Properties

Label 960.4.a.bc.1.1
Level $960$
Weight $4$
Character 960.1
Self dual yes
Analytic conductor $56.642$
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] = [960,4,Mod(1,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.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: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +5.00000 q^{5} -28.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +5.00000 q^{5} -28.0000 q^{7} +9.00000 q^{9} +24.0000 q^{11} +70.0000 q^{13} +15.0000 q^{15} +102.000 q^{17} -20.0000 q^{19} -84.0000 q^{21} -72.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -306.000 q^{29} -136.000 q^{31} +72.0000 q^{33} -140.000 q^{35} +214.000 q^{37} +210.000 q^{39} -150.000 q^{41} +292.000 q^{43} +45.0000 q^{45} -72.0000 q^{47} +441.000 q^{49} +306.000 q^{51} +414.000 q^{53} +120.000 q^{55} -60.0000 q^{57} +744.000 q^{59} +418.000 q^{61} -252.000 q^{63} +350.000 q^{65} -188.000 q^{67} -216.000 q^{69} +480.000 q^{71} +434.000 q^{73} +75.0000 q^{75} -672.000 q^{77} +1352.00 q^{79} +81.0000 q^{81} +612.000 q^{83} +510.000 q^{85} -918.000 q^{87} -30.0000 q^{89} -1960.00 q^{91} -408.000 q^{93} -100.000 q^{95} -286.000 q^{97} +216.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −28.0000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 24.0000 0.657843 0.328921 0.944357i \(-0.393315\pi\)
0.328921 + 0.944357i \(0.393315\pi\)
\(12\) 0 0
\(13\) 70.0000 1.49342 0.746712 0.665148i \(-0.231631\pi\)
0.746712 + 0.665148i \(0.231631\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) 102.000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 0 0
\(21\) −84.0000 −0.872872
\(22\) 0 0
\(23\) −72.0000 −0.652741 −0.326370 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −306.000 −1.95941 −0.979703 0.200455i \(-0.935758\pi\)
−0.979703 + 0.200455i \(0.935758\pi\)
\(30\) 0 0
\(31\) −136.000 −0.787946 −0.393973 0.919122i \(-0.628900\pi\)
−0.393973 + 0.919122i \(0.628900\pi\)
\(32\) 0 0
\(33\) 72.0000 0.379806
\(34\) 0 0
\(35\) −140.000 −0.676123
\(36\) 0 0
\(37\) 214.000 0.950848 0.475424 0.879757i \(-0.342295\pi\)
0.475424 + 0.879757i \(0.342295\pi\)
\(38\) 0 0
\(39\) 210.000 0.862229
\(40\) 0 0
\(41\) −150.000 −0.571367 −0.285684 0.958324i \(-0.592221\pi\)
−0.285684 + 0.958324i \(0.592221\pi\)
\(42\) 0 0
\(43\) 292.000 1.03557 0.517786 0.855510i \(-0.326756\pi\)
0.517786 + 0.855510i \(0.326756\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 0 0
\(47\) −72.0000 −0.223453 −0.111726 0.993739i \(-0.535638\pi\)
−0.111726 + 0.993739i \(0.535638\pi\)
\(48\) 0 0
\(49\) 441.000 1.28571
\(50\) 0 0
\(51\) 306.000 0.840168
\(52\) 0 0
\(53\) 414.000 1.07297 0.536484 0.843911i \(-0.319752\pi\)
0.536484 + 0.843911i \(0.319752\pi\)
\(54\) 0 0
\(55\) 120.000 0.294196
\(56\) 0 0
\(57\) −60.0000 −0.139424
\(58\) 0 0
\(59\) 744.000 1.64170 0.820852 0.571141i \(-0.193499\pi\)
0.820852 + 0.571141i \(0.193499\pi\)
\(60\) 0 0
\(61\) 418.000 0.877367 0.438684 0.898642i \(-0.355445\pi\)
0.438684 + 0.898642i \(0.355445\pi\)
\(62\) 0 0
\(63\) −252.000 −0.503953
\(64\) 0 0
\(65\) 350.000 0.667879
\(66\) 0 0
\(67\) −188.000 −0.342804 −0.171402 0.985201i \(-0.554830\pi\)
−0.171402 + 0.985201i \(0.554830\pi\)
\(68\) 0 0
\(69\) −216.000 −0.376860
\(70\) 0 0
\(71\) 480.000 0.802331 0.401166 0.916006i \(-0.368605\pi\)
0.401166 + 0.916006i \(0.368605\pi\)
\(72\) 0 0
\(73\) 434.000 0.695834 0.347917 0.937525i \(-0.386889\pi\)
0.347917 + 0.937525i \(0.386889\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −672.000 −0.994565
\(78\) 0 0
\(79\) 1352.00 1.92547 0.962733 0.270452i \(-0.0871732\pi\)
0.962733 + 0.270452i \(0.0871732\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 612.000 0.809346 0.404673 0.914461i \(-0.367385\pi\)
0.404673 + 0.914461i \(0.367385\pi\)
\(84\) 0 0
\(85\) 510.000 0.650791
\(86\) 0 0
\(87\) −918.000 −1.13126
\(88\) 0 0
\(89\) −30.0000 −0.0357303 −0.0178651 0.999840i \(-0.505687\pi\)
−0.0178651 + 0.999840i \(0.505687\pi\)
\(90\) 0 0
\(91\) −1960.00 −2.25784
\(92\) 0 0
\(93\) −408.000 −0.454921
\(94\) 0 0
\(95\) −100.000 −0.107998
\(96\) 0 0
\(97\) −286.000 −0.299370 −0.149685 0.988734i \(-0.547826\pi\)
−0.149685 + 0.988734i \(0.547826\pi\)
\(98\) 0 0
\(99\) 216.000 0.219281
\(100\) 0 0
\(101\) 1542.00 1.51916 0.759578 0.650416i \(-0.225406\pi\)
0.759578 + 0.650416i \(0.225406\pi\)
\(102\) 0 0
\(103\) 1172.00 1.12117 0.560585 0.828097i \(-0.310576\pi\)
0.560585 + 0.828097i \(0.310576\pi\)
\(104\) 0 0
\(105\) −420.000 −0.390360
\(106\) 0 0
\(107\) −1956.00 −1.76723 −0.883615 0.468214i \(-0.844898\pi\)
−0.883615 + 0.468214i \(0.844898\pi\)
\(108\) 0 0
\(109\) 1858.00 1.63270 0.816349 0.577559i \(-0.195995\pi\)
0.816349 + 0.577559i \(0.195995\pi\)
\(110\) 0 0
\(111\) 642.000 0.548972
\(112\) 0 0
\(113\) 174.000 0.144854 0.0724272 0.997374i \(-0.476926\pi\)
0.0724272 + 0.997374i \(0.476926\pi\)
\(114\) 0 0
\(115\) −360.000 −0.291915
\(116\) 0 0
\(117\) 630.000 0.497808
\(118\) 0 0
\(119\) −2856.00 −2.20008
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) 0 0
\(123\) −450.000 −0.329879
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −2068.00 −1.44492 −0.722462 0.691411i \(-0.756990\pi\)
−0.722462 + 0.691411i \(0.756990\pi\)
\(128\) 0 0
\(129\) 876.000 0.597888
\(130\) 0 0
\(131\) −312.000 −0.208088 −0.104044 0.994573i \(-0.533178\pi\)
−0.104044 + 0.994573i \(0.533178\pi\)
\(132\) 0 0
\(133\) 560.000 0.365099
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) 2646.00 1.65010 0.825048 0.565063i \(-0.191148\pi\)
0.825048 + 0.565063i \(0.191148\pi\)
\(138\) 0 0
\(139\) 1276.00 0.778625 0.389313 0.921106i \(-0.372713\pi\)
0.389313 + 0.921106i \(0.372713\pi\)
\(140\) 0 0
\(141\) −216.000 −0.129011
\(142\) 0 0
\(143\) 1680.00 0.982438
\(144\) 0 0
\(145\) −1530.00 −0.876273
\(146\) 0 0
\(147\) 1323.00 0.742307
\(148\) 0 0
\(149\) 3198.00 1.75832 0.879162 0.476522i \(-0.158103\pi\)
0.879162 + 0.476522i \(0.158103\pi\)
\(150\) 0 0
\(151\) −760.000 −0.409589 −0.204794 0.978805i \(-0.565653\pi\)
−0.204794 + 0.978805i \(0.565653\pi\)
\(152\) 0 0
\(153\) 918.000 0.485071
\(154\) 0 0
\(155\) −680.000 −0.352380
\(156\) 0 0
\(157\) 166.000 0.0843837 0.0421919 0.999110i \(-0.486566\pi\)
0.0421919 + 0.999110i \(0.486566\pi\)
\(158\) 0 0
\(159\) 1242.00 0.619478
\(160\) 0 0
\(161\) 2016.00 0.986851
\(162\) 0 0
\(163\) −3020.00 −1.45119 −0.725597 0.688120i \(-0.758436\pi\)
−0.725597 + 0.688120i \(0.758436\pi\)
\(164\) 0 0
\(165\) 360.000 0.169854
\(166\) 0 0
\(167\) −984.000 −0.455953 −0.227977 0.973667i \(-0.573211\pi\)
−0.227977 + 0.973667i \(0.573211\pi\)
\(168\) 0 0
\(169\) 2703.00 1.23031
\(170\) 0 0
\(171\) −180.000 −0.0804967
\(172\) 0 0
\(173\) −1962.00 −0.862243 −0.431122 0.902294i \(-0.641882\pi\)
−0.431122 + 0.902294i \(0.641882\pi\)
\(174\) 0 0
\(175\) −700.000 −0.302372
\(176\) 0 0
\(177\) 2232.00 0.947838
\(178\) 0 0
\(179\) −576.000 −0.240515 −0.120258 0.992743i \(-0.538372\pi\)
−0.120258 + 0.992743i \(0.538372\pi\)
\(180\) 0 0
\(181\) 1210.00 0.496898 0.248449 0.968645i \(-0.420079\pi\)
0.248449 + 0.968645i \(0.420079\pi\)
\(182\) 0 0
\(183\) 1254.00 0.506548
\(184\) 0 0
\(185\) 1070.00 0.425232
\(186\) 0 0
\(187\) 2448.00 0.957302
\(188\) 0 0
\(189\) −756.000 −0.290957
\(190\) 0 0
\(191\) 3384.00 1.28198 0.640989 0.767550i \(-0.278525\pi\)
0.640989 + 0.767550i \(0.278525\pi\)
\(192\) 0 0
\(193\) −2038.00 −0.760096 −0.380048 0.924967i \(-0.624092\pi\)
−0.380048 + 0.924967i \(0.624092\pi\)
\(194\) 0 0
\(195\) 1050.00 0.385600
\(196\) 0 0
\(197\) −4098.00 −1.48208 −0.741042 0.671459i \(-0.765668\pi\)
−0.741042 + 0.671459i \(0.765668\pi\)
\(198\) 0 0
\(199\) −2248.00 −0.800786 −0.400393 0.916343i \(-0.631126\pi\)
−0.400393 + 0.916343i \(0.631126\pi\)
\(200\) 0 0
\(201\) −564.000 −0.197918
\(202\) 0 0
\(203\) 8568.00 2.96234
\(204\) 0 0
\(205\) −750.000 −0.255523
\(206\) 0 0
\(207\) −648.000 −0.217580
\(208\) 0 0
\(209\) −480.000 −0.158863
\(210\) 0 0
\(211\) −3260.00 −1.06364 −0.531819 0.846858i \(-0.678491\pi\)
−0.531819 + 0.846858i \(0.678491\pi\)
\(212\) 0 0
\(213\) 1440.00 0.463226
\(214\) 0 0
\(215\) 1460.00 0.463122
\(216\) 0 0
\(217\) 3808.00 1.19126
\(218\) 0 0
\(219\) 1302.00 0.401740
\(220\) 0 0
\(221\) 7140.00 2.17325
\(222\) 0 0
\(223\) −2980.00 −0.894868 −0.447434 0.894317i \(-0.647662\pi\)
−0.447434 + 0.894317i \(0.647662\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) 3180.00 0.929797 0.464899 0.885364i \(-0.346091\pi\)
0.464899 + 0.885364i \(0.346091\pi\)
\(228\) 0 0
\(229\) −3374.00 −0.973625 −0.486813 0.873506i \(-0.661841\pi\)
−0.486813 + 0.873506i \(0.661841\pi\)
\(230\) 0 0
\(231\) −2016.00 −0.574212
\(232\) 0 0
\(233\) 1950.00 0.548278 0.274139 0.961690i \(-0.411607\pi\)
0.274139 + 0.961690i \(0.411607\pi\)
\(234\) 0 0
\(235\) −360.000 −0.0999311
\(236\) 0 0
\(237\) 4056.00 1.11167
\(238\) 0 0
\(239\) 2232.00 0.604084 0.302042 0.953295i \(-0.402332\pi\)
0.302042 + 0.953295i \(0.402332\pi\)
\(240\) 0 0
\(241\) −1822.00 −0.486993 −0.243497 0.969902i \(-0.578294\pi\)
−0.243497 + 0.969902i \(0.578294\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 2205.00 0.574989
\(246\) 0 0
\(247\) −1400.00 −0.360647
\(248\) 0 0
\(249\) 1836.00 0.467276
\(250\) 0 0
\(251\) −1488.00 −0.374190 −0.187095 0.982342i \(-0.559907\pi\)
−0.187095 + 0.982342i \(0.559907\pi\)
\(252\) 0 0
\(253\) −1728.00 −0.429401
\(254\) 0 0
\(255\) 1530.00 0.375735
\(256\) 0 0
\(257\) −2994.00 −0.726695 −0.363347 0.931654i \(-0.618366\pi\)
−0.363347 + 0.931654i \(0.618366\pi\)
\(258\) 0 0
\(259\) −5992.00 −1.43755
\(260\) 0 0
\(261\) −2754.00 −0.653135
\(262\) 0 0
\(263\) −2472.00 −0.579582 −0.289791 0.957090i \(-0.593586\pi\)
−0.289791 + 0.957090i \(0.593586\pi\)
\(264\) 0 0
\(265\) 2070.00 0.479846
\(266\) 0 0
\(267\) −90.0000 −0.0206289
\(268\) 0 0
\(269\) −3954.00 −0.896207 −0.448103 0.893982i \(-0.647900\pi\)
−0.448103 + 0.893982i \(0.647900\pi\)
\(270\) 0 0
\(271\) −2176.00 −0.487759 −0.243879 0.969806i \(-0.578420\pi\)
−0.243879 + 0.969806i \(0.578420\pi\)
\(272\) 0 0
\(273\) −5880.00 −1.30357
\(274\) 0 0
\(275\) 600.000 0.131569
\(276\) 0 0
\(277\) −1034.00 −0.224285 −0.112143 0.993692i \(-0.535771\pi\)
−0.112143 + 0.993692i \(0.535771\pi\)
\(278\) 0 0
\(279\) −1224.00 −0.262649
\(280\) 0 0
\(281\) −6654.00 −1.41261 −0.706307 0.707906i \(-0.749640\pi\)
−0.706307 + 0.707906i \(0.749640\pi\)
\(282\) 0 0
\(283\) 1756.00 0.368846 0.184423 0.982847i \(-0.440958\pi\)
0.184423 + 0.982847i \(0.440958\pi\)
\(284\) 0 0
\(285\) −300.000 −0.0623525
\(286\) 0 0
\(287\) 4200.00 0.863826
\(288\) 0 0
\(289\) 5491.00 1.11765
\(290\) 0 0
\(291\) −858.000 −0.172841
\(292\) 0 0
\(293\) −3234.00 −0.644820 −0.322410 0.946600i \(-0.604493\pi\)
−0.322410 + 0.946600i \(0.604493\pi\)
\(294\) 0 0
\(295\) 3720.00 0.734192
\(296\) 0 0
\(297\) 648.000 0.126602
\(298\) 0 0
\(299\) −5040.00 −0.974818
\(300\) 0 0
\(301\) −8176.00 −1.56564
\(302\) 0 0
\(303\) 4626.00 0.877085
\(304\) 0 0
\(305\) 2090.00 0.392371
\(306\) 0 0
\(307\) −2036.00 −0.378504 −0.189252 0.981929i \(-0.560606\pi\)
−0.189252 + 0.981929i \(0.560606\pi\)
\(308\) 0 0
\(309\) 3516.00 0.647308
\(310\) 0 0
\(311\) −96.0000 −0.0175037 −0.00875187 0.999962i \(-0.502786\pi\)
−0.00875187 + 0.999962i \(0.502786\pi\)
\(312\) 0 0
\(313\) 1202.00 0.217064 0.108532 0.994093i \(-0.465385\pi\)
0.108532 + 0.994093i \(0.465385\pi\)
\(314\) 0 0
\(315\) −1260.00 −0.225374
\(316\) 0 0
\(317\) 3798.00 0.672924 0.336462 0.941697i \(-0.390770\pi\)
0.336462 + 0.941697i \(0.390770\pi\)
\(318\) 0 0
\(319\) −7344.00 −1.28898
\(320\) 0 0
\(321\) −5868.00 −1.02031
\(322\) 0 0
\(323\) −2040.00 −0.351420
\(324\) 0 0
\(325\) 1750.00 0.298685
\(326\) 0 0
\(327\) 5574.00 0.942639
\(328\) 0 0
\(329\) 2016.00 0.337829
\(330\) 0 0
\(331\) 5668.00 0.941213 0.470606 0.882343i \(-0.344035\pi\)
0.470606 + 0.882343i \(0.344035\pi\)
\(332\) 0 0
\(333\) 1926.00 0.316949
\(334\) 0 0
\(335\) −940.000 −0.153307
\(336\) 0 0
\(337\) −454.000 −0.0733856 −0.0366928 0.999327i \(-0.511682\pi\)
−0.0366928 + 0.999327i \(0.511682\pi\)
\(338\) 0 0
\(339\) 522.000 0.0836317
\(340\) 0 0
\(341\) −3264.00 −0.518345
\(342\) 0 0
\(343\) −2744.00 −0.431959
\(344\) 0 0
\(345\) −1080.00 −0.168537
\(346\) 0 0
\(347\) 5604.00 0.866970 0.433485 0.901161i \(-0.357284\pi\)
0.433485 + 0.901161i \(0.357284\pi\)
\(348\) 0 0
\(349\) 11266.0 1.72795 0.863976 0.503533i \(-0.167967\pi\)
0.863976 + 0.503533i \(0.167967\pi\)
\(350\) 0 0
\(351\) 1890.00 0.287410
\(352\) 0 0
\(353\) −6426.00 −0.968899 −0.484450 0.874819i \(-0.660980\pi\)
−0.484450 + 0.874819i \(0.660980\pi\)
\(354\) 0 0
\(355\) 2400.00 0.358813
\(356\) 0 0
\(357\) −8568.00 −1.27021
\(358\) 0 0
\(359\) 6936.00 1.01969 0.509844 0.860267i \(-0.329703\pi\)
0.509844 + 0.860267i \(0.329703\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 0 0
\(363\) −2265.00 −0.327498
\(364\) 0 0
\(365\) 2170.00 0.311186
\(366\) 0 0
\(367\) −388.000 −0.0551865 −0.0275932 0.999619i \(-0.508784\pi\)
−0.0275932 + 0.999619i \(0.508784\pi\)
\(368\) 0 0
\(369\) −1350.00 −0.190456
\(370\) 0 0
\(371\) −11592.0 −1.62217
\(372\) 0 0
\(373\) 8062.00 1.11913 0.559564 0.828787i \(-0.310969\pi\)
0.559564 + 0.828787i \(0.310969\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) −21420.0 −2.92622
\(378\) 0 0
\(379\) 3388.00 0.459182 0.229591 0.973287i \(-0.426261\pi\)
0.229591 + 0.973287i \(0.426261\pi\)
\(380\) 0 0
\(381\) −6204.00 −0.834227
\(382\) 0 0
\(383\) 6984.00 0.931764 0.465882 0.884847i \(-0.345737\pi\)
0.465882 + 0.884847i \(0.345737\pi\)
\(384\) 0 0
\(385\) −3360.00 −0.444783
\(386\) 0 0
\(387\) 2628.00 0.345191
\(388\) 0 0
\(389\) 2526.00 0.329237 0.164619 0.986357i \(-0.447361\pi\)
0.164619 + 0.986357i \(0.447361\pi\)
\(390\) 0 0
\(391\) −7344.00 −0.949877
\(392\) 0 0
\(393\) −936.000 −0.120140
\(394\) 0 0
\(395\) 6760.00 0.861095
\(396\) 0 0
\(397\) −6146.00 −0.776975 −0.388487 0.921454i \(-0.627002\pi\)
−0.388487 + 0.921454i \(0.627002\pi\)
\(398\) 0 0
\(399\) 1680.00 0.210790
\(400\) 0 0
\(401\) 9786.00 1.21868 0.609339 0.792910i \(-0.291435\pi\)
0.609339 + 0.792910i \(0.291435\pi\)
\(402\) 0 0
\(403\) −9520.00 −1.17674
\(404\) 0 0
\(405\) 405.000 0.0496904
\(406\) 0 0
\(407\) 5136.00 0.625509
\(408\) 0 0
\(409\) −886.000 −0.107115 −0.0535573 0.998565i \(-0.517056\pi\)
−0.0535573 + 0.998565i \(0.517056\pi\)
\(410\) 0 0
\(411\) 7938.00 0.952683
\(412\) 0 0
\(413\) −20832.0 −2.48202
\(414\) 0 0
\(415\) 3060.00 0.361951
\(416\) 0 0
\(417\) 3828.00 0.449539
\(418\) 0 0
\(419\) 11352.0 1.32358 0.661792 0.749688i \(-0.269796\pi\)
0.661792 + 0.749688i \(0.269796\pi\)
\(420\) 0 0
\(421\) −10190.0 −1.17964 −0.589822 0.807533i \(-0.700802\pi\)
−0.589822 + 0.807533i \(0.700802\pi\)
\(422\) 0 0
\(423\) −648.000 −0.0744843
\(424\) 0 0
\(425\) 2550.00 0.291043
\(426\) 0 0
\(427\) −11704.0 −1.32645
\(428\) 0 0
\(429\) 5040.00 0.567211
\(430\) 0 0
\(431\) 2448.00 0.273587 0.136794 0.990600i \(-0.456320\pi\)
0.136794 + 0.990600i \(0.456320\pi\)
\(432\) 0 0
\(433\) −7078.00 −0.785559 −0.392779 0.919633i \(-0.628486\pi\)
−0.392779 + 0.919633i \(0.628486\pi\)
\(434\) 0 0
\(435\) −4590.00 −0.505916
\(436\) 0 0
\(437\) 1440.00 0.157631
\(438\) 0 0
\(439\) −18088.0 −1.96650 −0.983250 0.182264i \(-0.941657\pi\)
−0.983250 + 0.182264i \(0.941657\pi\)
\(440\) 0 0
\(441\) 3969.00 0.428571
\(442\) 0 0
\(443\) −3852.00 −0.413124 −0.206562 0.978433i \(-0.566228\pi\)
−0.206562 + 0.978433i \(0.566228\pi\)
\(444\) 0 0
\(445\) −150.000 −0.0159791
\(446\) 0 0
\(447\) 9594.00 1.01517
\(448\) 0 0
\(449\) 6522.00 0.685506 0.342753 0.939426i \(-0.388641\pi\)
0.342753 + 0.939426i \(0.388641\pi\)
\(450\) 0 0
\(451\) −3600.00 −0.375870
\(452\) 0 0
\(453\) −2280.00 −0.236476
\(454\) 0 0
\(455\) −9800.00 −1.00974
\(456\) 0 0
\(457\) 2090.00 0.213930 0.106965 0.994263i \(-0.465887\pi\)
0.106965 + 0.994263i \(0.465887\pi\)
\(458\) 0 0
\(459\) 2754.00 0.280056
\(460\) 0 0
\(461\) 9894.00 0.999587 0.499793 0.866145i \(-0.333409\pi\)
0.499793 + 0.866145i \(0.333409\pi\)
\(462\) 0 0
\(463\) 3044.00 0.305544 0.152772 0.988261i \(-0.451180\pi\)
0.152772 + 0.988261i \(0.451180\pi\)
\(464\) 0 0
\(465\) −2040.00 −0.203447
\(466\) 0 0
\(467\) −10236.0 −1.01427 −0.507137 0.861866i \(-0.669296\pi\)
−0.507137 + 0.861866i \(0.669296\pi\)
\(468\) 0 0
\(469\) 5264.00 0.518271
\(470\) 0 0
\(471\) 498.000 0.0487190
\(472\) 0 0
\(473\) 7008.00 0.681244
\(474\) 0 0
\(475\) −500.000 −0.0482980
\(476\) 0 0
\(477\) 3726.00 0.357656
\(478\) 0 0
\(479\) 11496.0 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 14980.0 1.42002
\(482\) 0 0
\(483\) 6048.00 0.569759
\(484\) 0 0
\(485\) −1430.00 −0.133882
\(486\) 0 0
\(487\) −15316.0 −1.42512 −0.712561 0.701610i \(-0.752465\pi\)
−0.712561 + 0.701610i \(0.752465\pi\)
\(488\) 0 0
\(489\) −9060.00 −0.837847
\(490\) 0 0
\(491\) 11616.0 1.06766 0.533832 0.845591i \(-0.320752\pi\)
0.533832 + 0.845591i \(0.320752\pi\)
\(492\) 0 0
\(493\) −31212.0 −2.85135
\(494\) 0 0
\(495\) 1080.00 0.0980654
\(496\) 0 0
\(497\) −13440.0 −1.21301
\(498\) 0 0
\(499\) −14996.0 −1.34532 −0.672658 0.739953i \(-0.734848\pi\)
−0.672658 + 0.739953i \(0.734848\pi\)
\(500\) 0 0
\(501\) −2952.00 −0.263245
\(502\) 0 0
\(503\) −21648.0 −1.91896 −0.959480 0.281778i \(-0.909076\pi\)
−0.959480 + 0.281778i \(0.909076\pi\)
\(504\) 0 0
\(505\) 7710.00 0.679387
\(506\) 0 0
\(507\) 8109.00 0.710322
\(508\) 0 0
\(509\) −3378.00 −0.294160 −0.147080 0.989125i \(-0.546987\pi\)
−0.147080 + 0.989125i \(0.546987\pi\)
\(510\) 0 0
\(511\) −12152.0 −1.05200
\(512\) 0 0
\(513\) −540.000 −0.0464748
\(514\) 0 0
\(515\) 5860.00 0.501403
\(516\) 0 0
\(517\) −1728.00 −0.146997
\(518\) 0 0
\(519\) −5886.00 −0.497816
\(520\) 0 0
\(521\) −16158.0 −1.35872 −0.679362 0.733804i \(-0.737743\pi\)
−0.679362 + 0.733804i \(0.737743\pi\)
\(522\) 0 0
\(523\) 76.0000 0.00635420 0.00317710 0.999995i \(-0.498989\pi\)
0.00317710 + 0.999995i \(0.498989\pi\)
\(524\) 0 0
\(525\) −2100.00 −0.174574
\(526\) 0 0
\(527\) −13872.0 −1.14663
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 6696.00 0.547235
\(532\) 0 0
\(533\) −10500.0 −0.853294
\(534\) 0 0
\(535\) −9780.00 −0.790329
\(536\) 0 0
\(537\) −1728.00 −0.138862
\(538\) 0 0
\(539\) 10584.0 0.845798
\(540\) 0 0
\(541\) −9278.00 −0.737324 −0.368662 0.929563i \(-0.620184\pi\)
−0.368662 + 0.929563i \(0.620184\pi\)
\(542\) 0 0
\(543\) 3630.00 0.286884
\(544\) 0 0
\(545\) 9290.00 0.730165
\(546\) 0 0
\(547\) −14564.0 −1.13841 −0.569206 0.822195i \(-0.692749\pi\)
−0.569206 + 0.822195i \(0.692749\pi\)
\(548\) 0 0
\(549\) 3762.00 0.292456
\(550\) 0 0
\(551\) 6120.00 0.473177
\(552\) 0 0
\(553\) −37856.0 −2.91103
\(554\) 0 0
\(555\) 3210.00 0.245508
\(556\) 0 0
\(557\) −2154.00 −0.163856 −0.0819281 0.996638i \(-0.526108\pi\)
−0.0819281 + 0.996638i \(0.526108\pi\)
\(558\) 0 0
\(559\) 20440.0 1.54655
\(560\) 0 0
\(561\) 7344.00 0.552699
\(562\) 0 0
\(563\) 8700.00 0.651263 0.325632 0.945497i \(-0.394423\pi\)
0.325632 + 0.945497i \(0.394423\pi\)
\(564\) 0 0
\(565\) 870.000 0.0647808
\(566\) 0 0
\(567\) −2268.00 −0.167984
\(568\) 0 0
\(569\) 4194.00 0.309001 0.154501 0.987993i \(-0.450623\pi\)
0.154501 + 0.987993i \(0.450623\pi\)
\(570\) 0 0
\(571\) 8020.00 0.587787 0.293894 0.955838i \(-0.405049\pi\)
0.293894 + 0.955838i \(0.405049\pi\)
\(572\) 0 0
\(573\) 10152.0 0.740150
\(574\) 0 0
\(575\) −1800.00 −0.130548
\(576\) 0 0
\(577\) −2686.00 −0.193795 −0.0968974 0.995294i \(-0.530892\pi\)
−0.0968974 + 0.995294i \(0.530892\pi\)
\(578\) 0 0
\(579\) −6114.00 −0.438841
\(580\) 0 0
\(581\) −17136.0 −1.22362
\(582\) 0 0
\(583\) 9936.00 0.705844
\(584\) 0 0
\(585\) 3150.00 0.222626
\(586\) 0 0
\(587\) −3012.00 −0.211786 −0.105893 0.994378i \(-0.533770\pi\)
−0.105893 + 0.994378i \(0.533770\pi\)
\(588\) 0 0
\(589\) 2720.00 0.190281
\(590\) 0 0
\(591\) −12294.0 −0.855681
\(592\) 0 0
\(593\) −15522.0 −1.07489 −0.537447 0.843298i \(-0.680611\pi\)
−0.537447 + 0.843298i \(0.680611\pi\)
\(594\) 0 0
\(595\) −14280.0 −0.983904
\(596\) 0 0
\(597\) −6744.00 −0.462334
\(598\) 0 0
\(599\) 19224.0 1.31130 0.655652 0.755063i \(-0.272394\pi\)
0.655652 + 0.755063i \(0.272394\pi\)
\(600\) 0 0
\(601\) −6502.00 −0.441301 −0.220651 0.975353i \(-0.570818\pi\)
−0.220651 + 0.975353i \(0.570818\pi\)
\(602\) 0 0
\(603\) −1692.00 −0.114268
\(604\) 0 0
\(605\) −3775.00 −0.253679
\(606\) 0 0
\(607\) 29396.0 1.96565 0.982823 0.184552i \(-0.0590834\pi\)
0.982823 + 0.184552i \(0.0590834\pi\)
\(608\) 0 0
\(609\) 25704.0 1.71031
\(610\) 0 0
\(611\) −5040.00 −0.333710
\(612\) 0 0
\(613\) 10006.0 0.659280 0.329640 0.944107i \(-0.393073\pi\)
0.329640 + 0.944107i \(0.393073\pi\)
\(614\) 0 0
\(615\) −2250.00 −0.147526
\(616\) 0 0
\(617\) 23118.0 1.50842 0.754210 0.656633i \(-0.228020\pi\)
0.754210 + 0.656633i \(0.228020\pi\)
\(618\) 0 0
\(619\) −14036.0 −0.911397 −0.455698 0.890134i \(-0.650610\pi\)
−0.455698 + 0.890134i \(0.650610\pi\)
\(620\) 0 0
\(621\) −1944.00 −0.125620
\(622\) 0 0
\(623\) 840.000 0.0540191
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −1440.00 −0.0917194
\(628\) 0 0
\(629\) 21828.0 1.38369
\(630\) 0 0
\(631\) −4288.00 −0.270527 −0.135264 0.990810i \(-0.543188\pi\)
−0.135264 + 0.990810i \(0.543188\pi\)
\(632\) 0 0
\(633\) −9780.00 −0.614092
\(634\) 0 0
\(635\) −10340.0 −0.646190
\(636\) 0 0
\(637\) 30870.0 1.92012
\(638\) 0 0
\(639\) 4320.00 0.267444
\(640\) 0 0
\(641\) 1314.00 0.0809671 0.0404835 0.999180i \(-0.487110\pi\)
0.0404835 + 0.999180i \(0.487110\pi\)
\(642\) 0 0
\(643\) 628.000 0.0385162 0.0192581 0.999815i \(-0.493870\pi\)
0.0192581 + 0.999815i \(0.493870\pi\)
\(644\) 0 0
\(645\) 4380.00 0.267383
\(646\) 0 0
\(647\) −10944.0 −0.664997 −0.332498 0.943104i \(-0.607892\pi\)
−0.332498 + 0.943104i \(0.607892\pi\)
\(648\) 0 0
\(649\) 17856.0 1.07998
\(650\) 0 0
\(651\) 11424.0 0.687776
\(652\) 0 0
\(653\) −1098.00 −0.0658010 −0.0329005 0.999459i \(-0.510474\pi\)
−0.0329005 + 0.999459i \(0.510474\pi\)
\(654\) 0 0
\(655\) −1560.00 −0.0930599
\(656\) 0 0
\(657\) 3906.00 0.231945
\(658\) 0 0
\(659\) −312.000 −0.0184428 −0.00922139 0.999957i \(-0.502935\pi\)
−0.00922139 + 0.999957i \(0.502935\pi\)
\(660\) 0 0
\(661\) −8678.00 −0.510643 −0.255322 0.966856i \(-0.582181\pi\)
−0.255322 + 0.966856i \(0.582181\pi\)
\(662\) 0 0
\(663\) 21420.0 1.25473
\(664\) 0 0
\(665\) 2800.00 0.163277
\(666\) 0 0
\(667\) 22032.0 1.27898
\(668\) 0 0
\(669\) −8940.00 −0.516652
\(670\) 0 0
\(671\) 10032.0 0.577170
\(672\) 0 0
\(673\) −14470.0 −0.828793 −0.414396 0.910097i \(-0.636007\pi\)
−0.414396 + 0.910097i \(0.636007\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 0 0
\(677\) 11838.0 0.672040 0.336020 0.941855i \(-0.390919\pi\)
0.336020 + 0.941855i \(0.390919\pi\)
\(678\) 0 0
\(679\) 8008.00 0.452605
\(680\) 0 0
\(681\) 9540.00 0.536819
\(682\) 0 0
\(683\) 25548.0 1.43128 0.715642 0.698467i \(-0.246134\pi\)
0.715642 + 0.698467i \(0.246134\pi\)
\(684\) 0 0
\(685\) 13230.0 0.737945
\(686\) 0 0
\(687\) −10122.0 −0.562123
\(688\) 0 0
\(689\) 28980.0 1.60239
\(690\) 0 0
\(691\) 18412.0 1.01364 0.506820 0.862052i \(-0.330821\pi\)
0.506820 + 0.862052i \(0.330821\pi\)
\(692\) 0 0
\(693\) −6048.00 −0.331522
\(694\) 0 0
\(695\) 6380.00 0.348212
\(696\) 0 0
\(697\) −15300.0 −0.831462
\(698\) 0 0
\(699\) 5850.00 0.316548
\(700\) 0 0
\(701\) 8814.00 0.474893 0.237447 0.971401i \(-0.423690\pi\)
0.237447 + 0.971401i \(0.423690\pi\)
\(702\) 0 0
\(703\) −4280.00 −0.229621
\(704\) 0 0
\(705\) −1080.00 −0.0576953
\(706\) 0 0
\(707\) −43176.0 −2.29675
\(708\) 0 0
\(709\) 17314.0 0.917124 0.458562 0.888662i \(-0.348365\pi\)
0.458562 + 0.888662i \(0.348365\pi\)
\(710\) 0 0
\(711\) 12168.0 0.641822
\(712\) 0 0
\(713\) 9792.00 0.514324
\(714\) 0 0
\(715\) 8400.00 0.439360
\(716\) 0 0
\(717\) 6696.00 0.348768
\(718\) 0 0
\(719\) −768.000 −0.0398353 −0.0199176 0.999802i \(-0.506340\pi\)
−0.0199176 + 0.999802i \(0.506340\pi\)
\(720\) 0 0
\(721\) −32816.0 −1.69505
\(722\) 0 0
\(723\) −5466.00 −0.281166
\(724\) 0 0
\(725\) −7650.00 −0.391881
\(726\) 0 0
\(727\) −18196.0 −0.928270 −0.464135 0.885764i \(-0.653635\pi\)
−0.464135 + 0.885764i \(0.653635\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 29784.0 1.50698
\(732\) 0 0
\(733\) 18142.0 0.914175 0.457087 0.889422i \(-0.348893\pi\)
0.457087 + 0.889422i \(0.348893\pi\)
\(734\) 0 0
\(735\) 6615.00 0.331970
\(736\) 0 0
\(737\) −4512.00 −0.225511
\(738\) 0 0
\(739\) 13660.0 0.679961 0.339981 0.940432i \(-0.389580\pi\)
0.339981 + 0.940432i \(0.389580\pi\)
\(740\) 0 0
\(741\) −4200.00 −0.208220
\(742\) 0 0
\(743\) −12768.0 −0.630434 −0.315217 0.949020i \(-0.602077\pi\)
−0.315217 + 0.949020i \(0.602077\pi\)
\(744\) 0 0
\(745\) 15990.0 0.786347
\(746\) 0 0
\(747\) 5508.00 0.269782
\(748\) 0 0
\(749\) 54768.0 2.67180
\(750\) 0 0
\(751\) 22952.0 1.11522 0.557610 0.830103i \(-0.311718\pi\)
0.557610 + 0.830103i \(0.311718\pi\)
\(752\) 0 0
\(753\) −4464.00 −0.216039
\(754\) 0 0
\(755\) −3800.00 −0.183174
\(756\) 0 0
\(757\) −15818.0 −0.759465 −0.379732 0.925096i \(-0.623984\pi\)
−0.379732 + 0.925096i \(0.623984\pi\)
\(758\) 0 0
\(759\) −5184.00 −0.247915
\(760\) 0 0
\(761\) −18558.0 −0.884004 −0.442002 0.897014i \(-0.645732\pi\)
−0.442002 + 0.897014i \(0.645732\pi\)
\(762\) 0 0
\(763\) −52024.0 −2.46841
\(764\) 0 0
\(765\) 4590.00 0.216930
\(766\) 0 0
\(767\) 52080.0 2.45176
\(768\) 0 0
\(769\) 14978.0 0.702367 0.351184 0.936307i \(-0.385779\pi\)
0.351184 + 0.936307i \(0.385779\pi\)
\(770\) 0 0
\(771\) −8982.00 −0.419557
\(772\) 0 0
\(773\) −8946.00 −0.416255 −0.208128 0.978102i \(-0.566737\pi\)
−0.208128 + 0.978102i \(0.566737\pi\)
\(774\) 0 0
\(775\) −3400.00 −0.157589
\(776\) 0 0
\(777\) −17976.0 −0.829968
\(778\) 0 0
\(779\) 3000.00 0.137980
\(780\) 0 0
\(781\) 11520.0 0.527808
\(782\) 0 0
\(783\) −8262.00 −0.377088
\(784\) 0 0
\(785\) 830.000 0.0377375
\(786\) 0 0
\(787\) 18436.0 0.835035 0.417517 0.908669i \(-0.362900\pi\)
0.417517 + 0.908669i \(0.362900\pi\)
\(788\) 0 0
\(789\) −7416.00 −0.334622
\(790\) 0 0
\(791\) −4872.00 −0.218999
\(792\) 0 0
\(793\) 29260.0 1.31028
\(794\) 0 0
\(795\) 6210.00 0.277039
\(796\) 0 0
\(797\) −16314.0 −0.725058 −0.362529 0.931972i \(-0.618087\pi\)
−0.362529 + 0.931972i \(0.618087\pi\)
\(798\) 0 0
\(799\) −7344.00 −0.325172
\(800\) 0 0
\(801\) −270.000 −0.0119101
\(802\) 0 0
\(803\) 10416.0 0.457749
\(804\) 0 0
\(805\) 10080.0 0.441333
\(806\) 0 0
\(807\) −11862.0 −0.517425
\(808\) 0 0
\(809\) −25446.0 −1.10585 −0.552926 0.833231i \(-0.686489\pi\)
−0.552926 + 0.833231i \(0.686489\pi\)
\(810\) 0 0
\(811\) −42740.0 −1.85056 −0.925280 0.379284i \(-0.876170\pi\)
−0.925280 + 0.379284i \(0.876170\pi\)
\(812\) 0 0
\(813\) −6528.00 −0.281608
\(814\) 0 0
\(815\) −15100.0 −0.648994
\(816\) 0 0
\(817\) −5840.00 −0.250080
\(818\) 0 0
\(819\) −17640.0 −0.752615
\(820\) 0 0
\(821\) −29946.0 −1.27299 −0.636494 0.771282i \(-0.719616\pi\)
−0.636494 + 0.771282i \(0.719616\pi\)
\(822\) 0 0
\(823\) −32596.0 −1.38059 −0.690295 0.723528i \(-0.742519\pi\)
−0.690295 + 0.723528i \(0.742519\pi\)
\(824\) 0 0
\(825\) 1800.00 0.0759612
\(826\) 0 0
\(827\) −3804.00 −0.159949 −0.0799746 0.996797i \(-0.525484\pi\)
−0.0799746 + 0.996797i \(0.525484\pi\)
\(828\) 0 0
\(829\) −3278.00 −0.137334 −0.0686669 0.997640i \(-0.521875\pi\)
−0.0686669 + 0.997640i \(0.521875\pi\)
\(830\) 0 0
\(831\) −3102.00 −0.129491
\(832\) 0 0
\(833\) 44982.0 1.87099
\(834\) 0 0
\(835\) −4920.00 −0.203909
\(836\) 0 0
\(837\) −3672.00 −0.151640
\(838\) 0 0
\(839\) −5784.00 −0.238005 −0.119002 0.992894i \(-0.537970\pi\)
−0.119002 + 0.992894i \(0.537970\pi\)
\(840\) 0 0
\(841\) 69247.0 2.83927
\(842\) 0 0
\(843\) −19962.0 −0.815573
\(844\) 0 0
\(845\) 13515.0 0.550213
\(846\) 0 0
\(847\) 21140.0 0.857590
\(848\) 0 0
\(849\) 5268.00 0.212953
\(850\) 0 0
\(851\) −15408.0 −0.620657
\(852\) 0 0
\(853\) −17306.0 −0.694661 −0.347331 0.937743i \(-0.612912\pi\)
−0.347331 + 0.937743i \(0.612912\pi\)
\(854\) 0 0
\(855\) −900.000 −0.0359992
\(856\) 0 0
\(857\) 31134.0 1.24098 0.620488 0.784216i \(-0.286934\pi\)
0.620488 + 0.784216i \(0.286934\pi\)
\(858\) 0 0
\(859\) 10780.0 0.428183 0.214091 0.976814i \(-0.431321\pi\)
0.214091 + 0.976814i \(0.431321\pi\)
\(860\) 0 0
\(861\) 12600.0 0.498730
\(862\) 0 0
\(863\) 3456.00 0.136319 0.0681597 0.997674i \(-0.478287\pi\)
0.0681597 + 0.997674i \(0.478287\pi\)
\(864\) 0 0
\(865\) −9810.00 −0.385607
\(866\) 0 0
\(867\) 16473.0 0.645274
\(868\) 0 0
\(869\) 32448.0 1.26665
\(870\) 0 0
\(871\) −13160.0 −0.511951
\(872\) 0 0
\(873\) −2574.00 −0.0997900
\(874\) 0 0
\(875\) −3500.00 −0.135225
\(876\) 0 0
\(877\) −2618.00 −0.100802 −0.0504011 0.998729i \(-0.516050\pi\)
−0.0504011 + 0.998729i \(0.516050\pi\)
\(878\) 0 0
\(879\) −9702.00 −0.372287
\(880\) 0 0
\(881\) −26550.0 −1.01531 −0.507657 0.861559i \(-0.669488\pi\)
−0.507657 + 0.861559i \(0.669488\pi\)
\(882\) 0 0
\(883\) −27596.0 −1.05173 −0.525866 0.850567i \(-0.676259\pi\)
−0.525866 + 0.850567i \(0.676259\pi\)
\(884\) 0 0
\(885\) 11160.0 0.423886
\(886\) 0 0
\(887\) −37848.0 −1.43271 −0.716354 0.697737i \(-0.754190\pi\)
−0.716354 + 0.697737i \(0.754190\pi\)
\(888\) 0 0
\(889\) 57904.0 2.18452
\(890\) 0 0
\(891\) 1944.00 0.0730937
\(892\) 0 0
\(893\) 1440.00 0.0539617
\(894\) 0 0
\(895\) −2880.00 −0.107562
\(896\) 0 0
\(897\) −15120.0 −0.562812
\(898\) 0 0
\(899\) 41616.0 1.54391
\(900\) 0 0
\(901\) 42228.0 1.56140
\(902\) 0 0
\(903\) −24528.0 −0.903921
\(904\) 0 0
\(905\) 6050.00 0.222220
\(906\) 0 0
\(907\) 4804.00 0.175870 0.0879351 0.996126i \(-0.471973\pi\)
0.0879351 + 0.996126i \(0.471973\pi\)
\(908\) 0 0
\(909\) 13878.0 0.506385
\(910\) 0 0
\(911\) 28608.0 1.04042 0.520211 0.854037i \(-0.325853\pi\)
0.520211 + 0.854037i \(0.325853\pi\)
\(912\) 0 0
\(913\) 14688.0 0.532423
\(914\) 0 0
\(915\) 6270.00 0.226535
\(916\) 0 0
\(917\) 8736.00 0.314600
\(918\) 0 0
\(919\) −40768.0 −1.46334 −0.731672 0.681657i \(-0.761259\pi\)
−0.731672 + 0.681657i \(0.761259\pi\)
\(920\) 0 0
\(921\) −6108.00 −0.218529
\(922\) 0 0
\(923\) 33600.0 1.19822
\(924\) 0 0
\(925\) 5350.00 0.190170
\(926\) 0 0
\(927\) 10548.0 0.373724
\(928\) 0 0
\(929\) 27642.0 0.976216 0.488108 0.872783i \(-0.337687\pi\)
0.488108 + 0.872783i \(0.337687\pi\)
\(930\) 0 0
\(931\) −8820.00 −0.310487
\(932\) 0 0
\(933\) −288.000 −0.0101058
\(934\) 0 0
\(935\) 12240.0 0.428119
\(936\) 0 0
\(937\) 28106.0 0.979918 0.489959 0.871746i \(-0.337012\pi\)
0.489959 + 0.871746i \(0.337012\pi\)
\(938\) 0 0
\(939\) 3606.00 0.125322
\(940\) 0 0
\(941\) −14730.0 −0.510291 −0.255146 0.966903i \(-0.582123\pi\)
−0.255146 + 0.966903i \(0.582123\pi\)
\(942\) 0 0
\(943\) 10800.0 0.372955
\(944\) 0 0
\(945\) −3780.00 −0.130120
\(946\) 0 0
\(947\) 9564.00 0.328182 0.164091 0.986445i \(-0.447531\pi\)
0.164091 + 0.986445i \(0.447531\pi\)
\(948\) 0 0
\(949\) 30380.0 1.03917
\(950\) 0 0
\(951\) 11394.0 0.388513
\(952\) 0 0
\(953\) −53898.0 −1.83203 −0.916017 0.401141i \(-0.868614\pi\)
−0.916017 + 0.401141i \(0.868614\pi\)
\(954\) 0 0
\(955\) 16920.0 0.573318
\(956\) 0 0
\(957\) −22032.0 −0.744194
\(958\) 0 0
\(959\) −74088.0 −2.49471
\(960\) 0 0
\(961\) −11295.0 −0.379141
\(962\) 0 0
\(963\) −17604.0 −0.589077
\(964\) 0 0
\(965\) −10190.0 −0.339925
\(966\) 0 0
\(967\) 15140.0 0.503485 0.251742 0.967794i \(-0.418996\pi\)
0.251742 + 0.967794i \(0.418996\pi\)
\(968\) 0 0
\(969\) −6120.00 −0.202892
\(970\) 0 0
\(971\) −23808.0 −0.786854 −0.393427 0.919356i \(-0.628711\pi\)
−0.393427 + 0.919356i \(0.628711\pi\)
\(972\) 0 0
\(973\) −35728.0 −1.17717
\(974\) 0 0
\(975\) 5250.00 0.172446
\(976\) 0 0
\(977\) 23094.0 0.756236 0.378118 0.925757i \(-0.376571\pi\)
0.378118 + 0.925757i \(0.376571\pi\)
\(978\) 0 0
\(979\) −720.000 −0.0235049
\(980\) 0 0
\(981\) 16722.0 0.544233
\(982\) 0 0
\(983\) 7584.00 0.246075 0.123038 0.992402i \(-0.460736\pi\)
0.123038 + 0.992402i \(0.460736\pi\)
\(984\) 0 0
\(985\) −20490.0 −0.662808
\(986\) 0 0
\(987\) 6048.00 0.195046
\(988\) 0 0
\(989\) −21024.0 −0.675960
\(990\) 0 0
\(991\) −26752.0 −0.857523 −0.428761 0.903418i \(-0.641050\pi\)
−0.428761 + 0.903418i \(0.641050\pi\)
\(992\) 0 0
\(993\) 17004.0 0.543409
\(994\) 0 0
\(995\) −11240.0 −0.358123
\(996\) 0 0
\(997\) −7778.00 −0.247073 −0.123536 0.992340i \(-0.539424\pi\)
−0.123536 + 0.992340i \(0.539424\pi\)
\(998\) 0 0
\(999\) 5778.00 0.182991
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 960.4.a.bc.1.1 1
4.3 odd 2 960.4.a.r.1.1 1
8.3 odd 2 240.4.a.i.1.1 1
8.5 even 2 60.4.a.a.1.1 1
24.5 odd 2 180.4.a.d.1.1 1
24.11 even 2 720.4.a.bb.1.1 1
40.3 even 4 1200.4.f.n.49.2 2
40.13 odd 4 300.4.d.b.49.1 2
40.19 odd 2 1200.4.a.a.1.1 1
40.27 even 4 1200.4.f.n.49.1 2
40.29 even 2 300.4.a.i.1.1 1
40.37 odd 4 300.4.d.b.49.2 2
72.5 odd 6 1620.4.i.f.541.1 2
72.13 even 6 1620.4.i.l.541.1 2
72.29 odd 6 1620.4.i.f.1081.1 2
72.61 even 6 1620.4.i.l.1081.1 2
120.29 odd 2 900.4.a.q.1.1 1
120.53 even 4 900.4.d.h.649.2 2
120.77 even 4 900.4.d.h.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.a.a.1.1 1 8.5 even 2
180.4.a.d.1.1 1 24.5 odd 2
240.4.a.i.1.1 1 8.3 odd 2
300.4.a.i.1.1 1 40.29 even 2
300.4.d.b.49.1 2 40.13 odd 4
300.4.d.b.49.2 2 40.37 odd 4
720.4.a.bb.1.1 1 24.11 even 2
900.4.a.q.1.1 1 120.29 odd 2
900.4.d.h.649.1 2 120.77 even 4
900.4.d.h.649.2 2 120.53 even 4
960.4.a.r.1.1 1 4.3 odd 2
960.4.a.bc.1.1 1 1.1 even 1 trivial
1200.4.a.a.1.1 1 40.19 odd 2
1200.4.f.n.49.1 2 40.27 even 4
1200.4.f.n.49.2 2 40.3 even 4
1620.4.i.f.541.1 2 72.5 odd 6
1620.4.i.f.1081.1 2 72.29 odd 6
1620.4.i.l.541.1 2 72.13 even 6
1620.4.i.l.1081.1 2 72.61 even 6