Properties

Label 15.4.a.b.1.1
Level $15$
Weight $4$
Character 15.1
Self dual yes
Analytic conductor $0.885$
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] = [15,4,Mod(1,15)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(15, 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("15.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 15.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: \(0.885028650086\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 15.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -5.00000 q^{5} -9.00000 q^{6} +20.0000 q^{7} -21.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -5.00000 q^{5} -9.00000 q^{6} +20.0000 q^{7} -21.0000 q^{8} +9.00000 q^{9} -15.0000 q^{10} -24.0000 q^{11} -3.00000 q^{12} +74.0000 q^{13} +60.0000 q^{14} +15.0000 q^{15} -71.0000 q^{16} +54.0000 q^{17} +27.0000 q^{18} -124.000 q^{19} -5.00000 q^{20} -60.0000 q^{21} -72.0000 q^{22} -120.000 q^{23} +63.0000 q^{24} +25.0000 q^{25} +222.000 q^{26} -27.0000 q^{27} +20.0000 q^{28} -78.0000 q^{29} +45.0000 q^{30} +200.000 q^{31} -45.0000 q^{32} +72.0000 q^{33} +162.000 q^{34} -100.000 q^{35} +9.00000 q^{36} -70.0000 q^{37} -372.000 q^{38} -222.000 q^{39} +105.000 q^{40} +330.000 q^{41} -180.000 q^{42} +92.0000 q^{43} -24.0000 q^{44} -45.0000 q^{45} -360.000 q^{46} -24.0000 q^{47} +213.000 q^{48} +57.0000 q^{49} +75.0000 q^{50} -162.000 q^{51} +74.0000 q^{52} +450.000 q^{53} -81.0000 q^{54} +120.000 q^{55} -420.000 q^{56} +372.000 q^{57} -234.000 q^{58} +24.0000 q^{59} +15.0000 q^{60} -322.000 q^{61} +600.000 q^{62} +180.000 q^{63} +433.000 q^{64} -370.000 q^{65} +216.000 q^{66} -196.000 q^{67} +54.0000 q^{68} +360.000 q^{69} -300.000 q^{70} -288.000 q^{71} -189.000 q^{72} -430.000 q^{73} -210.000 q^{74} -75.0000 q^{75} -124.000 q^{76} -480.000 q^{77} -666.000 q^{78} -520.000 q^{79} +355.000 q^{80} +81.0000 q^{81} +990.000 q^{82} +156.000 q^{83} -60.0000 q^{84} -270.000 q^{85} +276.000 q^{86} +234.000 q^{87} +504.000 q^{88} +1026.00 q^{89} -135.000 q^{90} +1480.00 q^{91} -120.000 q^{92} -600.000 q^{93} -72.0000 q^{94} +620.000 q^{95} +135.000 q^{96} -286.000 q^{97} +171.000 q^{98} -216.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000 1.06066 0.530330 0.847791i \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) −3.00000 −0.577350
\(4\) 1.00000 0.125000
\(5\) −5.00000 −0.447214
\(6\) −9.00000 −0.612372
\(7\) 20.0000 1.07990 0.539949 0.841698i \(-0.318443\pi\)
0.539949 + 0.841698i \(0.318443\pi\)
\(8\) −21.0000 −0.928078
\(9\) 9.00000 0.333333
\(10\) −15.0000 −0.474342
\(11\) −24.0000 −0.657843 −0.328921 0.944357i \(-0.606685\pi\)
−0.328921 + 0.944357i \(0.606685\pi\)
\(12\) −3.00000 −0.0721688
\(13\) 74.0000 1.57876 0.789381 0.613904i \(-0.210402\pi\)
0.789381 + 0.613904i \(0.210402\pi\)
\(14\) 60.0000 1.14541
\(15\) 15.0000 0.258199
\(16\) −71.0000 −1.10938
\(17\) 54.0000 0.770407 0.385204 0.922832i \(-0.374131\pi\)
0.385204 + 0.922832i \(0.374131\pi\)
\(18\) 27.0000 0.353553
\(19\) −124.000 −1.49724 −0.748620 0.663000i \(-0.769283\pi\)
−0.748620 + 0.663000i \(0.769283\pi\)
\(20\) −5.00000 −0.0559017
\(21\) −60.0000 −0.623480
\(22\) −72.0000 −0.697748
\(23\) −120.000 −1.08790 −0.543951 0.839117i \(-0.683072\pi\)
−0.543951 + 0.839117i \(0.683072\pi\)
\(24\) 63.0000 0.535826
\(25\) 25.0000 0.200000
\(26\) 222.000 1.67453
\(27\) −27.0000 −0.192450
\(28\) 20.0000 0.134987
\(29\) −78.0000 −0.499456 −0.249728 0.968316i \(-0.580341\pi\)
−0.249728 + 0.968316i \(0.580341\pi\)
\(30\) 45.0000 0.273861
\(31\) 200.000 1.15874 0.579372 0.815063i \(-0.303298\pi\)
0.579372 + 0.815063i \(0.303298\pi\)
\(32\) −45.0000 −0.248592
\(33\) 72.0000 0.379806
\(34\) 162.000 0.817140
\(35\) −100.000 −0.482945
\(36\) 9.00000 0.0416667
\(37\) −70.0000 −0.311025 −0.155513 0.987834i \(-0.549703\pi\)
−0.155513 + 0.987834i \(0.549703\pi\)
\(38\) −372.000 −1.58806
\(39\) −222.000 −0.911499
\(40\) 105.000 0.415049
\(41\) 330.000 1.25701 0.628504 0.777806i \(-0.283668\pi\)
0.628504 + 0.777806i \(0.283668\pi\)
\(42\) −180.000 −0.661300
\(43\) 92.0000 0.326276 0.163138 0.986603i \(-0.447838\pi\)
0.163138 + 0.986603i \(0.447838\pi\)
\(44\) −24.0000 −0.0822304
\(45\) −45.0000 −0.149071
\(46\) −360.000 −1.15389
\(47\) −24.0000 −0.0744843 −0.0372421 0.999306i \(-0.511857\pi\)
−0.0372421 + 0.999306i \(0.511857\pi\)
\(48\) 213.000 0.640498
\(49\) 57.0000 0.166181
\(50\) 75.0000 0.212132
\(51\) −162.000 −0.444795
\(52\) 74.0000 0.197345
\(53\) 450.000 1.16627 0.583134 0.812376i \(-0.301826\pi\)
0.583134 + 0.812376i \(0.301826\pi\)
\(54\) −81.0000 −0.204124
\(55\) 120.000 0.294196
\(56\) −420.000 −1.00223
\(57\) 372.000 0.864432
\(58\) −234.000 −0.529754
\(59\) 24.0000 0.0529582 0.0264791 0.999649i \(-0.491570\pi\)
0.0264791 + 0.999649i \(0.491570\pi\)
\(60\) 15.0000 0.0322749
\(61\) −322.000 −0.675867 −0.337933 0.941170i \(-0.609728\pi\)
−0.337933 + 0.941170i \(0.609728\pi\)
\(62\) 600.000 1.22903
\(63\) 180.000 0.359966
\(64\) 433.000 0.845703
\(65\) −370.000 −0.706044
\(66\) 216.000 0.402845
\(67\) −196.000 −0.357391 −0.178696 0.983904i \(-0.557188\pi\)
−0.178696 + 0.983904i \(0.557188\pi\)
\(68\) 54.0000 0.0963009
\(69\) 360.000 0.628100
\(70\) −300.000 −0.512241
\(71\) −288.000 −0.481399 −0.240699 0.970600i \(-0.577377\pi\)
−0.240699 + 0.970600i \(0.577377\pi\)
\(72\) −189.000 −0.309359
\(73\) −430.000 −0.689420 −0.344710 0.938709i \(-0.612023\pi\)
−0.344710 + 0.938709i \(0.612023\pi\)
\(74\) −210.000 −0.329892
\(75\) −75.0000 −0.115470
\(76\) −124.000 −0.187155
\(77\) −480.000 −0.710404
\(78\) −666.000 −0.966790
\(79\) −520.000 −0.740564 −0.370282 0.928919i \(-0.620739\pi\)
−0.370282 + 0.928919i \(0.620739\pi\)
\(80\) 355.000 0.496128
\(81\) 81.0000 0.111111
\(82\) 990.000 1.33326
\(83\) 156.000 0.206304 0.103152 0.994666i \(-0.467107\pi\)
0.103152 + 0.994666i \(0.467107\pi\)
\(84\) −60.0000 −0.0779350
\(85\) −270.000 −0.344537
\(86\) 276.000 0.346068
\(87\) 234.000 0.288361
\(88\) 504.000 0.610529
\(89\) 1026.00 1.22198 0.610988 0.791640i \(-0.290773\pi\)
0.610988 + 0.791640i \(0.290773\pi\)
\(90\) −135.000 −0.158114
\(91\) 1480.00 1.70490
\(92\) −120.000 −0.135988
\(93\) −600.000 −0.669001
\(94\) −72.0000 −0.0790025
\(95\) 620.000 0.669586
\(96\) 135.000 0.143525
\(97\) −286.000 −0.299370 −0.149685 0.988734i \(-0.547826\pi\)
−0.149685 + 0.988734i \(0.547826\pi\)
\(98\) 171.000 0.176261
\(99\) −216.000 −0.219281
\(100\) 25.0000 0.0250000
\(101\) −1734.00 −1.70831 −0.854156 0.520017i \(-0.825925\pi\)
−0.854156 + 0.520017i \(0.825925\pi\)
\(102\) −486.000 −0.471776
\(103\) 452.000 0.432397 0.216198 0.976349i \(-0.430634\pi\)
0.216198 + 0.976349i \(0.430634\pi\)
\(104\) −1554.00 −1.46521
\(105\) 300.000 0.278829
\(106\) 1350.00 1.23702
\(107\) −1404.00 −1.26850 −0.634251 0.773127i \(-0.718692\pi\)
−0.634251 + 0.773127i \(0.718692\pi\)
\(108\) −27.0000 −0.0240563
\(109\) −1474.00 −1.29526 −0.647631 0.761954i \(-0.724240\pi\)
−0.647631 + 0.761954i \(0.724240\pi\)
\(110\) 360.000 0.312042
\(111\) 210.000 0.179570
\(112\) −1420.00 −1.19801
\(113\) 1086.00 0.904091 0.452046 0.891995i \(-0.350694\pi\)
0.452046 + 0.891995i \(0.350694\pi\)
\(114\) 1116.00 0.916868
\(115\) 600.000 0.486524
\(116\) −78.0000 −0.0624321
\(117\) 666.000 0.526254
\(118\) 72.0000 0.0561707
\(119\) 1080.00 0.831962
\(120\) −315.000 −0.239629
\(121\) −755.000 −0.567243
\(122\) −966.000 −0.716865
\(123\) −990.000 −0.725734
\(124\) 200.000 0.144843
\(125\) −125.000 −0.0894427
\(126\) 540.000 0.381802
\(127\) 1244.00 0.869190 0.434595 0.900626i \(-0.356891\pi\)
0.434595 + 0.900626i \(0.356891\pi\)
\(128\) 1659.00 1.14560
\(129\) −276.000 −0.188376
\(130\) −1110.00 −0.748873
\(131\) 2328.00 1.55266 0.776329 0.630327i \(-0.217079\pi\)
0.776329 + 0.630327i \(0.217079\pi\)
\(132\) 72.0000 0.0474757
\(133\) −2480.00 −1.61687
\(134\) −588.000 −0.379071
\(135\) 135.000 0.0860663
\(136\) −1134.00 −0.714998
\(137\) 2118.00 1.32082 0.660412 0.750903i \(-0.270382\pi\)
0.660412 + 0.750903i \(0.270382\pi\)
\(138\) 1080.00 0.666201
\(139\) 2324.00 1.41812 0.709062 0.705147i \(-0.249119\pi\)
0.709062 + 0.705147i \(0.249119\pi\)
\(140\) −100.000 −0.0603682
\(141\) 72.0000 0.0430035
\(142\) −864.000 −0.510600
\(143\) −1776.00 −1.03858
\(144\) −639.000 −0.369792
\(145\) 390.000 0.223364
\(146\) −1290.00 −0.731241
\(147\) −171.000 −0.0959445
\(148\) −70.0000 −0.0388781
\(149\) 258.000 0.141854 0.0709268 0.997482i \(-0.477404\pi\)
0.0709268 + 0.997482i \(0.477404\pi\)
\(150\) −225.000 −0.122474
\(151\) −808.000 −0.435458 −0.217729 0.976009i \(-0.569865\pi\)
−0.217729 + 0.976009i \(0.569865\pi\)
\(152\) 2604.00 1.38955
\(153\) 486.000 0.256802
\(154\) −1440.00 −0.753497
\(155\) −1000.00 −0.518206
\(156\) −222.000 −0.113937
\(157\) 2378.00 1.20882 0.604411 0.796673i \(-0.293408\pi\)
0.604411 + 0.796673i \(0.293408\pi\)
\(158\) −1560.00 −0.785487
\(159\) −1350.00 −0.673346
\(160\) 225.000 0.111174
\(161\) −2400.00 −1.17482
\(162\) 243.000 0.117851
\(163\) −52.0000 −0.0249874 −0.0124937 0.999922i \(-0.503977\pi\)
−0.0124937 + 0.999922i \(0.503977\pi\)
\(164\) 330.000 0.157126
\(165\) −360.000 −0.169854
\(166\) 468.000 0.218818
\(167\) −3720.00 −1.72373 −0.861863 0.507141i \(-0.830702\pi\)
−0.861863 + 0.507141i \(0.830702\pi\)
\(168\) 1260.00 0.578638
\(169\) 3279.00 1.49249
\(170\) −810.000 −0.365436
\(171\) −1116.00 −0.499080
\(172\) 92.0000 0.0407845
\(173\) 426.000 0.187215 0.0936075 0.995609i \(-0.470160\pi\)
0.0936075 + 0.995609i \(0.470160\pi\)
\(174\) 702.000 0.305853
\(175\) 500.000 0.215980
\(176\) 1704.00 0.729795
\(177\) −72.0000 −0.0305754
\(178\) 3078.00 1.29610
\(179\) −1440.00 −0.601289 −0.300644 0.953736i \(-0.597202\pi\)
−0.300644 + 0.953736i \(0.597202\pi\)
\(180\) −45.0000 −0.0186339
\(181\) −3130.00 −1.28537 −0.642683 0.766133i \(-0.722179\pi\)
−0.642683 + 0.766133i \(0.722179\pi\)
\(182\) 4440.00 1.80832
\(183\) 966.000 0.390212
\(184\) 2520.00 1.00966
\(185\) 350.000 0.139095
\(186\) −1800.00 −0.709583
\(187\) −1296.00 −0.506807
\(188\) −24.0000 −0.00931053
\(189\) −540.000 −0.207827
\(190\) 1860.00 0.710203
\(191\) 3576.00 1.35471 0.677357 0.735655i \(-0.263125\pi\)
0.677357 + 0.735655i \(0.263125\pi\)
\(192\) −1299.00 −0.488267
\(193\) 2666.00 0.994315 0.497158 0.867660i \(-0.334377\pi\)
0.497158 + 0.867660i \(0.334377\pi\)
\(194\) −858.000 −0.317530
\(195\) 1110.00 0.407635
\(196\) 57.0000 0.0207726
\(197\) −2718.00 −0.982992 −0.491496 0.870880i \(-0.663550\pi\)
−0.491496 + 0.870880i \(0.663550\pi\)
\(198\) −648.000 −0.232583
\(199\) −3832.00 −1.36504 −0.682521 0.730866i \(-0.739116\pi\)
−0.682521 + 0.730866i \(0.739116\pi\)
\(200\) −525.000 −0.185616
\(201\) 588.000 0.206340
\(202\) −5202.00 −1.81194
\(203\) −1560.00 −0.539362
\(204\) −162.000 −0.0555994
\(205\) −1650.00 −0.562151
\(206\) 1356.00 0.458626
\(207\) −1080.00 −0.362634
\(208\) −5254.00 −1.75144
\(209\) 2976.00 0.984948
\(210\) 900.000 0.295742
\(211\) 1100.00 0.358896 0.179448 0.983767i \(-0.442569\pi\)
0.179448 + 0.983767i \(0.442569\pi\)
\(212\) 450.000 0.145784
\(213\) 864.000 0.277936
\(214\) −4212.00 −1.34545
\(215\) −460.000 −0.145915
\(216\) 567.000 0.178609
\(217\) 4000.00 1.25133
\(218\) −4422.00 −1.37383
\(219\) 1290.00 0.398037
\(220\) 120.000 0.0367745
\(221\) 3996.00 1.21629
\(222\) 630.000 0.190463
\(223\) 1964.00 0.589772 0.294886 0.955532i \(-0.404718\pi\)
0.294886 + 0.955532i \(0.404718\pi\)
\(224\) −900.000 −0.268454
\(225\) 225.000 0.0666667
\(226\) 3258.00 0.958933
\(227\) 660.000 0.192977 0.0964884 0.995334i \(-0.469239\pi\)
0.0964884 + 0.995334i \(0.469239\pi\)
\(228\) 372.000 0.108054
\(229\) −1906.00 −0.550009 −0.275004 0.961443i \(-0.588679\pi\)
−0.275004 + 0.961443i \(0.588679\pi\)
\(230\) 1800.00 0.516037
\(231\) 1440.00 0.410152
\(232\) 1638.00 0.463534
\(233\) −1458.00 −0.409943 −0.204972 0.978768i \(-0.565710\pi\)
−0.204972 + 0.978768i \(0.565710\pi\)
\(234\) 1998.00 0.558177
\(235\) 120.000 0.0333104
\(236\) 24.0000 0.00661978
\(237\) 1560.00 0.427565
\(238\) 3240.00 0.882429
\(239\) 1176.00 0.318281 0.159140 0.987256i \(-0.449128\pi\)
0.159140 + 0.987256i \(0.449128\pi\)
\(240\) −1065.00 −0.286439
\(241\) 866.000 0.231469 0.115734 0.993280i \(-0.463078\pi\)
0.115734 + 0.993280i \(0.463078\pi\)
\(242\) −2265.00 −0.601652
\(243\) −243.000 −0.0641500
\(244\) −322.000 −0.0844834
\(245\) −285.000 −0.0743183
\(246\) −2970.00 −0.769757
\(247\) −9176.00 −2.36379
\(248\) −4200.00 −1.07540
\(249\) −468.000 −0.119110
\(250\) −375.000 −0.0948683
\(251\) 432.000 0.108636 0.0543179 0.998524i \(-0.482702\pi\)
0.0543179 + 0.998524i \(0.482702\pi\)
\(252\) 180.000 0.0449958
\(253\) 2880.00 0.715668
\(254\) 3732.00 0.921915
\(255\) 810.000 0.198918
\(256\) 1513.00 0.369385
\(257\) 2526.00 0.613103 0.306552 0.951854i \(-0.400825\pi\)
0.306552 + 0.951854i \(0.400825\pi\)
\(258\) −828.000 −0.199802
\(259\) −1400.00 −0.335876
\(260\) −370.000 −0.0882555
\(261\) −702.000 −0.166485
\(262\) 6984.00 1.64684
\(263\) 5448.00 1.27733 0.638666 0.769484i \(-0.279487\pi\)
0.638666 + 0.769484i \(0.279487\pi\)
\(264\) −1512.00 −0.352489
\(265\) −2250.00 −0.521571
\(266\) −7440.00 −1.71495
\(267\) −3078.00 −0.705508
\(268\) −196.000 −0.0446739
\(269\) −2574.00 −0.583418 −0.291709 0.956507i \(-0.594224\pi\)
−0.291709 + 0.956507i \(0.594224\pi\)
\(270\) 405.000 0.0912871
\(271\) −3184.00 −0.713706 −0.356853 0.934161i \(-0.616150\pi\)
−0.356853 + 0.934161i \(0.616150\pi\)
\(272\) −3834.00 −0.854671
\(273\) −4440.00 −0.984326
\(274\) 6354.00 1.40095
\(275\) −600.000 −0.131569
\(276\) 360.000 0.0785125
\(277\) 3962.00 0.859399 0.429699 0.902972i \(-0.358620\pi\)
0.429699 + 0.902972i \(0.358620\pi\)
\(278\) 6972.00 1.50415
\(279\) 1800.00 0.386248
\(280\) 2100.00 0.448211
\(281\) −8286.00 −1.75908 −0.879540 0.475825i \(-0.842149\pi\)
−0.879540 + 0.475825i \(0.842149\pi\)
\(282\) 216.000 0.0456121
\(283\) −2716.00 −0.570493 −0.285246 0.958454i \(-0.592075\pi\)
−0.285246 + 0.958454i \(0.592075\pi\)
\(284\) −288.000 −0.0601748
\(285\) −1860.00 −0.386586
\(286\) −5328.00 −1.10158
\(287\) 6600.00 1.35744
\(288\) −405.000 −0.0828641
\(289\) −1997.00 −0.406473
\(290\) 1170.00 0.236913
\(291\) 858.000 0.172841
\(292\) −430.000 −0.0861776
\(293\) 6018.00 1.19992 0.599958 0.800032i \(-0.295184\pi\)
0.599958 + 0.800032i \(0.295184\pi\)
\(294\) −513.000 −0.101765
\(295\) −120.000 −0.0236836
\(296\) 1470.00 0.288655
\(297\) 648.000 0.126602
\(298\) 774.000 0.150458
\(299\) −8880.00 −1.71754
\(300\) −75.0000 −0.0144338
\(301\) 1840.00 0.352345
\(302\) −2424.00 −0.461873
\(303\) 5202.00 0.986294
\(304\) 8804.00 1.66100
\(305\) 1610.00 0.302257
\(306\) 1458.00 0.272380
\(307\) 9236.00 1.71702 0.858512 0.512793i \(-0.171389\pi\)
0.858512 + 0.512793i \(0.171389\pi\)
\(308\) −480.000 −0.0888004
\(309\) −1356.00 −0.249644
\(310\) −3000.00 −0.549640
\(311\) 1536.00 0.280060 0.140030 0.990147i \(-0.455280\pi\)
0.140030 + 0.990147i \(0.455280\pi\)
\(312\) 4662.00 0.845942
\(313\) −7342.00 −1.32586 −0.662930 0.748681i \(-0.730687\pi\)
−0.662930 + 0.748681i \(0.730687\pi\)
\(314\) 7134.00 1.28215
\(315\) −900.000 −0.160982
\(316\) −520.000 −0.0925705
\(317\) −3894.00 −0.689933 −0.344967 0.938615i \(-0.612110\pi\)
−0.344967 + 0.938615i \(0.612110\pi\)
\(318\) −4050.00 −0.714191
\(319\) 1872.00 0.328564
\(320\) −2165.00 −0.378210
\(321\) 4212.00 0.732370
\(322\) −7200.00 −1.24609
\(323\) −6696.00 −1.15348
\(324\) 81.0000 0.0138889
\(325\) 1850.00 0.315752
\(326\) −156.000 −0.0265032
\(327\) 4422.00 0.747820
\(328\) −6930.00 −1.16660
\(329\) −480.000 −0.0804354
\(330\) −1080.00 −0.180158
\(331\) 3692.00 0.613084 0.306542 0.951857i \(-0.400828\pi\)
0.306542 + 0.951857i \(0.400828\pi\)
\(332\) 156.000 0.0257880
\(333\) −630.000 −0.103675
\(334\) −11160.0 −1.82829
\(335\) 980.000 0.159830
\(336\) 4260.00 0.691673
\(337\) −8998.00 −1.45446 −0.727229 0.686395i \(-0.759192\pi\)
−0.727229 + 0.686395i \(0.759192\pi\)
\(338\) 9837.00 1.58302
\(339\) −3258.00 −0.521977
\(340\) −270.000 −0.0430671
\(341\) −4800.00 −0.762271
\(342\) −3348.00 −0.529354
\(343\) −5720.00 −0.900440
\(344\) −1932.00 −0.302809
\(345\) −1800.00 −0.280895
\(346\) 1278.00 0.198571
\(347\) 5244.00 0.811276 0.405638 0.914034i \(-0.367049\pi\)
0.405638 + 0.914034i \(0.367049\pi\)
\(348\) 234.000 0.0360452
\(349\) 6302.00 0.966585 0.483293 0.875459i \(-0.339441\pi\)
0.483293 + 0.875459i \(0.339441\pi\)
\(350\) 1500.00 0.229081
\(351\) −1998.00 −0.303833
\(352\) 1080.00 0.163535
\(353\) 3414.00 0.514756 0.257378 0.966311i \(-0.417141\pi\)
0.257378 + 0.966311i \(0.417141\pi\)
\(354\) −216.000 −0.0324301
\(355\) 1440.00 0.215288
\(356\) 1026.00 0.152747
\(357\) −3240.00 −0.480333
\(358\) −4320.00 −0.637763
\(359\) 4824.00 0.709195 0.354597 0.935019i \(-0.384618\pi\)
0.354597 + 0.935019i \(0.384618\pi\)
\(360\) 945.000 0.138350
\(361\) 8517.00 1.24173
\(362\) −9390.00 −1.36334
\(363\) 2265.00 0.327498
\(364\) 1480.00 0.213113
\(365\) 2150.00 0.308318
\(366\) 2898.00 0.413882
\(367\) −3508.00 −0.498954 −0.249477 0.968381i \(-0.580259\pi\)
−0.249477 + 0.968381i \(0.580259\pi\)
\(368\) 8520.00 1.20689
\(369\) 2970.00 0.419003
\(370\) 1050.00 0.147532
\(371\) 9000.00 1.25945
\(372\) −600.000 −0.0836251
\(373\) 10802.0 1.49948 0.749740 0.661732i \(-0.230178\pi\)
0.749740 + 0.661732i \(0.230178\pi\)
\(374\) −3888.00 −0.537550
\(375\) 375.000 0.0516398
\(376\) 504.000 0.0691272
\(377\) −5772.00 −0.788523
\(378\) −1620.00 −0.220433
\(379\) 1460.00 0.197876 0.0989382 0.995094i \(-0.468455\pi\)
0.0989382 + 0.995094i \(0.468455\pi\)
\(380\) 620.000 0.0836982
\(381\) −3732.00 −0.501827
\(382\) 10728.0 1.43689
\(383\) −4872.00 −0.649994 −0.324997 0.945715i \(-0.605363\pi\)
−0.324997 + 0.945715i \(0.605363\pi\)
\(384\) −4977.00 −0.661410
\(385\) 2400.00 0.317702
\(386\) 7998.00 1.05463
\(387\) 828.000 0.108759
\(388\) −286.000 −0.0374213
\(389\) −14046.0 −1.83075 −0.915373 0.402606i \(-0.868104\pi\)
−0.915373 + 0.402606i \(0.868104\pi\)
\(390\) 3330.00 0.432362
\(391\) −6480.00 −0.838127
\(392\) −1197.00 −0.154229
\(393\) −6984.00 −0.896428
\(394\) −8154.00 −1.04262
\(395\) 2600.00 0.331190
\(396\) −216.000 −0.0274101
\(397\) −2734.00 −0.345631 −0.172816 0.984954i \(-0.555286\pi\)
−0.172816 + 0.984954i \(0.555286\pi\)
\(398\) −11496.0 −1.44785
\(399\) 7440.00 0.933498
\(400\) −1775.00 −0.221875
\(401\) −15942.0 −1.98530 −0.992650 0.121019i \(-0.961384\pi\)
−0.992650 + 0.121019i \(0.961384\pi\)
\(402\) 1764.00 0.218857
\(403\) 14800.0 1.82938
\(404\) −1734.00 −0.213539
\(405\) −405.000 −0.0496904
\(406\) −4680.00 −0.572080
\(407\) 1680.00 0.204606
\(408\) 3402.00 0.412804
\(409\) 8714.00 1.05350 0.526748 0.850022i \(-0.323411\pi\)
0.526748 + 0.850022i \(0.323411\pi\)
\(410\) −4950.00 −0.596251
\(411\) −6354.00 −0.762578
\(412\) 452.000 0.0540496
\(413\) 480.000 0.0571895
\(414\) −3240.00 −0.384631
\(415\) −780.000 −0.0922619
\(416\) −3330.00 −0.392468
\(417\) −6972.00 −0.818754
\(418\) 8928.00 1.04470
\(419\) 11976.0 1.39634 0.698169 0.715933i \(-0.253998\pi\)
0.698169 + 0.715933i \(0.253998\pi\)
\(420\) 300.000 0.0348536
\(421\) 11054.0 1.27967 0.639833 0.768514i \(-0.279004\pi\)
0.639833 + 0.768514i \(0.279004\pi\)
\(422\) 3300.00 0.380667
\(423\) −216.000 −0.0248281
\(424\) −9450.00 −1.08239
\(425\) 1350.00 0.154081
\(426\) 2592.00 0.294795
\(427\) −6440.00 −0.729868
\(428\) −1404.00 −0.158563
\(429\) 5328.00 0.599623
\(430\) −1380.00 −0.154766
\(431\) 720.000 0.0804668 0.0402334 0.999190i \(-0.487190\pi\)
0.0402334 + 0.999190i \(0.487190\pi\)
\(432\) 1917.00 0.213499
\(433\) −15622.0 −1.73382 −0.866912 0.498462i \(-0.833898\pi\)
−0.866912 + 0.498462i \(0.833898\pi\)
\(434\) 12000.0 1.32723
\(435\) −1170.00 −0.128959
\(436\) −1474.00 −0.161908
\(437\) 14880.0 1.62885
\(438\) 3870.00 0.422182
\(439\) −9880.00 −1.07414 −0.537069 0.843538i \(-0.680469\pi\)
−0.537069 + 0.843538i \(0.680469\pi\)
\(440\) −2520.00 −0.273037
\(441\) 513.000 0.0553936
\(442\) 11988.0 1.29007
\(443\) −16116.0 −1.72843 −0.864215 0.503123i \(-0.832184\pi\)
−0.864215 + 0.503123i \(0.832184\pi\)
\(444\) 210.000 0.0224463
\(445\) −5130.00 −0.546484
\(446\) 5892.00 0.625548
\(447\) −774.000 −0.0818992
\(448\) 8660.00 0.913274
\(449\) 9018.00 0.947852 0.473926 0.880565i \(-0.342836\pi\)
0.473926 + 0.880565i \(0.342836\pi\)
\(450\) 675.000 0.0707107
\(451\) −7920.00 −0.826914
\(452\) 1086.00 0.113011
\(453\) 2424.00 0.251412
\(454\) 1980.00 0.204683
\(455\) −7400.00 −0.762456
\(456\) −7812.00 −0.802260
\(457\) −3670.00 −0.375657 −0.187829 0.982202i \(-0.560145\pi\)
−0.187829 + 0.982202i \(0.560145\pi\)
\(458\) −5718.00 −0.583372
\(459\) −1458.00 −0.148265
\(460\) 600.000 0.0608155
\(461\) 17562.0 1.77428 0.887141 0.461499i \(-0.152688\pi\)
0.887141 + 0.461499i \(0.152688\pi\)
\(462\) 4320.00 0.435032
\(463\) 1172.00 0.117640 0.0588202 0.998269i \(-0.481266\pi\)
0.0588202 + 0.998269i \(0.481266\pi\)
\(464\) 5538.00 0.554084
\(465\) 3000.00 0.299186
\(466\) −4374.00 −0.434810
\(467\) 6876.00 0.681335 0.340667 0.940184i \(-0.389347\pi\)
0.340667 + 0.940184i \(0.389347\pi\)
\(468\) 666.000 0.0657818
\(469\) −3920.00 −0.385946
\(470\) 360.000 0.0353310
\(471\) −7134.00 −0.697914
\(472\) −504.000 −0.0491493
\(473\) −2208.00 −0.214638
\(474\) 4680.00 0.453501
\(475\) −3100.00 −0.299448
\(476\) 1080.00 0.103995
\(477\) 4050.00 0.388756
\(478\) 3528.00 0.337588
\(479\) 2280.00 0.217486 0.108743 0.994070i \(-0.465317\pi\)
0.108743 + 0.994070i \(0.465317\pi\)
\(480\) −675.000 −0.0641862
\(481\) −5180.00 −0.491035
\(482\) 2598.00 0.245510
\(483\) 7200.00 0.678284
\(484\) −755.000 −0.0709053
\(485\) 1430.00 0.133882
\(486\) −729.000 −0.0680414
\(487\) −3076.00 −0.286215 −0.143108 0.989707i \(-0.545710\pi\)
−0.143108 + 0.989707i \(0.545710\pi\)
\(488\) 6762.00 0.627257
\(489\) 156.000 0.0144265
\(490\) −855.000 −0.0788265
\(491\) −18912.0 −1.73826 −0.869131 0.494582i \(-0.835321\pi\)
−0.869131 + 0.494582i \(0.835321\pi\)
\(492\) −990.000 −0.0907168
\(493\) −4212.00 −0.384785
\(494\) −27528.0 −2.50717
\(495\) 1080.00 0.0980654
\(496\) −14200.0 −1.28548
\(497\) −5760.00 −0.519862
\(498\) −1404.00 −0.126335
\(499\) 9956.00 0.893170 0.446585 0.894741i \(-0.352640\pi\)
0.446585 + 0.894741i \(0.352640\pi\)
\(500\) −125.000 −0.0111803
\(501\) 11160.0 0.995194
\(502\) 1296.00 0.115226
\(503\) −10656.0 −0.944588 −0.472294 0.881441i \(-0.656574\pi\)
−0.472294 + 0.881441i \(0.656574\pi\)
\(504\) −3780.00 −0.334077
\(505\) 8670.00 0.763980
\(506\) 8640.00 0.759081
\(507\) −9837.00 −0.861689
\(508\) 1244.00 0.108649
\(509\) −2766.00 −0.240866 −0.120433 0.992721i \(-0.538428\pi\)
−0.120433 + 0.992721i \(0.538428\pi\)
\(510\) 2430.00 0.210985
\(511\) −8600.00 −0.744504
\(512\) −8733.00 −0.753804
\(513\) 3348.00 0.288144
\(514\) 7578.00 0.650294
\(515\) −2260.00 −0.193374
\(516\) −276.000 −0.0235469
\(517\) 576.000 0.0489989
\(518\) −4200.00 −0.356250
\(519\) −1278.00 −0.108089
\(520\) 7770.00 0.655264
\(521\) 10530.0 0.885466 0.442733 0.896654i \(-0.354009\pi\)
0.442733 + 0.896654i \(0.354009\pi\)
\(522\) −2106.00 −0.176585
\(523\) 12692.0 1.06115 0.530576 0.847637i \(-0.321976\pi\)
0.530576 + 0.847637i \(0.321976\pi\)
\(524\) 2328.00 0.194082
\(525\) −1500.00 −0.124696
\(526\) 16344.0 1.35481
\(527\) 10800.0 0.892705
\(528\) −5112.00 −0.421347
\(529\) 2233.00 0.183529
\(530\) −6750.00 −0.553210
\(531\) 216.000 0.0176527
\(532\) −2480.00 −0.202108
\(533\) 24420.0 1.98452
\(534\) −9234.00 −0.748304
\(535\) 7020.00 0.567292
\(536\) 4116.00 0.331687
\(537\) 4320.00 0.347154
\(538\) −7722.00 −0.618809
\(539\) −1368.00 −0.109321
\(540\) 135.000 0.0107583
\(541\) 18110.0 1.43920 0.719602 0.694386i \(-0.244324\pi\)
0.719602 + 0.694386i \(0.244324\pi\)
\(542\) −9552.00 −0.756999
\(543\) 9390.00 0.742106
\(544\) −2430.00 −0.191517
\(545\) 7370.00 0.579259
\(546\) −13320.0 −1.04404
\(547\) 3620.00 0.282962 0.141481 0.989941i \(-0.454814\pi\)
0.141481 + 0.989941i \(0.454814\pi\)
\(548\) 2118.00 0.165103
\(549\) −2898.00 −0.225289
\(550\) −1800.00 −0.139550
\(551\) 9672.00 0.747806
\(552\) −7560.00 −0.582926
\(553\) −10400.0 −0.799734
\(554\) 11886.0 0.911530
\(555\) −1050.00 −0.0803063
\(556\) 2324.00 0.177265
\(557\) −14166.0 −1.07762 −0.538809 0.842428i \(-0.681125\pi\)
−0.538809 + 0.842428i \(0.681125\pi\)
\(558\) 5400.00 0.409678
\(559\) 6808.00 0.515112
\(560\) 7100.00 0.535767
\(561\) 3888.00 0.292605
\(562\) −24858.0 −1.86579
\(563\) −13404.0 −1.00339 −0.501697 0.865043i \(-0.667291\pi\)
−0.501697 + 0.865043i \(0.667291\pi\)
\(564\) 72.0000 0.00537544
\(565\) −5430.00 −0.404322
\(566\) −8148.00 −0.605099
\(567\) 1620.00 0.119989
\(568\) 6048.00 0.446775
\(569\) −18654.0 −1.37437 −0.687185 0.726483i \(-0.741154\pi\)
−0.687185 + 0.726483i \(0.741154\pi\)
\(570\) −5580.00 −0.410036
\(571\) −7684.00 −0.563162 −0.281581 0.959537i \(-0.590859\pi\)
−0.281581 + 0.959537i \(0.590859\pi\)
\(572\) −1776.00 −0.129822
\(573\) −10728.0 −0.782144
\(574\) 19800.0 1.43978
\(575\) −3000.00 −0.217580
\(576\) 3897.00 0.281901
\(577\) −1726.00 −0.124531 −0.0622654 0.998060i \(-0.519833\pi\)
−0.0622654 + 0.998060i \(0.519833\pi\)
\(578\) −5991.00 −0.431129
\(579\) −7998.00 −0.574068
\(580\) 390.000 0.0279205
\(581\) 3120.00 0.222787
\(582\) 2574.00 0.183326
\(583\) −10800.0 −0.767222
\(584\) 9030.00 0.639836
\(585\) −3330.00 −0.235348
\(586\) 18054.0 1.27270
\(587\) 10596.0 0.745049 0.372524 0.928022i \(-0.378492\pi\)
0.372524 + 0.928022i \(0.378492\pi\)
\(588\) −171.000 −0.0119931
\(589\) −24800.0 −1.73492
\(590\) −360.000 −0.0251203
\(591\) 8154.00 0.567531
\(592\) 4970.00 0.345043
\(593\) 2862.00 0.198193 0.0990963 0.995078i \(-0.468405\pi\)
0.0990963 + 0.995078i \(0.468405\pi\)
\(594\) 1944.00 0.134282
\(595\) −5400.00 −0.372065
\(596\) 258.000 0.0177317
\(597\) 11496.0 0.788107
\(598\) −26640.0 −1.82172
\(599\) −23592.0 −1.60925 −0.804627 0.593781i \(-0.797635\pi\)
−0.804627 + 0.593781i \(0.797635\pi\)
\(600\) 1575.00 0.107165
\(601\) −9574.00 −0.649803 −0.324902 0.945748i \(-0.605331\pi\)
−0.324902 + 0.945748i \(0.605331\pi\)
\(602\) 5520.00 0.373718
\(603\) −1764.00 −0.119130
\(604\) −808.000 −0.0544322
\(605\) 3775.00 0.253679
\(606\) 15606.0 1.04612
\(607\) 17444.0 1.16644 0.583221 0.812314i \(-0.301792\pi\)
0.583221 + 0.812314i \(0.301792\pi\)
\(608\) 5580.00 0.372202
\(609\) 4680.00 0.311401
\(610\) 4830.00 0.320592
\(611\) −1776.00 −0.117593
\(612\) 486.000 0.0321003
\(613\) −2374.00 −0.156419 −0.0782096 0.996937i \(-0.524920\pi\)
−0.0782096 + 0.996937i \(0.524920\pi\)
\(614\) 27708.0 1.82118
\(615\) 4950.00 0.324558
\(616\) 10080.0 0.659310
\(617\) −12162.0 −0.793555 −0.396778 0.917915i \(-0.629872\pi\)
−0.396778 + 0.917915i \(0.629872\pi\)
\(618\) −4068.00 −0.264788
\(619\) 8804.00 0.571668 0.285834 0.958279i \(-0.407729\pi\)
0.285834 + 0.958279i \(0.407729\pi\)
\(620\) −1000.00 −0.0647758
\(621\) 3240.00 0.209367
\(622\) 4608.00 0.297048
\(623\) 20520.0 1.31961
\(624\) 15762.0 1.01119
\(625\) 625.000 0.0400000
\(626\) −22026.0 −1.40629
\(627\) −8928.00 −0.568660
\(628\) 2378.00 0.151103
\(629\) −3780.00 −0.239616
\(630\) −2700.00 −0.170747
\(631\) −12688.0 −0.800478 −0.400239 0.916411i \(-0.631073\pi\)
−0.400239 + 0.916411i \(0.631073\pi\)
\(632\) 10920.0 0.687301
\(633\) −3300.00 −0.207209
\(634\) −11682.0 −0.731785
\(635\) −6220.00 −0.388714
\(636\) −1350.00 −0.0841682
\(637\) 4218.00 0.262360
\(638\) 5616.00 0.348495
\(639\) −2592.00 −0.160466
\(640\) −8295.00 −0.512326
\(641\) −9150.00 −0.563812 −0.281906 0.959442i \(-0.590967\pi\)
−0.281906 + 0.959442i \(0.590967\pi\)
\(642\) 12636.0 0.776796
\(643\) 25292.0 1.55120 0.775598 0.631227i \(-0.217448\pi\)
0.775598 + 0.631227i \(0.217448\pi\)
\(644\) −2400.00 −0.146853
\(645\) 1380.00 0.0842441
\(646\) −20088.0 −1.22345
\(647\) −2736.00 −0.166249 −0.0831246 0.996539i \(-0.526490\pi\)
−0.0831246 + 0.996539i \(0.526490\pi\)
\(648\) −1701.00 −0.103120
\(649\) −576.000 −0.0348382
\(650\) 5550.00 0.334906
\(651\) −12000.0 −0.722453
\(652\) −52.0000 −0.00312343
\(653\) 22218.0 1.33148 0.665741 0.746183i \(-0.268116\pi\)
0.665741 + 0.746183i \(0.268116\pi\)
\(654\) 13266.0 0.793183
\(655\) −11640.0 −0.694370
\(656\) −23430.0 −1.39449
\(657\) −3870.00 −0.229807
\(658\) −1440.00 −0.0853147
\(659\) 14520.0 0.858299 0.429149 0.903234i \(-0.358813\pi\)
0.429149 + 0.903234i \(0.358813\pi\)
\(660\) −360.000 −0.0212318
\(661\) −10618.0 −0.624799 −0.312400 0.949951i \(-0.601133\pi\)
−0.312400 + 0.949951i \(0.601133\pi\)
\(662\) 11076.0 0.650273
\(663\) −11988.0 −0.702225
\(664\) −3276.00 −0.191466
\(665\) 12400.0 0.723085
\(666\) −1890.00 −0.109964
\(667\) 9360.00 0.543359
\(668\) −3720.00 −0.215466
\(669\) −5892.00 −0.340505
\(670\) 2940.00 0.169526
\(671\) 7728.00 0.444614
\(672\) 2700.00 0.154992
\(673\) 1370.00 0.0784690 0.0392345 0.999230i \(-0.487508\pi\)
0.0392345 + 0.999230i \(0.487508\pi\)
\(674\) −26994.0 −1.54269
\(675\) −675.000 −0.0384900
\(676\) 3279.00 0.186561
\(677\) −13758.0 −0.781038 −0.390519 0.920595i \(-0.627704\pi\)
−0.390519 + 0.920595i \(0.627704\pi\)
\(678\) −9774.00 −0.553640
\(679\) −5720.00 −0.323289
\(680\) 5670.00 0.319757
\(681\) −1980.00 −0.111415
\(682\) −14400.0 −0.808511
\(683\) 11988.0 0.671608 0.335804 0.941932i \(-0.390992\pi\)
0.335804 + 0.941932i \(0.390992\pi\)
\(684\) −1116.00 −0.0623850
\(685\) −10590.0 −0.590691
\(686\) −17160.0 −0.955061
\(687\) 5718.00 0.317548
\(688\) −6532.00 −0.361962
\(689\) 33300.0 1.84126
\(690\) −5400.00 −0.297934
\(691\) 32996.0 1.81654 0.908268 0.418388i \(-0.137405\pi\)
0.908268 + 0.418388i \(0.137405\pi\)
\(692\) 426.000 0.0234019
\(693\) −4320.00 −0.236801
\(694\) 15732.0 0.860488
\(695\) −11620.0 −0.634204
\(696\) −4914.00 −0.267622
\(697\) 17820.0 0.968408
\(698\) 18906.0 1.02522
\(699\) 4374.00 0.236681
\(700\) 500.000 0.0269975
\(701\) −25902.0 −1.39558 −0.697792 0.716300i \(-0.745834\pi\)
−0.697792 + 0.716300i \(0.745834\pi\)
\(702\) −5994.00 −0.322263
\(703\) 8680.00 0.465679
\(704\) −10392.0 −0.556340
\(705\) −360.000 −0.0192318
\(706\) 10242.0 0.545981
\(707\) −34680.0 −1.84480
\(708\) −72.0000 −0.00382193
\(709\) −27394.0 −1.45106 −0.725531 0.688189i \(-0.758406\pi\)
−0.725531 + 0.688189i \(0.758406\pi\)
\(710\) 4320.00 0.228347
\(711\) −4680.00 −0.246855
\(712\) −21546.0 −1.13409
\(713\) −24000.0 −1.26060
\(714\) −9720.00 −0.509470
\(715\) 8880.00 0.464466
\(716\) −1440.00 −0.0751611
\(717\) −3528.00 −0.183760
\(718\) 14472.0 0.752215
\(719\) 34848.0 1.80753 0.903763 0.428033i \(-0.140793\pi\)
0.903763 + 0.428033i \(0.140793\pi\)
\(720\) 3195.00 0.165376
\(721\) 9040.00 0.466945
\(722\) 25551.0 1.31705
\(723\) −2598.00 −0.133639
\(724\) −3130.00 −0.160671
\(725\) −1950.00 −0.0998913
\(726\) 6795.00 0.347364
\(727\) 28028.0 1.42985 0.714925 0.699201i \(-0.246461\pi\)
0.714925 + 0.699201i \(0.246461\pi\)
\(728\) −31080.0 −1.58228
\(729\) 729.000 0.0370370
\(730\) 6450.00 0.327021
\(731\) 4968.00 0.251365
\(732\) 966.000 0.0487765
\(733\) 18002.0 0.907120 0.453560 0.891226i \(-0.350154\pi\)
0.453560 + 0.891226i \(0.350154\pi\)
\(734\) −10524.0 −0.529221
\(735\) 855.000 0.0429077
\(736\) 5400.00 0.270444
\(737\) 4704.00 0.235107
\(738\) 8910.00 0.444420
\(739\) 15284.0 0.760800 0.380400 0.924822i \(-0.375786\pi\)
0.380400 + 0.924822i \(0.375786\pi\)
\(740\) 350.000 0.0173868
\(741\) 27528.0 1.36473
\(742\) 27000.0 1.33585
\(743\) −18768.0 −0.926691 −0.463345 0.886178i \(-0.653351\pi\)
−0.463345 + 0.886178i \(0.653351\pi\)
\(744\) 12600.0 0.620885
\(745\) −1290.00 −0.0634388
\(746\) 32406.0 1.59044
\(747\) 1404.00 0.0687680
\(748\) −1296.00 −0.0633509
\(749\) −28080.0 −1.36985
\(750\) 1125.00 0.0547723
\(751\) 8696.00 0.422532 0.211266 0.977429i \(-0.432241\pi\)
0.211266 + 0.977429i \(0.432241\pi\)
\(752\) 1704.00 0.0826310
\(753\) −1296.00 −0.0627209
\(754\) −17316.0 −0.836355
\(755\) 4040.00 0.194743
\(756\) −540.000 −0.0259783
\(757\) −38662.0 −1.85627 −0.928134 0.372247i \(-0.878587\pi\)
−0.928134 + 0.372247i \(0.878587\pi\)
\(758\) 4380.00 0.209880
\(759\) −8640.00 −0.413191
\(760\) −13020.0 −0.621428
\(761\) 23874.0 1.13723 0.568615 0.822604i \(-0.307479\pi\)
0.568615 + 0.822604i \(0.307479\pi\)
\(762\) −11196.0 −0.532268
\(763\) −29480.0 −1.39875
\(764\) 3576.00 0.169339
\(765\) −2430.00 −0.114846
\(766\) −14616.0 −0.689422
\(767\) 1776.00 0.0836084
\(768\) −4539.00 −0.213264
\(769\) 23618.0 1.10753 0.553763 0.832675i \(-0.313192\pi\)
0.553763 + 0.832675i \(0.313192\pi\)
\(770\) 7200.00 0.336974
\(771\) −7578.00 −0.353975
\(772\) 2666.00 0.124289
\(773\) 11538.0 0.536860 0.268430 0.963299i \(-0.413495\pi\)
0.268430 + 0.963299i \(0.413495\pi\)
\(774\) 2484.00 0.115356
\(775\) 5000.00 0.231749
\(776\) 6006.00 0.277839
\(777\) 4200.00 0.193918
\(778\) −42138.0 −1.94180
\(779\) −40920.0 −1.88204
\(780\) 1110.00 0.0509543
\(781\) 6912.00 0.316685
\(782\) −19440.0 −0.888968
\(783\) 2106.00 0.0961204
\(784\) −4047.00 −0.184357
\(785\) −11890.0 −0.540602
\(786\) −20952.0 −0.950805
\(787\) −14884.0 −0.674152 −0.337076 0.941478i \(-0.609438\pi\)
−0.337076 + 0.941478i \(0.609438\pi\)
\(788\) −2718.00 −0.122874
\(789\) −16344.0 −0.737467
\(790\) 7800.00 0.351280
\(791\) 21720.0 0.976327
\(792\) 4536.00 0.203510
\(793\) −23828.0 −1.06703
\(794\) −8202.00 −0.366597
\(795\) 6750.00 0.301129
\(796\) −3832.00 −0.170630
\(797\) −11334.0 −0.503728 −0.251864 0.967763i \(-0.581043\pi\)
−0.251864 + 0.967763i \(0.581043\pi\)
\(798\) 22320.0 0.990125
\(799\) −1296.00 −0.0573832
\(800\) −1125.00 −0.0497184
\(801\) 9234.00 0.407325
\(802\) −47826.0 −2.10573
\(803\) 10320.0 0.453530
\(804\) 588.000 0.0257925
\(805\) 12000.0 0.525397
\(806\) 44400.0 1.94035
\(807\) 7722.00 0.336837
\(808\) 36414.0 1.58545
\(809\) 44730.0 1.94391 0.971955 0.235167i \(-0.0755638\pi\)
0.971955 + 0.235167i \(0.0755638\pi\)
\(810\) −1215.00 −0.0527046
\(811\) −42748.0 −1.85091 −0.925453 0.378862i \(-0.876316\pi\)
−0.925453 + 0.378862i \(0.876316\pi\)
\(812\) −1560.00 −0.0674203
\(813\) 9552.00 0.412058
\(814\) 5040.00 0.217017
\(815\) 260.000 0.0111747
\(816\) 11502.0 0.493444
\(817\) −11408.0 −0.488513
\(818\) 26142.0 1.11740
\(819\) 13320.0 0.568301
\(820\) −1650.00 −0.0702689
\(821\) −31686.0 −1.34695 −0.673477 0.739208i \(-0.735200\pi\)
−0.673477 + 0.739208i \(0.735200\pi\)
\(822\) −19062.0 −0.808836
\(823\) 11036.0 0.467425 0.233713 0.972306i \(-0.424913\pi\)
0.233713 + 0.972306i \(0.424913\pi\)
\(824\) −9492.00 −0.401298
\(825\) 1800.00 0.0759612
\(826\) 1440.00 0.0606586
\(827\) 25884.0 1.08836 0.544181 0.838968i \(-0.316841\pi\)
0.544181 + 0.838968i \(0.316841\pi\)
\(828\) −1080.00 −0.0453292
\(829\) 15950.0 0.668234 0.334117 0.942532i \(-0.391562\pi\)
0.334117 + 0.942532i \(0.391562\pi\)
\(830\) −2340.00 −0.0978585
\(831\) −11886.0 −0.496174
\(832\) 32042.0 1.33516
\(833\) 3078.00 0.128027
\(834\) −20916.0 −0.868419
\(835\) 18600.0 0.770874
\(836\) 2976.00 0.123119
\(837\) −5400.00 −0.223000
\(838\) 35928.0 1.48104
\(839\) 13800.0 0.567853 0.283927 0.958846i \(-0.408363\pi\)
0.283927 + 0.958846i \(0.408363\pi\)
\(840\) −6300.00 −0.258775
\(841\) −18305.0 −0.750543
\(842\) 33162.0 1.35729
\(843\) 24858.0 1.01560
\(844\) 1100.00 0.0448620
\(845\) −16395.0 −0.667462
\(846\) −648.000 −0.0263342
\(847\) −15100.0 −0.612565
\(848\) −31950.0 −1.29383
\(849\) 8148.00 0.329374
\(850\) 4050.00 0.163428
\(851\) 8400.00 0.338365
\(852\) 864.000 0.0347420
\(853\) −27862.0 −1.11838 −0.559189 0.829040i \(-0.688887\pi\)
−0.559189 + 0.829040i \(0.688887\pi\)
\(854\) −19320.0 −0.774141
\(855\) 5580.00 0.223195
\(856\) 29484.0 1.17727
\(857\) −7314.00 −0.291530 −0.145765 0.989319i \(-0.546564\pi\)
−0.145765 + 0.989319i \(0.546564\pi\)
\(858\) 15984.0 0.635996
\(859\) −28780.0 −1.14314 −0.571572 0.820552i \(-0.693666\pi\)
−0.571572 + 0.820552i \(0.693666\pi\)
\(860\) −460.000 −0.0182394
\(861\) −19800.0 −0.783719
\(862\) 2160.00 0.0853479
\(863\) −32688.0 −1.28935 −0.644677 0.764455i \(-0.723008\pi\)
−0.644677 + 0.764455i \(0.723008\pi\)
\(864\) 1215.00 0.0478416
\(865\) −2130.00 −0.0837251
\(866\) −46866.0 −1.83900
\(867\) 5991.00 0.234677
\(868\) 4000.00 0.156416
\(869\) 12480.0 0.487175
\(870\) −3510.00 −0.136782
\(871\) −14504.0 −0.564236
\(872\) 30954.0 1.20210
\(873\) −2574.00 −0.0997900
\(874\) 44640.0 1.72766
\(875\) −2500.00 −0.0965891
\(876\) 1290.00 0.0497546
\(877\) 36650.0 1.41115 0.705577 0.708633i \(-0.250688\pi\)
0.705577 + 0.708633i \(0.250688\pi\)
\(878\) −29640.0 −1.13930
\(879\) −18054.0 −0.692772
\(880\) −8520.00 −0.326374
\(881\) −2646.00 −0.101187 −0.0505936 0.998719i \(-0.516111\pi\)
−0.0505936 + 0.998719i \(0.516111\pi\)
\(882\) 1539.00 0.0587538
\(883\) 10892.0 0.415113 0.207557 0.978223i \(-0.433449\pi\)
0.207557 + 0.978223i \(0.433449\pi\)
\(884\) 3996.00 0.152036
\(885\) 360.000 0.0136737
\(886\) −48348.0 −1.83328
\(887\) −43464.0 −1.64530 −0.822648 0.568550i \(-0.807504\pi\)
−0.822648 + 0.568550i \(0.807504\pi\)
\(888\) −4410.00 −0.166655
\(889\) 24880.0 0.938637
\(890\) −15390.0 −0.579634
\(891\) −1944.00 −0.0730937
\(892\) 1964.00 0.0737215
\(893\) 2976.00 0.111521
\(894\) −2322.00 −0.0868672
\(895\) 7200.00 0.268904
\(896\) 33180.0 1.23713
\(897\) 26640.0 0.991621
\(898\) 27054.0 1.00535
\(899\) −15600.0 −0.578742
\(900\) 225.000 0.00833333
\(901\) 24300.0 0.898502
\(902\) −23760.0 −0.877075
\(903\) −5520.00 −0.203426
\(904\) −22806.0 −0.839067
\(905\) 15650.0 0.574833
\(906\) 7272.00 0.266662
\(907\) −14884.0 −0.544890 −0.272445 0.962171i \(-0.587832\pi\)
−0.272445 + 0.962171i \(0.587832\pi\)
\(908\) 660.000 0.0241221
\(909\) −15606.0 −0.569437
\(910\) −22200.0 −0.808706
\(911\) −1248.00 −0.0453876 −0.0226938 0.999742i \(-0.507224\pi\)
−0.0226938 + 0.999742i \(0.507224\pi\)
\(912\) −26412.0 −0.958979
\(913\) −3744.00 −0.135716
\(914\) −11010.0 −0.398445
\(915\) −4830.00 −0.174508
\(916\) −1906.00 −0.0687511
\(917\) 46560.0 1.67671
\(918\) −4374.00 −0.157259
\(919\) −6640.00 −0.238339 −0.119169 0.992874i \(-0.538023\pi\)
−0.119169 + 0.992874i \(0.538023\pi\)
\(920\) −12600.0 −0.451532
\(921\) −27708.0 −0.991324
\(922\) 52686.0 1.88191
\(923\) −21312.0 −0.760014
\(924\) 1440.00 0.0512690
\(925\) −1750.00 −0.0622050
\(926\) 3516.00 0.124776
\(927\) 4068.00 0.144132
\(928\) 3510.00 0.124161
\(929\) 29946.0 1.05758 0.528792 0.848751i \(-0.322645\pi\)
0.528792 + 0.848751i \(0.322645\pi\)
\(930\) 9000.00 0.317335
\(931\) −7068.00 −0.248812
\(932\) −1458.00 −0.0512429
\(933\) −4608.00 −0.161693
\(934\) 20628.0 0.722665
\(935\) 6480.00 0.226651
\(936\) −13986.0 −0.488405
\(937\) 45002.0 1.56900 0.784499 0.620130i \(-0.212920\pi\)
0.784499 + 0.620130i \(0.212920\pi\)
\(938\) −11760.0 −0.409358
\(939\) 22026.0 0.765486
\(940\) 120.000 0.00416380
\(941\) 6090.00 0.210976 0.105488 0.994421i \(-0.466360\pi\)
0.105488 + 0.994421i \(0.466360\pi\)
\(942\) −21402.0 −0.740249
\(943\) −39600.0 −1.36750
\(944\) −1704.00 −0.0587505
\(945\) 2700.00 0.0929429
\(946\) −6624.00 −0.227658
\(947\) 56388.0 1.93491 0.967457 0.253035i \(-0.0814288\pi\)
0.967457 + 0.253035i \(0.0814288\pi\)
\(948\) 1560.00 0.0534456
\(949\) −31820.0 −1.08843
\(950\) −9300.00 −0.317612
\(951\) 11682.0 0.398333
\(952\) −22680.0 −0.772125
\(953\) 10854.0 0.368936 0.184468 0.982839i \(-0.440944\pi\)
0.184468 + 0.982839i \(0.440944\pi\)
\(954\) 12150.0 0.412338
\(955\) −17880.0 −0.605846
\(956\) 1176.00 0.0397851
\(957\) −5616.00 −0.189696
\(958\) 6840.00 0.230679
\(959\) 42360.0 1.42636
\(960\) 6495.00 0.218360
\(961\) 10209.0 0.342687
\(962\) −15540.0 −0.520821
\(963\) −12636.0 −0.422834
\(964\) 866.000 0.0289336
\(965\) −13330.0 −0.444671
\(966\) 21600.0 0.719429
\(967\) −42316.0 −1.40723 −0.703615 0.710582i \(-0.748432\pi\)
−0.703615 + 0.710582i \(0.748432\pi\)
\(968\) 15855.0 0.526445
\(969\) 20088.0 0.665964
\(970\) 4290.00 0.142004
\(971\) 24480.0 0.809063 0.404532 0.914524i \(-0.367435\pi\)
0.404532 + 0.914524i \(0.367435\pi\)
\(972\) −243.000 −0.00801875
\(973\) 46480.0 1.53143
\(974\) −9228.00 −0.303577
\(975\) −5550.00 −0.182300
\(976\) 22862.0 0.749790
\(977\) −6906.00 −0.226144 −0.113072 0.993587i \(-0.536069\pi\)
−0.113072 + 0.993587i \(0.536069\pi\)
\(978\) 468.000 0.0153016
\(979\) −24624.0 −0.803868
\(980\) −285.000 −0.00928979
\(981\) −13266.0 −0.431754
\(982\) −56736.0 −1.84371
\(983\) 6960.00 0.225829 0.112914 0.993605i \(-0.463981\pi\)
0.112914 + 0.993605i \(0.463981\pi\)
\(984\) 20790.0 0.673538
\(985\) 13590.0 0.439608
\(986\) −12636.0 −0.408126
\(987\) 1440.00 0.0464394
\(988\) −9176.00 −0.295473
\(989\) −11040.0 −0.354956
\(990\) 3240.00 0.104014
\(991\) 47792.0 1.53195 0.765975 0.642870i \(-0.222256\pi\)
0.765975 + 0.642870i \(0.222256\pi\)
\(992\) −9000.00 −0.288055
\(993\) −11076.0 −0.353964
\(994\) −17280.0 −0.551397
\(995\) 19160.0 0.610465
\(996\) −468.000 −0.0148887
\(997\) 9938.00 0.315687 0.157843 0.987464i \(-0.449546\pi\)
0.157843 + 0.987464i \(0.449546\pi\)
\(998\) 29868.0 0.947350
\(999\) 1890.00 0.0598568
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 15.4.a.b.1.1 1
3.2 odd 2 45.4.a.b.1.1 1
4.3 odd 2 240.4.a.f.1.1 1
5.2 odd 4 75.4.b.a.49.2 2
5.3 odd 4 75.4.b.a.49.1 2
5.4 even 2 75.4.a.a.1.1 1
7.6 odd 2 735.4.a.i.1.1 1
8.3 odd 2 960.4.a.l.1.1 1
8.5 even 2 960.4.a.bi.1.1 1
9.2 odd 6 405.4.e.k.271.1 2
9.4 even 3 405.4.e.d.136.1 2
9.5 odd 6 405.4.e.k.136.1 2
9.7 even 3 405.4.e.d.271.1 2
11.10 odd 2 1815.4.a.a.1.1 1
12.11 even 2 720.4.a.r.1.1 1
15.2 even 4 225.4.b.d.199.1 2
15.8 even 4 225.4.b.d.199.2 2
15.14 odd 2 225.4.a.g.1.1 1
20.3 even 4 1200.4.f.m.49.2 2
20.7 even 4 1200.4.f.m.49.1 2
20.19 odd 2 1200.4.a.o.1.1 1
21.20 even 2 2205.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 1.1 even 1 trivial
45.4.a.b.1.1 1 3.2 odd 2
75.4.a.a.1.1 1 5.4 even 2
75.4.b.a.49.1 2 5.3 odd 4
75.4.b.a.49.2 2 5.2 odd 4
225.4.a.g.1.1 1 15.14 odd 2
225.4.b.d.199.1 2 15.2 even 4
225.4.b.d.199.2 2 15.8 even 4
240.4.a.f.1.1 1 4.3 odd 2
405.4.e.d.136.1 2 9.4 even 3
405.4.e.d.271.1 2 9.7 even 3
405.4.e.k.136.1 2 9.5 odd 6
405.4.e.k.271.1 2 9.2 odd 6
720.4.a.r.1.1 1 12.11 even 2
735.4.a.i.1.1 1 7.6 odd 2
960.4.a.l.1.1 1 8.3 odd 2
960.4.a.bi.1.1 1 8.5 even 2
1200.4.a.o.1.1 1 20.19 odd 2
1200.4.f.m.49.1 2 20.7 even 4
1200.4.f.m.49.2 2 20.3 even 4
1815.4.a.a.1.1 1 11.10 odd 2
2205.4.a.c.1.1 1 21.20 even 2