Properties

Label 528.4.a.e.1.1
Level $528$
Weight $4$
Character 528.1
Self dual yes
Analytic conductor $31.153$
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] = [528,4,Mod(1,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, 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("528.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 528.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: \(31.1530084830\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 132)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 528.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +10.0000 q^{5} -8.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +10.0000 q^{5} -8.00000 q^{7} +9.00000 q^{9} +11.0000 q^{11} +18.0000 q^{13} -30.0000 q^{15} +46.0000 q^{17} -40.0000 q^{19} +24.0000 q^{21} -44.0000 q^{23} -25.0000 q^{25} -27.0000 q^{27} +186.000 q^{29} +72.0000 q^{31} -33.0000 q^{33} -80.0000 q^{35} -114.000 q^{37} -54.0000 q^{39} +174.000 q^{41} +416.000 q^{43} +90.0000 q^{45} +156.000 q^{47} -279.000 q^{49} -138.000 q^{51} -62.0000 q^{53} +110.000 q^{55} +120.000 q^{57} +348.000 q^{59} -446.000 q^{61} -72.0000 q^{63} +180.000 q^{65} +956.000 q^{67} +132.000 q^{69} +444.000 q^{71} +306.000 q^{73} +75.0000 q^{75} -88.0000 q^{77} +664.000 q^{79} +81.0000 q^{81} +124.000 q^{83} +460.000 q^{85} -558.000 q^{87} +602.000 q^{89} -144.000 q^{91} -216.000 q^{93} -400.000 q^{95} +1522.00 q^{97} +99.0000 q^{99} +O(q^{100})\)

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
\(5\) 10.0000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −8.00000 −0.431959 −0.215980 0.976398i \(-0.569295\pi\)
−0.215980 + 0.976398i \(0.569295\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 18.0000 0.384023 0.192012 0.981393i \(-0.438499\pi\)
0.192012 + 0.981393i \(0.438499\pi\)
\(14\) 0 0
\(15\) −30.0000 −0.516398
\(16\) 0 0
\(17\) 46.0000 0.656273 0.328136 0.944630i \(-0.393579\pi\)
0.328136 + 0.944630i \(0.393579\pi\)
\(18\) 0 0
\(19\) −40.0000 −0.482980 −0.241490 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) 0 0
\(21\) 24.0000 0.249392
\(22\) 0 0
\(23\) −44.0000 −0.398897 −0.199449 0.979908i \(-0.563915\pi\)
−0.199449 + 0.979908i \(0.563915\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 186.000 1.19101 0.595506 0.803351i \(-0.296952\pi\)
0.595506 + 0.803351i \(0.296952\pi\)
\(30\) 0 0
\(31\) 72.0000 0.417148 0.208574 0.978007i \(-0.433118\pi\)
0.208574 + 0.978007i \(0.433118\pi\)
\(32\) 0 0
\(33\) −33.0000 −0.174078
\(34\) 0 0
\(35\) −80.0000 −0.386356
\(36\) 0 0
\(37\) −114.000 −0.506527 −0.253263 0.967397i \(-0.581504\pi\)
−0.253263 + 0.967397i \(0.581504\pi\)
\(38\) 0 0
\(39\) −54.0000 −0.221716
\(40\) 0 0
\(41\) 174.000 0.662786 0.331393 0.943493i \(-0.392481\pi\)
0.331393 + 0.943493i \(0.392481\pi\)
\(42\) 0 0
\(43\) 416.000 1.47534 0.737668 0.675164i \(-0.235927\pi\)
0.737668 + 0.675164i \(0.235927\pi\)
\(44\) 0 0
\(45\) 90.0000 0.298142
\(46\) 0 0
\(47\) 156.000 0.484148 0.242074 0.970258i \(-0.422172\pi\)
0.242074 + 0.970258i \(0.422172\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) −138.000 −0.378899
\(52\) 0 0
\(53\) −62.0000 −0.160686 −0.0803430 0.996767i \(-0.525602\pi\)
−0.0803430 + 0.996767i \(0.525602\pi\)
\(54\) 0 0
\(55\) 110.000 0.269680
\(56\) 0 0
\(57\) 120.000 0.278849
\(58\) 0 0
\(59\) 348.000 0.767894 0.383947 0.923355i \(-0.374565\pi\)
0.383947 + 0.923355i \(0.374565\pi\)
\(60\) 0 0
\(61\) −446.000 −0.936138 −0.468069 0.883692i \(-0.655050\pi\)
−0.468069 + 0.883692i \(0.655050\pi\)
\(62\) 0 0
\(63\) −72.0000 −0.143986
\(64\) 0 0
\(65\) 180.000 0.343481
\(66\) 0 0
\(67\) 956.000 1.74319 0.871597 0.490223i \(-0.163085\pi\)
0.871597 + 0.490223i \(0.163085\pi\)
\(68\) 0 0
\(69\) 132.000 0.230303
\(70\) 0 0
\(71\) 444.000 0.742156 0.371078 0.928602i \(-0.378988\pi\)
0.371078 + 0.928602i \(0.378988\pi\)
\(72\) 0 0
\(73\) 306.000 0.490611 0.245305 0.969446i \(-0.421112\pi\)
0.245305 + 0.969446i \(0.421112\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −88.0000 −0.130241
\(78\) 0 0
\(79\) 664.000 0.945644 0.472822 0.881158i \(-0.343235\pi\)
0.472822 + 0.881158i \(0.343235\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 124.000 0.163985 0.0819926 0.996633i \(-0.473872\pi\)
0.0819926 + 0.996633i \(0.473872\pi\)
\(84\) 0 0
\(85\) 460.000 0.586988
\(86\) 0 0
\(87\) −558.000 −0.687631
\(88\) 0 0
\(89\) 602.000 0.716987 0.358494 0.933532i \(-0.383290\pi\)
0.358494 + 0.933532i \(0.383290\pi\)
\(90\) 0 0
\(91\) −144.000 −0.165882
\(92\) 0 0
\(93\) −216.000 −0.240840
\(94\) 0 0
\(95\) −400.000 −0.431991
\(96\) 0 0
\(97\) 1522.00 1.59315 0.796576 0.604539i \(-0.206643\pi\)
0.796576 + 0.604539i \(0.206643\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) 1090.00 1.07385 0.536926 0.843629i \(-0.319585\pi\)
0.536926 + 0.843629i \(0.319585\pi\)
\(102\) 0 0
\(103\) 1392.00 1.33163 0.665815 0.746117i \(-0.268084\pi\)
0.665815 + 0.746117i \(0.268084\pi\)
\(104\) 0 0
\(105\) 240.000 0.223063
\(106\) 0 0
\(107\) −308.000 −0.278276 −0.139138 0.990273i \(-0.544433\pi\)
−0.139138 + 0.990273i \(0.544433\pi\)
\(108\) 0 0
\(109\) 978.000 0.859407 0.429704 0.902970i \(-0.358618\pi\)
0.429704 + 0.902970i \(0.358618\pi\)
\(110\) 0 0
\(111\) 342.000 0.292443
\(112\) 0 0
\(113\) 1162.00 0.967361 0.483680 0.875245i \(-0.339300\pi\)
0.483680 + 0.875245i \(0.339300\pi\)
\(114\) 0 0
\(115\) −440.000 −0.356784
\(116\) 0 0
\(117\) 162.000 0.128008
\(118\) 0 0
\(119\) −368.000 −0.283483
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −522.000 −0.382660
\(124\) 0 0
\(125\) −1500.00 −1.07331
\(126\) 0 0
\(127\) 984.000 0.687527 0.343763 0.939056i \(-0.388298\pi\)
0.343763 + 0.939056i \(0.388298\pi\)
\(128\) 0 0
\(129\) −1248.00 −0.851785
\(130\) 0 0
\(131\) −2012.00 −1.34190 −0.670951 0.741501i \(-0.734114\pi\)
−0.670951 + 0.741501i \(0.734114\pi\)
\(132\) 0 0
\(133\) 320.000 0.208628
\(134\) 0 0
\(135\) −270.000 −0.172133
\(136\) 0 0
\(137\) −1286.00 −0.801974 −0.400987 0.916084i \(-0.631333\pi\)
−0.400987 + 0.916084i \(0.631333\pi\)
\(138\) 0 0
\(139\) −464.000 −0.283136 −0.141568 0.989929i \(-0.545214\pi\)
−0.141568 + 0.989929i \(0.545214\pi\)
\(140\) 0 0
\(141\) −468.000 −0.279523
\(142\) 0 0
\(143\) 198.000 0.115787
\(144\) 0 0
\(145\) 1860.00 1.06527
\(146\) 0 0
\(147\) 837.000 0.469623
\(148\) 0 0
\(149\) −2894.00 −1.59118 −0.795590 0.605836i \(-0.792839\pi\)
−0.795590 + 0.605836i \(0.792839\pi\)
\(150\) 0 0
\(151\) −976.000 −0.525998 −0.262999 0.964796i \(-0.584712\pi\)
−0.262999 + 0.964796i \(0.584712\pi\)
\(152\) 0 0
\(153\) 414.000 0.218758
\(154\) 0 0
\(155\) 720.000 0.373108
\(156\) 0 0
\(157\) 1646.00 0.836720 0.418360 0.908281i \(-0.362605\pi\)
0.418360 + 0.908281i \(0.362605\pi\)
\(158\) 0 0
\(159\) 186.000 0.0927721
\(160\) 0 0
\(161\) 352.000 0.172307
\(162\) 0 0
\(163\) −3268.00 −1.57037 −0.785183 0.619264i \(-0.787431\pi\)
−0.785183 + 0.619264i \(0.787431\pi\)
\(164\) 0 0
\(165\) −330.000 −0.155700
\(166\) 0 0
\(167\) −1608.00 −0.745094 −0.372547 0.928013i \(-0.621516\pi\)
−0.372547 + 0.928013i \(0.621516\pi\)
\(168\) 0 0
\(169\) −1873.00 −0.852526
\(170\) 0 0
\(171\) −360.000 −0.160993
\(172\) 0 0
\(173\) −4070.00 −1.78865 −0.894325 0.447418i \(-0.852343\pi\)
−0.894325 + 0.447418i \(0.852343\pi\)
\(174\) 0 0
\(175\) 200.000 0.0863919
\(176\) 0 0
\(177\) −1044.00 −0.443344
\(178\) 0 0
\(179\) −52.0000 −0.0217132 −0.0108566 0.999941i \(-0.503456\pi\)
−0.0108566 + 0.999941i \(0.503456\pi\)
\(180\) 0 0
\(181\) 1798.00 0.738366 0.369183 0.929357i \(-0.379638\pi\)
0.369183 + 0.929357i \(0.379638\pi\)
\(182\) 0 0
\(183\) 1338.00 0.540480
\(184\) 0 0
\(185\) −1140.00 −0.453051
\(186\) 0 0
\(187\) 506.000 0.197874
\(188\) 0 0
\(189\) 216.000 0.0831306
\(190\) 0 0
\(191\) 3852.00 1.45927 0.729636 0.683836i \(-0.239690\pi\)
0.729636 + 0.683836i \(0.239690\pi\)
\(192\) 0 0
\(193\) −1958.00 −0.730259 −0.365129 0.930957i \(-0.618975\pi\)
−0.365129 + 0.930957i \(0.618975\pi\)
\(194\) 0 0
\(195\) −540.000 −0.198309
\(196\) 0 0
\(197\) −1630.00 −0.589506 −0.294753 0.955573i \(-0.595237\pi\)
−0.294753 + 0.955573i \(0.595237\pi\)
\(198\) 0 0
\(199\) 4504.00 1.60442 0.802211 0.597040i \(-0.203657\pi\)
0.802211 + 0.597040i \(0.203657\pi\)
\(200\) 0 0
\(201\) −2868.00 −1.00643
\(202\) 0 0
\(203\) −1488.00 −0.514469
\(204\) 0 0
\(205\) 1740.00 0.592814
\(206\) 0 0
\(207\) −396.000 −0.132966
\(208\) 0 0
\(209\) −440.000 −0.145624
\(210\) 0 0
\(211\) −3776.00 −1.23199 −0.615997 0.787749i \(-0.711246\pi\)
−0.615997 + 0.787749i \(0.711246\pi\)
\(212\) 0 0
\(213\) −1332.00 −0.428484
\(214\) 0 0
\(215\) 4160.00 1.31958
\(216\) 0 0
\(217\) −576.000 −0.180191
\(218\) 0 0
\(219\) −918.000 −0.283254
\(220\) 0 0
\(221\) 828.000 0.252024
\(222\) 0 0
\(223\) 1728.00 0.518903 0.259452 0.965756i \(-0.416458\pi\)
0.259452 + 0.965756i \(0.416458\pi\)
\(224\) 0 0
\(225\) −225.000 −0.0666667
\(226\) 0 0
\(227\) −4716.00 −1.37891 −0.689454 0.724330i \(-0.742149\pi\)
−0.689454 + 0.724330i \(0.742149\pi\)
\(228\) 0 0
\(229\) 1766.00 0.509609 0.254805 0.966993i \(-0.417989\pi\)
0.254805 + 0.966993i \(0.417989\pi\)
\(230\) 0 0
\(231\) 264.000 0.0751945
\(232\) 0 0
\(233\) 4382.00 1.23208 0.616039 0.787715i \(-0.288736\pi\)
0.616039 + 0.787715i \(0.288736\pi\)
\(234\) 0 0
\(235\) 1560.00 0.433035
\(236\) 0 0
\(237\) −1992.00 −0.545968
\(238\) 0 0
\(239\) −1264.00 −0.342098 −0.171049 0.985263i \(-0.554716\pi\)
−0.171049 + 0.985263i \(0.554716\pi\)
\(240\) 0 0
\(241\) 3010.00 0.804528 0.402264 0.915524i \(-0.368223\pi\)
0.402264 + 0.915524i \(0.368223\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −2790.00 −0.727537
\(246\) 0 0
\(247\) −720.000 −0.185476
\(248\) 0 0
\(249\) −372.000 −0.0946769
\(250\) 0 0
\(251\) −5404.00 −1.35895 −0.679477 0.733697i \(-0.737793\pi\)
−0.679477 + 0.733697i \(0.737793\pi\)
\(252\) 0 0
\(253\) −484.000 −0.120272
\(254\) 0 0
\(255\) −1380.00 −0.338898
\(256\) 0 0
\(257\) 1618.00 0.392716 0.196358 0.980532i \(-0.437088\pi\)
0.196358 + 0.980532i \(0.437088\pi\)
\(258\) 0 0
\(259\) 912.000 0.218799
\(260\) 0 0
\(261\) 1674.00 0.397004
\(262\) 0 0
\(263\) −48.0000 −0.0112540 −0.00562701 0.999984i \(-0.501791\pi\)
−0.00562701 + 0.999984i \(0.501791\pi\)
\(264\) 0 0
\(265\) −620.000 −0.143722
\(266\) 0 0
\(267\) −1806.00 −0.413953
\(268\) 0 0
\(269\) −1814.00 −0.411158 −0.205579 0.978641i \(-0.565908\pi\)
−0.205579 + 0.978641i \(0.565908\pi\)
\(270\) 0 0
\(271\) −1016.00 −0.227740 −0.113870 0.993496i \(-0.536325\pi\)
−0.113870 + 0.993496i \(0.536325\pi\)
\(272\) 0 0
\(273\) 432.000 0.0957723
\(274\) 0 0
\(275\) −275.000 −0.0603023
\(276\) 0 0
\(277\) 6914.00 1.49972 0.749859 0.661597i \(-0.230121\pi\)
0.749859 + 0.661597i \(0.230121\pi\)
\(278\) 0 0
\(279\) 648.000 0.139049
\(280\) 0 0
\(281\) −6098.00 −1.29458 −0.647289 0.762245i \(-0.724097\pi\)
−0.647289 + 0.762245i \(0.724097\pi\)
\(282\) 0 0
\(283\) −4192.00 −0.880525 −0.440262 0.897869i \(-0.645115\pi\)
−0.440262 + 0.897869i \(0.645115\pi\)
\(284\) 0 0
\(285\) 1200.00 0.249410
\(286\) 0 0
\(287\) −1392.00 −0.286297
\(288\) 0 0
\(289\) −2797.00 −0.569306
\(290\) 0 0
\(291\) −4566.00 −0.919806
\(292\) 0 0
\(293\) 9314.00 1.85710 0.928549 0.371210i \(-0.121057\pi\)
0.928549 + 0.371210i \(0.121057\pi\)
\(294\) 0 0
\(295\) 3480.00 0.686825
\(296\) 0 0
\(297\) −297.000 −0.0580259
\(298\) 0 0
\(299\) −792.000 −0.153186
\(300\) 0 0
\(301\) −3328.00 −0.637285
\(302\) 0 0
\(303\) −3270.00 −0.619989
\(304\) 0 0
\(305\) −4460.00 −0.837308
\(306\) 0 0
\(307\) −1384.00 −0.257293 −0.128647 0.991690i \(-0.541063\pi\)
−0.128647 + 0.991690i \(0.541063\pi\)
\(308\) 0 0
\(309\) −4176.00 −0.768817
\(310\) 0 0
\(311\) −7916.00 −1.44333 −0.721664 0.692243i \(-0.756623\pi\)
−0.721664 + 0.692243i \(0.756623\pi\)
\(312\) 0 0
\(313\) 218.000 0.0393677 0.0196838 0.999806i \(-0.493734\pi\)
0.0196838 + 0.999806i \(0.493734\pi\)
\(314\) 0 0
\(315\) −720.000 −0.128785
\(316\) 0 0
\(317\) 666.000 0.118001 0.0590005 0.998258i \(-0.481209\pi\)
0.0590005 + 0.998258i \(0.481209\pi\)
\(318\) 0 0
\(319\) 2046.00 0.359103
\(320\) 0 0
\(321\) 924.000 0.160662
\(322\) 0 0
\(323\) −1840.00 −0.316967
\(324\) 0 0
\(325\) −450.000 −0.0768046
\(326\) 0 0
\(327\) −2934.00 −0.496179
\(328\) 0 0
\(329\) −1248.00 −0.209132
\(330\) 0 0
\(331\) 2500.00 0.415143 0.207572 0.978220i \(-0.433444\pi\)
0.207572 + 0.978220i \(0.433444\pi\)
\(332\) 0 0
\(333\) −1026.00 −0.168842
\(334\) 0 0
\(335\) 9560.00 1.55916
\(336\) 0 0
\(337\) −6678.00 −1.07945 −0.539724 0.841842i \(-0.681471\pi\)
−0.539724 + 0.841842i \(0.681471\pi\)
\(338\) 0 0
\(339\) −3486.00 −0.558506
\(340\) 0 0
\(341\) 792.000 0.125775
\(342\) 0 0
\(343\) 4976.00 0.783320
\(344\) 0 0
\(345\) 1320.00 0.205990
\(346\) 0 0
\(347\) 1892.00 0.292703 0.146351 0.989233i \(-0.453247\pi\)
0.146351 + 0.989233i \(0.453247\pi\)
\(348\) 0 0
\(349\) −7350.00 −1.12733 −0.563663 0.826005i \(-0.690608\pi\)
−0.563663 + 0.826005i \(0.690608\pi\)
\(350\) 0 0
\(351\) −486.000 −0.0739053
\(352\) 0 0
\(353\) 2370.00 0.357344 0.178672 0.983909i \(-0.442820\pi\)
0.178672 + 0.983909i \(0.442820\pi\)
\(354\) 0 0
\(355\) 4440.00 0.663805
\(356\) 0 0
\(357\) 1104.00 0.163669
\(358\) 0 0
\(359\) −912.000 −0.134077 −0.0670383 0.997750i \(-0.521355\pi\)
−0.0670383 + 0.997750i \(0.521355\pi\)
\(360\) 0 0
\(361\) −5259.00 −0.766730
\(362\) 0 0
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) 3060.00 0.438816
\(366\) 0 0
\(367\) 8464.00 1.20386 0.601931 0.798548i \(-0.294398\pi\)
0.601931 + 0.798548i \(0.294398\pi\)
\(368\) 0 0
\(369\) 1566.00 0.220929
\(370\) 0 0
\(371\) 496.000 0.0694098
\(372\) 0 0
\(373\) 11890.0 1.65051 0.825256 0.564759i \(-0.191031\pi\)
0.825256 + 0.564759i \(0.191031\pi\)
\(374\) 0 0
\(375\) 4500.00 0.619677
\(376\) 0 0
\(377\) 3348.00 0.457376
\(378\) 0 0
\(379\) 9556.00 1.29514 0.647571 0.762005i \(-0.275785\pi\)
0.647571 + 0.762005i \(0.275785\pi\)
\(380\) 0 0
\(381\) −2952.00 −0.396944
\(382\) 0 0
\(383\) −5236.00 −0.698556 −0.349278 0.937019i \(-0.613573\pi\)
−0.349278 + 0.937019i \(0.613573\pi\)
\(384\) 0 0
\(385\) −880.000 −0.116491
\(386\) 0 0
\(387\) 3744.00 0.491778
\(388\) 0 0
\(389\) −8262.00 −1.07686 −0.538432 0.842669i \(-0.680983\pi\)
−0.538432 + 0.842669i \(0.680983\pi\)
\(390\) 0 0
\(391\) −2024.00 −0.261785
\(392\) 0 0
\(393\) 6036.00 0.774748
\(394\) 0 0
\(395\) 6640.00 0.845809
\(396\) 0 0
\(397\) −2402.00 −0.303660 −0.151830 0.988407i \(-0.548517\pi\)
−0.151830 + 0.988407i \(0.548517\pi\)
\(398\) 0 0
\(399\) −960.000 −0.120451
\(400\) 0 0
\(401\) 6290.00 0.783311 0.391655 0.920112i \(-0.371903\pi\)
0.391655 + 0.920112i \(0.371903\pi\)
\(402\) 0 0
\(403\) 1296.00 0.160194
\(404\) 0 0
\(405\) 810.000 0.0993808
\(406\) 0 0
\(407\) −1254.00 −0.152724
\(408\) 0 0
\(409\) −6950.00 −0.840233 −0.420117 0.907470i \(-0.638011\pi\)
−0.420117 + 0.907470i \(0.638011\pi\)
\(410\) 0 0
\(411\) 3858.00 0.463020
\(412\) 0 0
\(413\) −2784.00 −0.331699
\(414\) 0 0
\(415\) 1240.00 0.146673
\(416\) 0 0
\(417\) 1392.00 0.163469
\(418\) 0 0
\(419\) −12660.0 −1.47609 −0.738045 0.674752i \(-0.764251\pi\)
−0.738045 + 0.674752i \(0.764251\pi\)
\(420\) 0 0
\(421\) 5342.00 0.618416 0.309208 0.950994i \(-0.399936\pi\)
0.309208 + 0.950994i \(0.399936\pi\)
\(422\) 0 0
\(423\) 1404.00 0.161383
\(424\) 0 0
\(425\) −1150.00 −0.131255
\(426\) 0 0
\(427\) 3568.00 0.404374
\(428\) 0 0
\(429\) −594.000 −0.0668499
\(430\) 0 0
\(431\) 13560.0 1.51546 0.757729 0.652570i \(-0.226309\pi\)
0.757729 + 0.652570i \(0.226309\pi\)
\(432\) 0 0
\(433\) 4658.00 0.516973 0.258486 0.966015i \(-0.416776\pi\)
0.258486 + 0.966015i \(0.416776\pi\)
\(434\) 0 0
\(435\) −5580.00 −0.615036
\(436\) 0 0
\(437\) 1760.00 0.192660
\(438\) 0 0
\(439\) 6392.00 0.694928 0.347464 0.937693i \(-0.387043\pi\)
0.347464 + 0.937693i \(0.387043\pi\)
\(440\) 0 0
\(441\) −2511.00 −0.271137
\(442\) 0 0
\(443\) −10772.0 −1.15529 −0.577645 0.816288i \(-0.696028\pi\)
−0.577645 + 0.816288i \(0.696028\pi\)
\(444\) 0 0
\(445\) 6020.00 0.641293
\(446\) 0 0
\(447\) 8682.00 0.918668
\(448\) 0 0
\(449\) −5606.00 −0.589228 −0.294614 0.955616i \(-0.595191\pi\)
−0.294614 + 0.955616i \(0.595191\pi\)
\(450\) 0 0
\(451\) 1914.00 0.199838
\(452\) 0 0
\(453\) 2928.00 0.303685
\(454\) 0 0
\(455\) −1440.00 −0.148370
\(456\) 0 0
\(457\) 17050.0 1.74522 0.872610 0.488418i \(-0.162426\pi\)
0.872610 + 0.488418i \(0.162426\pi\)
\(458\) 0 0
\(459\) −1242.00 −0.126300
\(460\) 0 0
\(461\) −8438.00 −0.852488 −0.426244 0.904608i \(-0.640163\pi\)
−0.426244 + 0.904608i \(0.640163\pi\)
\(462\) 0 0
\(463\) −5064.00 −0.508302 −0.254151 0.967164i \(-0.581796\pi\)
−0.254151 + 0.967164i \(0.581796\pi\)
\(464\) 0 0
\(465\) −2160.00 −0.215414
\(466\) 0 0
\(467\) 14268.0 1.41380 0.706900 0.707314i \(-0.250093\pi\)
0.706900 + 0.707314i \(0.250093\pi\)
\(468\) 0 0
\(469\) −7648.00 −0.752989
\(470\) 0 0
\(471\) −4938.00 −0.483081
\(472\) 0 0
\(473\) 4576.00 0.444830
\(474\) 0 0
\(475\) 1000.00 0.0965961
\(476\) 0 0
\(477\) −558.000 −0.0535620
\(478\) 0 0
\(479\) 18312.0 1.74676 0.873379 0.487042i \(-0.161924\pi\)
0.873379 + 0.487042i \(0.161924\pi\)
\(480\) 0 0
\(481\) −2052.00 −0.194518
\(482\) 0 0
\(483\) −1056.00 −0.0994817
\(484\) 0 0
\(485\) 15220.0 1.42496
\(486\) 0 0
\(487\) −4376.00 −0.407178 −0.203589 0.979056i \(-0.565261\pi\)
−0.203589 + 0.979056i \(0.565261\pi\)
\(488\) 0 0
\(489\) 9804.00 0.906651
\(490\) 0 0
\(491\) −17380.0 −1.59745 −0.798725 0.601696i \(-0.794492\pi\)
−0.798725 + 0.601696i \(0.794492\pi\)
\(492\) 0 0
\(493\) 8556.00 0.781629
\(494\) 0 0
\(495\) 990.000 0.0898933
\(496\) 0 0
\(497\) −3552.00 −0.320581
\(498\) 0 0
\(499\) −11324.0 −1.01590 −0.507948 0.861388i \(-0.669596\pi\)
−0.507948 + 0.861388i \(0.669596\pi\)
\(500\) 0 0
\(501\) 4824.00 0.430180
\(502\) 0 0
\(503\) −2392.00 −0.212036 −0.106018 0.994364i \(-0.533810\pi\)
−0.106018 + 0.994364i \(0.533810\pi\)
\(504\) 0 0
\(505\) 10900.0 0.960482
\(506\) 0 0
\(507\) 5619.00 0.492206
\(508\) 0 0
\(509\) −1238.00 −0.107806 −0.0539031 0.998546i \(-0.517166\pi\)
−0.0539031 + 0.998546i \(0.517166\pi\)
\(510\) 0 0
\(511\) −2448.00 −0.211924
\(512\) 0 0
\(513\) 1080.00 0.0929496
\(514\) 0 0
\(515\) 13920.0 1.19105
\(516\) 0 0
\(517\) 1716.00 0.145976
\(518\) 0 0
\(519\) 12210.0 1.03268
\(520\) 0 0
\(521\) 21738.0 1.82794 0.913972 0.405777i \(-0.132999\pi\)
0.913972 + 0.405777i \(0.132999\pi\)
\(522\) 0 0
\(523\) −22016.0 −1.84071 −0.920356 0.391081i \(-0.872101\pi\)
−0.920356 + 0.391081i \(0.872101\pi\)
\(524\) 0 0
\(525\) −600.000 −0.0498784
\(526\) 0 0
\(527\) 3312.00 0.273763
\(528\) 0 0
\(529\) −10231.0 −0.840881
\(530\) 0 0
\(531\) 3132.00 0.255965
\(532\) 0 0
\(533\) 3132.00 0.254525
\(534\) 0 0
\(535\) −3080.00 −0.248897
\(536\) 0 0
\(537\) 156.000 0.0125361
\(538\) 0 0
\(539\) −3069.00 −0.245253
\(540\) 0 0
\(541\) 9490.00 0.754172 0.377086 0.926178i \(-0.376926\pi\)
0.377086 + 0.926178i \(0.376926\pi\)
\(542\) 0 0
\(543\) −5394.00 −0.426296
\(544\) 0 0
\(545\) 9780.00 0.768677
\(546\) 0 0
\(547\) −21632.0 −1.69089 −0.845446 0.534061i \(-0.820665\pi\)
−0.845446 + 0.534061i \(0.820665\pi\)
\(548\) 0 0
\(549\) −4014.00 −0.312046
\(550\) 0 0
\(551\) −7440.00 −0.575235
\(552\) 0 0
\(553\) −5312.00 −0.408480
\(554\) 0 0
\(555\) 3420.00 0.261569
\(556\) 0 0
\(557\) −4854.00 −0.369247 −0.184624 0.982809i \(-0.559107\pi\)
−0.184624 + 0.982809i \(0.559107\pi\)
\(558\) 0 0
\(559\) 7488.00 0.566563
\(560\) 0 0
\(561\) −1518.00 −0.114242
\(562\) 0 0
\(563\) 5308.00 0.397346 0.198673 0.980066i \(-0.436337\pi\)
0.198673 + 0.980066i \(0.436337\pi\)
\(564\) 0 0
\(565\) 11620.0 0.865234
\(566\) 0 0
\(567\) −648.000 −0.0479955
\(568\) 0 0
\(569\) −12490.0 −0.920225 −0.460113 0.887861i \(-0.652191\pi\)
−0.460113 + 0.887861i \(0.652191\pi\)
\(570\) 0 0
\(571\) 7448.00 0.545865 0.272933 0.962033i \(-0.412006\pi\)
0.272933 + 0.962033i \(0.412006\pi\)
\(572\) 0 0
\(573\) −11556.0 −0.842511
\(574\) 0 0
\(575\) 1100.00 0.0797794
\(576\) 0 0
\(577\) 10994.0 0.793217 0.396608 0.917988i \(-0.370187\pi\)
0.396608 + 0.917988i \(0.370187\pi\)
\(578\) 0 0
\(579\) 5874.00 0.421615
\(580\) 0 0
\(581\) −992.000 −0.0708349
\(582\) 0 0
\(583\) −682.000 −0.0484486
\(584\) 0 0
\(585\) 1620.00 0.114494
\(586\) 0 0
\(587\) 5148.00 0.361977 0.180989 0.983485i \(-0.442070\pi\)
0.180989 + 0.983485i \(0.442070\pi\)
\(588\) 0 0
\(589\) −2880.00 −0.201474
\(590\) 0 0
\(591\) 4890.00 0.340351
\(592\) 0 0
\(593\) −22314.0 −1.54524 −0.772619 0.634870i \(-0.781054\pi\)
−0.772619 + 0.634870i \(0.781054\pi\)
\(594\) 0 0
\(595\) −3680.00 −0.253555
\(596\) 0 0
\(597\) −13512.0 −0.926314
\(598\) 0 0
\(599\) 15588.0 1.06329 0.531643 0.846968i \(-0.321575\pi\)
0.531643 + 0.846968i \(0.321575\pi\)
\(600\) 0 0
\(601\) −21638.0 −1.46861 −0.734303 0.678822i \(-0.762491\pi\)
−0.734303 + 0.678822i \(0.762491\pi\)
\(602\) 0 0
\(603\) 8604.00 0.581065
\(604\) 0 0
\(605\) 1210.00 0.0813116
\(606\) 0 0
\(607\) 7496.00 0.501241 0.250620 0.968085i \(-0.419365\pi\)
0.250620 + 0.968085i \(0.419365\pi\)
\(608\) 0 0
\(609\) 4464.00 0.297029
\(610\) 0 0
\(611\) 2808.00 0.185924
\(612\) 0 0
\(613\) 2106.00 0.138761 0.0693805 0.997590i \(-0.477898\pi\)
0.0693805 + 0.997590i \(0.477898\pi\)
\(614\) 0 0
\(615\) −5220.00 −0.342261
\(616\) 0 0
\(617\) 26962.0 1.75924 0.879619 0.475680i \(-0.157798\pi\)
0.879619 + 0.475680i \(0.157798\pi\)
\(618\) 0 0
\(619\) 17740.0 1.15191 0.575954 0.817482i \(-0.304631\pi\)
0.575954 + 0.817482i \(0.304631\pi\)
\(620\) 0 0
\(621\) 1188.00 0.0767678
\(622\) 0 0
\(623\) −4816.00 −0.309709
\(624\) 0 0
\(625\) −11875.0 −0.760000
\(626\) 0 0
\(627\) 1320.00 0.0840761
\(628\) 0 0
\(629\) −5244.00 −0.332420
\(630\) 0 0
\(631\) −19360.0 −1.22141 −0.610705 0.791858i \(-0.709114\pi\)
−0.610705 + 0.791858i \(0.709114\pi\)
\(632\) 0 0
\(633\) 11328.0 0.711292
\(634\) 0 0
\(635\) 9840.00 0.614943
\(636\) 0 0
\(637\) −5022.00 −0.312369
\(638\) 0 0
\(639\) 3996.00 0.247385
\(640\) 0 0
\(641\) −19158.0 −1.18049 −0.590246 0.807223i \(-0.700969\pi\)
−0.590246 + 0.807223i \(0.700969\pi\)
\(642\) 0 0
\(643\) −11228.0 −0.688630 −0.344315 0.938854i \(-0.611889\pi\)
−0.344315 + 0.938854i \(0.611889\pi\)
\(644\) 0 0
\(645\) −12480.0 −0.761860
\(646\) 0 0
\(647\) 22628.0 1.37496 0.687480 0.726204i \(-0.258717\pi\)
0.687480 + 0.726204i \(0.258717\pi\)
\(648\) 0 0
\(649\) 3828.00 0.231529
\(650\) 0 0
\(651\) 1728.00 0.104033
\(652\) 0 0
\(653\) 28338.0 1.69824 0.849121 0.528199i \(-0.177132\pi\)
0.849121 + 0.528199i \(0.177132\pi\)
\(654\) 0 0
\(655\) −20120.0 −1.20023
\(656\) 0 0
\(657\) 2754.00 0.163537
\(658\) 0 0
\(659\) 17052.0 1.00797 0.503985 0.863713i \(-0.331867\pi\)
0.503985 + 0.863713i \(0.331867\pi\)
\(660\) 0 0
\(661\) −21354.0 −1.25654 −0.628271 0.777995i \(-0.716237\pi\)
−0.628271 + 0.777995i \(0.716237\pi\)
\(662\) 0 0
\(663\) −2484.00 −0.145506
\(664\) 0 0
\(665\) 3200.00 0.186603
\(666\) 0 0
\(667\) −8184.00 −0.475091
\(668\) 0 0
\(669\) −5184.00 −0.299589
\(670\) 0 0
\(671\) −4906.00 −0.282256
\(672\) 0 0
\(673\) −22198.0 −1.27143 −0.635713 0.771925i \(-0.719294\pi\)
−0.635713 + 0.771925i \(0.719294\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 0 0
\(677\) −32974.0 −1.87193 −0.935963 0.352099i \(-0.885468\pi\)
−0.935963 + 0.352099i \(0.885468\pi\)
\(678\) 0 0
\(679\) −12176.0 −0.688177
\(680\) 0 0
\(681\) 14148.0 0.796112
\(682\) 0 0
\(683\) −22572.0 −1.26456 −0.632279 0.774740i \(-0.717880\pi\)
−0.632279 + 0.774740i \(0.717880\pi\)
\(684\) 0 0
\(685\) −12860.0 −0.717307
\(686\) 0 0
\(687\) −5298.00 −0.294223
\(688\) 0 0
\(689\) −1116.00 −0.0617071
\(690\) 0 0
\(691\) −2700.00 −0.148644 −0.0743219 0.997234i \(-0.523679\pi\)
−0.0743219 + 0.997234i \(0.523679\pi\)
\(692\) 0 0
\(693\) −792.000 −0.0434136
\(694\) 0 0
\(695\) −4640.00 −0.253245
\(696\) 0 0
\(697\) 8004.00 0.434969
\(698\) 0 0
\(699\) −13146.0 −0.711341
\(700\) 0 0
\(701\) −17382.0 −0.936532 −0.468266 0.883587i \(-0.655121\pi\)
−0.468266 + 0.883587i \(0.655121\pi\)
\(702\) 0 0
\(703\) 4560.00 0.244642
\(704\) 0 0
\(705\) −4680.00 −0.250013
\(706\) 0 0
\(707\) −8720.00 −0.463860
\(708\) 0 0
\(709\) 20454.0 1.08345 0.541725 0.840556i \(-0.317771\pi\)
0.541725 + 0.840556i \(0.317771\pi\)
\(710\) 0 0
\(711\) 5976.00 0.315215
\(712\) 0 0
\(713\) −3168.00 −0.166399
\(714\) 0 0
\(715\) 1980.00 0.103563
\(716\) 0 0
\(717\) 3792.00 0.197510
\(718\) 0 0
\(719\) −10236.0 −0.530930 −0.265465 0.964121i \(-0.585525\pi\)
−0.265465 + 0.964121i \(0.585525\pi\)
\(720\) 0 0
\(721\) −11136.0 −0.575210
\(722\) 0 0
\(723\) −9030.00 −0.464494
\(724\) 0 0
\(725\) −4650.00 −0.238202
\(726\) 0 0
\(727\) −9672.00 −0.493418 −0.246709 0.969090i \(-0.579349\pi\)
−0.246709 + 0.969090i \(0.579349\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 19136.0 0.968222
\(732\) 0 0
\(733\) −28078.0 −1.41485 −0.707425 0.706789i \(-0.750143\pi\)
−0.707425 + 0.706789i \(0.750143\pi\)
\(734\) 0 0
\(735\) 8370.00 0.420044
\(736\) 0 0
\(737\) 10516.0 0.525593
\(738\) 0 0
\(739\) 26776.0 1.33284 0.666422 0.745575i \(-0.267825\pi\)
0.666422 + 0.745575i \(0.267825\pi\)
\(740\) 0 0
\(741\) 2160.00 0.107084
\(742\) 0 0
\(743\) −25280.0 −1.24823 −0.624114 0.781333i \(-0.714540\pi\)
−0.624114 + 0.781333i \(0.714540\pi\)
\(744\) 0 0
\(745\) −28940.0 −1.42319
\(746\) 0 0
\(747\) 1116.00 0.0546617
\(748\) 0 0
\(749\) 2464.00 0.120204
\(750\) 0 0
\(751\) −25160.0 −1.22251 −0.611253 0.791436i \(-0.709334\pi\)
−0.611253 + 0.791436i \(0.709334\pi\)
\(752\) 0 0
\(753\) 16212.0 0.784592
\(754\) 0 0
\(755\) −9760.00 −0.470467
\(756\) 0 0
\(757\) 28910.0 1.38805 0.694024 0.719952i \(-0.255836\pi\)
0.694024 + 0.719952i \(0.255836\pi\)
\(758\) 0 0
\(759\) 1452.00 0.0694391
\(760\) 0 0
\(761\) 10278.0 0.489589 0.244794 0.969575i \(-0.421280\pi\)
0.244794 + 0.969575i \(0.421280\pi\)
\(762\) 0 0
\(763\) −7824.00 −0.371229
\(764\) 0 0
\(765\) 4140.00 0.195663
\(766\) 0 0
\(767\) 6264.00 0.294889
\(768\) 0 0
\(769\) −29558.0 −1.38607 −0.693036 0.720903i \(-0.743727\pi\)
−0.693036 + 0.720903i \(0.743727\pi\)
\(770\) 0 0
\(771\) −4854.00 −0.226735
\(772\) 0 0
\(773\) 9762.00 0.454223 0.227112 0.973869i \(-0.427072\pi\)
0.227112 + 0.973869i \(0.427072\pi\)
\(774\) 0 0
\(775\) −1800.00 −0.0834296
\(776\) 0 0
\(777\) −2736.00 −0.126324
\(778\) 0 0
\(779\) −6960.00 −0.320113
\(780\) 0 0
\(781\) 4884.00 0.223769
\(782\) 0 0
\(783\) −5022.00 −0.229210
\(784\) 0 0
\(785\) 16460.0 0.748385
\(786\) 0 0
\(787\) −14824.0 −0.671434 −0.335717 0.941963i \(-0.608979\pi\)
−0.335717 + 0.941963i \(0.608979\pi\)
\(788\) 0 0
\(789\) 144.000 0.00649751
\(790\) 0 0
\(791\) −9296.00 −0.417861
\(792\) 0 0
\(793\) −8028.00 −0.359499
\(794\) 0 0
\(795\) 1860.00 0.0829779
\(796\) 0 0
\(797\) 14946.0 0.664259 0.332130 0.943234i \(-0.392233\pi\)
0.332130 + 0.943234i \(0.392233\pi\)
\(798\) 0 0
\(799\) 7176.00 0.317733
\(800\) 0 0
\(801\) 5418.00 0.238996
\(802\) 0 0
\(803\) 3366.00 0.147925
\(804\) 0 0
\(805\) 3520.00 0.154116
\(806\) 0 0
\(807\) 5442.00 0.237382
\(808\) 0 0
\(809\) 14774.0 0.642060 0.321030 0.947069i \(-0.395971\pi\)
0.321030 + 0.947069i \(0.395971\pi\)
\(810\) 0 0
\(811\) −22760.0 −0.985464 −0.492732 0.870181i \(-0.664002\pi\)
−0.492732 + 0.870181i \(0.664002\pi\)
\(812\) 0 0
\(813\) 3048.00 0.131486
\(814\) 0 0
\(815\) −32680.0 −1.40458
\(816\) 0 0
\(817\) −16640.0 −0.712558
\(818\) 0 0
\(819\) −1296.00 −0.0552941
\(820\) 0 0
\(821\) 16370.0 0.695879 0.347940 0.937517i \(-0.386881\pi\)
0.347940 + 0.937517i \(0.386881\pi\)
\(822\) 0 0
\(823\) −1784.00 −0.0755605 −0.0377803 0.999286i \(-0.512029\pi\)
−0.0377803 + 0.999286i \(0.512029\pi\)
\(824\) 0 0
\(825\) 825.000 0.0348155
\(826\) 0 0
\(827\) 22052.0 0.927235 0.463617 0.886036i \(-0.346551\pi\)
0.463617 + 0.886036i \(0.346551\pi\)
\(828\) 0 0
\(829\) −29738.0 −1.24589 −0.622945 0.782265i \(-0.714064\pi\)
−0.622945 + 0.782265i \(0.714064\pi\)
\(830\) 0 0
\(831\) −20742.0 −0.865863
\(832\) 0 0
\(833\) −12834.0 −0.533820
\(834\) 0 0
\(835\) −16080.0 −0.666433
\(836\) 0 0
\(837\) −1944.00 −0.0802801
\(838\) 0 0
\(839\) −43956.0 −1.80874 −0.904368 0.426753i \(-0.859657\pi\)
−0.904368 + 0.426753i \(0.859657\pi\)
\(840\) 0 0
\(841\) 10207.0 0.418508
\(842\) 0 0
\(843\) 18294.0 0.747424
\(844\) 0 0
\(845\) −18730.0 −0.762523
\(846\) 0 0
\(847\) −968.000 −0.0392690
\(848\) 0 0
\(849\) 12576.0 0.508371
\(850\) 0 0
\(851\) 5016.00 0.202052
\(852\) 0 0
\(853\) −6438.00 −0.258421 −0.129210 0.991617i \(-0.541244\pi\)
−0.129210 + 0.991617i \(0.541244\pi\)
\(854\) 0 0
\(855\) −3600.00 −0.143997
\(856\) 0 0
\(857\) −2282.00 −0.0909587 −0.0454794 0.998965i \(-0.514482\pi\)
−0.0454794 + 0.998965i \(0.514482\pi\)
\(858\) 0 0
\(859\) 29972.0 1.19049 0.595245 0.803544i \(-0.297055\pi\)
0.595245 + 0.803544i \(0.297055\pi\)
\(860\) 0 0
\(861\) 4176.00 0.165293
\(862\) 0 0
\(863\) 3716.00 0.146575 0.0732874 0.997311i \(-0.476651\pi\)
0.0732874 + 0.997311i \(0.476651\pi\)
\(864\) 0 0
\(865\) −40700.0 −1.59982
\(866\) 0 0
\(867\) 8391.00 0.328689
\(868\) 0 0
\(869\) 7304.00 0.285122
\(870\) 0 0
\(871\) 17208.0 0.669427
\(872\) 0 0
\(873\) 13698.0 0.531050
\(874\) 0 0
\(875\) 12000.0 0.463627
\(876\) 0 0
\(877\) 25194.0 0.970058 0.485029 0.874498i \(-0.338809\pi\)
0.485029 + 0.874498i \(0.338809\pi\)
\(878\) 0 0
\(879\) −27942.0 −1.07220
\(880\) 0 0
\(881\) 27194.0 1.03994 0.519971 0.854184i \(-0.325943\pi\)
0.519971 + 0.854184i \(0.325943\pi\)
\(882\) 0 0
\(883\) 14300.0 0.544998 0.272499 0.962156i \(-0.412150\pi\)
0.272499 + 0.962156i \(0.412150\pi\)
\(884\) 0 0
\(885\) −10440.0 −0.396539
\(886\) 0 0
\(887\) −4944.00 −0.187151 −0.0935757 0.995612i \(-0.529830\pi\)
−0.0935757 + 0.995612i \(0.529830\pi\)
\(888\) 0 0
\(889\) −7872.00 −0.296984
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) 0 0
\(893\) −6240.00 −0.233834
\(894\) 0 0
\(895\) −520.000 −0.0194209
\(896\) 0 0
\(897\) 2376.00 0.0884418
\(898\) 0 0
\(899\) 13392.0 0.496828
\(900\) 0 0
\(901\) −2852.00 −0.105454
\(902\) 0 0
\(903\) 9984.00 0.367937
\(904\) 0 0
\(905\) 17980.0 0.660415
\(906\) 0 0
\(907\) 24332.0 0.890773 0.445386 0.895338i \(-0.353066\pi\)
0.445386 + 0.895338i \(0.353066\pi\)
\(908\) 0 0
\(909\) 9810.00 0.357951
\(910\) 0 0
\(911\) −20796.0 −0.756314 −0.378157 0.925741i \(-0.623442\pi\)
−0.378157 + 0.925741i \(0.623442\pi\)
\(912\) 0 0
\(913\) 1364.00 0.0494434
\(914\) 0 0
\(915\) 13380.0 0.483420
\(916\) 0 0
\(917\) 16096.0 0.579647
\(918\) 0 0
\(919\) −15392.0 −0.552487 −0.276243 0.961088i \(-0.589090\pi\)
−0.276243 + 0.961088i \(0.589090\pi\)
\(920\) 0 0
\(921\) 4152.00 0.148548
\(922\) 0 0
\(923\) 7992.00 0.285005
\(924\) 0 0
\(925\) 2850.00 0.101305
\(926\) 0 0
\(927\) 12528.0 0.443876
\(928\) 0 0
\(929\) −15150.0 −0.535043 −0.267522 0.963552i \(-0.586205\pi\)
−0.267522 + 0.963552i \(0.586205\pi\)
\(930\) 0 0
\(931\) 11160.0 0.392862
\(932\) 0 0
\(933\) 23748.0 0.833306
\(934\) 0 0
\(935\) 5060.00 0.176984
\(936\) 0 0
\(937\) 15610.0 0.544244 0.272122 0.962263i \(-0.412275\pi\)
0.272122 + 0.962263i \(0.412275\pi\)
\(938\) 0 0
\(939\) −654.000 −0.0227289
\(940\) 0 0
\(941\) −45222.0 −1.56663 −0.783313 0.621627i \(-0.786472\pi\)
−0.783313 + 0.621627i \(0.786472\pi\)
\(942\) 0 0
\(943\) −7656.00 −0.264384
\(944\) 0 0
\(945\) 2160.00 0.0743543
\(946\) 0 0
\(947\) 14820.0 0.508538 0.254269 0.967134i \(-0.418165\pi\)
0.254269 + 0.967134i \(0.418165\pi\)
\(948\) 0 0
\(949\) 5508.00 0.188406
\(950\) 0 0
\(951\) −1998.00 −0.0681279
\(952\) 0 0
\(953\) 5334.00 0.181307 0.0906533 0.995883i \(-0.471104\pi\)
0.0906533 + 0.995883i \(0.471104\pi\)
\(954\) 0 0
\(955\) 38520.0 1.30521
\(956\) 0 0
\(957\) −6138.00 −0.207328
\(958\) 0 0
\(959\) 10288.0 0.346420
\(960\) 0 0
\(961\) −24607.0 −0.825988
\(962\) 0 0
\(963\) −2772.00 −0.0927585
\(964\) 0 0
\(965\) −19580.0 −0.653163
\(966\) 0 0
\(967\) −18400.0 −0.611897 −0.305948 0.952048i \(-0.598973\pi\)
−0.305948 + 0.952048i \(0.598973\pi\)
\(968\) 0 0
\(969\) 5520.00 0.183001
\(970\) 0 0
\(971\) 14460.0 0.477903 0.238951 0.971032i \(-0.423196\pi\)
0.238951 + 0.971032i \(0.423196\pi\)
\(972\) 0 0
\(973\) 3712.00 0.122303
\(974\) 0 0
\(975\) 1350.00 0.0443432
\(976\) 0 0
\(977\) −9998.00 −0.327394 −0.163697 0.986511i \(-0.552342\pi\)
−0.163697 + 0.986511i \(0.552342\pi\)
\(978\) 0 0
\(979\) 6622.00 0.216180
\(980\) 0 0
\(981\) 8802.00 0.286469
\(982\) 0 0
\(983\) 52548.0 1.70501 0.852503 0.522722i \(-0.175084\pi\)
0.852503 + 0.522722i \(0.175084\pi\)
\(984\) 0 0
\(985\) −16300.0 −0.527270
\(986\) 0 0
\(987\) 3744.00 0.120742
\(988\) 0 0
\(989\) −18304.0 −0.588507
\(990\) 0 0
\(991\) 7096.00 0.227459 0.113729 0.993512i \(-0.463720\pi\)
0.113729 + 0.993512i \(0.463720\pi\)
\(992\) 0 0
\(993\) −7500.00 −0.239683
\(994\) 0 0
\(995\) 45040.0 1.43504
\(996\) 0 0
\(997\) 45202.0 1.43587 0.717935 0.696111i \(-0.245088\pi\)
0.717935 + 0.696111i \(0.245088\pi\)
\(998\) 0 0
\(999\) 3078.00 0.0974811
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 528.4.a.e.1.1 1
3.2 odd 2 1584.4.a.f.1.1 1
4.3 odd 2 132.4.a.d.1.1 1
8.3 odd 2 2112.4.a.e.1.1 1
8.5 even 2 2112.4.a.q.1.1 1
12.11 even 2 396.4.a.c.1.1 1
44.43 even 2 1452.4.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
132.4.a.d.1.1 1 4.3 odd 2
396.4.a.c.1.1 1 12.11 even 2
528.4.a.e.1.1 1 1.1 even 1 trivial
1452.4.a.i.1.1 1 44.43 even 2
1584.4.a.f.1.1 1 3.2 odd 2
2112.4.a.e.1.1 1 8.3 odd 2
2112.4.a.q.1.1 1 8.5 even 2