Properties

Label 2205.4.a.k.1.1
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,4,Mod(1,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.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: \(130.099211563\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -7.00000 q^{4} -5.00000 q^{5} +15.0000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -7.00000 q^{4} -5.00000 q^{5} +15.0000 q^{8} +5.00000 q^{10} +44.0000 q^{11} -6.00000 q^{13} +41.0000 q^{16} -24.0000 q^{17} +114.000 q^{19} +35.0000 q^{20} -44.0000 q^{22} +52.0000 q^{23} +25.0000 q^{25} +6.00000 q^{26} -146.000 q^{29} +276.000 q^{31} -161.000 q^{32} +24.0000 q^{34} -210.000 q^{37} -114.000 q^{38} -75.0000 q^{40} +444.000 q^{41} +492.000 q^{43} -308.000 q^{44} -52.0000 q^{46} -612.000 q^{47} -25.0000 q^{50} +42.0000 q^{52} -50.0000 q^{53} -220.000 q^{55} +146.000 q^{58} +294.000 q^{59} -450.000 q^{61} -276.000 q^{62} -167.000 q^{64} +30.0000 q^{65} -668.000 q^{67} +168.000 q^{68} +308.000 q^{71} -12.0000 q^{73} +210.000 q^{74} -798.000 q^{76} +596.000 q^{79} -205.000 q^{80} -444.000 q^{82} -966.000 q^{83} +120.000 q^{85} -492.000 q^{86} +660.000 q^{88} -408.000 q^{89} -364.000 q^{92} +612.000 q^{94} -570.000 q^{95} +1200.00 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 0 0
\(4\) −7.00000 −0.875000
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 15.0000 0.662913
\(9\) 0 0
\(10\) 5.00000 0.158114
\(11\) 44.0000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −6.00000 −0.128008 −0.0640039 0.997950i \(-0.520387\pi\)
−0.0640039 + 0.997950i \(0.520387\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) −24.0000 −0.342403 −0.171202 0.985236i \(-0.554765\pi\)
−0.171202 + 0.985236i \(0.554765\pi\)
\(18\) 0 0
\(19\) 114.000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 35.0000 0.391312
\(21\) 0 0
\(22\) −44.0000 −0.426401
\(23\) 52.0000 0.471424 0.235712 0.971823i \(-0.424258\pi\)
0.235712 + 0.971823i \(0.424258\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 6.00000 0.0452576
\(27\) 0 0
\(28\) 0 0
\(29\) −146.000 −0.934880 −0.467440 0.884025i \(-0.654824\pi\)
−0.467440 + 0.884025i \(0.654824\pi\)
\(30\) 0 0
\(31\) 276.000 1.59907 0.799533 0.600622i \(-0.205080\pi\)
0.799533 + 0.600622i \(0.205080\pi\)
\(32\) −161.000 −0.889408
\(33\) 0 0
\(34\) 24.0000 0.121058
\(35\) 0 0
\(36\) 0 0
\(37\) −210.000 −0.933075 −0.466538 0.884501i \(-0.654499\pi\)
−0.466538 + 0.884501i \(0.654499\pi\)
\(38\) −114.000 −0.486664
\(39\) 0 0
\(40\) −75.0000 −0.296464
\(41\) 444.000 1.69125 0.845624 0.533779i \(-0.179229\pi\)
0.845624 + 0.533779i \(0.179229\pi\)
\(42\) 0 0
\(43\) 492.000 1.74487 0.872434 0.488733i \(-0.162541\pi\)
0.872434 + 0.488733i \(0.162541\pi\)
\(44\) −308.000 −1.05529
\(45\) 0 0
\(46\) −52.0000 −0.166674
\(47\) −612.000 −1.89935 −0.949674 0.313239i \(-0.898586\pi\)
−0.949674 + 0.313239i \(0.898586\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −25.0000 −0.0707107
\(51\) 0 0
\(52\) 42.0000 0.112007
\(53\) −50.0000 −0.129585 −0.0647927 0.997899i \(-0.520639\pi\)
−0.0647927 + 0.997899i \(0.520639\pi\)
\(54\) 0 0
\(55\) −220.000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 146.000 0.330530
\(59\) 294.000 0.648738 0.324369 0.945931i \(-0.394848\pi\)
0.324369 + 0.945931i \(0.394848\pi\)
\(60\) 0 0
\(61\) −450.000 −0.944534 −0.472267 0.881455i \(-0.656564\pi\)
−0.472267 + 0.881455i \(0.656564\pi\)
\(62\) −276.000 −0.565355
\(63\) 0 0
\(64\) −167.000 −0.326172
\(65\) 30.0000 0.0572468
\(66\) 0 0
\(67\) −668.000 −1.21805 −0.609024 0.793152i \(-0.708439\pi\)
−0.609024 + 0.793152i \(0.708439\pi\)
\(68\) 168.000 0.299603
\(69\) 0 0
\(70\) 0 0
\(71\) 308.000 0.514829 0.257415 0.966301i \(-0.417129\pi\)
0.257415 + 0.966301i \(0.417129\pi\)
\(72\) 0 0
\(73\) −12.0000 −0.0192396 −0.00961982 0.999954i \(-0.503062\pi\)
−0.00961982 + 0.999954i \(0.503062\pi\)
\(74\) 210.000 0.329892
\(75\) 0 0
\(76\) −798.000 −1.20443
\(77\) 0 0
\(78\) 0 0
\(79\) 596.000 0.848800 0.424400 0.905475i \(-0.360485\pi\)
0.424400 + 0.905475i \(0.360485\pi\)
\(80\) −205.000 −0.286496
\(81\) 0 0
\(82\) −444.000 −0.597946
\(83\) −966.000 −1.27750 −0.638749 0.769415i \(-0.720548\pi\)
−0.638749 + 0.769415i \(0.720548\pi\)
\(84\) 0 0
\(85\) 120.000 0.153127
\(86\) −492.000 −0.616904
\(87\) 0 0
\(88\) 660.000 0.799503
\(89\) −408.000 −0.485932 −0.242966 0.970035i \(-0.578120\pi\)
−0.242966 + 0.970035i \(0.578120\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −364.000 −0.412496
\(93\) 0 0
\(94\) 612.000 0.671521
\(95\) −570.000 −0.615587
\(96\) 0 0
\(97\) 1200.00 1.25610 0.628049 0.778174i \(-0.283854\pi\)
0.628049 + 0.778174i \(0.283854\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −175.000 −0.175000
\(101\) 1098.00 1.08173 0.540867 0.841108i \(-0.318096\pi\)
0.540867 + 0.841108i \(0.318096\pi\)
\(102\) 0 0
\(103\) −972.000 −0.929845 −0.464922 0.885351i \(-0.653918\pi\)
−0.464922 + 0.885351i \(0.653918\pi\)
\(104\) −90.0000 −0.0848579
\(105\) 0 0
\(106\) 50.0000 0.0458154
\(107\) −1516.00 −1.36969 −0.684847 0.728687i \(-0.740131\pi\)
−0.684847 + 0.728687i \(0.740131\pi\)
\(108\) 0 0
\(109\) 930.000 0.817228 0.408614 0.912707i \(-0.366012\pi\)
0.408614 + 0.912707i \(0.366012\pi\)
\(110\) 220.000 0.190693
\(111\) 0 0
\(112\) 0 0
\(113\) −1694.00 −1.41025 −0.705124 0.709084i \(-0.749109\pi\)
−0.705124 + 0.709084i \(0.749109\pi\)
\(114\) 0 0
\(115\) −260.000 −0.210827
\(116\) 1022.00 0.818020
\(117\) 0 0
\(118\) −294.000 −0.229364
\(119\) 0 0
\(120\) 0 0
\(121\) 605.000 0.454545
\(122\) 450.000 0.333943
\(123\) 0 0
\(124\) −1932.00 −1.39918
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 916.000 0.640015 0.320007 0.947415i \(-0.396315\pi\)
0.320007 + 0.947415i \(0.396315\pi\)
\(128\) 1455.00 1.00473
\(129\) 0 0
\(130\) −30.0000 −0.0202398
\(131\) −1002.00 −0.668284 −0.334142 0.942523i \(-0.608446\pi\)
−0.334142 + 0.942523i \(0.608446\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 668.000 0.430645
\(135\) 0 0
\(136\) −360.000 −0.226983
\(137\) 274.000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 270.000 0.164756 0.0823781 0.996601i \(-0.473749\pi\)
0.0823781 + 0.996601i \(0.473749\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −308.000 −0.182020
\(143\) −264.000 −0.154383
\(144\) 0 0
\(145\) 730.000 0.418091
\(146\) 12.0000 0.00680224
\(147\) 0 0
\(148\) 1470.00 0.816441
\(149\) 530.000 0.291405 0.145702 0.989328i \(-0.453456\pi\)
0.145702 + 0.989328i \(0.453456\pi\)
\(150\) 0 0
\(151\) −3120.00 −1.68147 −0.840735 0.541447i \(-0.817877\pi\)
−0.840735 + 0.541447i \(0.817877\pi\)
\(152\) 1710.00 0.912495
\(153\) 0 0
\(154\) 0 0
\(155\) −1380.00 −0.715124
\(156\) 0 0
\(157\) 2106.00 1.07055 0.535277 0.844676i \(-0.320207\pi\)
0.535277 + 0.844676i \(0.320207\pi\)
\(158\) −596.000 −0.300096
\(159\) 0 0
\(160\) 805.000 0.397755
\(161\) 0 0
\(162\) 0 0
\(163\) 628.000 0.301772 0.150886 0.988551i \(-0.451787\pi\)
0.150886 + 0.988551i \(0.451787\pi\)
\(164\) −3108.00 −1.47984
\(165\) 0 0
\(166\) 966.000 0.451663
\(167\) 1284.00 0.594963 0.297482 0.954728i \(-0.403853\pi\)
0.297482 + 0.954728i \(0.403853\pi\)
\(168\) 0 0
\(169\) −2161.00 −0.983614
\(170\) −120.000 −0.0541387
\(171\) 0 0
\(172\) −3444.00 −1.52676
\(173\) −906.000 −0.398161 −0.199081 0.979983i \(-0.563796\pi\)
−0.199081 + 0.979983i \(0.563796\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1804.00 0.772623
\(177\) 0 0
\(178\) 408.000 0.171803
\(179\) 2084.00 0.870198 0.435099 0.900383i \(-0.356713\pi\)
0.435099 + 0.900383i \(0.356713\pi\)
\(180\) 0 0
\(181\) 4674.00 1.91942 0.959712 0.280986i \(-0.0906615\pi\)
0.959712 + 0.280986i \(0.0906615\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 780.000 0.312513
\(185\) 1050.00 0.417284
\(186\) 0 0
\(187\) −1056.00 −0.412954
\(188\) 4284.00 1.66193
\(189\) 0 0
\(190\) 570.000 0.217643
\(191\) −2012.00 −0.762216 −0.381108 0.924531i \(-0.624457\pi\)
−0.381108 + 0.924531i \(0.624457\pi\)
\(192\) 0 0
\(193\) 4206.00 1.56868 0.784338 0.620334i \(-0.213003\pi\)
0.784338 + 0.620334i \(0.213003\pi\)
\(194\) −1200.00 −0.444098
\(195\) 0 0
\(196\) 0 0
\(197\) 1574.00 0.569253 0.284627 0.958638i \(-0.408130\pi\)
0.284627 + 0.958638i \(0.408130\pi\)
\(198\) 0 0
\(199\) 2724.00 0.970348 0.485174 0.874418i \(-0.338756\pi\)
0.485174 + 0.874418i \(0.338756\pi\)
\(200\) 375.000 0.132583
\(201\) 0 0
\(202\) −1098.00 −0.382451
\(203\) 0 0
\(204\) 0 0
\(205\) −2220.00 −0.756349
\(206\) 972.000 0.328750
\(207\) 0 0
\(208\) −246.000 −0.0820050
\(209\) 5016.00 1.66011
\(210\) 0 0
\(211\) −180.000 −0.0587285 −0.0293642 0.999569i \(-0.509348\pi\)
−0.0293642 + 0.999569i \(0.509348\pi\)
\(212\) 350.000 0.113387
\(213\) 0 0
\(214\) 1516.00 0.484260
\(215\) −2460.00 −0.780328
\(216\) 0 0
\(217\) 0 0
\(218\) −930.000 −0.288934
\(219\) 0 0
\(220\) 1540.00 0.471940
\(221\) 144.000 0.0438303
\(222\) 0 0
\(223\) 4584.00 1.37654 0.688268 0.725457i \(-0.258372\pi\)
0.688268 + 0.725457i \(0.258372\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1694.00 0.498598
\(227\) −1686.00 −0.492968 −0.246484 0.969147i \(-0.579275\pi\)
−0.246484 + 0.969147i \(0.579275\pi\)
\(228\) 0 0
\(229\) 4026.00 1.16177 0.580886 0.813985i \(-0.302706\pi\)
0.580886 + 0.813985i \(0.302706\pi\)
\(230\) 260.000 0.0745387
\(231\) 0 0
\(232\) −2190.00 −0.619744
\(233\) 6074.00 1.70782 0.853908 0.520425i \(-0.174226\pi\)
0.853908 + 0.520425i \(0.174226\pi\)
\(234\) 0 0
\(235\) 3060.00 0.849414
\(236\) −2058.00 −0.567646
\(237\) 0 0
\(238\) 0 0
\(239\) −3928.00 −1.06310 −0.531551 0.847027i \(-0.678390\pi\)
−0.531551 + 0.847027i \(0.678390\pi\)
\(240\) 0 0
\(241\) 1236.00 0.330364 0.165182 0.986263i \(-0.447179\pi\)
0.165182 + 0.986263i \(0.447179\pi\)
\(242\) −605.000 −0.160706
\(243\) 0 0
\(244\) 3150.00 0.826468
\(245\) 0 0
\(246\) 0 0
\(247\) −684.000 −0.176202
\(248\) 4140.00 1.06004
\(249\) 0 0
\(250\) 125.000 0.0316228
\(251\) −78.0000 −0.0196148 −0.00980740 0.999952i \(-0.503122\pi\)
−0.00980740 + 0.999952i \(0.503122\pi\)
\(252\) 0 0
\(253\) 2288.00 0.568559
\(254\) −916.000 −0.226279
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) −3276.00 −0.795141 −0.397571 0.917572i \(-0.630147\pi\)
−0.397571 + 0.917572i \(0.630147\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −210.000 −0.0500910
\(261\) 0 0
\(262\) 1002.00 0.236274
\(263\) −2240.00 −0.525188 −0.262594 0.964906i \(-0.584578\pi\)
−0.262594 + 0.964906i \(0.584578\pi\)
\(264\) 0 0
\(265\) 250.000 0.0579524
\(266\) 0 0
\(267\) 0 0
\(268\) 4676.00 1.06579
\(269\) 4494.00 1.01860 0.509301 0.860588i \(-0.329904\pi\)
0.509301 + 0.860588i \(0.329904\pi\)
\(270\) 0 0
\(271\) −3216.00 −0.720879 −0.360439 0.932783i \(-0.617373\pi\)
−0.360439 + 0.932783i \(0.617373\pi\)
\(272\) −984.000 −0.219352
\(273\) 0 0
\(274\) −274.000 −0.0604122
\(275\) 1100.00 0.241209
\(276\) 0 0
\(277\) 1514.00 0.328402 0.164201 0.986427i \(-0.447495\pi\)
0.164201 + 0.986427i \(0.447495\pi\)
\(278\) −270.000 −0.0582501
\(279\) 0 0
\(280\) 0 0
\(281\) 5690.00 1.20796 0.603980 0.796999i \(-0.293581\pi\)
0.603980 + 0.796999i \(0.293581\pi\)
\(282\) 0 0
\(283\) 7518.00 1.57915 0.789574 0.613656i \(-0.210302\pi\)
0.789574 + 0.613656i \(0.210302\pi\)
\(284\) −2156.00 −0.450476
\(285\) 0 0
\(286\) 264.000 0.0545827
\(287\) 0 0
\(288\) 0 0
\(289\) −4337.00 −0.882760
\(290\) −730.000 −0.147818
\(291\) 0 0
\(292\) 84.0000 0.0168347
\(293\) −702.000 −0.139970 −0.0699851 0.997548i \(-0.522295\pi\)
−0.0699851 + 0.997548i \(0.522295\pi\)
\(294\) 0 0
\(295\) −1470.00 −0.290124
\(296\) −3150.00 −0.618547
\(297\) 0 0
\(298\) −530.000 −0.103027
\(299\) −312.000 −0.0603459
\(300\) 0 0
\(301\) 0 0
\(302\) 3120.00 0.594489
\(303\) 0 0
\(304\) 4674.00 0.881817
\(305\) 2250.00 0.422409
\(306\) 0 0
\(307\) 10374.0 1.92858 0.964292 0.264840i \(-0.0853193\pi\)
0.964292 + 0.264840i \(0.0853193\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1380.00 0.252835
\(311\) −2784.00 −0.507608 −0.253804 0.967256i \(-0.581682\pi\)
−0.253804 + 0.967256i \(0.581682\pi\)
\(312\) 0 0
\(313\) −6216.00 −1.12252 −0.561261 0.827639i \(-0.689683\pi\)
−0.561261 + 0.827639i \(0.689683\pi\)
\(314\) −2106.00 −0.378498
\(315\) 0 0
\(316\) −4172.00 −0.742700
\(317\) −5066.00 −0.897586 −0.448793 0.893636i \(-0.648146\pi\)
−0.448793 + 0.893636i \(0.648146\pi\)
\(318\) 0 0
\(319\) −6424.00 −1.12751
\(320\) 835.000 0.145868
\(321\) 0 0
\(322\) 0 0
\(323\) −2736.00 −0.471316
\(324\) 0 0
\(325\) −150.000 −0.0256015
\(326\) −628.000 −0.106692
\(327\) 0 0
\(328\) 6660.00 1.12115
\(329\) 0 0
\(330\) 0 0
\(331\) −6468.00 −1.07406 −0.537029 0.843564i \(-0.680454\pi\)
−0.537029 + 0.843564i \(0.680454\pi\)
\(332\) 6762.00 1.11781
\(333\) 0 0
\(334\) −1284.00 −0.210351
\(335\) 3340.00 0.544727
\(336\) 0 0
\(337\) −3438.00 −0.555726 −0.277863 0.960621i \(-0.589626\pi\)
−0.277863 + 0.960621i \(0.589626\pi\)
\(338\) 2161.00 0.347760
\(339\) 0 0
\(340\) −840.000 −0.133986
\(341\) 12144.0 1.92855
\(342\) 0 0
\(343\) 0 0
\(344\) 7380.00 1.15669
\(345\) 0 0
\(346\) 906.000 0.140771
\(347\) −2212.00 −0.342209 −0.171104 0.985253i \(-0.554734\pi\)
−0.171104 + 0.985253i \(0.554734\pi\)
\(348\) 0 0
\(349\) −2910.00 −0.446329 −0.223164 0.974781i \(-0.571639\pi\)
−0.223164 + 0.974781i \(0.571639\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −7084.00 −1.07267
\(353\) 8364.00 1.26111 0.630554 0.776146i \(-0.282828\pi\)
0.630554 + 0.776146i \(0.282828\pi\)
\(354\) 0 0
\(355\) −1540.00 −0.230239
\(356\) 2856.00 0.425190
\(357\) 0 0
\(358\) −2084.00 −0.307662
\(359\) 6712.00 0.986757 0.493379 0.869815i \(-0.335762\pi\)
0.493379 + 0.869815i \(0.335762\pi\)
\(360\) 0 0
\(361\) 6137.00 0.894737
\(362\) −4674.00 −0.678619
\(363\) 0 0
\(364\) 0 0
\(365\) 60.0000 0.00860423
\(366\) 0 0
\(367\) −456.000 −0.0648583 −0.0324292 0.999474i \(-0.510324\pi\)
−0.0324292 + 0.999474i \(0.510324\pi\)
\(368\) 2132.00 0.302006
\(369\) 0 0
\(370\) −1050.00 −0.147532
\(371\) 0 0
\(372\) 0 0
\(373\) −2558.00 −0.355089 −0.177545 0.984113i \(-0.556815\pi\)
−0.177545 + 0.984113i \(0.556815\pi\)
\(374\) 1056.00 0.146001
\(375\) 0 0
\(376\) −9180.00 −1.25910
\(377\) 876.000 0.119672
\(378\) 0 0
\(379\) 2004.00 0.271606 0.135803 0.990736i \(-0.456639\pi\)
0.135803 + 0.990736i \(0.456639\pi\)
\(380\) 3990.00 0.538639
\(381\) 0 0
\(382\) 2012.00 0.269484
\(383\) 11340.0 1.51292 0.756458 0.654042i \(-0.226928\pi\)
0.756458 + 0.654042i \(0.226928\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4206.00 −0.554611
\(387\) 0 0
\(388\) −8400.00 −1.09909
\(389\) −10522.0 −1.37143 −0.685715 0.727870i \(-0.740511\pi\)
−0.685715 + 0.727870i \(0.740511\pi\)
\(390\) 0 0
\(391\) −1248.00 −0.161417
\(392\) 0 0
\(393\) 0 0
\(394\) −1574.00 −0.201261
\(395\) −2980.00 −0.379595
\(396\) 0 0
\(397\) 2898.00 0.366364 0.183182 0.983079i \(-0.441360\pi\)
0.183182 + 0.983079i \(0.441360\pi\)
\(398\) −2724.00 −0.343070
\(399\) 0 0
\(400\) 1025.00 0.128125
\(401\) −3026.00 −0.376836 −0.188418 0.982089i \(-0.560336\pi\)
−0.188418 + 0.982089i \(0.560336\pi\)
\(402\) 0 0
\(403\) −1656.00 −0.204693
\(404\) −7686.00 −0.946517
\(405\) 0 0
\(406\) 0 0
\(407\) −9240.00 −1.12533
\(408\) 0 0
\(409\) 8940.00 1.08082 0.540409 0.841402i \(-0.318270\pi\)
0.540409 + 0.841402i \(0.318270\pi\)
\(410\) 2220.00 0.267410
\(411\) 0 0
\(412\) 6804.00 0.813614
\(413\) 0 0
\(414\) 0 0
\(415\) 4830.00 0.571314
\(416\) 966.000 0.113851
\(417\) 0 0
\(418\) −5016.00 −0.586939
\(419\) 2994.00 0.349085 0.174542 0.984650i \(-0.444155\pi\)
0.174542 + 0.984650i \(0.444155\pi\)
\(420\) 0 0
\(421\) 15766.0 1.82515 0.912575 0.408910i \(-0.134091\pi\)
0.912575 + 0.408910i \(0.134091\pi\)
\(422\) 180.000 0.0207637
\(423\) 0 0
\(424\) −750.000 −0.0859038
\(425\) −600.000 −0.0684806
\(426\) 0 0
\(427\) 0 0
\(428\) 10612.0 1.19848
\(429\) 0 0
\(430\) 2460.00 0.275888
\(431\) −14720.0 −1.64510 −0.822549 0.568694i \(-0.807449\pi\)
−0.822549 + 0.568694i \(0.807449\pi\)
\(432\) 0 0
\(433\) −4440.00 −0.492778 −0.246389 0.969171i \(-0.579244\pi\)
−0.246389 + 0.969171i \(0.579244\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6510.00 −0.715074
\(437\) 5928.00 0.648912
\(438\) 0 0
\(439\) 1488.00 0.161773 0.0808865 0.996723i \(-0.474225\pi\)
0.0808865 + 0.996723i \(0.474225\pi\)
\(440\) −3300.00 −0.357548
\(441\) 0 0
\(442\) −144.000 −0.0154963
\(443\) 5828.00 0.625049 0.312524 0.949910i \(-0.398825\pi\)
0.312524 + 0.949910i \(0.398825\pi\)
\(444\) 0 0
\(445\) 2040.00 0.217315
\(446\) −4584.00 −0.486679
\(447\) 0 0
\(448\) 0 0
\(449\) 6958.00 0.731333 0.365666 0.930746i \(-0.380841\pi\)
0.365666 + 0.930746i \(0.380841\pi\)
\(450\) 0 0
\(451\) 19536.0 2.03972
\(452\) 11858.0 1.23397
\(453\) 0 0
\(454\) 1686.00 0.174291
\(455\) 0 0
\(456\) 0 0
\(457\) 18102.0 1.85290 0.926451 0.376416i \(-0.122844\pi\)
0.926451 + 0.376416i \(0.122844\pi\)
\(458\) −4026.00 −0.410748
\(459\) 0 0
\(460\) 1820.00 0.184474
\(461\) 2574.00 0.260050 0.130025 0.991511i \(-0.458494\pi\)
0.130025 + 0.991511i \(0.458494\pi\)
\(462\) 0 0
\(463\) −12832.0 −1.28802 −0.644010 0.765017i \(-0.722731\pi\)
−0.644010 + 0.765017i \(0.722731\pi\)
\(464\) −5986.00 −0.598907
\(465\) 0 0
\(466\) −6074.00 −0.603804
\(467\) 7170.00 0.710467 0.355233 0.934778i \(-0.384401\pi\)
0.355233 + 0.934778i \(0.384401\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −3060.00 −0.300313
\(471\) 0 0
\(472\) 4410.00 0.430057
\(473\) 21648.0 2.10439
\(474\) 0 0
\(475\) 2850.00 0.275299
\(476\) 0 0
\(477\) 0 0
\(478\) 3928.00 0.375863
\(479\) −10164.0 −0.969530 −0.484765 0.874644i \(-0.661095\pi\)
−0.484765 + 0.874644i \(0.661095\pi\)
\(480\) 0 0
\(481\) 1260.00 0.119441
\(482\) −1236.00 −0.116801
\(483\) 0 0
\(484\) −4235.00 −0.397727
\(485\) −6000.00 −0.561744
\(486\) 0 0
\(487\) 10212.0 0.950205 0.475103 0.879930i \(-0.342411\pi\)
0.475103 + 0.879930i \(0.342411\pi\)
\(488\) −6750.00 −0.626144
\(489\) 0 0
\(490\) 0 0
\(491\) 7972.00 0.732732 0.366366 0.930471i \(-0.380602\pi\)
0.366366 + 0.930471i \(0.380602\pi\)
\(492\) 0 0
\(493\) 3504.00 0.320106
\(494\) 684.000 0.0622968
\(495\) 0 0
\(496\) 11316.0 1.02440
\(497\) 0 0
\(498\) 0 0
\(499\) −1548.00 −0.138874 −0.0694369 0.997586i \(-0.522120\pi\)
−0.0694369 + 0.997586i \(0.522120\pi\)
\(500\) 875.000 0.0782624
\(501\) 0 0
\(502\) 78.0000 0.00693488
\(503\) 1368.00 0.121265 0.0606323 0.998160i \(-0.480688\pi\)
0.0606323 + 0.998160i \(0.480688\pi\)
\(504\) 0 0
\(505\) −5490.00 −0.483766
\(506\) −2288.00 −0.201016
\(507\) 0 0
\(508\) −6412.00 −0.560013
\(509\) −8274.00 −0.720508 −0.360254 0.932854i \(-0.617310\pi\)
−0.360254 + 0.932854i \(0.617310\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11521.0 −0.994455
\(513\) 0 0
\(514\) 3276.00 0.281125
\(515\) 4860.00 0.415839
\(516\) 0 0
\(517\) −26928.0 −2.29070
\(518\) 0 0
\(519\) 0 0
\(520\) 450.000 0.0379496
\(521\) 20268.0 1.70433 0.852166 0.523271i \(-0.175289\pi\)
0.852166 + 0.523271i \(0.175289\pi\)
\(522\) 0 0
\(523\) 1302.00 0.108858 0.0544288 0.998518i \(-0.482666\pi\)
0.0544288 + 0.998518i \(0.482666\pi\)
\(524\) 7014.00 0.584748
\(525\) 0 0
\(526\) 2240.00 0.185682
\(527\) −6624.00 −0.547526
\(528\) 0 0
\(529\) −9463.00 −0.777760
\(530\) −250.000 −0.0204893
\(531\) 0 0
\(532\) 0 0
\(533\) −2664.00 −0.216493
\(534\) 0 0
\(535\) 7580.00 0.612546
\(536\) −10020.0 −0.807459
\(537\) 0 0
\(538\) −4494.00 −0.360130
\(539\) 0 0
\(540\) 0 0
\(541\) −8314.00 −0.660715 −0.330357 0.943856i \(-0.607169\pi\)
−0.330357 + 0.943856i \(0.607169\pi\)
\(542\) 3216.00 0.254869
\(543\) 0 0
\(544\) 3864.00 0.304536
\(545\) −4650.00 −0.365475
\(546\) 0 0
\(547\) −9484.00 −0.741328 −0.370664 0.928767i \(-0.620870\pi\)
−0.370664 + 0.928767i \(0.620870\pi\)
\(548\) −1918.00 −0.149513
\(549\) 0 0
\(550\) −1100.00 −0.0852803
\(551\) −16644.0 −1.28686
\(552\) 0 0
\(553\) 0 0
\(554\) −1514.00 −0.116108
\(555\) 0 0
\(556\) −1890.00 −0.144162
\(557\) −23218.0 −1.76621 −0.883104 0.469177i \(-0.844551\pi\)
−0.883104 + 0.469177i \(0.844551\pi\)
\(558\) 0 0
\(559\) −2952.00 −0.223357
\(560\) 0 0
\(561\) 0 0
\(562\) −5690.00 −0.427079
\(563\) 5334.00 0.399292 0.199646 0.979868i \(-0.436021\pi\)
0.199646 + 0.979868i \(0.436021\pi\)
\(564\) 0 0
\(565\) 8470.00 0.630682
\(566\) −7518.00 −0.558313
\(567\) 0 0
\(568\) 4620.00 0.341287
\(569\) 182.000 0.0134092 0.00670460 0.999978i \(-0.497866\pi\)
0.00670460 + 0.999978i \(0.497866\pi\)
\(570\) 0 0
\(571\) 14164.0 1.03808 0.519041 0.854749i \(-0.326289\pi\)
0.519041 + 0.854749i \(0.326289\pi\)
\(572\) 1848.00 0.135085
\(573\) 0 0
\(574\) 0 0
\(575\) 1300.00 0.0942848
\(576\) 0 0
\(577\) 13740.0 0.991341 0.495670 0.868511i \(-0.334922\pi\)
0.495670 + 0.868511i \(0.334922\pi\)
\(578\) 4337.00 0.312103
\(579\) 0 0
\(580\) −5110.00 −0.365830
\(581\) 0 0
\(582\) 0 0
\(583\) −2200.00 −0.156286
\(584\) −180.000 −0.0127542
\(585\) 0 0
\(586\) 702.000 0.0494870
\(587\) 9174.00 0.645062 0.322531 0.946559i \(-0.395466\pi\)
0.322531 + 0.946559i \(0.395466\pi\)
\(588\) 0 0
\(589\) 31464.0 2.20111
\(590\) 1470.00 0.102574
\(591\) 0 0
\(592\) −8610.00 −0.597751
\(593\) 14580.0 1.00966 0.504830 0.863219i \(-0.331555\pi\)
0.504830 + 0.863219i \(0.331555\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3710.00 −0.254979
\(597\) 0 0
\(598\) 312.000 0.0213355
\(599\) −1988.00 −0.135605 −0.0678026 0.997699i \(-0.521599\pi\)
−0.0678026 + 0.997699i \(0.521599\pi\)
\(600\) 0 0
\(601\) 7800.00 0.529399 0.264699 0.964331i \(-0.414727\pi\)
0.264699 + 0.964331i \(0.414727\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 21840.0 1.47129
\(605\) −3025.00 −0.203279
\(606\) 0 0
\(607\) −24288.0 −1.62408 −0.812042 0.583598i \(-0.801644\pi\)
−0.812042 + 0.583598i \(0.801644\pi\)
\(608\) −18354.0 −1.22426
\(609\) 0 0
\(610\) −2250.00 −0.149344
\(611\) 3672.00 0.243131
\(612\) 0 0
\(613\) −9866.00 −0.650055 −0.325028 0.945704i \(-0.605374\pi\)
−0.325028 + 0.945704i \(0.605374\pi\)
\(614\) −10374.0 −0.681858
\(615\) 0 0
\(616\) 0 0
\(617\) −22858.0 −1.49146 −0.745728 0.666250i \(-0.767898\pi\)
−0.745728 + 0.666250i \(0.767898\pi\)
\(618\) 0 0
\(619\) 19074.0 1.23853 0.619264 0.785183i \(-0.287431\pi\)
0.619264 + 0.785183i \(0.287431\pi\)
\(620\) 9660.00 0.625734
\(621\) 0 0
\(622\) 2784.00 0.179467
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 6216.00 0.396871
\(627\) 0 0
\(628\) −14742.0 −0.936735
\(629\) 5040.00 0.319488
\(630\) 0 0
\(631\) 22084.0 1.39326 0.696632 0.717428i \(-0.254681\pi\)
0.696632 + 0.717428i \(0.254681\pi\)
\(632\) 8940.00 0.562681
\(633\) 0 0
\(634\) 5066.00 0.317345
\(635\) −4580.00 −0.286223
\(636\) 0 0
\(637\) 0 0
\(638\) 6424.00 0.398634
\(639\) 0 0
\(640\) −7275.00 −0.449328
\(641\) 16622.0 1.02423 0.512114 0.858918i \(-0.328863\pi\)
0.512114 + 0.858918i \(0.328863\pi\)
\(642\) 0 0
\(643\) 12906.0 0.791544 0.395772 0.918349i \(-0.370477\pi\)
0.395772 + 0.918349i \(0.370477\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 2736.00 0.166635
\(647\) 3156.00 0.191770 0.0958850 0.995392i \(-0.469432\pi\)
0.0958850 + 0.995392i \(0.469432\pi\)
\(648\) 0 0
\(649\) 12936.0 0.782407
\(650\) 150.000 0.00905151
\(651\) 0 0
\(652\) −4396.00 −0.264050
\(653\) 3658.00 0.219217 0.109608 0.993975i \(-0.465040\pi\)
0.109608 + 0.993975i \(0.465040\pi\)
\(654\) 0 0
\(655\) 5010.00 0.298865
\(656\) 18204.0 1.08346
\(657\) 0 0
\(658\) 0 0
\(659\) 12316.0 0.728017 0.364009 0.931396i \(-0.381408\pi\)
0.364009 + 0.931396i \(0.381408\pi\)
\(660\) 0 0
\(661\) −32298.0 −1.90052 −0.950262 0.311451i \(-0.899185\pi\)
−0.950262 + 0.311451i \(0.899185\pi\)
\(662\) 6468.00 0.379737
\(663\) 0 0
\(664\) −14490.0 −0.846869
\(665\) 0 0
\(666\) 0 0
\(667\) −7592.00 −0.440725
\(668\) −8988.00 −0.520593
\(669\) 0 0
\(670\) −3340.00 −0.192590
\(671\) −19800.0 −1.13915
\(672\) 0 0
\(673\) −23274.0 −1.33306 −0.666528 0.745480i \(-0.732220\pi\)
−0.666528 + 0.745480i \(0.732220\pi\)
\(674\) 3438.00 0.196479
\(675\) 0 0
\(676\) 15127.0 0.860662
\(677\) 4518.00 0.256486 0.128243 0.991743i \(-0.459066\pi\)
0.128243 + 0.991743i \(0.459066\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1800.00 0.101510
\(681\) 0 0
\(682\) −12144.0 −0.681844
\(683\) 19636.0 1.10007 0.550037 0.835140i \(-0.314614\pi\)
0.550037 + 0.835140i \(0.314614\pi\)
\(684\) 0 0
\(685\) −1370.00 −0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 20172.0 1.11781
\(689\) 300.000 0.0165879
\(690\) 0 0
\(691\) 17226.0 0.948347 0.474174 0.880431i \(-0.342747\pi\)
0.474174 + 0.880431i \(0.342747\pi\)
\(692\) 6342.00 0.348391
\(693\) 0 0
\(694\) 2212.00 0.120989
\(695\) −1350.00 −0.0736812
\(696\) 0 0
\(697\) −10656.0 −0.579089
\(698\) 2910.00 0.157801
\(699\) 0 0
\(700\) 0 0
\(701\) −21362.0 −1.15097 −0.575486 0.817812i \(-0.695187\pi\)
−0.575486 + 0.817812i \(0.695187\pi\)
\(702\) 0 0
\(703\) −23940.0 −1.28437
\(704\) −7348.00 −0.393378
\(705\) 0 0
\(706\) −8364.00 −0.445869
\(707\) 0 0
\(708\) 0 0
\(709\) 8658.00 0.458615 0.229307 0.973354i \(-0.426354\pi\)
0.229307 + 0.973354i \(0.426354\pi\)
\(710\) 1540.00 0.0814016
\(711\) 0 0
\(712\) −6120.00 −0.322130
\(713\) 14352.0 0.753838
\(714\) 0 0
\(715\) 1320.00 0.0690422
\(716\) −14588.0 −0.761423
\(717\) 0 0
\(718\) −6712.00 −0.348871
\(719\) 29484.0 1.52930 0.764651 0.644445i \(-0.222912\pi\)
0.764651 + 0.644445i \(0.222912\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −6137.00 −0.316337
\(723\) 0 0
\(724\) −32718.0 −1.67950
\(725\) −3650.00 −0.186976
\(726\) 0 0
\(727\) 28260.0 1.44169 0.720843 0.693099i \(-0.243755\pi\)
0.720843 + 0.693099i \(0.243755\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −60.0000 −0.00304205
\(731\) −11808.0 −0.597448
\(732\) 0 0
\(733\) 10950.0 0.551770 0.275885 0.961191i \(-0.411029\pi\)
0.275885 + 0.961191i \(0.411029\pi\)
\(734\) 456.000 0.0229309
\(735\) 0 0
\(736\) −8372.00 −0.419288
\(737\) −29392.0 −1.46902
\(738\) 0 0
\(739\) −9772.00 −0.486426 −0.243213 0.969973i \(-0.578201\pi\)
−0.243213 + 0.969973i \(0.578201\pi\)
\(740\) −7350.00 −0.365123
\(741\) 0 0
\(742\) 0 0
\(743\) −7844.00 −0.387306 −0.193653 0.981070i \(-0.562034\pi\)
−0.193653 + 0.981070i \(0.562034\pi\)
\(744\) 0 0
\(745\) −2650.00 −0.130320
\(746\) 2558.00 0.125543
\(747\) 0 0
\(748\) 7392.00 0.361335
\(749\) 0 0
\(750\) 0 0
\(751\) 1800.00 0.0874606 0.0437303 0.999043i \(-0.486076\pi\)
0.0437303 + 0.999043i \(0.486076\pi\)
\(752\) −25092.0 −1.21677
\(753\) 0 0
\(754\) −876.000 −0.0423104
\(755\) 15600.0 0.751976
\(756\) 0 0
\(757\) −11274.0 −0.541295 −0.270648 0.962678i \(-0.587238\pi\)
−0.270648 + 0.962678i \(0.587238\pi\)
\(758\) −2004.00 −0.0960271
\(759\) 0 0
\(760\) −8550.00 −0.408080
\(761\) 22668.0 1.07978 0.539891 0.841735i \(-0.318465\pi\)
0.539891 + 0.841735i \(0.318465\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 14084.0 0.666939
\(765\) 0 0
\(766\) −11340.0 −0.534897
\(767\) −1764.00 −0.0830435
\(768\) 0 0
\(769\) 15468.0 0.725345 0.362673 0.931917i \(-0.381864\pi\)
0.362673 + 0.931917i \(0.381864\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −29442.0 −1.37259
\(773\) 28986.0 1.34871 0.674356 0.738407i \(-0.264421\pi\)
0.674356 + 0.738407i \(0.264421\pi\)
\(774\) 0 0
\(775\) 6900.00 0.319813
\(776\) 18000.0 0.832683
\(777\) 0 0
\(778\) 10522.0 0.484874
\(779\) 50616.0 2.32799
\(780\) 0 0
\(781\) 13552.0 0.620907
\(782\) 1248.00 0.0570696
\(783\) 0 0
\(784\) 0 0
\(785\) −10530.0 −0.478767
\(786\) 0 0
\(787\) −20562.0 −0.931329 −0.465665 0.884961i \(-0.654185\pi\)
−0.465665 + 0.884961i \(0.654185\pi\)
\(788\) −11018.0 −0.498096
\(789\) 0 0
\(790\) 2980.00 0.134207
\(791\) 0 0
\(792\) 0 0
\(793\) 2700.00 0.120908
\(794\) −2898.00 −0.129529
\(795\) 0 0
\(796\) −19068.0 −0.849054
\(797\) −20826.0 −0.925589 −0.462795 0.886465i \(-0.653153\pi\)
−0.462795 + 0.886465i \(0.653153\pi\)
\(798\) 0 0
\(799\) 14688.0 0.650343
\(800\) −4025.00 −0.177882
\(801\) 0 0
\(802\) 3026.00 0.133232
\(803\) −528.000 −0.0232039
\(804\) 0 0
\(805\) 0 0
\(806\) 1656.00 0.0723699
\(807\) 0 0
\(808\) 16470.0 0.717095
\(809\) 4250.00 0.184700 0.0923498 0.995727i \(-0.470562\pi\)
0.0923498 + 0.995727i \(0.470562\pi\)
\(810\) 0 0
\(811\) −24342.0 −1.05396 −0.526981 0.849877i \(-0.676676\pi\)
−0.526981 + 0.849877i \(0.676676\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 9240.00 0.397865
\(815\) −3140.00 −0.134956
\(816\) 0 0
\(817\) 56088.0 2.40180
\(818\) −8940.00 −0.382127
\(819\) 0 0
\(820\) 15540.0 0.661805
\(821\) 2138.00 0.0908852 0.0454426 0.998967i \(-0.485530\pi\)
0.0454426 + 0.998967i \(0.485530\pi\)
\(822\) 0 0
\(823\) 9560.00 0.404910 0.202455 0.979292i \(-0.435108\pi\)
0.202455 + 0.979292i \(0.435108\pi\)
\(824\) −14580.0 −0.616406
\(825\) 0 0
\(826\) 0 0
\(827\) −7748.00 −0.325785 −0.162893 0.986644i \(-0.552082\pi\)
−0.162893 + 0.986644i \(0.552082\pi\)
\(828\) 0 0
\(829\) 5334.00 0.223471 0.111736 0.993738i \(-0.464359\pi\)
0.111736 + 0.993738i \(0.464359\pi\)
\(830\) −4830.00 −0.201990
\(831\) 0 0
\(832\) 1002.00 0.0417525
\(833\) 0 0
\(834\) 0 0
\(835\) −6420.00 −0.266076
\(836\) −35112.0 −1.45260
\(837\) 0 0
\(838\) −2994.00 −0.123420
\(839\) −36444.0 −1.49963 −0.749813 0.661650i \(-0.769857\pi\)
−0.749813 + 0.661650i \(0.769857\pi\)
\(840\) 0 0
\(841\) −3073.00 −0.125999
\(842\) −15766.0 −0.645288
\(843\) 0 0
\(844\) 1260.00 0.0513874
\(845\) 10805.0 0.439886
\(846\) 0 0
\(847\) 0 0
\(848\) −2050.00 −0.0830157
\(849\) 0 0
\(850\) 600.000 0.0242116
\(851\) −10920.0 −0.439874
\(852\) 0 0
\(853\) −21030.0 −0.844142 −0.422071 0.906563i \(-0.638697\pi\)
−0.422071 + 0.906563i \(0.638697\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −22740.0 −0.907987
\(857\) −32328.0 −1.28857 −0.644284 0.764786i \(-0.722845\pi\)
−0.644284 + 0.764786i \(0.722845\pi\)
\(858\) 0 0
\(859\) 43518.0 1.72854 0.864269 0.503029i \(-0.167781\pi\)
0.864269 + 0.503029i \(0.167781\pi\)
\(860\) 17220.0 0.682787
\(861\) 0 0
\(862\) 14720.0 0.581630
\(863\) 5032.00 0.198484 0.0992418 0.995063i \(-0.468358\pi\)
0.0992418 + 0.995063i \(0.468358\pi\)
\(864\) 0 0
\(865\) 4530.00 0.178063
\(866\) 4440.00 0.174223
\(867\) 0 0
\(868\) 0 0
\(869\) 26224.0 1.02369
\(870\) 0 0
\(871\) 4008.00 0.155920
\(872\) 13950.0 0.541751
\(873\) 0 0
\(874\) −5928.00 −0.229425
\(875\) 0 0
\(876\) 0 0
\(877\) 4286.00 0.165026 0.0825131 0.996590i \(-0.473705\pi\)
0.0825131 + 0.996590i \(0.473705\pi\)
\(878\) −1488.00 −0.0571954
\(879\) 0 0
\(880\) −9020.00 −0.345527
\(881\) 19080.0 0.729650 0.364825 0.931076i \(-0.381129\pi\)
0.364825 + 0.931076i \(0.381129\pi\)
\(882\) 0 0
\(883\) 26580.0 1.01301 0.506505 0.862237i \(-0.330937\pi\)
0.506505 + 0.862237i \(0.330937\pi\)
\(884\) −1008.00 −0.0383515
\(885\) 0 0
\(886\) −5828.00 −0.220988
\(887\) 12588.0 0.476509 0.238255 0.971203i \(-0.423425\pi\)
0.238255 + 0.971203i \(0.423425\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2040.00 −0.0768325
\(891\) 0 0
\(892\) −32088.0 −1.20447
\(893\) −69768.0 −2.61444
\(894\) 0 0
\(895\) −10420.0 −0.389164
\(896\) 0 0
\(897\) 0 0
\(898\) −6958.00 −0.258565
\(899\) −40296.0 −1.49494
\(900\) 0 0
\(901\) 1200.00 0.0443705
\(902\) −19536.0 −0.721150
\(903\) 0 0
\(904\) −25410.0 −0.934872
\(905\) −23370.0 −0.858392
\(906\) 0 0
\(907\) −19332.0 −0.707727 −0.353864 0.935297i \(-0.615132\pi\)
−0.353864 + 0.935297i \(0.615132\pi\)
\(908\) 11802.0 0.431347
\(909\) 0 0
\(910\) 0 0
\(911\) 43640.0 1.58711 0.793555 0.608498i \(-0.208228\pi\)
0.793555 + 0.608498i \(0.208228\pi\)
\(912\) 0 0
\(913\) −42504.0 −1.54072
\(914\) −18102.0 −0.655099
\(915\) 0 0
\(916\) −28182.0 −1.01655
\(917\) 0 0
\(918\) 0 0
\(919\) 9084.00 0.326065 0.163032 0.986621i \(-0.447872\pi\)
0.163032 + 0.986621i \(0.447872\pi\)
\(920\) −3900.00 −0.139760
\(921\) 0 0
\(922\) −2574.00 −0.0919416
\(923\) −1848.00 −0.0659021
\(924\) 0 0
\(925\) −5250.00 −0.186615
\(926\) 12832.0 0.455384
\(927\) 0 0
\(928\) 23506.0 0.831489
\(929\) −48228.0 −1.70324 −0.851620 0.524160i \(-0.824379\pi\)
−0.851620 + 0.524160i \(0.824379\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −42518.0 −1.49434
\(933\) 0 0
\(934\) −7170.00 −0.251188
\(935\) 5280.00 0.184679
\(936\) 0 0
\(937\) −39204.0 −1.36685 −0.683425 0.730021i \(-0.739510\pi\)
−0.683425 + 0.730021i \(0.739510\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −21420.0 −0.743238
\(941\) 4770.00 0.165247 0.0826236 0.996581i \(-0.473670\pi\)
0.0826236 + 0.996581i \(0.473670\pi\)
\(942\) 0 0
\(943\) 23088.0 0.797295
\(944\) 12054.0 0.415598
\(945\) 0 0
\(946\) −21648.0 −0.744014
\(947\) −56236.0 −1.92970 −0.964849 0.262804i \(-0.915353\pi\)
−0.964849 + 0.262804i \(0.915353\pi\)
\(948\) 0 0
\(949\) 72.0000 0.00246282
\(950\) −2850.00 −0.0973329
\(951\) 0 0
\(952\) 0 0
\(953\) 52814.0 1.79519 0.897594 0.440824i \(-0.145314\pi\)
0.897594 + 0.440824i \(0.145314\pi\)
\(954\) 0 0
\(955\) 10060.0 0.340873
\(956\) 27496.0 0.930214
\(957\) 0 0
\(958\) 10164.0 0.342781
\(959\) 0 0
\(960\) 0 0
\(961\) 46385.0 1.55701
\(962\) −1260.00 −0.0422287
\(963\) 0 0
\(964\) −8652.00 −0.289069
\(965\) −21030.0 −0.701533
\(966\) 0 0
\(967\) −39364.0 −1.30906 −0.654530 0.756036i \(-0.727133\pi\)
−0.654530 + 0.756036i \(0.727133\pi\)
\(968\) 9075.00 0.301324
\(969\) 0 0
\(970\) 6000.00 0.198607
\(971\) −29322.0 −0.969091 −0.484546 0.874766i \(-0.661015\pi\)
−0.484546 + 0.874766i \(0.661015\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −10212.0 −0.335948
\(975\) 0 0
\(976\) −18450.0 −0.605092
\(977\) 35882.0 1.17499 0.587496 0.809227i \(-0.300114\pi\)
0.587496 + 0.809227i \(0.300114\pi\)
\(978\) 0 0
\(979\) −17952.0 −0.586056
\(980\) 0 0
\(981\) 0 0
\(982\) −7972.00 −0.259060
\(983\) −32580.0 −1.05711 −0.528556 0.848899i \(-0.677266\pi\)
−0.528556 + 0.848899i \(0.677266\pi\)
\(984\) 0 0
\(985\) −7870.00 −0.254578
\(986\) −3504.00 −0.113175
\(987\) 0 0
\(988\) 4788.00 0.154177
\(989\) 25584.0 0.822572
\(990\) 0 0
\(991\) −30036.0 −0.962790 −0.481395 0.876504i \(-0.659870\pi\)
−0.481395 + 0.876504i \(0.659870\pi\)
\(992\) −44436.0 −1.42222
\(993\) 0 0
\(994\) 0 0
\(995\) −13620.0 −0.433953
\(996\) 0 0
\(997\) 25134.0 0.798397 0.399198 0.916865i \(-0.369288\pi\)
0.399198 + 0.916865i \(0.369288\pi\)
\(998\) 1548.00 0.0490993
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2205.4.a.k.1.1 1
3.2 odd 2 245.4.a.b.1.1 1
7.6 odd 2 2205.4.a.n.1.1 1
15.14 odd 2 1225.4.a.g.1.1 1
21.2 odd 6 245.4.e.d.116.1 2
21.5 even 6 245.4.e.c.116.1 2
21.11 odd 6 245.4.e.d.226.1 2
21.17 even 6 245.4.e.c.226.1 2
21.20 even 2 245.4.a.c.1.1 yes 1
105.104 even 2 1225.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.b.1.1 1 3.2 odd 2
245.4.a.c.1.1 yes 1 21.20 even 2
245.4.e.c.116.1 2 21.5 even 6
245.4.e.c.226.1 2 21.17 even 6
245.4.e.d.116.1 2 21.2 odd 6
245.4.e.d.226.1 2 21.11 odd 6
1225.4.a.f.1.1 1 105.104 even 2
1225.4.a.g.1.1 1 15.14 odd 2
2205.4.a.k.1.1 1 1.1 even 1 trivial
2205.4.a.n.1.1 1 7.6 odd 2