Properties

Label 150.4.a.f.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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} -2.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} -2.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +70.0000 q^{11} -12.0000 q^{12} +54.0000 q^{13} -4.00000 q^{14} +16.0000 q^{16} -22.0000 q^{17} +18.0000 q^{18} +24.0000 q^{19} +6.00000 q^{21} +140.000 q^{22} -100.000 q^{23} -24.0000 q^{24} +108.000 q^{26} -27.0000 q^{27} -8.00000 q^{28} +216.000 q^{29} +208.000 q^{31} +32.0000 q^{32} -210.000 q^{33} -44.0000 q^{34} +36.0000 q^{36} -254.000 q^{37} +48.0000 q^{38} -162.000 q^{39} -206.000 q^{41} +12.0000 q^{42} +292.000 q^{43} +280.000 q^{44} -200.000 q^{46} -320.000 q^{47} -48.0000 q^{48} -339.000 q^{49} +66.0000 q^{51} +216.000 q^{52} -402.000 q^{53} -54.0000 q^{54} -16.0000 q^{56} -72.0000 q^{57} +432.000 q^{58} -370.000 q^{59} -550.000 q^{61} +416.000 q^{62} -18.0000 q^{63} +64.0000 q^{64} -420.000 q^{66} +728.000 q^{67} -88.0000 q^{68} +300.000 q^{69} -540.000 q^{71} +72.0000 q^{72} +604.000 q^{73} -508.000 q^{74} +96.0000 q^{76} -140.000 q^{77} -324.000 q^{78} +792.000 q^{79} +81.0000 q^{81} -412.000 q^{82} +404.000 q^{83} +24.0000 q^{84} +584.000 q^{86} -648.000 q^{87} +560.000 q^{88} -938.000 q^{89} -108.000 q^{91} -400.000 q^{92} -624.000 q^{93} -640.000 q^{94} -96.0000 q^{96} +56.0000 q^{97} -678.000 q^{98} +630.000 q^{99} +O(q^{100})\)

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
\(7\) −2.00000 −0.107990 −0.0539949 0.998541i \(-0.517195\pi\)
−0.0539949 + 0.998541i \(0.517195\pi\)
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 70.0000 1.91871 0.959354 0.282204i \(-0.0910657\pi\)
0.959354 + 0.282204i \(0.0910657\pi\)
\(12\) −12.0000 −0.288675
\(13\) 54.0000 1.15207 0.576035 0.817425i \(-0.304599\pi\)
0.576035 + 0.817425i \(0.304599\pi\)
\(14\) −4.00000 −0.0763604
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −22.0000 −0.313870 −0.156935 0.987609i \(-0.550161\pi\)
−0.156935 + 0.987609i \(0.550161\pi\)
\(18\) 18.0000 0.235702
\(19\) 24.0000 0.289788 0.144894 0.989447i \(-0.453716\pi\)
0.144894 + 0.989447i \(0.453716\pi\)
\(20\) 0 0
\(21\) 6.00000 0.0623480
\(22\) 140.000 1.35673
\(23\) −100.000 −0.906584 −0.453292 0.891362i \(-0.649751\pi\)
−0.453292 + 0.891362i \(0.649751\pi\)
\(24\) −24.0000 −0.204124
\(25\) 0 0
\(26\) 108.000 0.814636
\(27\) −27.0000 −0.192450
\(28\) −8.00000 −0.0539949
\(29\) 216.000 1.38311 0.691555 0.722324i \(-0.256926\pi\)
0.691555 + 0.722324i \(0.256926\pi\)
\(30\) 0 0
\(31\) 208.000 1.20509 0.602547 0.798084i \(-0.294153\pi\)
0.602547 + 0.798084i \(0.294153\pi\)
\(32\) 32.0000 0.176777
\(33\) −210.000 −1.10777
\(34\) −44.0000 −0.221939
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) −254.000 −1.12858 −0.564288 0.825578i \(-0.690849\pi\)
−0.564288 + 0.825578i \(0.690849\pi\)
\(38\) 48.0000 0.204911
\(39\) −162.000 −0.665148
\(40\) 0 0
\(41\) −206.000 −0.784678 −0.392339 0.919821i \(-0.628334\pi\)
−0.392339 + 0.919821i \(0.628334\pi\)
\(42\) 12.0000 0.0440867
\(43\) 292.000 1.03557 0.517786 0.855510i \(-0.326756\pi\)
0.517786 + 0.855510i \(0.326756\pi\)
\(44\) 280.000 0.959354
\(45\) 0 0
\(46\) −200.000 −0.641052
\(47\) −320.000 −0.993123 −0.496562 0.868001i \(-0.665404\pi\)
−0.496562 + 0.868001i \(0.665404\pi\)
\(48\) −48.0000 −0.144338
\(49\) −339.000 −0.988338
\(50\) 0 0
\(51\) 66.0000 0.181213
\(52\) 216.000 0.576035
\(53\) −402.000 −1.04187 −0.520933 0.853597i \(-0.674416\pi\)
−0.520933 + 0.853597i \(0.674416\pi\)
\(54\) −54.0000 −0.136083
\(55\) 0 0
\(56\) −16.0000 −0.0381802
\(57\) −72.0000 −0.167309
\(58\) 432.000 0.978007
\(59\) −370.000 −0.816439 −0.408219 0.912884i \(-0.633850\pi\)
−0.408219 + 0.912884i \(0.633850\pi\)
\(60\) 0 0
\(61\) −550.000 −1.15443 −0.577215 0.816592i \(-0.695861\pi\)
−0.577215 + 0.816592i \(0.695861\pi\)
\(62\) 416.000 0.852130
\(63\) −18.0000 −0.0359966
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −420.000 −0.783309
\(67\) 728.000 1.32745 0.663727 0.747975i \(-0.268974\pi\)
0.663727 + 0.747975i \(0.268974\pi\)
\(68\) −88.0000 −0.156935
\(69\) 300.000 0.523417
\(70\) 0 0
\(71\) −540.000 −0.902623 −0.451311 0.892367i \(-0.649044\pi\)
−0.451311 + 0.892367i \(0.649044\pi\)
\(72\) 72.0000 0.117851
\(73\) 604.000 0.968395 0.484198 0.874959i \(-0.339112\pi\)
0.484198 + 0.874959i \(0.339112\pi\)
\(74\) −508.000 −0.798024
\(75\) 0 0
\(76\) 96.0000 0.144894
\(77\) −140.000 −0.207201
\(78\) −324.000 −0.470330
\(79\) 792.000 1.12794 0.563968 0.825797i \(-0.309274\pi\)
0.563968 + 0.825797i \(0.309274\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −412.000 −0.554851
\(83\) 404.000 0.534274 0.267137 0.963659i \(-0.413922\pi\)
0.267137 + 0.963659i \(0.413922\pi\)
\(84\) 24.0000 0.0311740
\(85\) 0 0
\(86\) 584.000 0.732260
\(87\) −648.000 −0.798539
\(88\) 560.000 0.678366
\(89\) −938.000 −1.11717 −0.558583 0.829449i \(-0.688655\pi\)
−0.558583 + 0.829449i \(0.688655\pi\)
\(90\) 0 0
\(91\) −108.000 −0.124412
\(92\) −400.000 −0.453292
\(93\) −624.000 −0.695761
\(94\) −640.000 −0.702244
\(95\) 0 0
\(96\) −96.0000 −0.102062
\(97\) 56.0000 0.0586179 0.0293090 0.999570i \(-0.490669\pi\)
0.0293090 + 0.999570i \(0.490669\pi\)
\(98\) −678.000 −0.698861
\(99\) 630.000 0.639570
\(100\) 0 0
\(101\) −592.000 −0.583230 −0.291615 0.956536i \(-0.594193\pi\)
−0.291615 + 0.956536i \(0.594193\pi\)
\(102\) 132.000 0.128137
\(103\) 62.0000 0.0593111 0.0296555 0.999560i \(-0.490559\pi\)
0.0296555 + 0.999560i \(0.490559\pi\)
\(104\) 432.000 0.407318
\(105\) 0 0
\(106\) −804.000 −0.736711
\(107\) 84.0000 0.0758933 0.0379467 0.999280i \(-0.487918\pi\)
0.0379467 + 0.999280i \(0.487918\pi\)
\(108\) −108.000 −0.0962250
\(109\) 370.000 0.325134 0.162567 0.986698i \(-0.448023\pi\)
0.162567 + 0.986698i \(0.448023\pi\)
\(110\) 0 0
\(111\) 762.000 0.651584
\(112\) −32.0000 −0.0269975
\(113\) −1746.00 −1.45354 −0.726769 0.686882i \(-0.758979\pi\)
−0.726769 + 0.686882i \(0.758979\pi\)
\(114\) −144.000 −0.118306
\(115\) 0 0
\(116\) 864.000 0.691555
\(117\) 486.000 0.384023
\(118\) −740.000 −0.577310
\(119\) 44.0000 0.0338947
\(120\) 0 0
\(121\) 3569.00 2.68144
\(122\) −1100.00 −0.816306
\(123\) 618.000 0.453034
\(124\) 832.000 0.602547
\(125\) 0 0
\(126\) −36.0000 −0.0254535
\(127\) 1630.00 1.13889 0.569445 0.822029i \(-0.307158\pi\)
0.569445 + 0.822029i \(0.307158\pi\)
\(128\) 128.000 0.0883883
\(129\) −876.000 −0.597888
\(130\) 0 0
\(131\) −870.000 −0.580246 −0.290123 0.956989i \(-0.593696\pi\)
−0.290123 + 0.956989i \(0.593696\pi\)
\(132\) −840.000 −0.553883
\(133\) −48.0000 −0.0312942
\(134\) 1456.00 0.938651
\(135\) 0 0
\(136\) −176.000 −0.110970
\(137\) 918.000 0.572482 0.286241 0.958158i \(-0.407594\pi\)
0.286241 + 0.958158i \(0.407594\pi\)
\(138\) 600.000 0.370112
\(139\) −596.000 −0.363684 −0.181842 0.983328i \(-0.558206\pi\)
−0.181842 + 0.983328i \(0.558206\pi\)
\(140\) 0 0
\(141\) 960.000 0.573380
\(142\) −1080.00 −0.638251
\(143\) 3780.00 2.21049
\(144\) 144.000 0.0833333
\(145\) 0 0
\(146\) 1208.00 0.684759
\(147\) 1017.00 0.570617
\(148\) −1016.00 −0.564288
\(149\) 1076.00 0.591606 0.295803 0.955249i \(-0.404413\pi\)
0.295803 + 0.955249i \(0.404413\pi\)
\(150\) 0 0
\(151\) −32.0000 −0.0172458 −0.00862292 0.999963i \(-0.502745\pi\)
−0.00862292 + 0.999963i \(0.502745\pi\)
\(152\) 192.000 0.102456
\(153\) −198.000 −0.104623
\(154\) −280.000 −0.146513
\(155\) 0 0
\(156\) −648.000 −0.332574
\(157\) −2554.00 −1.29829 −0.649145 0.760665i \(-0.724873\pi\)
−0.649145 + 0.760665i \(0.724873\pi\)
\(158\) 1584.00 0.797571
\(159\) 1206.00 0.601522
\(160\) 0 0
\(161\) 200.000 0.0979019
\(162\) 162.000 0.0785674
\(163\) 752.000 0.361357 0.180678 0.983542i \(-0.442171\pi\)
0.180678 + 0.983542i \(0.442171\pi\)
\(164\) −824.000 −0.392339
\(165\) 0 0
\(166\) 808.000 0.377789
\(167\) 2700.00 1.25109 0.625546 0.780188i \(-0.284876\pi\)
0.625546 + 0.780188i \(0.284876\pi\)
\(168\) 48.0000 0.0220433
\(169\) 719.000 0.327264
\(170\) 0 0
\(171\) 216.000 0.0965961
\(172\) 1168.00 0.517786
\(173\) −1334.00 −0.586255 −0.293128 0.956073i \(-0.594696\pi\)
−0.293128 + 0.956073i \(0.594696\pi\)
\(174\) −1296.00 −0.564652
\(175\) 0 0
\(176\) 1120.00 0.479677
\(177\) 1110.00 0.471371
\(178\) −1876.00 −0.789956
\(179\) −1714.00 −0.715700 −0.357850 0.933779i \(-0.616490\pi\)
−0.357850 + 0.933779i \(0.616490\pi\)
\(180\) 0 0
\(181\) −4006.00 −1.64510 −0.822551 0.568691i \(-0.807450\pi\)
−0.822551 + 0.568691i \(0.807450\pi\)
\(182\) −216.000 −0.0879724
\(183\) 1650.00 0.666511
\(184\) −800.000 −0.320526
\(185\) 0 0
\(186\) −1248.00 −0.491977
\(187\) −1540.00 −0.602224
\(188\) −1280.00 −0.496562
\(189\) 54.0000 0.0207827
\(190\) 0 0
\(191\) −684.000 −0.259123 −0.129562 0.991571i \(-0.541357\pi\)
−0.129562 + 0.991571i \(0.541357\pi\)
\(192\) −192.000 −0.0721688
\(193\) −4484.00 −1.67236 −0.836180 0.548455i \(-0.815216\pi\)
−0.836180 + 0.548455i \(0.815216\pi\)
\(194\) 112.000 0.0414491
\(195\) 0 0
\(196\) −1356.00 −0.494169
\(197\) −1058.00 −0.382636 −0.191318 0.981528i \(-0.561276\pi\)
−0.191318 + 0.981528i \(0.561276\pi\)
\(198\) 1260.00 0.452244
\(199\) −1128.00 −0.401818 −0.200909 0.979610i \(-0.564390\pi\)
−0.200909 + 0.979610i \(0.564390\pi\)
\(200\) 0 0
\(201\) −2184.00 −0.766405
\(202\) −1184.00 −0.412406
\(203\) −432.000 −0.149362
\(204\) 264.000 0.0906064
\(205\) 0 0
\(206\) 124.000 0.0419393
\(207\) −900.000 −0.302195
\(208\) 864.000 0.288017
\(209\) 1680.00 0.556019
\(210\) 0 0
\(211\) 780.000 0.254490 0.127245 0.991871i \(-0.459387\pi\)
0.127245 + 0.991871i \(0.459387\pi\)
\(212\) −1608.00 −0.520933
\(213\) 1620.00 0.521129
\(214\) 168.000 0.0536647
\(215\) 0 0
\(216\) −216.000 −0.0680414
\(217\) −416.000 −0.130138
\(218\) 740.000 0.229904
\(219\) −1812.00 −0.559103
\(220\) 0 0
\(221\) −1188.00 −0.361600
\(222\) 1524.00 0.460740
\(223\) −2570.00 −0.771749 −0.385874 0.922551i \(-0.626100\pi\)
−0.385874 + 0.922551i \(0.626100\pi\)
\(224\) −64.0000 −0.0190901
\(225\) 0 0
\(226\) −3492.00 −1.02781
\(227\) −2836.00 −0.829216 −0.414608 0.910000i \(-0.636081\pi\)
−0.414608 + 0.910000i \(0.636081\pi\)
\(228\) −288.000 −0.0836547
\(229\) −610.000 −0.176026 −0.0880130 0.996119i \(-0.528052\pi\)
−0.0880130 + 0.996119i \(0.528052\pi\)
\(230\) 0 0
\(231\) 420.000 0.119628
\(232\) 1728.00 0.489003
\(233\) 3514.00 0.988025 0.494012 0.869455i \(-0.335530\pi\)
0.494012 + 0.869455i \(0.335530\pi\)
\(234\) 972.000 0.271545
\(235\) 0 0
\(236\) −1480.00 −0.408219
\(237\) −2376.00 −0.651214
\(238\) 88.0000 0.0239672
\(239\) −1844.00 −0.499073 −0.249536 0.968365i \(-0.580278\pi\)
−0.249536 + 0.968365i \(0.580278\pi\)
\(240\) 0 0
\(241\) 982.000 0.262474 0.131237 0.991351i \(-0.458105\pi\)
0.131237 + 0.991351i \(0.458105\pi\)
\(242\) 7138.00 1.89607
\(243\) −243.000 −0.0641500
\(244\) −2200.00 −0.577215
\(245\) 0 0
\(246\) 1236.00 0.320343
\(247\) 1296.00 0.333856
\(248\) 1664.00 0.426065
\(249\) −1212.00 −0.308463
\(250\) 0 0
\(251\) −3174.00 −0.798172 −0.399086 0.916914i \(-0.630672\pi\)
−0.399086 + 0.916914i \(0.630672\pi\)
\(252\) −72.0000 −0.0179983
\(253\) −7000.00 −1.73947
\(254\) 3260.00 0.805317
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −1194.00 −0.289804 −0.144902 0.989446i \(-0.546287\pi\)
−0.144902 + 0.989446i \(0.546287\pi\)
\(258\) −1752.00 −0.422770
\(259\) 508.000 0.121875
\(260\) 0 0
\(261\) 1944.00 0.461037
\(262\) −1740.00 −0.410296
\(263\) 140.000 0.0328242 0.0164121 0.999865i \(-0.494776\pi\)
0.0164121 + 0.999865i \(0.494776\pi\)
\(264\) −1680.00 −0.391655
\(265\) 0 0
\(266\) −96.0000 −0.0221283
\(267\) 2814.00 0.644996
\(268\) 2912.00 0.663727
\(269\) 5256.00 1.19132 0.595658 0.803238i \(-0.296891\pi\)
0.595658 + 0.803238i \(0.296891\pi\)
\(270\) 0 0
\(271\) 544.000 0.121940 0.0609698 0.998140i \(-0.480581\pi\)
0.0609698 + 0.998140i \(0.480581\pi\)
\(272\) −352.000 −0.0784674
\(273\) 324.000 0.0718292
\(274\) 1836.00 0.404806
\(275\) 0 0
\(276\) 1200.00 0.261708
\(277\) 946.000 0.205197 0.102599 0.994723i \(-0.467284\pi\)
0.102599 + 0.994723i \(0.467284\pi\)
\(278\) −1192.00 −0.257163
\(279\) 1872.00 0.401698
\(280\) 0 0
\(281\) 1278.00 0.271313 0.135657 0.990756i \(-0.456686\pi\)
0.135657 + 0.990756i \(0.456686\pi\)
\(282\) 1920.00 0.405441
\(283\) 7424.00 1.55940 0.779701 0.626152i \(-0.215371\pi\)
0.779701 + 0.626152i \(0.215371\pi\)
\(284\) −2160.00 −0.451311
\(285\) 0 0
\(286\) 7560.00 1.56305
\(287\) 412.000 0.0847373
\(288\) 288.000 0.0589256
\(289\) −4429.00 −0.901486
\(290\) 0 0
\(291\) −168.000 −0.0338431
\(292\) 2416.00 0.484198
\(293\) −1362.00 −0.271566 −0.135783 0.990739i \(-0.543355\pi\)
−0.135783 + 0.990739i \(0.543355\pi\)
\(294\) 2034.00 0.403487
\(295\) 0 0
\(296\) −2032.00 −0.399012
\(297\) −1890.00 −0.369256
\(298\) 2152.00 0.418329
\(299\) −5400.00 −1.04445
\(300\) 0 0
\(301\) −584.000 −0.111831
\(302\) −64.0000 −0.0121947
\(303\) 1776.00 0.336728
\(304\) 384.000 0.0724471
\(305\) 0 0
\(306\) −396.000 −0.0739798
\(307\) −7740.00 −1.43891 −0.719455 0.694539i \(-0.755608\pi\)
−0.719455 + 0.694539i \(0.755608\pi\)
\(308\) −560.000 −0.103601
\(309\) −186.000 −0.0342433
\(310\) 0 0
\(311\) 4980.00 0.908006 0.454003 0.891000i \(-0.349996\pi\)
0.454003 + 0.891000i \(0.349996\pi\)
\(312\) −1296.00 −0.235165
\(313\) 604.000 0.109074 0.0545369 0.998512i \(-0.482632\pi\)
0.0545369 + 0.998512i \(0.482632\pi\)
\(314\) −5108.00 −0.918029
\(315\) 0 0
\(316\) 3168.00 0.563968
\(317\) −8566.00 −1.51771 −0.758856 0.651259i \(-0.774241\pi\)
−0.758856 + 0.651259i \(0.774241\pi\)
\(318\) 2412.00 0.425340
\(319\) 15120.0 2.65379
\(320\) 0 0
\(321\) −252.000 −0.0438170
\(322\) 400.000 0.0692271
\(323\) −528.000 −0.0909557
\(324\) 324.000 0.0555556
\(325\) 0 0
\(326\) 1504.00 0.255518
\(327\) −1110.00 −0.187716
\(328\) −1648.00 −0.277426
\(329\) 640.000 0.107247
\(330\) 0 0
\(331\) 3472.00 0.576551 0.288275 0.957548i \(-0.406918\pi\)
0.288275 + 0.957548i \(0.406918\pi\)
\(332\) 1616.00 0.267137
\(333\) −2286.00 −0.376192
\(334\) 5400.00 0.884655
\(335\) 0 0
\(336\) 96.0000 0.0155870
\(337\) 5668.00 0.916189 0.458094 0.888904i \(-0.348532\pi\)
0.458094 + 0.888904i \(0.348532\pi\)
\(338\) 1438.00 0.231411
\(339\) 5238.00 0.839201
\(340\) 0 0
\(341\) 14560.0 2.31222
\(342\) 432.000 0.0683038
\(343\) 1364.00 0.214720
\(344\) 2336.00 0.366130
\(345\) 0 0
\(346\) −2668.00 −0.414545
\(347\) 10836.0 1.67639 0.838194 0.545371i \(-0.183611\pi\)
0.838194 + 0.545371i \(0.183611\pi\)
\(348\) −2592.00 −0.399269
\(349\) −8990.00 −1.37886 −0.689432 0.724350i \(-0.742140\pi\)
−0.689432 + 0.724350i \(0.742140\pi\)
\(350\) 0 0
\(351\) −1458.00 −0.221716
\(352\) 2240.00 0.339183
\(353\) −5078.00 −0.765651 −0.382825 0.923821i \(-0.625049\pi\)
−0.382825 + 0.923821i \(0.625049\pi\)
\(354\) 2220.00 0.333310
\(355\) 0 0
\(356\) −3752.00 −0.558583
\(357\) −132.000 −0.0195691
\(358\) −3428.00 −0.506077
\(359\) −3696.00 −0.543363 −0.271682 0.962387i \(-0.587580\pi\)
−0.271682 + 0.962387i \(0.587580\pi\)
\(360\) 0 0
\(361\) −6283.00 −0.916023
\(362\) −8012.00 −1.16326
\(363\) −10707.0 −1.54813
\(364\) −432.000 −0.0622059
\(365\) 0 0
\(366\) 3300.00 0.471294
\(367\) 286.000 0.0406787 0.0203393 0.999793i \(-0.493525\pi\)
0.0203393 + 0.999793i \(0.493525\pi\)
\(368\) −1600.00 −0.226646
\(369\) −1854.00 −0.261559
\(370\) 0 0
\(371\) 804.000 0.112511
\(372\) −2496.00 −0.347881
\(373\) 8262.00 1.14689 0.573445 0.819244i \(-0.305607\pi\)
0.573445 + 0.819244i \(0.305607\pi\)
\(374\) −3080.00 −0.425837
\(375\) 0 0
\(376\) −2560.00 −0.351122
\(377\) 11664.0 1.59344
\(378\) 108.000 0.0146956
\(379\) −2956.00 −0.400632 −0.200316 0.979731i \(-0.564197\pi\)
−0.200316 + 0.979731i \(0.564197\pi\)
\(380\) 0 0
\(381\) −4890.00 −0.657539
\(382\) −1368.00 −0.183228
\(383\) −5240.00 −0.699090 −0.349545 0.936920i \(-0.613664\pi\)
−0.349545 + 0.936920i \(0.613664\pi\)
\(384\) −384.000 −0.0510310
\(385\) 0 0
\(386\) −8968.00 −1.18254
\(387\) 2628.00 0.345191
\(388\) 224.000 0.0293090
\(389\) −884.000 −0.115220 −0.0576100 0.998339i \(-0.518348\pi\)
−0.0576100 + 0.998339i \(0.518348\pi\)
\(390\) 0 0
\(391\) 2200.00 0.284549
\(392\) −2712.00 −0.349430
\(393\) 2610.00 0.335005
\(394\) −2116.00 −0.270565
\(395\) 0 0
\(396\) 2520.00 0.319785
\(397\) 3394.00 0.429068 0.214534 0.976717i \(-0.431177\pi\)
0.214534 + 0.976717i \(0.431177\pi\)
\(398\) −2256.00 −0.284128
\(399\) 144.000 0.0180677
\(400\) 0 0
\(401\) −6826.00 −0.850060 −0.425030 0.905179i \(-0.639737\pi\)
−0.425030 + 0.905179i \(0.639737\pi\)
\(402\) −4368.00 −0.541930
\(403\) 11232.0 1.38835
\(404\) −2368.00 −0.291615
\(405\) 0 0
\(406\) −864.000 −0.105615
\(407\) −17780.0 −2.16541
\(408\) 528.000 0.0640684
\(409\) 7814.00 0.944688 0.472344 0.881414i \(-0.343408\pi\)
0.472344 + 0.881414i \(0.343408\pi\)
\(410\) 0 0
\(411\) −2754.00 −0.330523
\(412\) 248.000 0.0296555
\(413\) 740.000 0.0881671
\(414\) −1800.00 −0.213684
\(415\) 0 0
\(416\) 1728.00 0.203659
\(417\) 1788.00 0.209973
\(418\) 3360.00 0.393165
\(419\) 8290.00 0.966570 0.483285 0.875463i \(-0.339443\pi\)
0.483285 + 0.875463i \(0.339443\pi\)
\(420\) 0 0
\(421\) 2110.00 0.244264 0.122132 0.992514i \(-0.461027\pi\)
0.122132 + 0.992514i \(0.461027\pi\)
\(422\) 1560.00 0.179952
\(423\) −2880.00 −0.331041
\(424\) −3216.00 −0.368356
\(425\) 0 0
\(426\) 3240.00 0.368494
\(427\) 1100.00 0.124667
\(428\) 336.000 0.0379467
\(429\) −11340.0 −1.27622
\(430\) 0 0
\(431\) −12080.0 −1.35005 −0.675027 0.737793i \(-0.735868\pi\)
−0.675027 + 0.737793i \(0.735868\pi\)
\(432\) −432.000 −0.0481125
\(433\) 16492.0 1.83038 0.915190 0.403022i \(-0.132040\pi\)
0.915190 + 0.403022i \(0.132040\pi\)
\(434\) −832.000 −0.0920214
\(435\) 0 0
\(436\) 1480.00 0.162567
\(437\) −2400.00 −0.262718
\(438\) −3624.00 −0.395346
\(439\) −15048.0 −1.63600 −0.817998 0.575222i \(-0.804916\pi\)
−0.817998 + 0.575222i \(0.804916\pi\)
\(440\) 0 0
\(441\) −3051.00 −0.329446
\(442\) −2376.00 −0.255690
\(443\) −9876.00 −1.05919 −0.529597 0.848249i \(-0.677657\pi\)
−0.529597 + 0.848249i \(0.677657\pi\)
\(444\) 3048.00 0.325792
\(445\) 0 0
\(446\) −5140.00 −0.545709
\(447\) −3228.00 −0.341564
\(448\) −128.000 −0.0134987
\(449\) 17166.0 1.80426 0.902131 0.431462i \(-0.142002\pi\)
0.902131 + 0.431462i \(0.142002\pi\)
\(450\) 0 0
\(451\) −14420.0 −1.50557
\(452\) −6984.00 −0.726769
\(453\) 96.0000 0.00995690
\(454\) −5672.00 −0.586344
\(455\) 0 0
\(456\) −576.000 −0.0591528
\(457\) 14848.0 1.51983 0.759913 0.650025i \(-0.225242\pi\)
0.759913 + 0.650025i \(0.225242\pi\)
\(458\) −1220.00 −0.124469
\(459\) 594.000 0.0604042
\(460\) 0 0
\(461\) −1260.00 −0.127297 −0.0636486 0.997972i \(-0.520274\pi\)
−0.0636486 + 0.997972i \(0.520274\pi\)
\(462\) 840.000 0.0845895
\(463\) −11238.0 −1.12802 −0.564011 0.825767i \(-0.690742\pi\)
−0.564011 + 0.825767i \(0.690742\pi\)
\(464\) 3456.00 0.345778
\(465\) 0 0
\(466\) 7028.00 0.698639
\(467\) 14772.0 1.46374 0.731870 0.681444i \(-0.238648\pi\)
0.731870 + 0.681444i \(0.238648\pi\)
\(468\) 1944.00 0.192012
\(469\) −1456.00 −0.143351
\(470\) 0 0
\(471\) 7662.00 0.749568
\(472\) −2960.00 −0.288655
\(473\) 20440.0 1.98696
\(474\) −4752.00 −0.460478
\(475\) 0 0
\(476\) 176.000 0.0169474
\(477\) −3618.00 −0.347289
\(478\) −3688.00 −0.352898
\(479\) −6116.00 −0.583397 −0.291699 0.956510i \(-0.594220\pi\)
−0.291699 + 0.956510i \(0.594220\pi\)
\(480\) 0 0
\(481\) −13716.0 −1.30020
\(482\) 1964.00 0.185597
\(483\) −600.000 −0.0565237
\(484\) 14276.0 1.34072
\(485\) 0 0
\(486\) −486.000 −0.0453609
\(487\) 15906.0 1.48002 0.740010 0.672596i \(-0.234821\pi\)
0.740010 + 0.672596i \(0.234821\pi\)
\(488\) −4400.00 −0.408153
\(489\) −2256.00 −0.208630
\(490\) 0 0
\(491\) 18714.0 1.72006 0.860032 0.510241i \(-0.170444\pi\)
0.860032 + 0.510241i \(0.170444\pi\)
\(492\) 2472.00 0.226517
\(493\) −4752.00 −0.434116
\(494\) 2592.00 0.236072
\(495\) 0 0
\(496\) 3328.00 0.301273
\(497\) 1080.00 0.0974741
\(498\) −2424.00 −0.218117
\(499\) −4056.00 −0.363871 −0.181935 0.983310i \(-0.558236\pi\)
−0.181935 + 0.983310i \(0.558236\pi\)
\(500\) 0 0
\(501\) −8100.00 −0.722318
\(502\) −6348.00 −0.564393
\(503\) 6288.00 0.557392 0.278696 0.960379i \(-0.410098\pi\)
0.278696 + 0.960379i \(0.410098\pi\)
\(504\) −144.000 −0.0127267
\(505\) 0 0
\(506\) −14000.0 −1.22999
\(507\) −2157.00 −0.188946
\(508\) 6520.00 0.569445
\(509\) 2856.00 0.248703 0.124352 0.992238i \(-0.460315\pi\)
0.124352 + 0.992238i \(0.460315\pi\)
\(510\) 0 0
\(511\) −1208.00 −0.104577
\(512\) 512.000 0.0441942
\(513\) −648.000 −0.0557698
\(514\) −2388.00 −0.204922
\(515\) 0 0
\(516\) −3504.00 −0.298944
\(517\) −22400.0 −1.90551
\(518\) 1016.00 0.0861785
\(519\) 4002.00 0.338475
\(520\) 0 0
\(521\) 17078.0 1.43609 0.718043 0.695999i \(-0.245038\pi\)
0.718043 + 0.695999i \(0.245038\pi\)
\(522\) 3888.00 0.326002
\(523\) −8560.00 −0.715684 −0.357842 0.933782i \(-0.616487\pi\)
−0.357842 + 0.933782i \(0.616487\pi\)
\(524\) −3480.00 −0.290123
\(525\) 0 0
\(526\) 280.000 0.0232102
\(527\) −4576.00 −0.378242
\(528\) −3360.00 −0.276942
\(529\) −2167.00 −0.178105
\(530\) 0 0
\(531\) −3330.00 −0.272146
\(532\) −192.000 −0.0156471
\(533\) −11124.0 −0.904004
\(534\) 5628.00 0.456081
\(535\) 0 0
\(536\) 5824.00 0.469326
\(537\) 5142.00 0.413210
\(538\) 10512.0 0.842388
\(539\) −23730.0 −1.89633
\(540\) 0 0
\(541\) 15970.0 1.26914 0.634569 0.772866i \(-0.281178\pi\)
0.634569 + 0.772866i \(0.281178\pi\)
\(542\) 1088.00 0.0862244
\(543\) 12018.0 0.949801
\(544\) −704.000 −0.0554848
\(545\) 0 0
\(546\) 648.000 0.0507909
\(547\) −15524.0 −1.21345 −0.606726 0.794911i \(-0.707518\pi\)
−0.606726 + 0.794911i \(0.707518\pi\)
\(548\) 3672.00 0.286241
\(549\) −4950.00 −0.384810
\(550\) 0 0
\(551\) 5184.00 0.400809
\(552\) 2400.00 0.185056
\(553\) −1584.00 −0.121806
\(554\) 1892.00 0.145096
\(555\) 0 0
\(556\) −2384.00 −0.181842
\(557\) 6774.00 0.515303 0.257651 0.966238i \(-0.417051\pi\)
0.257651 + 0.966238i \(0.417051\pi\)
\(558\) 3744.00 0.284043
\(559\) 15768.0 1.19305
\(560\) 0 0
\(561\) 4620.00 0.347694
\(562\) 2556.00 0.191848
\(563\) −10484.0 −0.784810 −0.392405 0.919793i \(-0.628357\pi\)
−0.392405 + 0.919793i \(0.628357\pi\)
\(564\) 3840.00 0.286690
\(565\) 0 0
\(566\) 14848.0 1.10266
\(567\) −162.000 −0.0119989
\(568\) −4320.00 −0.319125
\(569\) 23302.0 1.71682 0.858410 0.512964i \(-0.171453\pi\)
0.858410 + 0.512964i \(0.171453\pi\)
\(570\) 0 0
\(571\) 21520.0 1.57720 0.788602 0.614903i \(-0.210805\pi\)
0.788602 + 0.614903i \(0.210805\pi\)
\(572\) 15120.0 1.10524
\(573\) 2052.00 0.149605
\(574\) 824.000 0.0599183
\(575\) 0 0
\(576\) 576.000 0.0416667
\(577\) 3856.00 0.278210 0.139105 0.990278i \(-0.455577\pi\)
0.139105 + 0.990278i \(0.455577\pi\)
\(578\) −8858.00 −0.637447
\(579\) 13452.0 0.965537
\(580\) 0 0
\(581\) −808.000 −0.0576962
\(582\) −336.000 −0.0239307
\(583\) −28140.0 −1.99904
\(584\) 4832.00 0.342379
\(585\) 0 0
\(586\) −2724.00 −0.192026
\(587\) −26796.0 −1.88414 −0.942069 0.335418i \(-0.891122\pi\)
−0.942069 + 0.335418i \(0.891122\pi\)
\(588\) 4068.00 0.285309
\(589\) 4992.00 0.349222
\(590\) 0 0
\(591\) 3174.00 0.220915
\(592\) −4064.00 −0.282144
\(593\) 9870.00 0.683495 0.341747 0.939792i \(-0.388981\pi\)
0.341747 + 0.939792i \(0.388981\pi\)
\(594\) −3780.00 −0.261103
\(595\) 0 0
\(596\) 4304.00 0.295803
\(597\) 3384.00 0.231990
\(598\) −10800.0 −0.738537
\(599\) −13296.0 −0.906945 −0.453472 0.891270i \(-0.649815\pi\)
−0.453472 + 0.891270i \(0.649815\pi\)
\(600\) 0 0
\(601\) −9262.00 −0.628627 −0.314314 0.949319i \(-0.601774\pi\)
−0.314314 + 0.949319i \(0.601774\pi\)
\(602\) −1168.00 −0.0790766
\(603\) 6552.00 0.442484
\(604\) −128.000 −0.00862292
\(605\) 0 0
\(606\) 3552.00 0.238103
\(607\) 5498.00 0.367639 0.183820 0.982960i \(-0.441154\pi\)
0.183820 + 0.982960i \(0.441154\pi\)
\(608\) 768.000 0.0512278
\(609\) 1296.00 0.0862341
\(610\) 0 0
\(611\) −17280.0 −1.14415
\(612\) −792.000 −0.0523116
\(613\) 394.000 0.0259600 0.0129800 0.999916i \(-0.495868\pi\)
0.0129800 + 0.999916i \(0.495868\pi\)
\(614\) −15480.0 −1.01746
\(615\) 0 0
\(616\) −1120.00 −0.0732566
\(617\) 7370.00 0.480883 0.240442 0.970664i \(-0.422708\pi\)
0.240442 + 0.970664i \(0.422708\pi\)
\(618\) −372.000 −0.0242136
\(619\) 25316.0 1.64384 0.821919 0.569604i \(-0.192903\pi\)
0.821919 + 0.569604i \(0.192903\pi\)
\(620\) 0 0
\(621\) 2700.00 0.174472
\(622\) 9960.00 0.642057
\(623\) 1876.00 0.120643
\(624\) −2592.00 −0.166287
\(625\) 0 0
\(626\) 1208.00 0.0771268
\(627\) −5040.00 −0.321018
\(628\) −10216.0 −0.649145
\(629\) 5588.00 0.354226
\(630\) 0 0
\(631\) 2552.00 0.161004 0.0805020 0.996754i \(-0.474348\pi\)
0.0805020 + 0.996754i \(0.474348\pi\)
\(632\) 6336.00 0.398786
\(633\) −2340.00 −0.146930
\(634\) −17132.0 −1.07318
\(635\) 0 0
\(636\) 4824.00 0.300761
\(637\) −18306.0 −1.13863
\(638\) 30240.0 1.87651
\(639\) −4860.00 −0.300874
\(640\) 0 0
\(641\) 8050.00 0.496031 0.248016 0.968756i \(-0.420222\pi\)
0.248016 + 0.968756i \(0.420222\pi\)
\(642\) −504.000 −0.0309833
\(643\) −19368.0 −1.18787 −0.593934 0.804514i \(-0.702426\pi\)
−0.593934 + 0.804514i \(0.702426\pi\)
\(644\) 800.000 0.0489510
\(645\) 0 0
\(646\) −1056.00 −0.0643154
\(647\) 9912.00 0.602289 0.301144 0.953579i \(-0.402631\pi\)
0.301144 + 0.953579i \(0.402631\pi\)
\(648\) 648.000 0.0392837
\(649\) −25900.0 −1.56651
\(650\) 0 0
\(651\) 1248.00 0.0751351
\(652\) 3008.00 0.180678
\(653\) 27986.0 1.67715 0.838573 0.544789i \(-0.183390\pi\)
0.838573 + 0.544789i \(0.183390\pi\)
\(654\) −2220.00 −0.132735
\(655\) 0 0
\(656\) −3296.00 −0.196169
\(657\) 5436.00 0.322798
\(658\) 1280.00 0.0758353
\(659\) 7562.00 0.447001 0.223501 0.974704i \(-0.428252\pi\)
0.223501 + 0.974704i \(0.428252\pi\)
\(660\) 0 0
\(661\) 20234.0 1.19064 0.595319 0.803490i \(-0.297026\pi\)
0.595319 + 0.803490i \(0.297026\pi\)
\(662\) 6944.00 0.407683
\(663\) 3564.00 0.208770
\(664\) 3232.00 0.188894
\(665\) 0 0
\(666\) −4572.00 −0.266008
\(667\) −21600.0 −1.25391
\(668\) 10800.0 0.625546
\(669\) 7710.00 0.445569
\(670\) 0 0
\(671\) −38500.0 −2.21502
\(672\) 192.000 0.0110217
\(673\) 25332.0 1.45093 0.725466 0.688258i \(-0.241624\pi\)
0.725466 + 0.688258i \(0.241624\pi\)
\(674\) 11336.0 0.647843
\(675\) 0 0
\(676\) 2876.00 0.163632
\(677\) −18358.0 −1.04218 −0.521090 0.853502i \(-0.674474\pi\)
−0.521090 + 0.853502i \(0.674474\pi\)
\(678\) 10476.0 0.593405
\(679\) −112.000 −0.00633014
\(680\) 0 0
\(681\) 8508.00 0.478748
\(682\) 29120.0 1.63499
\(683\) −124.000 −0.00694689 −0.00347345 0.999994i \(-0.501106\pi\)
−0.00347345 + 0.999994i \(0.501106\pi\)
\(684\) 864.000 0.0482980
\(685\) 0 0
\(686\) 2728.00 0.151830
\(687\) 1830.00 0.101629
\(688\) 4672.00 0.258893
\(689\) −21708.0 −1.20030
\(690\) 0 0
\(691\) −17456.0 −0.961009 −0.480505 0.876992i \(-0.659547\pi\)
−0.480505 + 0.876992i \(0.659547\pi\)
\(692\) −5336.00 −0.293128
\(693\) −1260.00 −0.0690670
\(694\) 21672.0 1.18539
\(695\) 0 0
\(696\) −5184.00 −0.282326
\(697\) 4532.00 0.246287
\(698\) −17980.0 −0.975004
\(699\) −10542.0 −0.570436
\(700\) 0 0
\(701\) −17816.0 −0.959916 −0.479958 0.877291i \(-0.659348\pi\)
−0.479958 + 0.877291i \(0.659348\pi\)
\(702\) −2916.00 −0.156777
\(703\) −6096.00 −0.327048
\(704\) 4480.00 0.239839
\(705\) 0 0
\(706\) −10156.0 −0.541397
\(707\) 1184.00 0.0629829
\(708\) 4440.00 0.235686
\(709\) −14298.0 −0.757366 −0.378683 0.925526i \(-0.623623\pi\)
−0.378683 + 0.925526i \(0.623623\pi\)
\(710\) 0 0
\(711\) 7128.00 0.375979
\(712\) −7504.00 −0.394978
\(713\) −20800.0 −1.09252
\(714\) −264.000 −0.0138375
\(715\) 0 0
\(716\) −6856.00 −0.357850
\(717\) 5532.00 0.288140
\(718\) −7392.00 −0.384216
\(719\) −18440.0 −0.956462 −0.478231 0.878234i \(-0.658722\pi\)
−0.478231 + 0.878234i \(0.658722\pi\)
\(720\) 0 0
\(721\) −124.000 −0.00640499
\(722\) −12566.0 −0.647726
\(723\) −2946.00 −0.151539
\(724\) −16024.0 −0.822551
\(725\) 0 0
\(726\) −21414.0 −1.09469
\(727\) 9666.00 0.493112 0.246556 0.969129i \(-0.420701\pi\)
0.246556 + 0.969129i \(0.420701\pi\)
\(728\) −864.000 −0.0439862
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −6424.00 −0.325035
\(732\) 6600.00 0.333255
\(733\) −6094.00 −0.307076 −0.153538 0.988143i \(-0.549067\pi\)
−0.153538 + 0.988143i \(0.549067\pi\)
\(734\) 572.000 0.0287642
\(735\) 0 0
\(736\) −3200.00 −0.160263
\(737\) 50960.0 2.54700
\(738\) −3708.00 −0.184950
\(739\) 9952.00 0.495386 0.247693 0.968839i \(-0.420328\pi\)
0.247693 + 0.968839i \(0.420328\pi\)
\(740\) 0 0
\(741\) −3888.00 −0.192752
\(742\) 1608.00 0.0795573
\(743\) −2208.00 −0.109022 −0.0545112 0.998513i \(-0.517360\pi\)
−0.0545112 + 0.998513i \(0.517360\pi\)
\(744\) −4992.00 −0.245989
\(745\) 0 0
\(746\) 16524.0 0.810974
\(747\) 3636.00 0.178091
\(748\) −6160.00 −0.301112
\(749\) −168.000 −0.00819571
\(750\) 0 0
\(751\) −9400.00 −0.456739 −0.228369 0.973575i \(-0.573339\pi\)
−0.228369 + 0.973575i \(0.573339\pi\)
\(752\) −5120.00 −0.248281
\(753\) 9522.00 0.460825
\(754\) 23328.0 1.12673
\(755\) 0 0
\(756\) 216.000 0.0103913
\(757\) −22574.0 −1.08384 −0.541919 0.840430i \(-0.682302\pi\)
−0.541919 + 0.840430i \(0.682302\pi\)
\(758\) −5912.00 −0.283290
\(759\) 21000.0 1.00428
\(760\) 0 0
\(761\) 7278.00 0.346685 0.173343 0.984862i \(-0.444543\pi\)
0.173343 + 0.984862i \(0.444543\pi\)
\(762\) −9780.00 −0.464950
\(763\) −740.000 −0.0351111
\(764\) −2736.00 −0.129562
\(765\) 0 0
\(766\) −10480.0 −0.494331
\(767\) −19980.0 −0.940595
\(768\) −768.000 −0.0360844
\(769\) −16542.0 −0.775708 −0.387854 0.921721i \(-0.626784\pi\)
−0.387854 + 0.921721i \(0.626784\pi\)
\(770\) 0 0
\(771\) 3582.00 0.167319
\(772\) −17936.0 −0.836180
\(773\) −28926.0 −1.34592 −0.672960 0.739679i \(-0.734977\pi\)
−0.672960 + 0.739679i \(0.734977\pi\)
\(774\) 5256.00 0.244087
\(775\) 0 0
\(776\) 448.000 0.0207246
\(777\) −1524.00 −0.0703645
\(778\) −1768.00 −0.0814728
\(779\) −4944.00 −0.227390
\(780\) 0 0
\(781\) −37800.0 −1.73187
\(782\) 4400.00 0.201207
\(783\) −5832.00 −0.266180
\(784\) −5424.00 −0.247085
\(785\) 0 0
\(786\) 5220.00 0.236885
\(787\) 20608.0 0.933413 0.466706 0.884412i \(-0.345440\pi\)
0.466706 + 0.884412i \(0.345440\pi\)
\(788\) −4232.00 −0.191318
\(789\) −420.000 −0.0189511
\(790\) 0 0
\(791\) 3492.00 0.156967
\(792\) 5040.00 0.226122
\(793\) −29700.0 −1.32998
\(794\) 6788.00 0.303397
\(795\) 0 0
\(796\) −4512.00 −0.200909
\(797\) −41350.0 −1.83776 −0.918878 0.394541i \(-0.870904\pi\)
−0.918878 + 0.394541i \(0.870904\pi\)
\(798\) 288.000 0.0127758
\(799\) 7040.00 0.311711
\(800\) 0 0
\(801\) −8442.00 −0.372389
\(802\) −13652.0 −0.601083
\(803\) 42280.0 1.85807
\(804\) −8736.00 −0.383203
\(805\) 0 0
\(806\) 22464.0 0.981713
\(807\) −15768.0 −0.687807
\(808\) −4736.00 −0.206203
\(809\) 1794.00 0.0779650 0.0389825 0.999240i \(-0.487588\pi\)
0.0389825 + 0.999240i \(0.487588\pi\)
\(810\) 0 0
\(811\) 22756.0 0.985291 0.492646 0.870230i \(-0.336030\pi\)
0.492646 + 0.870230i \(0.336030\pi\)
\(812\) −1728.00 −0.0746809
\(813\) −1632.00 −0.0704019
\(814\) −35560.0 −1.53118
\(815\) 0 0
\(816\) 1056.00 0.0453032
\(817\) 7008.00 0.300097
\(818\) 15628.0 0.667995
\(819\) −972.000 −0.0414706
\(820\) 0 0
\(821\) −23632.0 −1.00458 −0.502291 0.864698i \(-0.667510\pi\)
−0.502291 + 0.864698i \(0.667510\pi\)
\(822\) −5508.00 −0.233715
\(823\) 33210.0 1.40660 0.703298 0.710896i \(-0.251710\pi\)
0.703298 + 0.710896i \(0.251710\pi\)
\(824\) 496.000 0.0209696
\(825\) 0 0
\(826\) 1480.00 0.0623436
\(827\) 30476.0 1.28144 0.640722 0.767773i \(-0.278635\pi\)
0.640722 + 0.767773i \(0.278635\pi\)
\(828\) −3600.00 −0.151097
\(829\) 29802.0 1.24857 0.624286 0.781196i \(-0.285390\pi\)
0.624286 + 0.781196i \(0.285390\pi\)
\(830\) 0 0
\(831\) −2838.00 −0.118471
\(832\) 3456.00 0.144009
\(833\) 7458.00 0.310209
\(834\) 3576.00 0.148473
\(835\) 0 0
\(836\) 6720.00 0.278010
\(837\) −5616.00 −0.231920
\(838\) 16580.0 0.683468
\(839\) −28024.0 −1.15315 −0.576577 0.817043i \(-0.695612\pi\)
−0.576577 + 0.817043i \(0.695612\pi\)
\(840\) 0 0
\(841\) 22267.0 0.912994
\(842\) 4220.00 0.172721
\(843\) −3834.00 −0.156643
\(844\) 3120.00 0.127245
\(845\) 0 0
\(846\) −5760.00 −0.234081
\(847\) −7138.00 −0.289569
\(848\) −6432.00 −0.260467
\(849\) −22272.0 −0.900322
\(850\) 0 0
\(851\) 25400.0 1.02315
\(852\) 6480.00 0.260565
\(853\) 3938.00 0.158071 0.0790355 0.996872i \(-0.474816\pi\)
0.0790355 + 0.996872i \(0.474816\pi\)
\(854\) 2200.00 0.0881528
\(855\) 0 0
\(856\) 672.000 0.0268323
\(857\) −8094.00 −0.322621 −0.161310 0.986904i \(-0.551572\pi\)
−0.161310 + 0.986904i \(0.551572\pi\)
\(858\) −22680.0 −0.902427
\(859\) 9044.00 0.359229 0.179614 0.983737i \(-0.442515\pi\)
0.179614 + 0.983737i \(0.442515\pi\)
\(860\) 0 0
\(861\) −1236.00 −0.0489231
\(862\) −24160.0 −0.954632
\(863\) −6252.00 −0.246606 −0.123303 0.992369i \(-0.539349\pi\)
−0.123303 + 0.992369i \(0.539349\pi\)
\(864\) −864.000 −0.0340207
\(865\) 0 0
\(866\) 32984.0 1.29427
\(867\) 13287.0 0.520473
\(868\) −1664.00 −0.0650689
\(869\) 55440.0 2.16418
\(870\) 0 0
\(871\) 39312.0 1.52932
\(872\) 2960.00 0.114952
\(873\) 504.000 0.0195393
\(874\) −4800.00 −0.185769
\(875\) 0 0
\(876\) −7248.00 −0.279552
\(877\) 40166.0 1.54653 0.773267 0.634081i \(-0.218621\pi\)
0.773267 + 0.634081i \(0.218621\pi\)
\(878\) −30096.0 −1.15682
\(879\) 4086.00 0.156789
\(880\) 0 0
\(881\) −12834.0 −0.490793 −0.245396 0.969423i \(-0.578918\pi\)
−0.245396 + 0.969423i \(0.578918\pi\)
\(882\) −6102.00 −0.232954
\(883\) 27192.0 1.03633 0.518167 0.855279i \(-0.326614\pi\)
0.518167 + 0.855279i \(0.326614\pi\)
\(884\) −4752.00 −0.180800
\(885\) 0 0
\(886\) −19752.0 −0.748963
\(887\) 42060.0 1.59215 0.796075 0.605198i \(-0.206906\pi\)
0.796075 + 0.605198i \(0.206906\pi\)
\(888\) 6096.00 0.230370
\(889\) −3260.00 −0.122989
\(890\) 0 0
\(891\) 5670.00 0.213190
\(892\) −10280.0 −0.385874
\(893\) −7680.00 −0.287796
\(894\) −6456.00 −0.241522
\(895\) 0 0
\(896\) −256.000 −0.00954504
\(897\) 16200.0 0.603013
\(898\) 34332.0 1.27581
\(899\) 44928.0 1.66678
\(900\) 0 0
\(901\) 8844.00 0.327010
\(902\) −28840.0 −1.06460
\(903\) 1752.00 0.0645658
\(904\) −13968.0 −0.513904
\(905\) 0 0
\(906\) 192.000 0.00704059
\(907\) 41172.0 1.50727 0.753635 0.657293i \(-0.228299\pi\)
0.753635 + 0.657293i \(0.228299\pi\)
\(908\) −11344.0 −0.414608
\(909\) −5328.00 −0.194410
\(910\) 0 0
\(911\) −48.0000 −0.00174568 −0.000872838 1.00000i \(-0.500278\pi\)
−0.000872838 1.00000i \(0.500278\pi\)
\(912\) −1152.00 −0.0418273
\(913\) 28280.0 1.02512
\(914\) 29696.0 1.07468
\(915\) 0 0
\(916\) −2440.00 −0.0880130
\(917\) 1740.00 0.0626607
\(918\) 1188.00 0.0427122
\(919\) −34584.0 −1.24137 −0.620686 0.784059i \(-0.713146\pi\)
−0.620686 + 0.784059i \(0.713146\pi\)
\(920\) 0 0
\(921\) 23220.0 0.830755
\(922\) −2520.00 −0.0900128
\(923\) −29160.0 −1.03988
\(924\) 1680.00 0.0598138
\(925\) 0 0
\(926\) −22476.0 −0.797632
\(927\) 558.000 0.0197704
\(928\) 6912.00 0.244502
\(929\) −3474.00 −0.122689 −0.0613446 0.998117i \(-0.519539\pi\)
−0.0613446 + 0.998117i \(0.519539\pi\)
\(930\) 0 0
\(931\) −8136.00 −0.286409
\(932\) 14056.0 0.494012
\(933\) −14940.0 −0.524238
\(934\) 29544.0 1.03502
\(935\) 0 0
\(936\) 3888.00 0.135773
\(937\) 44408.0 1.54829 0.774144 0.633009i \(-0.218181\pi\)
0.774144 + 0.633009i \(0.218181\pi\)
\(938\) −2912.00 −0.101365
\(939\) −1812.00 −0.0629738
\(940\) 0 0
\(941\) 20188.0 0.699373 0.349686 0.936867i \(-0.386288\pi\)
0.349686 + 0.936867i \(0.386288\pi\)
\(942\) 15324.0 0.530024
\(943\) 20600.0 0.711377
\(944\) −5920.00 −0.204110
\(945\) 0 0
\(946\) 40880.0 1.40499
\(947\) −31212.0 −1.07102 −0.535509 0.844530i \(-0.679880\pi\)
−0.535509 + 0.844530i \(0.679880\pi\)
\(948\) −9504.00 −0.325607
\(949\) 32616.0 1.11566
\(950\) 0 0
\(951\) 25698.0 0.876251
\(952\) 352.000 0.0119836
\(953\) 20182.0 0.686001 0.343001 0.939335i \(-0.388557\pi\)
0.343001 + 0.939335i \(0.388557\pi\)
\(954\) −7236.00 −0.245570
\(955\) 0 0
\(956\) −7376.00 −0.249536
\(957\) −45360.0 −1.53216
\(958\) −12232.0 −0.412524
\(959\) −1836.00 −0.0618222
\(960\) 0 0
\(961\) 13473.0 0.452251
\(962\) −27432.0 −0.919380
\(963\) 756.000 0.0252978
\(964\) 3928.00 0.131237
\(965\) 0 0
\(966\) −1200.00 −0.0399683
\(967\) −53722.0 −1.78654 −0.893269 0.449522i \(-0.851594\pi\)
−0.893269 + 0.449522i \(0.851594\pi\)
\(968\) 28552.0 0.948033
\(969\) 1584.00 0.0525133
\(970\) 0 0
\(971\) 22554.0 0.745409 0.372705 0.927950i \(-0.378430\pi\)
0.372705 + 0.927950i \(0.378430\pi\)
\(972\) −972.000 −0.0320750
\(973\) 1192.00 0.0392742
\(974\) 31812.0 1.04653
\(975\) 0 0
\(976\) −8800.00 −0.288608
\(977\) −18126.0 −0.593554 −0.296777 0.954947i \(-0.595912\pi\)
−0.296777 + 0.954947i \(0.595912\pi\)
\(978\) −4512.00 −0.147523
\(979\) −65660.0 −2.14352
\(980\) 0 0
\(981\) 3330.00 0.108378
\(982\) 37428.0 1.21627
\(983\) 6232.00 0.202207 0.101104 0.994876i \(-0.467763\pi\)
0.101104 + 0.994876i \(0.467763\pi\)
\(984\) 4944.00 0.160172
\(985\) 0 0
\(986\) −9504.00 −0.306967
\(987\) −1920.00 −0.0619192
\(988\) 5184.00 0.166928
\(989\) −29200.0 −0.938833
\(990\) 0 0
\(991\) 15184.0 0.486716 0.243358 0.969937i \(-0.421751\pi\)
0.243358 + 0.969937i \(0.421751\pi\)
\(992\) 6656.00 0.213032
\(993\) −10416.0 −0.332872
\(994\) 2160.00 0.0689246
\(995\) 0 0
\(996\) −4848.00 −0.154232
\(997\) −29922.0 −0.950491 −0.475245 0.879853i \(-0.657641\pi\)
−0.475245 + 0.879853i \(0.657641\pi\)
\(998\) −8112.00 −0.257295
\(999\) 6858.00 0.217195
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.f.1.1 1
3.2 odd 2 450.4.a.e.1.1 1
4.3 odd 2 1200.4.a.bc.1.1 1
5.2 odd 4 30.4.c.a.19.2 yes 2
5.3 odd 4 30.4.c.a.19.1 2
5.4 even 2 150.4.a.d.1.1 1
15.2 even 4 90.4.c.a.19.1 2
15.8 even 4 90.4.c.a.19.2 2
15.14 odd 2 450.4.a.p.1.1 1
20.3 even 4 240.4.f.d.49.2 2
20.7 even 4 240.4.f.d.49.1 2
20.19 odd 2 1200.4.a.h.1.1 1
40.3 even 4 960.4.f.d.769.1 2
40.13 odd 4 960.4.f.c.769.2 2
40.27 even 4 960.4.f.d.769.2 2
40.37 odd 4 960.4.f.c.769.1 2
60.23 odd 4 720.4.f.c.289.2 2
60.47 odd 4 720.4.f.c.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.c.a.19.1 2 5.3 odd 4
30.4.c.a.19.2 yes 2 5.2 odd 4
90.4.c.a.19.1 2 15.2 even 4
90.4.c.a.19.2 2 15.8 even 4
150.4.a.d.1.1 1 5.4 even 2
150.4.a.f.1.1 1 1.1 even 1 trivial
240.4.f.d.49.1 2 20.7 even 4
240.4.f.d.49.2 2 20.3 even 4
450.4.a.e.1.1 1 3.2 odd 2
450.4.a.p.1.1 1 15.14 odd 2
720.4.f.c.289.1 2 60.47 odd 4
720.4.f.c.289.2 2 60.23 odd 4
960.4.f.c.769.1 2 40.37 odd 4
960.4.f.c.769.2 2 40.13 odd 4
960.4.f.d.769.1 2 40.3 even 4
960.4.f.d.769.2 2 40.27 even 4
1200.4.a.h.1.1 1 20.19 odd 2
1200.4.a.bc.1.1 1 4.3 odd 2