Properties

Label 960.4.a.r.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 \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −24.0000 −0.657843 −0.328921 0.944357i \(-0.606685\pi\)
−0.328921 + 0.944357i \(0.606685\pi\)
\(12\) 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 \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 0 0
\(21\) −84.0000 −0.872872
\(22\) 0 0
\(23\) 72.0000 0.652741 0.326370 0.945242i \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 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 \(-0.371100\pi\)
0.393973 + 0.919122i \(0.371100\pi\)
\(32\) 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 \(-0.673244\pi\)
−0.517786 + 0.855510i \(0.673244\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 0 0
\(47\) 72.0000 0.223453 0.111726 0.993739i \(-0.464362\pi\)
0.111726 + 0.993739i \(0.464362\pi\)
\(48\) 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 \(-0.806501\pi\)
−0.820852 + 0.571141i \(0.806501\pi\)
\(60\) 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 \(-0.445170\pi\)
0.171402 + 0.985201i \(0.445170\pi\)
\(68\) 0 0
\(69\) −216.000 −0.376860
\(70\) 0 0
\(71\) −480.000 −0.802331 −0.401166 0.916006i \(-0.631395\pi\)
−0.401166 + 0.916006i \(0.631395\pi\)
\(72\) 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 \(-0.912827\pi\)
−0.962733 + 0.270452i \(0.912827\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −612.000 −0.809346 −0.404673 0.914461i \(-0.632615\pi\)
−0.404673 + 0.914461i \(0.632615\pi\)
\(84\) 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 \(-0.689424\pi\)
−0.560585 + 0.828097i \(0.689424\pi\)
\(104\) 0 0
\(105\) −420.000 −0.390360
\(106\) 0 0
\(107\) 1956.00 1.76723 0.883615 0.468214i \(-0.155102\pi\)
0.883615 + 0.468214i \(0.155102\pi\)
\(108\) 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 \(-0.243010\pi\)
0.722462 + 0.691411i \(0.243010\pi\)
\(128\) 0 0
\(129\) 876.000 0.597888
\(130\) 0 0
\(131\) 312.000 0.208088 0.104044 0.994573i \(-0.466822\pi\)
0.104044 + 0.994573i \(0.466822\pi\)
\(132\) 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 \(-0.627287\pi\)
−0.389313 + 0.921106i \(0.627287\pi\)
\(140\) 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 \(-0.434347\pi\)
0.204794 + 0.978805i \(0.434347\pi\)
\(152\) 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 \(-0.241564\pi\)
0.725597 + 0.688120i \(0.241564\pi\)
\(164\) 0 0
\(165\) 360.000 0.169854
\(166\) 0 0
\(167\) 984.000 0.455953 0.227977 0.973667i \(-0.426789\pi\)
0.227977 + 0.973667i \(0.426789\pi\)
\(168\) 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 \(-0.461628\pi\)
0.120258 + 0.992743i \(0.461628\pi\)
\(180\) 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 \(-0.721475\pi\)
−0.640989 + 0.767550i \(0.721475\pi\)
\(192\) 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 \(-0.368874\pi\)
0.400393 + 0.916343i \(0.368874\pi\)
\(200\) 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 \(-0.321509\pi\)
0.531819 + 0.846858i \(0.321509\pi\)
\(212\) 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 \(-0.352338\pi\)
0.447434 + 0.894317i \(0.352338\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −3180.00 −0.929797 −0.464899 0.885364i \(-0.653909\pi\)
−0.464899 + 0.885364i \(0.653909\pi\)
\(228\) 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 \(-0.597668\pi\)
−0.302042 + 0.953295i \(0.597668\pi\)
\(240\) 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 \(-0.440093\pi\)
0.187095 + 0.982342i \(0.440093\pi\)
\(252\) 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 \(-0.406414\pi\)
0.289791 + 0.957090i \(0.406414\pi\)
\(264\) 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 \(-0.421580\pi\)
0.243879 + 0.969806i \(0.421580\pi\)
\(272\) 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 \(-0.559042\pi\)
−0.184423 + 0.982847i \(0.559042\pi\)
\(284\) 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 \(-0.439394\pi\)
0.189252 + 0.981929i \(0.439394\pi\)
\(308\) 0 0
\(309\) 3516.00 0.647308
\(310\) 0 0
\(311\) 96.0000 0.0175037 0.00875187 0.999962i \(-0.497214\pi\)
0.00875187 + 0.999962i \(0.497214\pi\)
\(312\) 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 \(-0.655965\pi\)
−0.470606 + 0.882343i \(0.655965\pi\)
\(332\) 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 \(-0.642716\pi\)
−0.433485 + 0.901161i \(0.642716\pi\)
\(348\) 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 \(-0.670297\pi\)
−0.509844 + 0.860267i \(0.670297\pi\)
\(360\) 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 \(-0.491216\pi\)
0.0275932 + 0.999619i \(0.491216\pi\)
\(368\) 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 \(-0.573739\pi\)
−0.229591 + 0.973287i \(0.573739\pi\)
\(380\) 0 0
\(381\) −6204.00 −0.834227
\(382\) 0 0
\(383\) −6984.00 −0.931764 −0.465882 0.884847i \(-0.654263\pi\)
−0.465882 + 0.884847i \(0.654263\pi\)
\(384\) 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 \(-0.730204\pi\)
−0.661792 + 0.749688i \(0.730204\pi\)
\(420\) 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 \(-0.543680\pi\)
−0.136794 + 0.990600i \(0.543680\pi\)
\(432\) 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 \(-0.0583426\pi\)
0.983250 + 0.182264i \(0.0583426\pi\)
\(440\) 0 0
\(441\) 3969.00 0.428571
\(442\) 0 0
\(443\) 3852.00 0.413124 0.206562 0.978433i \(-0.433772\pi\)
0.206562 + 0.978433i \(0.433772\pi\)
\(444\) 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 \(-0.548820\pi\)
−0.152772 + 0.988261i \(0.548820\pi\)
\(464\) 0 0
\(465\) −2040.00 −0.203447
\(466\) 0 0
\(467\) 10236.0 1.01427 0.507137 0.861866i \(-0.330704\pi\)
0.507137 + 0.861866i \(0.330704\pi\)
\(468\) 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 \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 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 \(-0.247535\pi\)
0.712561 + 0.701610i \(0.247535\pi\)
\(488\) 0 0
\(489\) −9060.00 −0.837847
\(490\) 0 0
\(491\) −11616.0 −1.06766 −0.533832 0.845591i \(-0.679248\pi\)
−0.533832 + 0.845591i \(0.679248\pi\)
\(492\) 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 \(-0.265152\pi\)
0.672658 + 0.739953i \(0.265152\pi\)
\(500\) 0 0
\(501\) −2952.00 −0.263245
\(502\) 0 0
\(503\) 21648.0 1.91896 0.959480 0.281778i \(-0.0909240\pi\)
0.959480 + 0.281778i \(0.0909240\pi\)
\(504\) 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 \(-0.501011\pi\)
−0.00317710 + 0.999995i \(0.501011\pi\)
\(524\) 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 \(-0.307251\pi\)
0.569206 + 0.822195i \(0.307251\pi\)
\(548\) 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 \(-0.605577\pi\)
−0.325632 + 0.945497i \(0.605577\pi\)
\(564\) 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 \(-0.594951\pi\)
−0.293894 + 0.955838i \(0.594951\pi\)
\(572\) 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 \(-0.466230\pi\)
0.105893 + 0.994378i \(0.466230\pi\)
\(588\) 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 \(-0.727606\pi\)
−0.655652 + 0.755063i \(0.727606\pi\)
\(600\) 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 \(-0.940917\pi\)
−0.982823 + 0.184552i \(0.940917\pi\)
\(608\) 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 \(-0.349390\pi\)
0.455698 + 0.890134i \(0.349390\pi\)
\(620\) 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 \(-0.456812\pi\)
0.135264 + 0.990810i \(0.456812\pi\)
\(632\) 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 \(-0.506130\pi\)
−0.0192581 + 0.999815i \(0.506130\pi\)
\(644\) 0 0
\(645\) 4380.00 0.267383
\(646\) 0 0
\(647\) 10944.0 0.664997 0.332498 0.943104i \(-0.392108\pi\)
0.332498 + 0.943104i \(0.392108\pi\)
\(648\) 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 \(-0.497065\pi\)
0.00922139 + 0.999957i \(0.497065\pi\)
\(660\) 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 \(-0.753866\pi\)
−0.715642 + 0.698467i \(0.753866\pi\)
\(684\) 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 \(-0.669179\pi\)
−0.506820 + 0.862052i \(0.669179\pi\)
\(692\) 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 \(-0.493660\pi\)
0.0199176 + 0.999802i \(0.493660\pi\)
\(720\) 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 \(-0.346365\pi\)
0.464135 + 0.885764i \(0.346365\pi\)
\(728\) 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 \(-0.610420\pi\)
−0.339981 + 0.940432i \(0.610420\pi\)
\(740\) 0 0
\(741\) −4200.00 −0.208220
\(742\) 0 0
\(743\) 12768.0 0.630434 0.315217 0.949020i \(-0.397923\pi\)
0.315217 + 0.949020i \(0.397923\pi\)
\(744\) 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 \(-0.688282\pi\)
−0.557610 + 0.830103i \(0.688282\pi\)
\(752\) 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 \(-0.637100\pi\)
−0.417517 + 0.908669i \(0.637100\pi\)
\(788\) 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 \(-0.123830\pi\)
0.925280 + 0.379284i \(0.123830\pi\)
\(812\) 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 \(-0.257481\pi\)
0.690295 + 0.723528i \(0.257481\pi\)
\(824\) 0 0
\(825\) 1800.00 0.0759612
\(826\) 0 0
\(827\) 3804.00 0.159949 0.0799746 0.996797i \(-0.474516\pi\)
0.0799746 + 0.996797i \(0.474516\pi\)
\(828\) 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 \(-0.462030\pi\)
0.119002 + 0.992894i \(0.462030\pi\)
\(840\) 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 \(-0.568679\pi\)
−0.214091 + 0.976814i \(0.568679\pi\)
\(860\) 0 0
\(861\) 12600.0 0.498730
\(862\) 0 0
\(863\) −3456.00 −0.136319 −0.0681597 0.997674i \(-0.521713\pi\)
−0.0681597 + 0.997674i \(0.521713\pi\)
\(864\) 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 \(-0.323741\pi\)
0.525866 + 0.850567i \(0.323741\pi\)
\(884\) 0 0
\(885\) 11160.0 0.423886
\(886\) 0 0
\(887\) 37848.0 1.43271 0.716354 0.697737i \(-0.245810\pi\)
0.716354 + 0.697737i \(0.245810\pi\)
\(888\) 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 \(-0.528027\pi\)
−0.0879351 + 0.996126i \(0.528027\pi\)
\(908\) 0 0
\(909\) 13878.0 0.506385
\(910\) 0 0
\(911\) −28608.0 −1.04042 −0.520211 0.854037i \(-0.674147\pi\)
−0.520211 + 0.854037i \(0.674147\pi\)
\(912\) 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 \(-0.238741\pi\)
0.731672 + 0.681657i \(0.238741\pi\)
\(920\) 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 \(-0.552469\pi\)
−0.164091 + 0.986445i \(0.552469\pi\)
\(948\) 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 \(-0.581004\pi\)
−0.251742 + 0.967794i \(0.581004\pi\)
\(968\) 0 0
\(969\) −6120.00 −0.202892
\(970\) 0 0
\(971\) 23808.0 0.786854 0.393427 0.919356i \(-0.371289\pi\)
0.393427 + 0.919356i \(0.371289\pi\)
\(972\) 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 \(-0.539264\pi\)
−0.123038 + 0.992402i \(0.539264\pi\)
\(984\) 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 \(-0.358950\pi\)
0.428761 + 0.903418i \(0.358950\pi\)
\(992\) 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.r.1.1 1
4.3 odd 2 960.4.a.bc.1.1 1
8.3 odd 2 60.4.a.a.1.1 1
8.5 even 2 240.4.a.i.1.1 1
24.5 odd 2 720.4.a.bb.1.1 1
24.11 even 2 180.4.a.d.1.1 1
40.3 even 4 300.4.d.b.49.1 2
40.13 odd 4 1200.4.f.n.49.2 2
40.19 odd 2 300.4.a.i.1.1 1
40.27 even 4 300.4.d.b.49.2 2
40.29 even 2 1200.4.a.a.1.1 1
40.37 odd 4 1200.4.f.n.49.1 2
72.11 even 6 1620.4.i.f.1081.1 2
72.43 odd 6 1620.4.i.l.1081.1 2
72.59 even 6 1620.4.i.f.541.1 2
72.67 odd 6 1620.4.i.l.541.1 2
120.59 even 2 900.4.a.q.1.1 1
120.83 odd 4 900.4.d.h.649.2 2
120.107 odd 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 8.3 odd 2
180.4.a.d.1.1 1 24.11 even 2
240.4.a.i.1.1 1 8.5 even 2
300.4.a.i.1.1 1 40.19 odd 2
300.4.d.b.49.1 2 40.3 even 4
300.4.d.b.49.2 2 40.27 even 4
720.4.a.bb.1.1 1 24.5 odd 2
900.4.a.q.1.1 1 120.59 even 2
900.4.d.h.649.1 2 120.107 odd 4
900.4.d.h.649.2 2 120.83 odd 4
960.4.a.r.1.1 1 1.1 even 1 trivial
960.4.a.bc.1.1 1 4.3 odd 2
1200.4.a.a.1.1 1 40.29 even 2
1200.4.f.n.49.1 2 40.37 odd 4
1200.4.f.n.49.2 2 40.13 odd 4
1620.4.i.f.541.1 2 72.59 even 6
1620.4.i.f.1081.1 2 72.11 even 6
1620.4.i.l.541.1 2 72.67 odd 6
1620.4.i.l.1081.1 2 72.43 odd 6