Properties

Label 150.4.a.b.1.1
Level $150$
Weight $4$
Character 150.1
Self dual yes
Analytic conductor $8.850$
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] = [150,4,Mod(1,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 150.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: \(8.85028650086\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 150.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +4.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +4.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -48.0000 q^{11} -12.0000 q^{12} -2.00000 q^{13} -8.00000 q^{14} +16.0000 q^{16} +114.000 q^{17} -18.0000 q^{18} +140.000 q^{19} -12.0000 q^{21} +96.0000 q^{22} -72.0000 q^{23} +24.0000 q^{24} +4.00000 q^{26} -27.0000 q^{27} +16.0000 q^{28} +210.000 q^{29} +272.000 q^{31} -32.0000 q^{32} +144.000 q^{33} -228.000 q^{34} +36.0000 q^{36} +334.000 q^{37} -280.000 q^{38} +6.00000 q^{39} -198.000 q^{41} +24.0000 q^{42} +268.000 q^{43} -192.000 q^{44} +144.000 q^{46} -216.000 q^{47} -48.0000 q^{48} -327.000 q^{49} -342.000 q^{51} -8.00000 q^{52} +78.0000 q^{53} +54.0000 q^{54} -32.0000 q^{56} -420.000 q^{57} -420.000 q^{58} +240.000 q^{59} +302.000 q^{61} -544.000 q^{62} +36.0000 q^{63} +64.0000 q^{64} -288.000 q^{66} -596.000 q^{67} +456.000 q^{68} +216.000 q^{69} -768.000 q^{71} -72.0000 q^{72} +478.000 q^{73} -668.000 q^{74} +560.000 q^{76} -192.000 q^{77} -12.0000 q^{78} -640.000 q^{79} +81.0000 q^{81} +396.000 q^{82} +348.000 q^{83} -48.0000 q^{84} -536.000 q^{86} -630.000 q^{87} +384.000 q^{88} +210.000 q^{89} -8.00000 q^{91} -288.000 q^{92} -816.000 q^{93} +432.000 q^{94} +96.0000 q^{96} +1534.00 q^{97} +654.000 q^{98} -432.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 6.00000 0.408248
\(7\) 4.00000 0.215980 0.107990 0.994152i \(-0.465559\pi\)
0.107990 + 0.994152i \(0.465559\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −48.0000 −1.31569 −0.657843 0.753155i \(-0.728531\pi\)
−0.657843 + 0.753155i \(0.728531\pi\)
\(12\) −12.0000 −0.288675
\(13\) −2.00000 −0.0426692 −0.0213346 0.999772i \(-0.506792\pi\)
−0.0213346 + 0.999772i \(0.506792\pi\)
\(14\) −8.00000 −0.152721
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 114.000 1.62642 0.813208 0.581974i \(-0.197719\pi\)
0.813208 + 0.581974i \(0.197719\pi\)
\(18\) −18.0000 −0.235702
\(19\) 140.000 1.69043 0.845216 0.534425i \(-0.179472\pi\)
0.845216 + 0.534425i \(0.179472\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.124696
\(22\) 96.0000 0.930330
\(23\) −72.0000 −0.652741 −0.326370 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) 24.0000 0.204124
\(25\) 0 0
\(26\) 4.00000 0.0301717
\(27\) −27.0000 −0.192450
\(28\) 16.0000 0.107990
\(29\) 210.000 1.34469 0.672345 0.740238i \(-0.265287\pi\)
0.672345 + 0.740238i \(0.265287\pi\)
\(30\) 0 0
\(31\) 272.000 1.57589 0.787946 0.615745i \(-0.211145\pi\)
0.787946 + 0.615745i \(0.211145\pi\)
\(32\) −32.0000 −0.176777
\(33\) 144.000 0.759612
\(34\) −228.000 −1.15005
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 334.000 1.48403 0.742017 0.670381i \(-0.233869\pi\)
0.742017 + 0.670381i \(0.233869\pi\)
\(38\) −280.000 −1.19532
\(39\) 6.00000 0.0246351
\(40\) 0 0
\(41\) −198.000 −0.754205 −0.377102 0.926172i \(-0.623080\pi\)
−0.377102 + 0.926172i \(0.623080\pi\)
\(42\) 24.0000 0.0881733
\(43\) 268.000 0.950456 0.475228 0.879863i \(-0.342366\pi\)
0.475228 + 0.879863i \(0.342366\pi\)
\(44\) −192.000 −0.657843
\(45\) 0 0
\(46\) 144.000 0.461557
\(47\) −216.000 −0.670358 −0.335179 0.942154i \(-0.608797\pi\)
−0.335179 + 0.942154i \(0.608797\pi\)
\(48\) −48.0000 −0.144338
\(49\) −327.000 −0.953353
\(50\) 0 0
\(51\) −342.000 −0.939011
\(52\) −8.00000 −0.0213346
\(53\) 78.0000 0.202153 0.101077 0.994879i \(-0.467771\pi\)
0.101077 + 0.994879i \(0.467771\pi\)
\(54\) 54.0000 0.136083
\(55\) 0 0
\(56\) −32.0000 −0.0763604
\(57\) −420.000 −0.975971
\(58\) −420.000 −0.950840
\(59\) 240.000 0.529582 0.264791 0.964306i \(-0.414697\pi\)
0.264791 + 0.964306i \(0.414697\pi\)
\(60\) 0 0
\(61\) 302.000 0.633888 0.316944 0.948444i \(-0.397343\pi\)
0.316944 + 0.948444i \(0.397343\pi\)
\(62\) −544.000 −1.11432
\(63\) 36.0000 0.0719932
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −288.000 −0.537127
\(67\) −596.000 −1.08676 −0.543381 0.839487i \(-0.682856\pi\)
−0.543381 + 0.839487i \(0.682856\pi\)
\(68\) 456.000 0.813208
\(69\) 216.000 0.376860
\(70\) 0 0
\(71\) −768.000 −1.28373 −0.641865 0.766818i \(-0.721839\pi\)
−0.641865 + 0.766818i \(0.721839\pi\)
\(72\) −72.0000 −0.117851
\(73\) 478.000 0.766379 0.383190 0.923670i \(-0.374826\pi\)
0.383190 + 0.923670i \(0.374826\pi\)
\(74\) −668.000 −1.04937
\(75\) 0 0
\(76\) 560.000 0.845216
\(77\) −192.000 −0.284161
\(78\) −12.0000 −0.0174196
\(79\) −640.000 −0.911464 −0.455732 0.890117i \(-0.650622\pi\)
−0.455732 + 0.890117i \(0.650622\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 396.000 0.533303
\(83\) 348.000 0.460216 0.230108 0.973165i \(-0.426092\pi\)
0.230108 + 0.973165i \(0.426092\pi\)
\(84\) −48.0000 −0.0623480
\(85\) 0 0
\(86\) −536.000 −0.672074
\(87\) −630.000 −0.776357
\(88\) 384.000 0.465165
\(89\) 210.000 0.250112 0.125056 0.992150i \(-0.460089\pi\)
0.125056 + 0.992150i \(0.460089\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.00921569
\(92\) −288.000 −0.326370
\(93\) −816.000 −0.909841
\(94\) 432.000 0.474015
\(95\) 0 0
\(96\) 96.0000 0.102062
\(97\) 1534.00 1.60571 0.802856 0.596173i \(-0.203313\pi\)
0.802856 + 0.596173i \(0.203313\pi\)
\(98\) 654.000 0.674122
\(99\) −432.000 −0.438562
\(100\) 0 0
\(101\) 1722.00 1.69649 0.848245 0.529605i \(-0.177660\pi\)
0.848245 + 0.529605i \(0.177660\pi\)
\(102\) 684.000 0.663981
\(103\) −1052.00 −1.00638 −0.503188 0.864177i \(-0.667840\pi\)
−0.503188 + 0.864177i \(0.667840\pi\)
\(104\) 16.0000 0.0150859
\(105\) 0 0
\(106\) −156.000 −0.142944
\(107\) 564.000 0.509570 0.254785 0.966998i \(-0.417995\pi\)
0.254785 + 0.966998i \(0.417995\pi\)
\(108\) −108.000 −0.0962250
\(109\) −610.000 −0.536031 −0.268016 0.963415i \(-0.586368\pi\)
−0.268016 + 0.963415i \(0.586368\pi\)
\(110\) 0 0
\(111\) −1002.00 −0.856807
\(112\) 64.0000 0.0539949
\(113\) −1302.00 −1.08391 −0.541955 0.840407i \(-0.682316\pi\)
−0.541955 + 0.840407i \(0.682316\pi\)
\(114\) 840.000 0.690116
\(115\) 0 0
\(116\) 840.000 0.672345
\(117\) −18.0000 −0.0142231
\(118\) −480.000 −0.374471
\(119\) 456.000 0.351273
\(120\) 0 0
\(121\) 973.000 0.731029
\(122\) −604.000 −0.448226
\(123\) 594.000 0.435440
\(124\) 1088.00 0.787946
\(125\) 0 0
\(126\) −72.0000 −0.0509069
\(127\) 124.000 0.0866395 0.0433198 0.999061i \(-0.486207\pi\)
0.0433198 + 0.999061i \(0.486207\pi\)
\(128\) −128.000 −0.0883883
\(129\) −804.000 −0.548746
\(130\) 0 0
\(131\) 192.000 0.128054 0.0640272 0.997948i \(-0.479606\pi\)
0.0640272 + 0.997948i \(0.479606\pi\)
\(132\) 576.000 0.379806
\(133\) 560.000 0.365099
\(134\) 1192.00 0.768456
\(135\) 0 0
\(136\) −912.000 −0.575025
\(137\) 2514.00 1.56778 0.783889 0.620901i \(-0.213233\pi\)
0.783889 + 0.620901i \(0.213233\pi\)
\(138\) −432.000 −0.266480
\(139\) 1340.00 0.817679 0.408839 0.912606i \(-0.365934\pi\)
0.408839 + 0.912606i \(0.365934\pi\)
\(140\) 0 0
\(141\) 648.000 0.387032
\(142\) 1536.00 0.907734
\(143\) 96.0000 0.0561393
\(144\) 144.000 0.0833333
\(145\) 0 0
\(146\) −956.000 −0.541912
\(147\) 981.000 0.550418
\(148\) 1336.00 0.742017
\(149\) 1410.00 0.775246 0.387623 0.921818i \(-0.373296\pi\)
0.387623 + 0.921818i \(0.373296\pi\)
\(150\) 0 0
\(151\) −2128.00 −1.14685 −0.573424 0.819258i \(-0.694385\pi\)
−0.573424 + 0.819258i \(0.694385\pi\)
\(152\) −1120.00 −0.597658
\(153\) 1026.00 0.542138
\(154\) 384.000 0.200932
\(155\) 0 0
\(156\) 24.0000 0.0123176
\(157\) −3026.00 −1.53822 −0.769112 0.639114i \(-0.779301\pi\)
−0.769112 + 0.639114i \(0.779301\pi\)
\(158\) 1280.00 0.644502
\(159\) −234.000 −0.116713
\(160\) 0 0
\(161\) −288.000 −0.140979
\(162\) −162.000 −0.0785674
\(163\) −2612.00 −1.25514 −0.627569 0.778561i \(-0.715950\pi\)
−0.627569 + 0.778561i \(0.715950\pi\)
\(164\) −792.000 −0.377102
\(165\) 0 0
\(166\) −696.000 −0.325422
\(167\) 24.0000 0.0111208 0.00556041 0.999985i \(-0.498230\pi\)
0.00556041 + 0.999985i \(0.498230\pi\)
\(168\) 96.0000 0.0440867
\(169\) −2193.00 −0.998179
\(170\) 0 0
\(171\) 1260.00 0.563477
\(172\) 1072.00 0.475228
\(173\) −1962.00 −0.862243 −0.431122 0.902294i \(-0.641882\pi\)
−0.431122 + 0.902294i \(0.641882\pi\)
\(174\) 1260.00 0.548968
\(175\) 0 0
\(176\) −768.000 −0.328921
\(177\) −720.000 −0.305754
\(178\) −420.000 −0.176856
\(179\) −120.000 −0.0501074 −0.0250537 0.999686i \(-0.507976\pi\)
−0.0250537 + 0.999686i \(0.507976\pi\)
\(180\) 0 0
\(181\) 902.000 0.370415 0.185208 0.982699i \(-0.440704\pi\)
0.185208 + 0.982699i \(0.440704\pi\)
\(182\) 16.0000 0.00651648
\(183\) −906.000 −0.365975
\(184\) 576.000 0.230779
\(185\) 0 0
\(186\) 1632.00 0.643355
\(187\) −5472.00 −2.13985
\(188\) −864.000 −0.335179
\(189\) −108.000 −0.0415653
\(190\) 0 0
\(191\) −168.000 −0.0636443 −0.0318221 0.999494i \(-0.510131\pi\)
−0.0318221 + 0.999494i \(0.510131\pi\)
\(192\) −192.000 −0.0721688
\(193\) 1318.00 0.491563 0.245782 0.969325i \(-0.420955\pi\)
0.245782 + 0.969325i \(0.420955\pi\)
\(194\) −3068.00 −1.13541
\(195\) 0 0
\(196\) −1308.00 −0.476676
\(197\) 4014.00 1.45170 0.725852 0.687851i \(-0.241446\pi\)
0.725852 + 0.687851i \(0.241446\pi\)
\(198\) 864.000 0.310110
\(199\) 2000.00 0.712443 0.356222 0.934401i \(-0.384065\pi\)
0.356222 + 0.934401i \(0.384065\pi\)
\(200\) 0 0
\(201\) 1788.00 0.627442
\(202\) −3444.00 −1.19960
\(203\) 840.000 0.290426
\(204\) −1368.00 −0.469506
\(205\) 0 0
\(206\) 2104.00 0.711615
\(207\) −648.000 −0.217580
\(208\) −32.0000 −0.0106673
\(209\) −6720.00 −2.22408
\(210\) 0 0
\(211\) −3868.00 −1.26201 −0.631005 0.775779i \(-0.717357\pi\)
−0.631005 + 0.775779i \(0.717357\pi\)
\(212\) 312.000 0.101077
\(213\) 2304.00 0.741162
\(214\) −1128.00 −0.360320
\(215\) 0 0
\(216\) 216.000 0.0680414
\(217\) 1088.00 0.340361
\(218\) 1220.00 0.379031
\(219\) −1434.00 −0.442469
\(220\) 0 0
\(221\) −228.000 −0.0693979
\(222\) 2004.00 0.605854
\(223\) 3148.00 0.945317 0.472658 0.881246i \(-0.343294\pi\)
0.472658 + 0.881246i \(0.343294\pi\)
\(224\) −128.000 −0.0381802
\(225\) 0 0
\(226\) 2604.00 0.766440
\(227\) −2556.00 −0.747347 −0.373673 0.927560i \(-0.621902\pi\)
−0.373673 + 0.927560i \(0.621902\pi\)
\(228\) −1680.00 −0.487986
\(229\) −610.000 −0.176026 −0.0880130 0.996119i \(-0.528052\pi\)
−0.0880130 + 0.996119i \(0.528052\pi\)
\(230\) 0 0
\(231\) 576.000 0.164061
\(232\) −1680.00 −0.475420
\(233\) 2058.00 0.578644 0.289322 0.957232i \(-0.406570\pi\)
0.289322 + 0.957232i \(0.406570\pi\)
\(234\) 36.0000 0.0100572
\(235\) 0 0
\(236\) 960.000 0.264791
\(237\) 1920.00 0.526234
\(238\) −912.000 −0.248387
\(239\) 4920.00 1.33158 0.665792 0.746138i \(-0.268094\pi\)
0.665792 + 0.746138i \(0.268094\pi\)
\(240\) 0 0
\(241\) −1438.00 −0.384356 −0.192178 0.981360i \(-0.561555\pi\)
−0.192178 + 0.981360i \(0.561555\pi\)
\(242\) −1946.00 −0.516916
\(243\) −243.000 −0.0641500
\(244\) 1208.00 0.316944
\(245\) 0 0
\(246\) −1188.00 −0.307903
\(247\) −280.000 −0.0721294
\(248\) −2176.00 −0.557162
\(249\) −1044.00 −0.265706
\(250\) 0 0
\(251\) 792.000 0.199166 0.0995829 0.995029i \(-0.468249\pi\)
0.0995829 + 0.995029i \(0.468249\pi\)
\(252\) 144.000 0.0359966
\(253\) 3456.00 0.858802
\(254\) −248.000 −0.0612634
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −2166.00 −0.525725 −0.262863 0.964833i \(-0.584667\pi\)
−0.262863 + 0.964833i \(0.584667\pi\)
\(258\) 1608.00 0.388022
\(259\) 1336.00 0.320521
\(260\) 0 0
\(261\) 1890.00 0.448230
\(262\) −384.000 −0.0905481
\(263\) −3192.00 −0.748392 −0.374196 0.927350i \(-0.622081\pi\)
−0.374196 + 0.927350i \(0.622081\pi\)
\(264\) −1152.00 −0.268563
\(265\) 0 0
\(266\) −1120.00 −0.258164
\(267\) −630.000 −0.144402
\(268\) −2384.00 −0.543381
\(269\) 5490.00 1.24435 0.622177 0.782877i \(-0.286248\pi\)
0.622177 + 0.782877i \(0.286248\pi\)
\(270\) 0 0
\(271\) −6328.00 −1.41845 −0.709223 0.704985i \(-0.750954\pi\)
−0.709223 + 0.704985i \(0.750954\pi\)
\(272\) 1824.00 0.406604
\(273\) 24.0000 0.00532068
\(274\) −5028.00 −1.10859
\(275\) 0 0
\(276\) 864.000 0.188430
\(277\) 574.000 0.124507 0.0622533 0.998060i \(-0.480171\pi\)
0.0622533 + 0.998060i \(0.480171\pi\)
\(278\) −2680.00 −0.578186
\(279\) 2448.00 0.525297
\(280\) 0 0
\(281\) 4242.00 0.900557 0.450278 0.892888i \(-0.351325\pi\)
0.450278 + 0.892888i \(0.351325\pi\)
\(282\) −1296.00 −0.273673
\(283\) 628.000 0.131911 0.0659553 0.997823i \(-0.478991\pi\)
0.0659553 + 0.997823i \(0.478991\pi\)
\(284\) −3072.00 −0.641865
\(285\) 0 0
\(286\) −192.000 −0.0396965
\(287\) −792.000 −0.162893
\(288\) −288.000 −0.0589256
\(289\) 8083.00 1.64523
\(290\) 0 0
\(291\) −4602.00 −0.927058
\(292\) 1912.00 0.383190
\(293\) 558.000 0.111258 0.0556292 0.998451i \(-0.482284\pi\)
0.0556292 + 0.998451i \(0.482284\pi\)
\(294\) −1962.00 −0.389205
\(295\) 0 0
\(296\) −2672.00 −0.524685
\(297\) 1296.00 0.253204
\(298\) −2820.00 −0.548182
\(299\) 144.000 0.0278520
\(300\) 0 0
\(301\) 1072.00 0.205279
\(302\) 4256.00 0.810945
\(303\) −5166.00 −0.979468
\(304\) 2240.00 0.422608
\(305\) 0 0
\(306\) −2052.00 −0.383350
\(307\) 6964.00 1.29465 0.647323 0.762216i \(-0.275888\pi\)
0.647323 + 0.762216i \(0.275888\pi\)
\(308\) −768.000 −0.142081
\(309\) 3156.00 0.581031
\(310\) 0 0
\(311\) 2832.00 0.516360 0.258180 0.966097i \(-0.416877\pi\)
0.258180 + 0.966097i \(0.416877\pi\)
\(312\) −48.0000 −0.00870982
\(313\) −8642.00 −1.56062 −0.780311 0.625392i \(-0.784939\pi\)
−0.780311 + 0.625392i \(0.784939\pi\)
\(314\) 6052.00 1.08769
\(315\) 0 0
\(316\) −2560.00 −0.455732
\(317\) 2214.00 0.392273 0.196137 0.980577i \(-0.437160\pi\)
0.196137 + 0.980577i \(0.437160\pi\)
\(318\) 468.000 0.0825287
\(319\) −10080.0 −1.76919
\(320\) 0 0
\(321\) −1692.00 −0.294200
\(322\) 576.000 0.0996870
\(323\) 15960.0 2.74934
\(324\) 324.000 0.0555556
\(325\) 0 0
\(326\) 5224.00 0.887517
\(327\) 1830.00 0.309478
\(328\) 1584.00 0.266652
\(329\) −864.000 −0.144784
\(330\) 0 0
\(331\) 10772.0 1.78877 0.894385 0.447299i \(-0.147614\pi\)
0.894385 + 0.447299i \(0.147614\pi\)
\(332\) 1392.00 0.230108
\(333\) 3006.00 0.494678
\(334\) −48.0000 −0.00786360
\(335\) 0 0
\(336\) −192.000 −0.0311740
\(337\) 1654.00 0.267356 0.133678 0.991025i \(-0.457321\pi\)
0.133678 + 0.991025i \(0.457321\pi\)
\(338\) 4386.00 0.705819
\(339\) 3906.00 0.625796
\(340\) 0 0
\(341\) −13056.0 −2.07338
\(342\) −2520.00 −0.398439
\(343\) −2680.00 −0.421885
\(344\) −2144.00 −0.336037
\(345\) 0 0
\(346\) 3924.00 0.609698
\(347\) −2196.00 −0.339733 −0.169867 0.985467i \(-0.554334\pi\)
−0.169867 + 0.985467i \(0.554334\pi\)
\(348\) −2520.00 −0.388179
\(349\) 8270.00 1.26843 0.634216 0.773156i \(-0.281323\pi\)
0.634216 + 0.773156i \(0.281323\pi\)
\(350\) 0 0
\(351\) 54.0000 0.00821170
\(352\) 1536.00 0.232583
\(353\) −10302.0 −1.55331 −0.776657 0.629923i \(-0.783086\pi\)
−0.776657 + 0.629923i \(0.783086\pi\)
\(354\) 1440.00 0.216201
\(355\) 0 0
\(356\) 840.000 0.125056
\(357\) −1368.00 −0.202807
\(358\) 240.000 0.0354313
\(359\) −2280.00 −0.335192 −0.167596 0.985856i \(-0.553600\pi\)
−0.167596 + 0.985856i \(0.553600\pi\)
\(360\) 0 0
\(361\) 12741.0 1.85756
\(362\) −1804.00 −0.261923
\(363\) −2919.00 −0.422060
\(364\) −32.0000 −0.00460785
\(365\) 0 0
\(366\) 1812.00 0.258783
\(367\) 8764.00 1.24653 0.623266 0.782010i \(-0.285805\pi\)
0.623266 + 0.782010i \(0.285805\pi\)
\(368\) −1152.00 −0.163185
\(369\) −1782.00 −0.251402
\(370\) 0 0
\(371\) 312.000 0.0436610
\(372\) −3264.00 −0.454921
\(373\) 1318.00 0.182958 0.0914792 0.995807i \(-0.470841\pi\)
0.0914792 + 0.995807i \(0.470841\pi\)
\(374\) 10944.0 1.51310
\(375\) 0 0
\(376\) 1728.00 0.237007
\(377\) −420.000 −0.0573769
\(378\) 216.000 0.0293911
\(379\) 1100.00 0.149085 0.0745425 0.997218i \(-0.476250\pi\)
0.0745425 + 0.997218i \(0.476250\pi\)
\(380\) 0 0
\(381\) −372.000 −0.0500214
\(382\) 336.000 0.0450033
\(383\) 3528.00 0.470685 0.235343 0.971912i \(-0.424379\pi\)
0.235343 + 0.971912i \(0.424379\pi\)
\(384\) 384.000 0.0510310
\(385\) 0 0
\(386\) −2636.00 −0.347588
\(387\) 2412.00 0.316819
\(388\) 6136.00 0.802856
\(389\) −9630.00 −1.25517 −0.627584 0.778549i \(-0.715956\pi\)
−0.627584 + 0.778549i \(0.715956\pi\)
\(390\) 0 0
\(391\) −8208.00 −1.06163
\(392\) 2616.00 0.337061
\(393\) −576.000 −0.0739322
\(394\) −8028.00 −1.02651
\(395\) 0 0
\(396\) −1728.00 −0.219281
\(397\) 3094.00 0.391142 0.195571 0.980690i \(-0.437344\pi\)
0.195571 + 0.980690i \(0.437344\pi\)
\(398\) −4000.00 −0.503774
\(399\) −1680.00 −0.210790
\(400\) 0 0
\(401\) −1638.00 −0.203985 −0.101992 0.994785i \(-0.532522\pi\)
−0.101992 + 0.994785i \(0.532522\pi\)
\(402\) −3576.00 −0.443668
\(403\) −544.000 −0.0672421
\(404\) 6888.00 0.848245
\(405\) 0 0
\(406\) −1680.00 −0.205362
\(407\) −16032.0 −1.95252
\(408\) 2736.00 0.331991
\(409\) −13750.0 −1.66233 −0.831166 0.556024i \(-0.812326\pi\)
−0.831166 + 0.556024i \(0.812326\pi\)
\(410\) 0 0
\(411\) −7542.00 −0.905157
\(412\) −4208.00 −0.503188
\(413\) 960.000 0.114379
\(414\) 1296.00 0.153852
\(415\) 0 0
\(416\) 64.0000 0.00754293
\(417\) −4020.00 −0.472087
\(418\) 13440.0 1.57266
\(419\) −12480.0 −1.45510 −0.727551 0.686053i \(-0.759342\pi\)
−0.727551 + 0.686053i \(0.759342\pi\)
\(420\) 0 0
\(421\) 7262.00 0.840685 0.420342 0.907366i \(-0.361910\pi\)
0.420342 + 0.907366i \(0.361910\pi\)
\(422\) 7736.00 0.892376
\(423\) −1944.00 −0.223453
\(424\) −624.000 −0.0714720
\(425\) 0 0
\(426\) −4608.00 −0.524081
\(427\) 1208.00 0.136907
\(428\) 2256.00 0.254785
\(429\) −288.000 −0.0324121
\(430\) 0 0
\(431\) 9792.00 1.09435 0.547174 0.837019i \(-0.315704\pi\)
0.547174 + 0.837019i \(0.315704\pi\)
\(432\) −432.000 −0.0481125
\(433\) −1802.00 −0.199997 −0.0999984 0.994988i \(-0.531884\pi\)
−0.0999984 + 0.994988i \(0.531884\pi\)
\(434\) −2176.00 −0.240671
\(435\) 0 0
\(436\) −2440.00 −0.268016
\(437\) −10080.0 −1.10341
\(438\) 2868.00 0.312873
\(439\) −2320.00 −0.252227 −0.126113 0.992016i \(-0.540250\pi\)
−0.126113 + 0.992016i \(0.540250\pi\)
\(440\) 0 0
\(441\) −2943.00 −0.317784
\(442\) 456.000 0.0490717
\(443\) −11172.0 −1.19819 −0.599095 0.800678i \(-0.704473\pi\)
−0.599095 + 0.800678i \(0.704473\pi\)
\(444\) −4008.00 −0.428404
\(445\) 0 0
\(446\) −6296.00 −0.668440
\(447\) −4230.00 −0.447589
\(448\) 256.000 0.0269975
\(449\) 6810.00 0.715777 0.357888 0.933764i \(-0.383497\pi\)
0.357888 + 0.933764i \(0.383497\pi\)
\(450\) 0 0
\(451\) 9504.00 0.992297
\(452\) −5208.00 −0.541955
\(453\) 6384.00 0.662134
\(454\) 5112.00 0.528454
\(455\) 0 0
\(456\) 3360.00 0.345058
\(457\) −17066.0 −1.74686 −0.873429 0.486952i \(-0.838109\pi\)
−0.873429 + 0.486952i \(0.838109\pi\)
\(458\) 1220.00 0.124469
\(459\) −3078.00 −0.313004
\(460\) 0 0
\(461\) −18918.0 −1.91128 −0.955639 0.294541i \(-0.904833\pi\)
−0.955639 + 0.294541i \(0.904833\pi\)
\(462\) −1152.00 −0.116008
\(463\) −1052.00 −0.105595 −0.0527976 0.998605i \(-0.516814\pi\)
−0.0527976 + 0.998605i \(0.516814\pi\)
\(464\) 3360.00 0.336173
\(465\) 0 0
\(466\) −4116.00 −0.409163
\(467\) −11076.0 −1.09751 −0.548754 0.835984i \(-0.684898\pi\)
−0.548754 + 0.835984i \(0.684898\pi\)
\(468\) −72.0000 −0.00711154
\(469\) −2384.00 −0.234718
\(470\) 0 0
\(471\) 9078.00 0.888094
\(472\) −1920.00 −0.187236
\(473\) −12864.0 −1.25050
\(474\) −3840.00 −0.372103
\(475\) 0 0
\(476\) 1824.00 0.175636
\(477\) 702.000 0.0673844
\(478\) −9840.00 −0.941571
\(479\) −9000.00 −0.858498 −0.429249 0.903186i \(-0.641222\pi\)
−0.429249 + 0.903186i \(0.641222\pi\)
\(480\) 0 0
\(481\) −668.000 −0.0633226
\(482\) 2876.00 0.271781
\(483\) 864.000 0.0813941
\(484\) 3892.00 0.365515
\(485\) 0 0
\(486\) 486.000 0.0453609
\(487\) 8764.00 0.815472 0.407736 0.913100i \(-0.366318\pi\)
0.407736 + 0.913100i \(0.366318\pi\)
\(488\) −2416.00 −0.224113
\(489\) 7836.00 0.724655
\(490\) 0 0
\(491\) 5592.00 0.513978 0.256989 0.966414i \(-0.417270\pi\)
0.256989 + 0.966414i \(0.417270\pi\)
\(492\) 2376.00 0.217720
\(493\) 23940.0 2.18703
\(494\) 560.000 0.0510032
\(495\) 0 0
\(496\) 4352.00 0.393973
\(497\) −3072.00 −0.277260
\(498\) 2088.00 0.187883
\(499\) 4700.00 0.421645 0.210823 0.977524i \(-0.432386\pi\)
0.210823 + 0.977524i \(0.432386\pi\)
\(500\) 0 0
\(501\) −72.0000 −0.00642060
\(502\) −1584.00 −0.140831
\(503\) 11808.0 1.04671 0.523353 0.852116i \(-0.324681\pi\)
0.523353 + 0.852116i \(0.324681\pi\)
\(504\) −288.000 −0.0254535
\(505\) 0 0
\(506\) −6912.00 −0.607265
\(507\) 6579.00 0.576299
\(508\) 496.000 0.0433198
\(509\) 1170.00 0.101885 0.0509424 0.998702i \(-0.483778\pi\)
0.0509424 + 0.998702i \(0.483778\pi\)
\(510\) 0 0
\(511\) 1912.00 0.165522
\(512\) −512.000 −0.0441942
\(513\) −3780.00 −0.325324
\(514\) 4332.00 0.371744
\(515\) 0 0
\(516\) −3216.00 −0.274373
\(517\) 10368.0 0.881981
\(518\) −2672.00 −0.226643
\(519\) 5886.00 0.497816
\(520\) 0 0
\(521\) −16638.0 −1.39909 −0.699543 0.714590i \(-0.746613\pi\)
−0.699543 + 0.714590i \(0.746613\pi\)
\(522\) −3780.00 −0.316947
\(523\) −15692.0 −1.31198 −0.655988 0.754771i \(-0.727748\pi\)
−0.655988 + 0.754771i \(0.727748\pi\)
\(524\) 768.000 0.0640272
\(525\) 0 0
\(526\) 6384.00 0.529193
\(527\) 31008.0 2.56305
\(528\) 2304.00 0.189903
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 2160.00 0.176527
\(532\) 2240.00 0.182549
\(533\) 396.000 0.0321814
\(534\) 1260.00 0.102108
\(535\) 0 0
\(536\) 4768.00 0.384228
\(537\) 360.000 0.0289295
\(538\) −10980.0 −0.879891
\(539\) 15696.0 1.25431
\(540\) 0 0
\(541\) −22018.0 −1.74977 −0.874887 0.484327i \(-0.839064\pi\)
−0.874887 + 0.484327i \(0.839064\pi\)
\(542\) 12656.0 1.00299
\(543\) −2706.00 −0.213859
\(544\) −3648.00 −0.287512
\(545\) 0 0
\(546\) −48.0000 −0.00376229
\(547\) 4564.00 0.356751 0.178375 0.983963i \(-0.442916\pi\)
0.178375 + 0.983963i \(0.442916\pi\)
\(548\) 10056.0 0.783889
\(549\) 2718.00 0.211296
\(550\) 0 0
\(551\) 29400.0 2.27311
\(552\) −1728.00 −0.133240
\(553\) −2560.00 −0.196858
\(554\) −1148.00 −0.0880394
\(555\) 0 0
\(556\) 5360.00 0.408839
\(557\) 7734.00 0.588331 0.294165 0.955755i \(-0.404958\pi\)
0.294165 + 0.955755i \(0.404958\pi\)
\(558\) −4896.00 −0.371441
\(559\) −536.000 −0.0405552
\(560\) 0 0
\(561\) 16416.0 1.23544
\(562\) −8484.00 −0.636790
\(563\) 20148.0 1.50824 0.754118 0.656739i \(-0.228065\pi\)
0.754118 + 0.656739i \(0.228065\pi\)
\(564\) 2592.00 0.193516
\(565\) 0 0
\(566\) −1256.00 −0.0932749
\(567\) 324.000 0.0239977
\(568\) 6144.00 0.453867
\(569\) −24030.0 −1.77046 −0.885228 0.465156i \(-0.845998\pi\)
−0.885228 + 0.465156i \(0.845998\pi\)
\(570\) 0 0
\(571\) 2372.00 0.173844 0.0869222 0.996215i \(-0.472297\pi\)
0.0869222 + 0.996215i \(0.472297\pi\)
\(572\) 384.000 0.0280697
\(573\) 504.000 0.0367450
\(574\) 1584.00 0.115183
\(575\) 0 0
\(576\) 576.000 0.0416667
\(577\) −8546.00 −0.616594 −0.308297 0.951290i \(-0.599759\pi\)
−0.308297 + 0.951290i \(0.599759\pi\)
\(578\) −16166.0 −1.16335
\(579\) −3954.00 −0.283804
\(580\) 0 0
\(581\) 1392.00 0.0993974
\(582\) 9204.00 0.655529
\(583\) −3744.00 −0.265970
\(584\) −3824.00 −0.270956
\(585\) 0 0
\(586\) −1116.00 −0.0786716
\(587\) 15444.0 1.08593 0.542966 0.839755i \(-0.317301\pi\)
0.542966 + 0.839755i \(0.317301\pi\)
\(588\) 3924.00 0.275209
\(589\) 38080.0 2.66394
\(590\) 0 0
\(591\) −12042.0 −0.838142
\(592\) 5344.00 0.371009
\(593\) −18342.0 −1.27018 −0.635089 0.772439i \(-0.719037\pi\)
−0.635089 + 0.772439i \(0.719037\pi\)
\(594\) −2592.00 −0.179042
\(595\) 0 0
\(596\) 5640.00 0.387623
\(597\) −6000.00 −0.411329
\(598\) −288.000 −0.0196943
\(599\) 24600.0 1.67801 0.839006 0.544123i \(-0.183137\pi\)
0.839006 + 0.544123i \(0.183137\pi\)
\(600\) 0 0
\(601\) −8998.00 −0.610709 −0.305354 0.952239i \(-0.598775\pi\)
−0.305354 + 0.952239i \(0.598775\pi\)
\(602\) −2144.00 −0.145154
\(603\) −5364.00 −0.362254
\(604\) −8512.00 −0.573424
\(605\) 0 0
\(606\) 10332.0 0.692589
\(607\) −4076.00 −0.272553 −0.136277 0.990671i \(-0.543514\pi\)
−0.136277 + 0.990671i \(0.543514\pi\)
\(608\) −4480.00 −0.298829
\(609\) −2520.00 −0.167677
\(610\) 0 0
\(611\) 432.000 0.0286037
\(612\) 4104.00 0.271069
\(613\) 4078.00 0.268693 0.134347 0.990934i \(-0.457106\pi\)
0.134347 + 0.990934i \(0.457106\pi\)
\(614\) −13928.0 −0.915453
\(615\) 0 0
\(616\) 1536.00 0.100466
\(617\) −10086.0 −0.658099 −0.329049 0.944313i \(-0.606728\pi\)
−0.329049 + 0.944313i \(0.606728\pi\)
\(618\) −6312.00 −0.410851
\(619\) 8780.00 0.570110 0.285055 0.958511i \(-0.407988\pi\)
0.285055 + 0.958511i \(0.407988\pi\)
\(620\) 0 0
\(621\) 1944.00 0.125620
\(622\) −5664.00 −0.365122
\(623\) 840.000 0.0540191
\(624\) 96.0000 0.00615878
\(625\) 0 0
\(626\) 17284.0 1.10353
\(627\) 20160.0 1.28407
\(628\) −12104.0 −0.769112
\(629\) 38076.0 2.41366
\(630\) 0 0
\(631\) 2792.00 0.176145 0.0880727 0.996114i \(-0.471929\pi\)
0.0880727 + 0.996114i \(0.471929\pi\)
\(632\) 5120.00 0.322251
\(633\) 11604.0 0.728622
\(634\) −4428.00 −0.277379
\(635\) 0 0
\(636\) −936.000 −0.0583566
\(637\) 654.000 0.0406788
\(638\) 20160.0 1.25101
\(639\) −6912.00 −0.427910
\(640\) 0 0
\(641\) 7602.00 0.468426 0.234213 0.972185i \(-0.424749\pi\)
0.234213 + 0.972185i \(0.424749\pi\)
\(642\) 3384.00 0.208031
\(643\) −24212.0 −1.48496 −0.742479 0.669869i \(-0.766350\pi\)
−0.742479 + 0.669869i \(0.766350\pi\)
\(644\) −1152.00 −0.0704894
\(645\) 0 0
\(646\) −31920.0 −1.94408
\(647\) −9456.00 −0.574581 −0.287290 0.957844i \(-0.592754\pi\)
−0.287290 + 0.957844i \(0.592754\pi\)
\(648\) −648.000 −0.0392837
\(649\) −11520.0 −0.696764
\(650\) 0 0
\(651\) −3264.00 −0.196507
\(652\) −10448.0 −0.627569
\(653\) 9558.00 0.572792 0.286396 0.958111i \(-0.407543\pi\)
0.286396 + 0.958111i \(0.407543\pi\)
\(654\) −3660.00 −0.218834
\(655\) 0 0
\(656\) −3168.00 −0.188551
\(657\) 4302.00 0.255460
\(658\) 1728.00 0.102378
\(659\) −29280.0 −1.73078 −0.865392 0.501095i \(-0.832931\pi\)
−0.865392 + 0.501095i \(0.832931\pi\)
\(660\) 0 0
\(661\) −29098.0 −1.71223 −0.856113 0.516789i \(-0.827127\pi\)
−0.856113 + 0.516789i \(0.827127\pi\)
\(662\) −21544.0 −1.26485
\(663\) 684.000 0.0400669
\(664\) −2784.00 −0.162711
\(665\) 0 0
\(666\) −6012.00 −0.349790
\(667\) −15120.0 −0.877734
\(668\) 96.0000 0.00556041
\(669\) −9444.00 −0.545779
\(670\) 0 0
\(671\) −14496.0 −0.833997
\(672\) 384.000 0.0220433
\(673\) 11638.0 0.666585 0.333293 0.942823i \(-0.391840\pi\)
0.333293 + 0.942823i \(0.391840\pi\)
\(674\) −3308.00 −0.189050
\(675\) 0 0
\(676\) −8772.00 −0.499090
\(677\) −3426.00 −0.194493 −0.0972466 0.995260i \(-0.531004\pi\)
−0.0972466 + 0.995260i \(0.531004\pi\)
\(678\) −7812.00 −0.442505
\(679\) 6136.00 0.346801
\(680\) 0 0
\(681\) 7668.00 0.431481
\(682\) 26112.0 1.46610
\(683\) 20148.0 1.12876 0.564379 0.825516i \(-0.309116\pi\)
0.564379 + 0.825516i \(0.309116\pi\)
\(684\) 5040.00 0.281739
\(685\) 0 0
\(686\) 5360.00 0.298317
\(687\) 1830.00 0.101629
\(688\) 4288.00 0.237614
\(689\) −156.000 −0.00862573
\(690\) 0 0
\(691\) −29428.0 −1.62011 −0.810053 0.586356i \(-0.800562\pi\)
−0.810053 + 0.586356i \(0.800562\pi\)
\(692\) −7848.00 −0.431122
\(693\) −1728.00 −0.0947205
\(694\) 4392.00 0.240228
\(695\) 0 0
\(696\) 5040.00 0.274484
\(697\) −22572.0 −1.22665
\(698\) −16540.0 −0.896917
\(699\) −6174.00 −0.334080
\(700\) 0 0
\(701\) 16242.0 0.875110 0.437555 0.899192i \(-0.355845\pi\)
0.437555 + 0.899192i \(0.355845\pi\)
\(702\) −108.000 −0.00580655
\(703\) 46760.0 2.50866
\(704\) −3072.00 −0.164461
\(705\) 0 0
\(706\) 20604.0 1.09836
\(707\) 6888.00 0.366407
\(708\) −2880.00 −0.152877
\(709\) 2030.00 0.107529 0.0537646 0.998554i \(-0.482878\pi\)
0.0537646 + 0.998554i \(0.482878\pi\)
\(710\) 0 0
\(711\) −5760.00 −0.303821
\(712\) −1680.00 −0.0884279
\(713\) −19584.0 −1.02865
\(714\) 2736.00 0.143406
\(715\) 0 0
\(716\) −480.000 −0.0250537
\(717\) −14760.0 −0.768790
\(718\) 4560.00 0.237016
\(719\) 6960.00 0.361007 0.180504 0.983574i \(-0.442227\pi\)
0.180504 + 0.983574i \(0.442227\pi\)
\(720\) 0 0
\(721\) −4208.00 −0.217357
\(722\) −25482.0 −1.31349
\(723\) 4314.00 0.221908
\(724\) 3608.00 0.185208
\(725\) 0 0
\(726\) 5838.00 0.298441
\(727\) −18596.0 −0.948676 −0.474338 0.880343i \(-0.657313\pi\)
−0.474338 + 0.880343i \(0.657313\pi\)
\(728\) 64.0000 0.00325824
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 30552.0 1.54584
\(732\) −3624.00 −0.182988
\(733\) −21242.0 −1.07038 −0.535192 0.844731i \(-0.679761\pi\)
−0.535192 + 0.844731i \(0.679761\pi\)
\(734\) −17528.0 −0.881431
\(735\) 0 0
\(736\) 2304.00 0.115389
\(737\) 28608.0 1.42984
\(738\) 3564.00 0.177768
\(739\) −340.000 −0.0169244 −0.00846218 0.999964i \(-0.502694\pi\)
−0.00846218 + 0.999964i \(0.502694\pi\)
\(740\) 0 0
\(741\) 840.000 0.0416440
\(742\) −624.000 −0.0308730
\(743\) 21888.0 1.08074 0.540372 0.841426i \(-0.318284\pi\)
0.540372 + 0.841426i \(0.318284\pi\)
\(744\) 6528.00 0.321678
\(745\) 0 0
\(746\) −2636.00 −0.129371
\(747\) 3132.00 0.153405
\(748\) −21888.0 −1.06993
\(749\) 2256.00 0.110057
\(750\) 0 0
\(751\) 17792.0 0.864500 0.432250 0.901754i \(-0.357720\pi\)
0.432250 + 0.901754i \(0.357720\pi\)
\(752\) −3456.00 −0.167590
\(753\) −2376.00 −0.114988
\(754\) 840.000 0.0405716
\(755\) 0 0
\(756\) −432.000 −0.0207827
\(757\) −37346.0 −1.79308 −0.896541 0.442960i \(-0.853928\pi\)
−0.896541 + 0.442960i \(0.853928\pi\)
\(758\) −2200.00 −0.105419
\(759\) −10368.0 −0.495829
\(760\) 0 0
\(761\) −11358.0 −0.541034 −0.270517 0.962715i \(-0.587195\pi\)
−0.270517 + 0.962715i \(0.587195\pi\)
\(762\) 744.000 0.0353704
\(763\) −2440.00 −0.115772
\(764\) −672.000 −0.0318221
\(765\) 0 0
\(766\) −7056.00 −0.332825
\(767\) −480.000 −0.0225969
\(768\) −768.000 −0.0360844
\(769\) −34270.0 −1.60703 −0.803516 0.595283i \(-0.797040\pi\)
−0.803516 + 0.595283i \(0.797040\pi\)
\(770\) 0 0
\(771\) 6498.00 0.303528
\(772\) 5272.00 0.245782
\(773\) 13278.0 0.617822 0.308911 0.951091i \(-0.400035\pi\)
0.308911 + 0.951091i \(0.400035\pi\)
\(774\) −4824.00 −0.224025
\(775\) 0 0
\(776\) −12272.0 −0.567705
\(777\) −4008.00 −0.185053
\(778\) 19260.0 0.887538
\(779\) −27720.0 −1.27493
\(780\) 0 0
\(781\) 36864.0 1.68899
\(782\) 16416.0 0.750684
\(783\) −5670.00 −0.258786
\(784\) −5232.00 −0.238338
\(785\) 0 0
\(786\) 1152.00 0.0522780
\(787\) 11164.0 0.505659 0.252829 0.967511i \(-0.418639\pi\)
0.252829 + 0.967511i \(0.418639\pi\)
\(788\) 16056.0 0.725852
\(789\) 9576.00 0.432084
\(790\) 0 0
\(791\) −5208.00 −0.234103
\(792\) 3456.00 0.155055
\(793\) −604.000 −0.0270475
\(794\) −6188.00 −0.276579
\(795\) 0 0
\(796\) 8000.00 0.356222
\(797\) 5094.00 0.226397 0.113199 0.993572i \(-0.463890\pi\)
0.113199 + 0.993572i \(0.463890\pi\)
\(798\) 3360.00 0.149051
\(799\) −24624.0 −1.09028
\(800\) 0 0
\(801\) 1890.00 0.0833706
\(802\) 3276.00 0.144239
\(803\) −22944.0 −1.00831
\(804\) 7152.00 0.313721
\(805\) 0 0
\(806\) 1088.00 0.0475474
\(807\) −16470.0 −0.718428
\(808\) −13776.0 −0.599799
\(809\) −8790.00 −0.382002 −0.191001 0.981590i \(-0.561173\pi\)
−0.191001 + 0.981590i \(0.561173\pi\)
\(810\) 0 0
\(811\) 5852.00 0.253380 0.126690 0.991942i \(-0.459565\pi\)
0.126690 + 0.991942i \(0.459565\pi\)
\(812\) 3360.00 0.145213
\(813\) 18984.0 0.818940
\(814\) 32064.0 1.38064
\(815\) 0 0
\(816\) −5472.00 −0.234753
\(817\) 37520.0 1.60668
\(818\) 27500.0 1.17545
\(819\) −72.0000 −0.00307190
\(820\) 0 0
\(821\) −29478.0 −1.25309 −0.626546 0.779384i \(-0.715532\pi\)
−0.626546 + 0.779384i \(0.715532\pi\)
\(822\) 15084.0 0.640042
\(823\) −39332.0 −1.66589 −0.832945 0.553356i \(-0.813347\pi\)
−0.832945 + 0.553356i \(0.813347\pi\)
\(824\) 8416.00 0.355807
\(825\) 0 0
\(826\) −1920.00 −0.0808781
\(827\) −6756.00 −0.284074 −0.142037 0.989861i \(-0.545365\pi\)
−0.142037 + 0.989861i \(0.545365\pi\)
\(828\) −2592.00 −0.108790
\(829\) 3950.00 0.165488 0.0827438 0.996571i \(-0.473632\pi\)
0.0827438 + 0.996571i \(0.473632\pi\)
\(830\) 0 0
\(831\) −1722.00 −0.0718839
\(832\) −128.000 −0.00533366
\(833\) −37278.0 −1.55055
\(834\) 8040.00 0.333816
\(835\) 0 0
\(836\) −26880.0 −1.11204
\(837\) −7344.00 −0.303280
\(838\) 24960.0 1.02891
\(839\) 12360.0 0.508599 0.254300 0.967126i \(-0.418155\pi\)
0.254300 + 0.967126i \(0.418155\pi\)
\(840\) 0 0
\(841\) 19711.0 0.808192
\(842\) −14524.0 −0.594454
\(843\) −12726.0 −0.519937
\(844\) −15472.0 −0.631005
\(845\) 0 0
\(846\) 3888.00 0.158005
\(847\) 3892.00 0.157887
\(848\) 1248.00 0.0505383
\(849\) −1884.00 −0.0761587
\(850\) 0 0
\(851\) −24048.0 −0.968690
\(852\) 9216.00 0.370581
\(853\) 35998.0 1.44496 0.722478 0.691394i \(-0.243003\pi\)
0.722478 + 0.691394i \(0.243003\pi\)
\(854\) −2416.00 −0.0968077
\(855\) 0 0
\(856\) −4512.00 −0.180160
\(857\) 21594.0 0.860720 0.430360 0.902657i \(-0.358387\pi\)
0.430360 + 0.902657i \(0.358387\pi\)
\(858\) 576.000 0.0229188
\(859\) 9260.00 0.367808 0.183904 0.982944i \(-0.441126\pi\)
0.183904 + 0.982944i \(0.441126\pi\)
\(860\) 0 0
\(861\) 2376.00 0.0940463
\(862\) −19584.0 −0.773821
\(863\) −31632.0 −1.24770 −0.623850 0.781544i \(-0.714433\pi\)
−0.623850 + 0.781544i \(0.714433\pi\)
\(864\) 864.000 0.0340207
\(865\) 0 0
\(866\) 3604.00 0.141419
\(867\) −24249.0 −0.949872
\(868\) 4352.00 0.170180
\(869\) 30720.0 1.19920
\(870\) 0 0
\(871\) 1192.00 0.0463713
\(872\) 4880.00 0.189516
\(873\) 13806.0 0.535237
\(874\) 20160.0 0.780231
\(875\) 0 0
\(876\) −5736.00 −0.221235
\(877\) 39694.0 1.52836 0.764180 0.645003i \(-0.223144\pi\)
0.764180 + 0.645003i \(0.223144\pi\)
\(878\) 4640.00 0.178351
\(879\) −1674.00 −0.0642351
\(880\) 0 0
\(881\) 1242.00 0.0474961 0.0237480 0.999718i \(-0.492440\pi\)
0.0237480 + 0.999718i \(0.492440\pi\)
\(882\) 5886.00 0.224707
\(883\) 2668.00 0.101682 0.0508411 0.998707i \(-0.483810\pi\)
0.0508411 + 0.998707i \(0.483810\pi\)
\(884\) −912.000 −0.0346990
\(885\) 0 0
\(886\) 22344.0 0.847248
\(887\) 4344.00 0.164439 0.0822194 0.996614i \(-0.473799\pi\)
0.0822194 + 0.996614i \(0.473799\pi\)
\(888\) 8016.00 0.302927
\(889\) 496.000 0.0187124
\(890\) 0 0
\(891\) −3888.00 −0.146187
\(892\) 12592.0 0.472658
\(893\) −30240.0 −1.13319
\(894\) 8460.00 0.316493
\(895\) 0 0
\(896\) −512.000 −0.0190901
\(897\) −432.000 −0.0160803
\(898\) −13620.0 −0.506131
\(899\) 57120.0 2.11909
\(900\) 0 0
\(901\) 8892.00 0.328785
\(902\) −19008.0 −0.701660
\(903\) −3216.00 −0.118518
\(904\) 10416.0 0.383220
\(905\) 0 0
\(906\) −12768.0 −0.468199
\(907\) −4436.00 −0.162398 −0.0811990 0.996698i \(-0.525875\pi\)
−0.0811990 + 0.996698i \(0.525875\pi\)
\(908\) −10224.0 −0.373673
\(909\) 15498.0 0.565496
\(910\) 0 0
\(911\) 22752.0 0.827450 0.413725 0.910402i \(-0.364227\pi\)
0.413725 + 0.910402i \(0.364227\pi\)
\(912\) −6720.00 −0.243993
\(913\) −16704.0 −0.605500
\(914\) 34132.0 1.23521
\(915\) 0 0
\(916\) −2440.00 −0.0880130
\(917\) 768.000 0.0276571
\(918\) 6156.00 0.221327
\(919\) −27160.0 −0.974892 −0.487446 0.873153i \(-0.662071\pi\)
−0.487446 + 0.873153i \(0.662071\pi\)
\(920\) 0 0
\(921\) −20892.0 −0.747465
\(922\) 37836.0 1.35148
\(923\) 1536.00 0.0547758
\(924\) 2304.00 0.0820303
\(925\) 0 0
\(926\) 2104.00 0.0746671
\(927\) −9468.00 −0.335458
\(928\) −6720.00 −0.237710
\(929\) −33030.0 −1.16650 −0.583250 0.812292i \(-0.698219\pi\)
−0.583250 + 0.812292i \(0.698219\pi\)
\(930\) 0 0
\(931\) −45780.0 −1.61158
\(932\) 8232.00 0.289322
\(933\) −8496.00 −0.298121
\(934\) 22152.0 0.776055
\(935\) 0 0
\(936\) 144.000 0.00502862
\(937\) 29974.0 1.04505 0.522523 0.852625i \(-0.324991\pi\)
0.522523 + 0.852625i \(0.324991\pi\)
\(938\) 4768.00 0.165971
\(939\) 25926.0 0.901026
\(940\) 0 0
\(941\) 13962.0 0.483686 0.241843 0.970315i \(-0.422248\pi\)
0.241843 + 0.970315i \(0.422248\pi\)
\(942\) −18156.0 −0.627977
\(943\) 14256.0 0.492300
\(944\) 3840.00 0.132396
\(945\) 0 0
\(946\) 25728.0 0.884238
\(947\) −35196.0 −1.20773 −0.603863 0.797088i \(-0.706373\pi\)
−0.603863 + 0.797088i \(0.706373\pi\)
\(948\) 7680.00 0.263117
\(949\) −956.000 −0.0327008
\(950\) 0 0
\(951\) −6642.00 −0.226479
\(952\) −3648.00 −0.124194
\(953\) 28338.0 0.963230 0.481615 0.876383i \(-0.340050\pi\)
0.481615 + 0.876383i \(0.340050\pi\)
\(954\) −1404.00 −0.0476480
\(955\) 0 0
\(956\) 19680.0 0.665792
\(957\) 30240.0 1.02144
\(958\) 18000.0 0.607050
\(959\) 10056.0 0.338608
\(960\) 0 0
\(961\) 44193.0 1.48343
\(962\) 1336.00 0.0447759
\(963\) 5076.00 0.169857
\(964\) −5752.00 −0.192178
\(965\) 0 0
\(966\) −1728.00 −0.0575543
\(967\) 17524.0 0.582765 0.291383 0.956607i \(-0.405885\pi\)
0.291383 + 0.956607i \(0.405885\pi\)
\(968\) −7784.00 −0.258458
\(969\) −47880.0 −1.58733
\(970\) 0 0
\(971\) −26808.0 −0.886004 −0.443002 0.896521i \(-0.646087\pi\)
−0.443002 + 0.896521i \(0.646087\pi\)
\(972\) −972.000 −0.0320750
\(973\) 5360.00 0.176602
\(974\) −17528.0 −0.576626
\(975\) 0 0
\(976\) 4832.00 0.158472
\(977\) 10914.0 0.357390 0.178695 0.983905i \(-0.442813\pi\)
0.178695 + 0.983905i \(0.442813\pi\)
\(978\) −15672.0 −0.512408
\(979\) −10080.0 −0.329069
\(980\) 0 0
\(981\) −5490.00 −0.178677
\(982\) −11184.0 −0.363438
\(983\) −22272.0 −0.722652 −0.361326 0.932440i \(-0.617676\pi\)
−0.361326 + 0.932440i \(0.617676\pi\)
\(984\) −4752.00 −0.153951
\(985\) 0 0
\(986\) −47880.0 −1.54646
\(987\) 2592.00 0.0835910
\(988\) −1120.00 −0.0360647
\(989\) −19296.0 −0.620402
\(990\) 0 0
\(991\) 14072.0 0.451071 0.225536 0.974235i \(-0.427587\pi\)
0.225536 + 0.974235i \(0.427587\pi\)
\(992\) −8704.00 −0.278581
\(993\) −32316.0 −1.03275
\(994\) 6144.00 0.196052
\(995\) 0 0
\(996\) −4176.00 −0.132853
\(997\) −4826.00 −0.153301 −0.0766504 0.997058i \(-0.524423\pi\)
−0.0766504 + 0.997058i \(0.524423\pi\)
\(998\) −9400.00 −0.298148
\(999\) −9018.00 −0.285602
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 150.4.a.b.1.1 1
3.2 odd 2 450.4.a.r.1.1 1
4.3 odd 2 1200.4.a.ba.1.1 1
5.2 odd 4 150.4.c.c.49.1 2
5.3 odd 4 150.4.c.c.49.2 2
5.4 even 2 30.4.a.b.1.1 1
15.2 even 4 450.4.c.j.199.2 2
15.8 even 4 450.4.c.j.199.1 2
15.14 odd 2 90.4.a.c.1.1 1
20.3 even 4 1200.4.f.r.49.2 2
20.7 even 4 1200.4.f.r.49.1 2
20.19 odd 2 240.4.a.b.1.1 1
35.34 odd 2 1470.4.a.r.1.1 1
40.19 odd 2 960.4.a.bg.1.1 1
40.29 even 2 960.4.a.n.1.1 1
45.4 even 6 810.4.e.i.541.1 2
45.14 odd 6 810.4.e.p.541.1 2
45.29 odd 6 810.4.e.p.271.1 2
45.34 even 6 810.4.e.i.271.1 2
60.59 even 2 720.4.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.a.b.1.1 1 5.4 even 2
90.4.a.c.1.1 1 15.14 odd 2
150.4.a.b.1.1 1 1.1 even 1 trivial
150.4.c.c.49.1 2 5.2 odd 4
150.4.c.c.49.2 2 5.3 odd 4
240.4.a.b.1.1 1 20.19 odd 2
450.4.a.r.1.1 1 3.2 odd 2
450.4.c.j.199.1 2 15.8 even 4
450.4.c.j.199.2 2 15.2 even 4
720.4.a.y.1.1 1 60.59 even 2
810.4.e.i.271.1 2 45.34 even 6
810.4.e.i.541.1 2 45.4 even 6
810.4.e.p.271.1 2 45.29 odd 6
810.4.e.p.541.1 2 45.14 odd 6
960.4.a.n.1.1 1 40.29 even 2
960.4.a.bg.1.1 1 40.19 odd 2
1200.4.a.ba.1.1 1 4.3 odd 2
1200.4.f.r.49.1 2 20.7 even 4
1200.4.f.r.49.2 2 20.3 even 4
1470.4.a.r.1.1 1 35.34 odd 2