39\) 0 0
\(40\) 0 0
\(41\) 240.000 0.914188 0.457094 0.889418i \(-0.348890\pi\)
0.457094 + 0.889418i \(0.348890\pi\)
\(42\) 0 0
\(43\) −367.000 −1.30156 −0.650779 0.759267i \(-0.725557\pi\)
−0.650779 + 0.759267i \(0.725557\pi\)
\(44\) 48.0000 0.164461
\(45\) 0 0
\(46\) 36.0000 0.115389
\(47\) −72.0000 −0.223453 −0.111726 0.993739i \(-0.535638\pi\)
−0.111726 + 0.993739i \(0.535638\pi\)
\(48\) 0 0
\(49\) 18.0000 0.0524781
\(50\) 0 0
\(51\) 0 0
\(52\) 200.000 0.533366
\(53\) −636.000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −152.000 −0.362712
\(57\) 0 0
\(58\) 204.000 0.461836
\(59\) 102.000 0.225072 0.112536 0.993648i \(-0.464103\pi\)
0.112536 + 0.993648i \(0.464103\pi\)
\(60\) 0 0
\(61\) −103.000 −0.216193 −0.108097 0.994140i \(-0.534476\pi\)
−0.108097 + 0.994140i \(0.534476\pi\)
\(62\) −530.000 −1.08565
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −52.0000 −0.0948181 −0.0474090 0.998876i \(-0.515096\pi\)
−0.0474090 + 0.998876i \(0.515096\pi\)
\(68\) −504.000 −0.898808
\(69\) 0 0
\(70\) 0 0
\(71\) −582.000 −0.972827 −0.486413 0.873729i \(-0.661695\pi\)
−0.486413 + 0.873729i \(0.661695\pi\)
\(72\) 0 0
\(73\) 65.0000 0.104215 0.0521074 0.998641i \(-0.483406\pi\)
0.0521074 + 0.998641i \(0.483406\pi\)
\(74\) 130.000 0.204219
\(75\) 0 0
\(76\) 116.000 0.175080
\(77\) −228.000 −0.337442
\(78\) 0 0
\(79\) 173.000 0.246380 0.123190 0.992383i \(-0.460688\pi\)
0.123190 + 0.992383i \(0.460688\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 480.000 0.646428
\(83\) 498.000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −734.000 −0.920340
\(87\) 0 0
\(88\) 96.0000 0.116291
\(89\) −822.000 −0.979009 −0.489505 0.872001i \(-0.662822\pi\)
−0.489505 + 0.872001i \(0.662822\pi\)
\(90\) 0 0
\(91\) −950.000 −1.09436
\(92\) 72.0000 0.0815926
\(93\) 0 0
\(94\) −144.000 −0.158005
\(95\) 0 0
\(96\) 0 0
\(97\) 821.000 0.859381 0.429690 0.902976i \(-0.358623\pi\)
0.429690 + 0.902976i \(0.358623\pi\)
\(98\) 36.0000 0.0371076
\(99\) 0 0
\(100\) 0 0
\(101\) −1200.00 −1.18222 −0.591111 0.806590i \(-0.701311\pi\)
−0.591111 + 0.806590i \(0.701311\pi\)
\(102\) 0 0
\(103\) −2041.00 −1.95248 −0.976241 0.216686i \(-0.930475\pi\)
−0.976241 + 0.216686i \(0.930475\pi\)
\(104\) 400.000 0.377146
\(105\) 0 0
\(106\) −1272.00 −1.16554
\(107\) −1278.00 −1.15466 −0.577331 0.816510i \(-0.695906\pi\)
−0.577331 + 0.816510i \(0.695906\pi\)
\(108\) 0 0
\(109\) −205.000 −0.180142 −0.0900708 0.995935i \(-0.528709\pi\)
−0.0900708 + 0.995935i \(0.528709\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −304.000 −0.256476
\(113\) −1500.00 −1.24874 −0.624372 0.781127i \(-0.714645\pi\)
−0.624372 + 0.781127i \(0.714645\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 408.000 0.326568
\(117\) 0 0
\(118\) 204.000 0.159150
\(119\) 2394.00 1.84418
\(120\) 0 0
\(121\) −1187.00 −0.891811
\(122\) −206.000 −0.152872
\(123\) 0 0
\(124\) −1060.00 −0.767668
\(125\) 0 0
\(126\) 0 0
\(127\) 416.000 0.290662 0.145331 0.989383i \(-0.453575\pi\)
0.145331 + 0.989383i \(0.453575\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −1902.00 −1.26854 −0.634269 0.773112i \(-0.718699\pi\)
−0.634269 + 0.773112i \(0.718699\pi\)
\(132\) 0 0
\(133\) −551.000 −0.359231
\(134\) −104.000 −0.0670465
\(135\) 0 0
\(136\) −1008.00 −0.635554
\(137\) −2712.00 −1.69125 −0.845627 0.533774i \(-0.820773\pi\)
−0.845627 + 0.533774i \(0.820773\pi\)
\(138\) 0 0
\(139\) −367.000 −0.223946 −0.111973 0.993711i \(-0.535717\pi\)
−0.111973 + 0.993711i \(0.535717\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1164.00 −0.687892
\(143\) 600.000 0.350871
\(144\) 0 0
\(145\) 0 0
\(146\) 130.000 0.0736909
\(147\) 0 0
\(148\) 260.000 0.144405
\(149\) 3006.00 1.65276 0.826380 0.563113i \(-0.190397\pi\)
0.826380 + 0.563113i \(0.190397\pi\)
\(150\) 0 0
\(151\) 2651.00 1.42871 0.714355 0.699783i \(-0.246720\pi\)
0.714355 + 0.699783i \(0.246720\pi\)
\(152\) 232.000 0.123801
\(153\) 0 0
\(154\) −456.000 −0.238607
\(155\) 0 0
\(156\) 0 0
\(157\) 2801.00 1.42385 0.711924 0.702257i \(-0.247824\pi\)
0.711924 + 0.702257i \(0.247824\pi\)
\(158\) 346.000 0.174217
\(159\) 0 0
\(160\) 0 0
\(161\) −342.000 −0.167412
\(162\) 0 0
\(163\) 1412.00 0.678505 0.339253 0.940695i \(-0.389826\pi\)
0.339253 + 0.940695i \(0.389826\pi\)
\(164\) 960.000 0.457094
\(165\) 0 0
\(166\) 996.000 0.465690
\(167\) −2154.00 −0.998093 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(168\) 0 0
\(169\) 303.000 0.137915
\(170\) 0 0
\(171\) 0 0
\(172\) −1468.00 −0.650779
\(173\) 234.000 0.102836 0.0514182 0.998677i \(-0.483626\pi\)
0.0514182 + 0.998677i \(0.483626\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 192.000 0.0822304
\(177\) 0 0
\(178\) −1644.00 −0.692264
\(179\) 2472.00 1.03221 0.516106 0.856525i \(-0.327381\pi\)
0.516106 + 0.856525i \(0.327381\pi\)
\(180\) 0 0
\(181\) −190.000 −0.0780254 −0.0390127 0.999239i \(-0.512421\pi\)
−0.0390127 + 0.999239i \(0.512421\pi\)
\(182\) −1900.00 −0.773832
\(183\) 0 0
\(184\) 144.000 0.0576947
\(185\) 0 0
\(186\) 0 0
\(187\) −1512.00 −0.591275
\(188\) −288.000 −0.111726
\(189\) 0 0
\(190\) 0 0
\(191\) 1524.00 0.577344 0.288672 0.957428i \(-0.406786\pi\)
0.288672 + 0.957428i \(0.406786\pi\)
\(192\) 0 0
\(193\) 3335.00 1.24383 0.621913 0.783086i \(-0.286356\pi\)
0.621913 + 0.783086i \(0.286356\pi\)
\(194\) 1642.00 0.607674
\(195\) 0 0
\(196\) 72.0000 0.0262391
\(197\) −4440.00 −1.60577 −0.802886 0.596133i \(-0.796703\pi\)
−0.802886 + 0.596133i \(0.796703\pi\)
\(198\) 0 0
\(199\) 2192.00 0.780838 0.390419 0.920637i \(-0.372330\pi\)
0.390419 + 0.920637i \(0.372330\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2400.00 −0.835957
\(203\) −1938.00 −0.670054
\(204\) 0 0
\(205\) 0 0
\(206\) −4082.00 −1.38061
\(207\) 0 0
\(208\) 800.000 0.266683
\(209\) 348.000 0.115175
\(210\) 0 0
\(211\) −1732.00 −0.565099 −0.282549 0.959253i \(-0.591180\pi\)
−0.282549 + 0.959253i \(0.591180\pi\)
\(212\) −2544.00 −0.824163
\(213\) 0 0
\(214\) −2556.00 −0.816470
\(215\) 0 0
\(216\) 0 0
\(217\) 5035.00 1.57511
\(218\) −410.000 −0.127379
\(219\) 0 0
\(220\) 0 0
\(221\) −6300.00 −1.91757
\(222\) 0 0
\(223\) −5989.00 −1.79844 −0.899222 0.437492i \(-0.855867\pi\)
−0.899222 + 0.437492i \(0.855867\pi\)
\(224\) −608.000 −0.181356
\(225\) 0 0
\(226\) −3000.00 −0.882996
\(227\) −3300.00 −0.964884 −0.482442 0.875928i \(-0.660250\pi\)
−0.482442 + 0.875928i \(0.660250\pi\)
\(228\) 0 0
\(229\) 4379.00 1.26364 0.631818 0.775117i \(-0.282309\pi\)
0.631818 + 0.775117i \(0.282309\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 816.000 0.230918
\(233\) 2958.00 0.831695 0.415848 0.909434i \(-0.363485\pi\)
0.415848 + 0.909434i \(0.363485\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 408.000 0.112536
\(237\) 0 0
\(238\) 4788.00 1.30403
\(239\) −2676.00 −0.724251 −0.362126 0.932129i \(-0.617949\pi\)
−0.362126 + 0.932129i \(0.617949\pi\)
\(240\) 0 0
\(241\) 3626.00 0.969175 0.484588 0.874743i \(-0.338970\pi\)
0.484588 + 0.874743i \(0.338970\pi\)
\(242\) −2374.00 −0.630605
\(243\) 0 0
\(244\) −412.000 −0.108097
\(245\) 0 0
\(246\) 0 0
\(247\) 1450.00 0.373527
\(248\) −2120.00 −0.542823
\(249\) 0 0
\(250\) 0 0
\(251\) −2046.00 −0.514511 −0.257256 0.966343i \(-0.582818\pi\)
−0.257256 + 0.966343i \(0.582818\pi\)
\(252\) 0 0
\(253\) 216.000 0.0536751
\(254\) 832.000 0.205529
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 5700.00 1.38349 0.691744 0.722143i \(-0.256843\pi\)
0.691744 + 0.722143i \(0.256843\pi\)
\(258\) 0 0
\(259\) −1235.00 −0.296290
\(260\) 0 0
\(261\) 0 0
\(262\) −3804.00 −0.896992
\(263\) −7674.00 −1.79924 −0.899618 0.436678i \(-0.856155\pi\)
−0.899618 + 0.436678i \(0.856155\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1102.00 −0.254015
\(267\) 0 0
\(268\) −208.000 −0.0474090
\(269\) 4326.00 0.980524 0.490262 0.871575i \(-0.336901\pi\)
0.490262 + 0.871575i \(0.336901\pi\)
\(270\) 0 0
\(271\) 6473.00 1.45095 0.725474 0.688250i \(-0.241621\pi\)
0.725474 + 0.688250i \(0.241621\pi\)
\(272\) −2016.00 −0.449404
\(273\) 0 0
\(274\) −5424.00 −1.19590
\(275\) 0 0
\(276\) 0 0
\(277\) −7825.00 −1.69732 −0.848662 0.528936i \(-0.822591\pi\)
−0.848662 + 0.528936i \(0.822591\pi\)
\(278\) −734.000 −0.158354
\(279\) 0 0
\(280\) 0 0
\(281\) 2760.00 0.585935 0.292968 0.956122i \(-0.405357\pi\)
0.292968 + 0.956122i \(0.405357\pi\)
\(282\) 0 0
\(283\) −4801.00 −1.00844 −0.504222 0.863574i \(-0.668221\pi\)
−0.504222 + 0.863574i \(0.668221\pi\)
\(284\) −2328.00 −0.486413
\(285\) 0 0
\(286\) 1200.00 0.248103
\(287\) −4560.00 −0.937869
\(288\) 0 0
\(289\) 10963.0 2.23143
\(290\) 0 0
\(291\) 0 0
\(292\) 260.000 0.0521074
\(293\) −8166.00 −1.62820 −0.814100 0.580724i \(-0.802769\pi\)
−0.814100 + 0.580724i \(0.802769\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 520.000 0.102109
\(297\) 0 0
\(298\) 6012.00 1.16868
\(299\) 900.000 0.174075
\(300\) 0 0
\(301\) 6973.00 1.33527
\(302\) 5302.00 1.01025
\(303\) 0 0
\(304\) 464.000 0.0875402
\(305\) 0 0
\(306\) 0 0
\(307\) −577.000 −0.107268 −0.0536338 0.998561i \(-0.517080\pi\)
−0.0536338 + 0.998561i \(0.517080\pi\)
\(308\) −912.000 −0.168721
\(309\) 0 0
\(310\) 0 0
\(311\) −7656.00 −1.39592 −0.697961 0.716135i \(-0.745909\pi\)
−0.697961 + 0.716135i \(0.745909\pi\)
\(312\) 0 0
\(313\) 2726.00 0.492277 0.246138 0.969235i \(-0.420838\pi\)
0.246138 + 0.969235i \(0.420838\pi\)
\(314\) 5602.00 1.00681
\(315\) 0 0
\(316\) 692.000 0.123190
\(317\) −9450.00 −1.67434 −0.837169 0.546945i \(-0.815791\pi\)
−0.837169 + 0.546945i \(0.815791\pi\)
\(318\) 0 0
\(319\) 1224.00 0.214830
\(320\) 0 0
\(321\) 0 0
\(322\) −684.000 −0.118378
\(323\) −3654.00 −0.629455
\(324\) 0 0
\(325\) 0 0
\(326\) 2824.00 0.479776
\(327\) 0 0
\(328\) 1920.00 0.323214
\(329\) 1368.00 0.229241
\(330\) 0 0
\(331\) 3089.00 0.512951 0.256476 0.966551i \(-0.417439\pi\)
0.256476 + 0.966551i \(0.417439\pi\)
\(332\) 1992.00 0.329293
\(333\) 0 0
\(334\) −4308.00 −0.705758
\(335\) 0 0
\(336\) 0 0
\(337\) −7894.00 −1.27600 −0.638002 0.770034i \(-0.720239\pi\)
−0.638002 + 0.770034i \(0.720239\pi\)
\(338\) 606.000 0.0975209
\(339\) 0 0
\(340\) 0 0
\(341\) −3180.00 −0.505005
\(342\) 0 0
\(343\) 6175.00 0.972066
\(344\) −2936.00 −0.460170
\(345\) 0 0
\(346\) 468.000 0.0727163
\(347\) −3450.00 −0.533734 −0.266867 0.963733i \(-0.585988\pi\)
−0.266867 + 0.963733i \(0.585988\pi\)
\(348\) 0 0
\(349\) 2261.00 0.346787 0.173393 0.984853i \(-0.444527\pi\)
0.173393 + 0.984853i \(0.444527\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 384.000 0.0581456
\(353\) 11544.0 1.74058 0.870291 0.492539i \(-0.163931\pi\)
0.870291 + 0.492539i \(0.163931\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3288.00 −0.489505
\(357\) 0 0
\(358\) 4944.00 0.729884
\(359\) −198.000 −0.0291087 −0.0145544 0.999894i \(-0.504633\pi\)
−0.0145544 + 0.999894i \(0.504633\pi\)
\(360\) 0 0
\(361\) −6018.00 −0.877387
\(362\) −380.000 −0.0551723
\(363\) 0 0
\(364\) −3800.00 −0.547182
\(365\) 0 0
\(366\) 0 0
\(367\) 10388.0 1.47752 0.738759 0.673970i \(-0.235412\pi\)
0.738759 + 0.673970i \(0.235412\pi\)
\(368\) 288.000 0.0407963
\(369\) 0 0
\(370\) 0 0
\(371\) 12084.0 1.69102
\(372\) 0 0
\(373\) −4867.00 −0.675613 −0.337807 0.941216i \(-0.609685\pi\)
−0.337807 + 0.941216i \(0.609685\pi\)
\(374\) −3024.00 −0.418094
\(375\) 0 0
\(376\) −576.000 −0.0790025
\(377\) 5100.00 0.696720
\(378\) 0 0
\(379\) 656.000 0.0889089 0.0444544 0.999011i \(-0.485845\pi\)
0.0444544 + 0.999011i \(0.485845\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 3048.00 0.408244
\(383\) 3078.00 0.410649 0.205324 0.978694i \(-0.434175\pi\)
0.205324 + 0.978694i \(0.434175\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 6670.00 0.879518
\(387\) 0 0
\(388\) 3284.00 0.429690
\(389\) 1644.00 0.214278 0.107139 0.994244i \(-0.465831\pi\)
0.107139 + 0.994244i \(0.465831\pi\)
\(390\) 0 0
\(391\) −2268.00 −0.293344
\(392\) 144.000 0.0185538
\(393\) 0 0
\(394\) −8880.00 −1.13545
\(395\) 0 0
\(396\) 0 0
\(397\) 5729.00 0.724258 0.362129 0.932128i \(-0.382050\pi\)
0.362129 + 0.932128i \(0.382050\pi\)
\(398\) 4384.00 0.552136
\(399\) 0 0
\(400\) 0 0
\(401\) −1776.00 −0.221170 −0.110585 0.993867i \(-0.535272\pi\)
−0.110585 + 0.993867i \(0.535272\pi\)
\(402\) 0 0
\(403\) −13250.0 −1.63779
\(404\) −4800.00 −0.591111
\(405\) 0 0
\(406\) −3876.00 −0.473800
\(407\) 780.000 0.0949955
\(408\) 0 0
\(409\) 12842.0 1.55256 0.776279 0.630390i \(-0.217105\pi\)
0.776279 + 0.630390i \(0.217105\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8164.00 −0.976241
\(413\) −1938.00 −0.230903
\(414\) 0 0
\(415\) 0 0
\(416\) 1600.00 0.188573
\(417\) 0 0
\(418\) 696.000 0.0814413
\(419\) −11298.0 −1.31729 −0.658644 0.752455i \(-0.728870\pi\)
−0.658644 + 0.752455i \(0.728870\pi\)
\(420\) 0 0
\(421\) 3431.00 0.397189 0.198595 0.980082i \(-0.436362\pi\)
0.198595 + 0.980082i \(0.436362\pi\)
\(422\) −3464.00 −0.399585
\(423\) 0 0
\(424\) −5088.00 −0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) 1957.00 0.221794
\(428\) −5112.00 −0.577331
\(429\) 0 0
\(430\) 0 0
\(431\) 10530.0 1.17683 0.588413 0.808560i \(-0.299753\pi\)
0.588413 + 0.808560i \(0.299753\pi\)
\(432\) 0 0
\(433\) 3179.00 0.352824 0.176412 0.984316i \(-0.443551\pi\)
0.176412 + 0.984316i \(0.443551\pi\)
\(434\) 10070.0 1.11377
\(435\) 0 0
\(436\) −820.000 −0.0900708
\(437\) 522.000 0.0571411
\(438\) 0 0
\(439\) 6047.00 0.657420 0.328710 0.944431i \(-0.393386\pi\)
0.328710 + 0.944431i \(0.393386\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −12600.0 −1.35593
\(443\) 16554.0 1.77540 0.887702 0.460418i \(-0.152300\pi\)
0.887702 + 0.460418i \(0.152300\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −11978.0 −1.27169
\(447\) 0 0
\(448\) −1216.00 −0.128238
\(449\) −4446.00 −0.467304 −0.233652 0.972320i \(-0.575068\pi\)
−0.233652 + 0.972320i \(0.575068\pi\)
\(450\) 0 0
\(451\) 2880.00 0.300696
\(452\) −6000.00 −0.624372
\(453\) 0 0
\(454\) −6600.00 −0.682276
\(455\) 0 0
\(456\) 0 0
\(457\) 146.000 0.0149444 0.00747220 0.999972i \(-0.497622\pi\)
0.00747220 + 0.999972i \(0.497622\pi\)
\(458\) 8758.00 0.893525
\(459\) 0 0
\(460\) 0 0
\(461\) 9822.00 0.992313 0.496156 0.868233i \(-0.334744\pi\)
0.496156 + 0.868233i \(0.334744\pi\)
\(462\) 0 0
\(463\) 7931.00 0.796080 0.398040 0.917368i \(-0.369691\pi\)
0.398040 + 0.917368i \(0.369691\pi\)
\(464\) 1632.00 0.163284
\(465\) 0 0
\(466\) 5916.00 0.588097
\(467\) 18714.0 1.85435 0.927174 0.374631i \(-0.122230\pi\)
0.927174 + 0.374631i \(0.122230\pi\)
\(468\) 0 0
\(469\) 988.000 0.0972742
\(470\) 0 0
\(471\) 0 0
\(472\) 816.000 0.0795751
\(473\) −4404.00 −0.428110
\(474\) 0 0
\(475\) 0 0
\(476\) 9576.00 0.922091
\(477\) 0 0
\(478\) −5352.00 −0.512123
\(479\) 12786.0 1.21964 0.609820 0.792540i \(-0.291242\pi\)
0.609820 + 0.792540i \(0.291242\pi\)
\(480\) 0 0
\(481\) 3250.00 0.308082
\(482\) 7252.00 0.685310
\(483\) 0 0
\(484\) −4748.00 −0.445905
\(485\) 0 0
\(486\) 0 0
\(487\) 10400.0 0.967698 0.483849 0.875151i \(-0.339238\pi\)
0.483849 + 0.875151i \(0.339238\pi\)
\(488\) −824.000 −0.0764359
\(489\) 0 0
\(490\) 0 0
\(491\) −19548.0 −1.79672 −0.898359 0.439261i \(-0.855240\pi\)
−0.898359 + 0.439261i \(0.855240\pi\)
\(492\) 0 0
\(493\) −12852.0 −1.17409
\(494\) 2900.00 0.264124
\(495\) 0 0
\(496\) −4240.00 −0.383834
\(497\) 11058.0 0.998026
\(498\) 0 0
\(499\) 14939.0 1.34020 0.670102 0.742269i \(-0.266250\pi\)
0.670102 + 0.742269i \(0.266250\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −4092.00 −0.363815
\(503\) 4098.00 0.363262 0.181631 0.983367i \(-0.441862\pi\)
0.181631 + 0.983367i \(0.441862\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 432.000 0.0379540
\(507\) 0 0
\(508\) 1664.00 0.145331
\(509\) −12216.0 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) −1235.00 −0.106914
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 11400.0 0.978273
\(515\) 0 0
\(516\) 0 0
\(517\) −864.000 −0.0734984
\(518\) −2470.00 −0.209509
\(519\) 0 0
\(520\) 0 0
\(521\) 18492.0 1.55499 0.777494 0.628890i \(-0.216490\pi\)
0.777494 + 0.628890i \(0.216490\pi\)
\(522\) 0 0
\(523\) −5467.00 −0.457085 −0.228542 0.973534i \(-0.573396\pi\)
−0.228542 + 0.973534i \(0.573396\pi\)
\(524\) −7608.00 −0.634269
\(525\) 0 0
\(526\) −15348.0 −1.27225
\(527\) 33390.0 2.75995
\(528\) 0 0
\(529\) −11843.0 −0.973371
\(530\) 0 0
\(531\) 0 0
\(532\) −2204.00 −0.179616
\(533\) 12000.0 0.975193
\(534\) 0 0
\(535\) 0 0
\(536\) −416.000 −0.0335233
\(537\) 0 0
\(538\) 8652.00 0.693335
\(539\) 216.000 0.0172612
\(540\) 0 0
\(541\) 6983.00 0.554940 0.277470 0.960734i \(-0.410504\pi\)
0.277470 + 0.960734i \(0.410504\pi\)
\(542\) 12946.0 1.02597
\(543\) 0 0
\(544\) −4032.00 −0.317777
\(545\) 0 0
\(546\) 0 0
\(547\) 9281.00 0.725461 0.362730 0.931894i \(-0.381845\pi\)
0.362730 + 0.931894i \(0.381845\pi\)
\(548\) −10848.0 −0.845627
\(549\) 0 0
\(550\) 0 0
\(551\) 2958.00 0.228702
\(552\) 0 0
\(553\) −3287.00 −0.252762
\(554\) −15650.0 −1.20019
\(555\) 0 0
\(556\) −1468.00 −0.111973
\(557\) −1986.00 −0.151076 −0.0755382 0.997143i \(-0.524067\pi\)
−0.0755382 + 0.997143i \(0.524067\pi\)
\(558\) 0 0
\(559\) −18350.0 −1.38841
\(560\) 0 0
\(561\) 0 0
\(562\) 5520.00 0.414319
\(563\) −12672.0 −0.948599 −0.474299 0.880364i \(-0.657299\pi\)
−0.474299 + 0.880364i \(0.657299\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −9602.00 −0.713078
\(567\) 0 0
\(568\) −4656.00 −0.343946
\(569\) −8370.00 −0.616676 −0.308338 0.951277i \(-0.599773\pi\)
−0.308338 + 0.951277i \(0.599773\pi\)
\(570\) 0 0
\(571\) −19675.0 −1.44198 −0.720992 0.692943i \(-0.756314\pi\)
−0.720992 + 0.692943i \(0.756314\pi\)
\(572\) 2400.00 0.175435
\(573\) 0 0
\(574\) −9120.00 −0.663173
\(575\) 0 0
\(576\) 0 0
\(577\) 26951.0 1.94451 0.972257 0.233914i \(-0.0751535\pi\)
0.972257 + 0.233914i \(0.0751535\pi\)
\(578\) 21926.0 1.57786
\(579\) 0 0
\(580\) 0 0
\(581\) −9462.00 −0.675645
\(582\) 0 0
\(583\) −7632.00 −0.542170
\(584\) 520.000 0.0368455
\(585\) 0 0
\(586\) −16332.0 −1.15131
\(587\) −10986.0 −0.772471 −0.386236 0.922400i \(-0.626225\pi\)
−0.386236 + 0.922400i \(0.626225\pi\)
\(588\) 0 0
\(589\) −7685.00 −0.537614
\(590\) 0 0
\(591\) 0 0
\(592\) 1040.00 0.0722023
\(593\) −11226.0 −0.777397 −0.388699 0.921365i \(-0.627075\pi\)
−0.388699 + 0.921365i \(0.627075\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12024.0 0.826380
\(597\) 0 0
\(598\) 1800.00 0.123089
\(599\) −11400.0 −0.777615 −0.388807 0.921319i \(-0.627113\pi\)
−0.388807 + 0.921319i \(0.627113\pi\)
\(600\) 0 0
\(601\) −6469.00 −0.439062 −0.219531 0.975606i \(-0.570453\pi\)
−0.219531 + 0.975606i \(0.570453\pi\)
\(602\) 13946.0 0.944180
\(603\) 0 0
\(604\) 10604.0 0.714355
\(605\) 0 0
\(606\) 0 0
\(607\) 6383.00 0.426817 0.213409 0.976963i \(-0.431543\pi\)
0.213409 + 0.976963i \(0.431543\pi\)
\(608\) 928.000 0.0619003
\(609\) 0 0
\(610\) 0 0
\(611\) −3600.00 −0.238364
\(612\) 0 0
\(613\) −10843.0 −0.714428 −0.357214 0.934022i \(-0.616273\pi\)
−0.357214 + 0.934022i \(0.616273\pi\)
\(614\) −1154.00 −0.0758496
\(615\) 0 0
\(616\) −1824.00 −0.119304
\(617\) 25236.0 1.64662 0.823309 0.567594i \(-0.192126\pi\)
0.823309 + 0.567594i \(0.192126\pi\)
\(618\) 0 0
\(619\) 24809.0 1.61092 0.805459 0.592652i \(-0.201919\pi\)
0.805459 + 0.592652i \(0.201919\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −15312.0 −0.987066
\(623\) 15618.0 1.00437
\(624\) 0 0
\(625\) 0 0
\(626\) 5452.00 0.348092
\(627\) 0 0
\(628\) 11204.0 0.711924
\(629\) −8190.00 −0.519168
\(630\) 0 0
\(631\) −11476.0 −0.724013 −0.362007 0.932176i \(-0.617908\pi\)
−0.362007 + 0.932176i \(0.617908\pi\)
\(632\) 1384.00 0.0871085
\(633\) 0 0
\(634\) −18900.0 −1.18394
\(635\) 0 0
\(636\) 0 0
\(637\) 900.000 0.0559801
\(638\) 2448.00 0.151908
\(639\) 0 0
\(640\) 0 0
\(641\) 25350.0 1.56204 0.781018 0.624509i \(-0.214701\pi\)
0.781018 + 0.624509i \(0.214701\pi\)
\(642\) 0 0
\(643\) −14164.0 −0.868699 −0.434350 0.900744i \(-0.643022\pi\)
−0.434350 + 0.900744i \(0.643022\pi\)
\(644\) −1368.00 −0.0837061
\(645\) 0 0
\(646\) −7308.00 −0.445092
\(647\) −7536.00 −0.457915 −0.228957 0.973436i \(-0.573532\pi\)
−0.228957 + 0.973436i \(0.573532\pi\)
\(648\) 0 0
\(649\) 1224.00 0.0740311
\(650\) 0 0
\(651\) 0 0
\(652\) 5648.00 0.339253
\(653\) −432.000 −0.0258889 −0.0129445 0.999916i \(-0.504120\pi\)
−0.0129445 + 0.999916i \(0.504120\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3840.00 0.228547
\(657\) 0 0
\(658\) 2736.00 0.162098
\(659\) −10782.0 −0.637340 −0.318670 0.947866i \(-0.603236\pi\)
−0.318670 + 0.947866i \(0.603236\pi\)
\(660\) 0 0
\(661\) 26057.0 1.53328 0.766641 0.642076i \(-0.221926\pi\)
0.766641 + 0.642076i \(0.221926\pi\)
\(662\) 6178.00 0.362711
\(663\) 0 0
\(664\) 3984.00 0.232845
\(665\) 0 0
\(666\) 0 0
\(667\) 1836.00 0.106582
\(668\) −8616.00 −0.499046
\(669\) 0 0
\(670\) 0 0
\(671\) −1236.00 −0.0711107
\(672\) 0 0
\(673\) 19091.0 1.09347 0.546734 0.837306i \(-0.315871\pi\)
0.546734 + 0.837306i \(0.315871\pi\)
\(674\) −15788.0 −0.902272
\(675\) 0 0
\(676\) 1212.00 0.0689577
\(677\) 1044.00 0.0592676 0.0296338 0.999561i \(-0.490566\pi\)
0.0296338 + 0.999561i \(0.490566\pi\)
\(678\) 0 0
\(679\) −15599.0 −0.881642
\(680\) 0 0
\(681\) 0 0
\(682\) −6360.00 −0.357092
\(683\) 26046.0 1.45918 0.729592 0.683883i \(-0.239710\pi\)
0.729592 + 0.683883i \(0.239710\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 12350.0 0.687355
\(687\) 0 0
\(688\) −5872.00 −0.325389
\(689\) −31800.0 −1.75832
\(690\) 0 0
\(691\) −28060.0 −1.54479 −0.772397 0.635140i \(-0.780942\pi\)
−0.772397 + 0.635140i \(0.780942\pi\)
\(692\) 936.000 0.0514182
\(693\) 0 0
\(694\) −6900.00 −0.377407
\(695\) 0 0
\(696\) 0 0
\(697\) −30240.0 −1.64336
\(698\) 4522.00 0.245215
\(699\) 0 0
\(700\) 0 0
\(701\) 16794.0 0.904851 0.452426 0.891802i \(-0.350559\pi\)
0.452426 + 0.891802i \(0.350559\pi\)
\(702\) 0 0
\(703\) 1885.00 0.101130
\(704\) 768.000 0.0411152
\(705\) 0 0
\(706\) 23088.0 1.23078
\(707\) 22800.0 1.21285
\(708\) 0 0
\(709\) 2495.00 0.132160 0.0660802 0.997814i \(-0.478951\pi\)
0.0660802 + 0.997814i \(0.478951\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −6576.00 −0.346132
\(713\) −4770.00 −0.250544
\(714\) 0 0
\(715\) 0 0
\(716\) 9888.00 0.516106
\(717\) 0 0
\(718\) −396.000 −0.0205830
\(719\) 1950.00 0.101144 0.0505721 0.998720i \(-0.483896\pi\)
0.0505721 + 0.998720i \(0.483896\pi\)
\(720\) 0 0
\(721\) 38779.0 2.00306
\(722\) −12036.0 −0.620407
\(723\) 0 0
\(724\) −760.000 −0.0390127
\(725\) 0 0
\(726\) 0 0
\(727\) 16127.0 0.822720 0.411360 0.911473i \(-0.365054\pi\)
0.411360 + 0.911473i \(0.365054\pi\)
\(728\) −7600.00 −0.386916
\(729\) 0 0
\(730\) 0 0
\(731\) 46242.0 2.33970
\(732\) 0 0
\(733\) −12634.0 −0.636627 −0.318313 0.947986i \(-0.603116\pi\)
−0.318313 + 0.947986i \(0.603116\pi\)
\(734\) 20776.0 1.04476
\(735\) 0 0
\(736\) 576.000 0.0288473
\(737\) −624.000 −0.0311877
\(738\) 0 0
\(739\) −10420.0 −0.518682 −0.259341 0.965786i \(-0.583505\pi\)
−0.259341 + 0.965786i \(0.583505\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 24168.0 1.19573
\(743\) −21936.0 −1.08311 −0.541557 0.840664i \(-0.682165\pi\)
−0.541557 + 0.840664i \(0.682165\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −9734.00 −0.477731
\(747\) 0 0
\(748\) −6048.00 −0.295637
\(749\) 24282.0 1.18457
\(750\) 0 0
\(751\) −21715.0 −1.05512 −0.527558 0.849519i \(-0.676892\pi\)
−0.527558 + 0.849519i \(0.676892\pi\)
\(752\) −1152.00 −0.0558632
\(753\) 0 0
\(754\) 10200.0 0.492655
\(755\) 0 0
\(756\) 0 0
\(757\) −19849.0 −0.953004 −0.476502 0.879173i \(-0.658096\pi\)
−0.476502 + 0.879173i \(0.658096\pi\)
\(758\) 1312.00 0.0628681
\(759\) 0 0
\(760\) 0 0
\(761\) −29952.0 −1.42675 −0.713377 0.700781i \(-0.752835\pi\)
−0.713377 + 0.700781i \(0.752835\pi\)
\(762\) 0 0
\(763\) 3895.00 0.184808
\(764\) 6096.00 0.288672
\(765\) 0 0
\(766\) 6156.00 0.290372
\(767\) 5100.00 0.240092
\(768\) 0 0
\(769\) 17582.0 0.824477 0.412239 0.911076i \(-0.364747\pi\)
0.412239 + 0.911076i \(0.364747\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 13340.0 0.621913
\(773\) 19194.0 0.893092 0.446546 0.894761i \(-0.352654\pi\)
0.446546 + 0.894761i \(0.352654\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 6568.00 0.303837
\(777\) 0 0
\(778\) 3288.00 0.151517
\(779\) 6960.00 0.320113
\(780\) 0 0
\(781\) −6984.00 −0.319984
\(782\) −4536.00 −0.207426
\(783\) 0 0
\(784\) 288.000 0.0131195
\(785\) 0 0
\(786\) 0 0
\(787\) −9823.00 −0.444920 −0.222460 0.974942i \(-0.571409\pi\)
−0.222460 + 0.974942i \(0.571409\pi\)
\(788\) −17760.0 −0.802886
\(789\) 0 0
\(790\) 0 0
\(791\) 28500.0 1.28109
\(792\) 0 0
\(793\) −5150.00 −0.230620
\(794\) 11458.0 0.512127
\(795\) 0 0
\(796\) 8768.00 0.390419
\(797\) 22188.0 0.986122 0.493061 0.869995i \(-0.335878\pi\)
0.493061 + 0.869995i \(0.335878\pi\)
\(798\) 0 0
\(799\) 9072.00 0.401682
\(800\) 0 0
\(801\) 0 0
\(802\) −3552.00 −0.156391
\(803\) 780.000 0.0342785
\(804\) 0 0
\(805\) 0 0
\(806\) −26500.0 −1.15809
\(807\) 0 0
\(808\) −9600.00 −0.417979
\(809\) 9654.00 0.419551 0.209775 0.977750i \(-0.432727\pi\)
0.209775 + 0.977750i \(0.432727\pi\)
\(810\) 0 0
\(811\) −32377.0 −1.40186 −0.700931 0.713229i \(-0.747232\pi\)
−0.700931 + 0.713229i \(0.747232\pi\)
\(812\) −7752.00 −0.335027
\(813\) 0 0
\(814\) 1560.00 0.0671720
\(815\) 0 0
\(816\) 0 0
\(817\) −10643.0 −0.455755
\(818\) 25684.0 1.09782
\(819\) 0 0
\(820\) 0 0
\(821\) 23952.0 1.01819 0.509093 0.860712i \(-0.329981\pi\)
0.509093 + 0.860712i \(0.329981\pi\)
\(822\) 0 0
\(823\) 5072.00 0.214822 0.107411 0.994215i \(-0.465744\pi\)
0.107411 + 0.994215i \(0.465744\pi\)
\(824\) −16328.0 −0.690307
\(825\) 0 0
\(826\) −3876.00 −0.163273
\(827\) −3024.00 −0.127152 −0.0635760 0.997977i \(-0.520251\pi\)
−0.0635760 + 0.997977i \(0.520251\pi\)
\(828\) 0 0
\(829\) −3049.00 −0.127740 −0.0638698 0.997958i \(-0.520344\pi\)
−0.0638698 + 0.997958i \(0.520344\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3200.00 0.133341
\(833\) −2268.00 −0.0943356
\(834\) 0 0
\(835\) 0 0
\(836\) 1392.00 0.0575877
\(837\) 0 0
\(838\) −22596.0 −0.931463
\(839\) −40374.0 −1.66134 −0.830671 0.556764i \(-0.812043\pi\)
−0.830671 + 0.556764i \(0.812043\pi\)
\(840\) 0 0
\(841\) −13985.0 −0.573414
\(842\) 6862.00 0.280855
\(843\) 0 0
\(844\) −6928.00 −0.282549
\(845\) 0 0
\(846\) 0 0
\(847\) 22553.0 0.914912
\(848\) −10176.0 −0.412082
\(849\) 0 0
\(850\) 0 0
\(851\) 1170.00 0.0471294
\(852\) 0 0
\(853\) 40322.0 1.61852 0.809261 0.587449i \(-0.199868\pi\)
0.809261 + 0.587449i \(0.199868\pi\)
\(854\) 3914.00 0.156832
\(855\) 0 0
\(856\) −10224.0 −0.408235
\(857\) −19932.0 −0.794474 −0.397237 0.917716i \(-0.630031\pi\)
−0.397237 + 0.917716i \(0.630031\pi\)
\(858\) 0 0
\(859\) 34553.0 1.37245 0.686224 0.727390i \(-0.259267\pi\)
0.686224 + 0.727390i \(0.259267\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 21060.0 0.832142
\(863\) 6918.00 0.272875 0.136438 0.990649i \(-0.456435\pi\)
0.136438 + 0.990649i \(0.456435\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 6358.00 0.249485
\(867\) 0 0
\(868\) 20140.0 0.787553
\(869\) 2076.00 0.0810397
\(870\) 0 0
\(871\) −2600.00 −0.101145
\(872\) −1640.00 −0.0636897
\(873\) 0 0
\(874\) 1044.00 0.0404048
\(875\) 0 0
\(876\) 0 0
\(877\) 6497.00 0.250157 0.125079 0.992147i \(-0.460082\pi\)
0.125079 + 0.992147i \(0.460082\pi\)
\(878\) 12094.0 0.464866
\(879\) 0 0
\(880\) 0 0
\(881\) −44082.0 −1.68577 −0.842883 0.538096i \(-0.819144\pi\)
−0.842883 + 0.538096i \(0.819144\pi\)
\(882\) 0 0
\(883\) 10127.0 0.385958 0.192979 0.981203i \(-0.438185\pi\)
0.192979 + 0.981203i \(0.438185\pi\)
\(884\) −25200.0 −0.958787
\(885\) 0 0
\(886\) 33108.0 1.25540
\(887\) −12936.0 −0.489682 −0.244841 0.969563i \(-0.578736\pi\)
−0.244841 + 0.969563i \(0.578736\pi\)
\(888\) 0 0
\(889\) −7904.00 −0.298191
\(890\) 0 0
\(891\) 0 0
\(892\) −23956.0 −0.899222
\(893\) −2088.00 −0.0782444
\(894\) 0 0
\(895\) 0 0
\(896\) −2432.00 −0.0906779
\(897\) 0 0
\(898\) −8892.00 −0.330434
\(899\) −27030.0 −1.00278
\(900\) 0 0
\(901\) 80136.0 2.96306
\(902\) 5760.00 0.212624
\(903\) 0 0
\(904\) −12000.0 −0.441498
\(905\) 0 0
\(906\) 0 0
\(907\) −45853.0 −1.67864 −0.839319 0.543640i \(-0.817046\pi\)
−0.839319 + 0.543640i \(0.817046\pi\)
\(908\) −13200.0 −0.482442
\(909\) 0 0
\(910\) 0 0
\(911\) −34530.0 −1.25580 −0.627898 0.778296i \(-0.716084\pi\)
−0.627898 + 0.778296i \(0.716084\pi\)
\(912\) 0 0
\(913\) 5976.00 0.216623
\(914\) 292.000 0.0105673
\(915\) 0 0
\(916\) 17516.0 0.631818
\(917\) 36138.0 1.30140
\(918\) 0 0
\(919\) −31015.0 −1.11326 −0.556632 0.830759i \(-0.687907\pi\)
−0.556632 + 0.830759i \(0.687907\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 19644.0 0.701671
\(923\) −29100.0 −1.03774
\(924\) 0 0
\(925\) 0 0
\(926\) 15862.0 0.562913
\(927\) 0 0
\(928\) 3264.00 0.115459
\(929\) 36186.0 1.27796 0.638980 0.769224i \(-0.279357\pi\)
0.638980 + 0.769224i \(0.279357\pi\)
\(930\) 0 0
\(931\) 522.000 0.0183758
\(932\) 11832.0 0.415848
\(933\) 0 0
\(934\) 37428.0 1.31122
\(935\) 0 0
\(936\) 0 0
\(937\) −1441.00 −0.0502406 −0.0251203 0.999684i \(-0.507997\pi\)
−0.0251203 + 0.999684i \(0.507997\pi\)
\(938\) 1976.00 0.0687832
\(939\) 0 0
\(940\) 0 0
\(941\) −33168.0 −1.14904 −0.574520 0.818491i \(-0.694811\pi\)
−0.574520 + 0.818491i \(0.694811\pi\)
\(942\) 0 0
\(943\) 4320.00 0.149182
\(944\) 1632.00 0.0562681
\(945\) 0 0
\(946\) −8808.00 −0.302720
\(947\) −42054.0 −1.44305 −0.721527 0.692387i \(-0.756559\pi\)
−0.721527 + 0.692387i \(0.756559\pi\)
\(948\) 0 0
\(949\) 3250.00 0.111169
\(950\) 0 0
\(951\) 0 0
\(952\) 19152.0 0.652017
\(953\) 26460.0 0.899395 0.449698 0.893181i \(-0.351532\pi\)
0.449698 + 0.893181i \(0.351532\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −10704.0 −0.362126
\(957\) 0 0
\(958\) 25572.0 0.862415
\(959\) 51528.0 1.73506
\(960\) 0 0
\(961\) 40434.0 1.35726
\(962\) 6500.00 0.217847
\(963\) 0 0
\(964\) 14504.0 0.484588
\(965\) 0 0
\(966\) 0 0
\(967\) −29239.0 −0.972350 −0.486175 0.873861i \(-0.661608\pi\)
−0.486175 + 0.873861i \(0.661608\pi\)
\(968\) −9496.00 −0.315303
\(969\) 0 0
\(970\) 0 0
\(971\) −54288.0 −1.79422 −0.897109 0.441810i \(-0.854336\pi\)
−0.897109 + 0.441810i \(0.854336\pi\)
\(972\) 0 0
\(973\) 6973.00 0.229747
\(974\) 20800.0 0.684266
\(975\) 0 0
\(976\) −1648.00 −0.0540484
\(977\) −38634.0 −1.26511 −0.632554 0.774516i \(-0.717993\pi\)
−0.632554 + 0.774516i \(0.717993\pi\)
\(978\) 0 0
\(979\) −9864.00 −0.322017
\(980\) 0 0
\(981\) 0 0
\(982\) −39096.0 −1.27047
\(983\) −19548.0 −0.634267 −0.317133 0.948381i \(-0.602720\pi\)
−0.317133 + 0.948381i \(0.602720\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −25704.0 −0.830205
\(987\) 0 0
\(988\) 5800.00 0.186764
\(989\) −6606.00 −0.212395
\(990\) 0 0
\(991\) 28445.0 0.911791 0.455896 0.890033i \(-0.349319\pi\)
0.455896 + 0.890033i \(0.349319\pi\)
\(992\) −8480.00 −0.271412
\(993\) 0 0
\(994\) 22116.0 0.705711
\(995\) 0 0
\(996\) 0 0
\(997\) −48706.0 −1.54718 −0.773588 0.633689i \(-0.781540\pi\)
−0.773588 + 0.633689i \(0.781540\pi\)
\(998\) 29878.0 0.947667
\(999\) 0 0
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 1350.4.a.q.1.1 yes 1
3.2 odd 2 1350.4.a.c.1.1 1
5.2 odd 4 1350.4.c.m.649.2 2
5.3 odd 4 1350.4.c.m.649.1 2
5.4 even 2 1350.4.a.k.1.1 yes 1
15.2 even 4 1350.4.c.h.649.1 2
15.8 even 4 1350.4.c.h.649.2 2
15.14 odd 2 1350.4.a.y.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1350.4.a.c.1.1 1 3.2 odd 2
1350.4.a.k.1.1 yes 1 5.4 even 2
1350.4.a.q.1.1 yes 1 1.1 even 1 trivial
1350.4.a.y.1.1 yes 1 15.14 odd 2
1350.4.c.h.649.1 2 15.2 even 4
1350.4.c.h.649.2 2 15.8 even 4
1350.4.c.m.649.1 2 5.3 odd 4
1350.4.c.m.649.2 2 5.2 odd 4