Properties

Label 735.4.a.i.1.1
Level $735$
Weight $4$
Character 735.1
Self dual yes
Analytic conductor $43.366$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.3664038542\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 735.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +5.00000 q^{5} +9.00000 q^{6} -21.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +5.00000 q^{5} +9.00000 q^{6} -21.0000 q^{8} +9.00000 q^{9} +15.0000 q^{10} -24.0000 q^{11} +3.00000 q^{12} -74.0000 q^{13} +15.0000 q^{15} -71.0000 q^{16} -54.0000 q^{17} +27.0000 q^{18} +124.000 q^{19} +5.00000 q^{20} -72.0000 q^{22} -120.000 q^{23} -63.0000 q^{24} +25.0000 q^{25} -222.000 q^{26} +27.0000 q^{27} -78.0000 q^{29} +45.0000 q^{30} -200.000 q^{31} -45.0000 q^{32} -72.0000 q^{33} -162.000 q^{34} +9.00000 q^{36} -70.0000 q^{37} +372.000 q^{38} -222.000 q^{39} -105.000 q^{40} -330.000 q^{41} +92.0000 q^{43} -24.0000 q^{44} +45.0000 q^{45} -360.000 q^{46} +24.0000 q^{47} -213.000 q^{48} +75.0000 q^{50} -162.000 q^{51} -74.0000 q^{52} +450.000 q^{53} +81.0000 q^{54} -120.000 q^{55} +372.000 q^{57} -234.000 q^{58} -24.0000 q^{59} +15.0000 q^{60} +322.000 q^{61} -600.000 q^{62} +433.000 q^{64} -370.000 q^{65} -216.000 q^{66} -196.000 q^{67} -54.0000 q^{68} -360.000 q^{69} -288.000 q^{71} -189.000 q^{72} +430.000 q^{73} -210.000 q^{74} +75.0000 q^{75} +124.000 q^{76} -666.000 q^{78} -520.000 q^{79} -355.000 q^{80} +81.0000 q^{81} -990.000 q^{82} -156.000 q^{83} -270.000 q^{85} +276.000 q^{86} -234.000 q^{87} +504.000 q^{88} -1026.00 q^{89} +135.000 q^{90} -120.000 q^{92} -600.000 q^{93} +72.0000 q^{94} +620.000 q^{95} -135.000 q^{96} +286.000 q^{97} -216.000 q^{99} +O(q^{100})\)

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
\(7\) 0 0
\(8\) −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 \(-0.789598\pi\)
−0.789381 + 0.613904i \(0.789598\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) −71.0000 −1.10938
\(17\) −54.0000 −0.770407 −0.385204 0.922832i \(-0.625869\pi\)
−0.385204 + 0.922832i \(0.625869\pi\)
\(18\) 27.0000 0.353553
\(19\) 124.000 1.49724 0.748620 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) 5.00000 0.0559017
\(21\) 0 0
\(22\) −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
\(28\) 0 0
\(29\) −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 \(-0.696702\pi\)
−0.579372 + 0.815063i \(0.696702\pi\)
\(32\) −45.0000 −0.248592
\(33\) −72.0000 −0.379806
\(34\) −162.000 −0.817140
\(35\) 0 0
\(36\) 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 \(-0.716332\pi\)
−0.628504 + 0.777806i \(0.716332\pi\)
\(42\) 0 0
\(43\) 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 \(-0.488143\pi\)
0.0372421 + 0.999306i \(0.488143\pi\)
\(48\) −213.000 −0.640498
\(49\) 0 0
\(50\) 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
\(56\) 0 0
\(57\) 372.000 0.864432
\(58\) −234.000 −0.529754
\(59\) −24.0000 −0.0529582 −0.0264791 0.999649i \(-0.508430\pi\)
−0.0264791 + 0.999649i \(0.508430\pi\)
\(60\) 15.0000 0.0322749
\(61\) 322.000 0.675867 0.337933 0.941170i \(-0.390272\pi\)
0.337933 + 0.941170i \(0.390272\pi\)
\(62\) −600.000 −1.22903
\(63\) 0 0
\(64\) 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
\(70\) 0 0
\(71\) −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 \(-0.387977\pi\)
0.344710 + 0.938709i \(0.387977\pi\)
\(74\) −210.000 −0.329892
\(75\) 75.0000 0.115470
\(76\) 124.000 0.187155
\(77\) 0 0
\(78\) −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 \(-0.532893\pi\)
−0.103152 + 0.994666i \(0.532893\pi\)
\(84\) 0 0
\(85\) −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 \(-0.709227\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 135.000 0.158114
\(91\) 0 0
\(92\) −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 \(-0.452174\pi\)
0.149685 + 0.988734i \(0.452174\pi\)
\(98\) 0 0
\(99\) −216.000 −0.219281
\(100\) 25.0000 0.0250000
\(101\) 1734.00 1.70831 0.854156 0.520017i \(-0.174075\pi\)
0.854156 + 0.520017i \(0.174075\pi\)
\(102\) −486.000 −0.471776
\(103\) −452.000 −0.432397 −0.216198 0.976349i \(-0.569366\pi\)
−0.216198 + 0.976349i \(0.569366\pi\)
\(104\) 1554.00 1.46521
\(105\) 0 0
\(106\) 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
\(112\) 0 0
\(113\) 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
\(119\) 0 0
\(120\) −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
\(126\) 0 0
\(127\) 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 \(-0.782921\pi\)
−0.776329 + 0.630327i \(0.782921\pi\)
\(132\) −72.0000 −0.0474757
\(133\) 0 0
\(134\) −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 \(-0.750881\pi\)
−0.709062 + 0.705147i \(0.750881\pi\)
\(140\) 0 0
\(141\) 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
\(147\) 0 0
\(148\) −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
\(154\) 0 0
\(155\) −1000.00 −0.518206
\(156\) −222.000 −0.113937
\(157\) −2378.00 −1.20882 −0.604411 0.796673i \(-0.706592\pi\)
−0.604411 + 0.796673i \(0.706592\pi\)
\(158\) −1560.00 −0.785487
\(159\) 1350.00 0.673346
\(160\) −225.000 −0.111174
\(161\) 0 0
\(162\) 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 \(-0.169298\pi\)
0.861863 + 0.507141i \(0.169298\pi\)
\(168\) 0 0
\(169\) 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 \(-0.529840\pi\)
−0.0936075 + 0.995609i \(0.529840\pi\)
\(174\) −702.000 −0.305853
\(175\) 0 0
\(176\) 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 \(-0.277821\pi\)
0.642683 + 0.766133i \(0.277821\pi\)
\(182\) 0 0
\(183\) 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
\(189\) 0 0
\(190\) 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
\(196\) 0 0
\(197\) −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 \(-0.260884\pi\)
0.682521 + 0.730866i \(0.260884\pi\)
\(200\) −525.000 −0.185616
\(201\) −588.000 −0.206340
\(202\) 5202.00 1.81194
\(203\) 0 0
\(204\) −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
\(210\) 0 0
\(211\) 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
\(217\) 0 0
\(218\) −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 \(-0.595282\pi\)
−0.294886 + 0.955532i \(0.595282\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 3258.00 0.958933
\(227\) −660.000 −0.192977 −0.0964884 0.995334i \(-0.530761\pi\)
−0.0964884 + 0.995334i \(0.530761\pi\)
\(228\) 372.000 0.108054
\(229\) 1906.00 0.550009 0.275004 0.961443i \(-0.411321\pi\)
0.275004 + 0.961443i \(0.411321\pi\)
\(230\) −1800.00 −0.516037
\(231\) 0 0
\(232\) 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
\(238\) 0 0
\(239\) 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 \(-0.536922\pi\)
−0.115734 + 0.993280i \(0.536922\pi\)
\(242\) −2265.00 −0.601652
\(243\) 243.000 0.0641500
\(244\) 322.000 0.0844834
\(245\) 0 0
\(246\) −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 \(-0.517298\pi\)
−0.0543179 + 0.998524i \(0.517298\pi\)
\(252\) 0 0
\(253\) 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 \(-0.599175\pi\)
−0.306552 + 0.951854i \(0.599175\pi\)
\(258\) 828.000 0.199802
\(259\) 0 0
\(260\) −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
\(266\) 0 0
\(267\) −3078.00 −0.705508
\(268\) −196.000 −0.0446739
\(269\) 2574.00 0.583418 0.291709 0.956507i \(-0.405776\pi\)
0.291709 + 0.956507i \(0.405776\pi\)
\(270\) 405.000 0.0912871
\(271\) 3184.00 0.713706 0.356853 0.934161i \(-0.383850\pi\)
0.356853 + 0.934161i \(0.383850\pi\)
\(272\) 3834.00 0.854671
\(273\) 0 0
\(274\) 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
\(280\) 0 0
\(281\) −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 \(-0.407925\pi\)
0.285246 + 0.958454i \(0.407925\pi\)
\(284\) −288.000 −0.0601748
\(285\) 1860.00 0.386586
\(286\) 5328.00 1.10158
\(287\) 0 0
\(288\) −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 \(-0.704816\pi\)
−0.599958 + 0.800032i \(0.704816\pi\)
\(294\) 0 0
\(295\) −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
\(301\) 0 0
\(302\) −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 \(-0.828611\pi\)
−0.858512 + 0.512793i \(0.828611\pi\)
\(308\) 0 0
\(309\) −1356.00 −0.249644
\(310\) −3000.00 −0.549640
\(311\) −1536.00 −0.280060 −0.140030 0.990147i \(-0.544720\pi\)
−0.140030 + 0.990147i \(0.544720\pi\)
\(312\) 4662.00 0.845942
\(313\) 7342.00 1.32586 0.662930 0.748681i \(-0.269313\pi\)
0.662930 + 0.748681i \(0.269313\pi\)
\(314\) −7134.00 −1.28215
\(315\) 0 0
\(316\) −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
\(322\) 0 0
\(323\) −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
\(329\) 0 0
\(330\) −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
\(336\) 0 0
\(337\) −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
\(343\) 0 0
\(344\) −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 \(-0.660559\pi\)
−0.483293 + 0.875459i \(0.660559\pi\)
\(350\) 0 0
\(351\) −1998.00 −0.303833
\(352\) 1080.00 0.163535
\(353\) −3414.00 −0.514756 −0.257378 0.966311i \(-0.582859\pi\)
−0.257378 + 0.966311i \(0.582859\pi\)
\(354\) −216.000 −0.0324301
\(355\) −1440.00 −0.215288
\(356\) −1026.00 −0.152747
\(357\) 0 0
\(358\) −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
\(364\) 0 0
\(365\) 2150.00 0.308318
\(366\) 2898.00 0.413882
\(367\) 3508.00 0.498954 0.249477 0.968381i \(-0.419741\pi\)
0.249477 + 0.968381i \(0.419741\pi\)
\(368\) 8520.00 1.20689
\(369\) −2970.00 −0.419003
\(370\) −1050.00 −0.147532
\(371\) 0 0
\(372\) −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
\(378\) 0 0
\(379\) 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 \(-0.394637\pi\)
0.324997 + 0.945715i \(0.394637\pi\)
\(384\) 4977.00 0.661410
\(385\) 0 0
\(386\) 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
\(392\) 0 0
\(393\) −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 \(-0.444714\pi\)
0.172816 + 0.984954i \(0.444714\pi\)
\(398\) 11496.0 1.44785
\(399\) 0 0
\(400\) −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
\(406\) 0 0
\(407\) 1680.00 0.204606
\(408\) 3402.00 0.412804
\(409\) −8714.00 −1.05350 −0.526748 0.850022i \(-0.676589\pi\)
−0.526748 + 0.850022i \(0.676589\pi\)
\(410\) −4950.00 −0.596251
\(411\) 6354.00 0.762578
\(412\) −452.000 −0.0540496
\(413\) 0 0
\(414\) −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 \(-0.746002\pi\)
−0.698169 + 0.715933i \(0.746002\pi\)
\(420\) 0 0
\(421\) 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
\(427\) 0 0
\(428\) −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 \(-0.166102\pi\)
0.866912 + 0.498462i \(0.166102\pi\)
\(434\) 0 0
\(435\) −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 \(-0.319531\pi\)
0.537069 + 0.843538i \(0.319531\pi\)
\(440\) 2520.00 0.273037
\(441\) 0 0
\(442\) 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
\(448\) 0 0
\(449\) 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
\(455\) 0 0
\(456\) −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 \(-0.847312\pi\)
−0.887141 + 0.461499i \(0.847312\pi\)
\(462\) 0 0
\(463\) 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 \(-0.610653\pi\)
−0.340667 + 0.940184i \(0.610653\pi\)
\(468\) −666.000 −0.0657818
\(469\) 0 0
\(470\) 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
\(476\) 0 0
\(477\) 4050.00 0.388756
\(478\) 3528.00 0.337588
\(479\) −2280.00 −0.217486 −0.108743 0.994070i \(-0.534683\pi\)
−0.108743 + 0.994070i \(0.534683\pi\)
\(480\) −675.000 −0.0641862
\(481\) 5180.00 0.491035
\(482\) −2598.00 −0.245510
\(483\) 0 0
\(484\) −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
\(490\) 0 0
\(491\) −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
\(497\) 0 0
\(498\) −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 \(-0.343426\pi\)
0.472294 + 0.881441i \(0.343426\pi\)
\(504\) 0 0
\(505\) 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 \(-0.461572\pi\)
0.120433 + 0.992721i \(0.461572\pi\)
\(510\) −2430.00 −0.210985
\(511\) 0 0
\(512\) −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
\(518\) 0 0
\(519\) −1278.00 −0.108089
\(520\) 7770.00 0.655264
\(521\) −10530.0 −0.885466 −0.442733 0.896654i \(-0.645991\pi\)
−0.442733 + 0.896654i \(0.645991\pi\)
\(522\) −2106.00 −0.176585
\(523\) −12692.0 −1.06115 −0.530576 0.847637i \(-0.678024\pi\)
−0.530576 + 0.847637i \(0.678024\pi\)
\(524\) −2328.00 −0.194082
\(525\) 0 0
\(526\) 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
\(532\) 0 0
\(533\) 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
\(539\) 0 0
\(540\) 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
\(546\) 0 0
\(547\) 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
\(553\) 0 0
\(554\) 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
\(560\) 0 0
\(561\) 3888.00 0.292605
\(562\) −24858.0 −1.86579
\(563\) 13404.0 1.00339 0.501697 0.865043i \(-0.332709\pi\)
0.501697 + 0.865043i \(0.332709\pi\)
\(564\) 72.0000 0.00537544
\(565\) 5430.00 0.404322
\(566\) 8148.00 0.605099
\(567\) 0 0
\(568\) 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
\(574\) 0 0
\(575\) −3000.00 −0.217580
\(576\) 3897.00 0.281901
\(577\) 1726.00 0.124531 0.0622654 0.998060i \(-0.480167\pi\)
0.0622654 + 0.998060i \(0.480167\pi\)
\(578\) −5991.00 −0.431129
\(579\) 7998.00 0.574068
\(580\) −390.000 −0.0279205
\(581\) 0 0
\(582\) 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 \(-0.621508\pi\)
−0.372524 + 0.928022i \(0.621508\pi\)
\(588\) 0 0
\(589\) −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 \(-0.531595\pi\)
−0.0990963 + 0.995078i \(0.531595\pi\)
\(594\) −1944.00 −0.134282
\(595\) 0 0
\(596\) 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 \(-0.394669\pi\)
0.324902 + 0.945748i \(0.394669\pi\)
\(602\) 0 0
\(603\) −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 \(-0.698208\pi\)
−0.583221 + 0.812314i \(0.698208\pi\)
\(608\) −5580.00 −0.372202
\(609\) 0 0
\(610\) 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
\(616\) 0 0
\(617\) −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 \(-0.592271\pi\)
−0.285834 + 0.958279i \(0.592271\pi\)
\(620\) −1000.00 −0.0647758
\(621\) −3240.00 −0.209367
\(622\) −4608.00 −0.297048
\(623\) 0 0
\(624\) 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
\(630\) 0 0
\(631\) −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
\(637\) 0 0
\(638\) 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 \(-0.782552\pi\)
−0.775598 + 0.631227i \(0.782552\pi\)
\(644\) 0 0
\(645\) 1380.00 0.0842441
\(646\) −20088.0 −1.22345
\(647\) 2736.00 0.166249 0.0831246 0.996539i \(-0.473510\pi\)
0.0831246 + 0.996539i \(0.473510\pi\)
\(648\) −1701.00 −0.103120
\(649\) 576.000 0.0348382
\(650\) −5550.00 −0.334906
\(651\) 0 0
\(652\) −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
\(658\) 0 0
\(659\) 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 \(-0.398867\pi\)
0.312400 + 0.949951i \(0.398867\pi\)
\(662\) 11076.0 0.650273
\(663\) 11988.0 0.702225
\(664\) 3276.00 0.191466
\(665\) 0 0
\(666\) −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
\(672\) 0 0
\(673\) 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 \(-0.372296\pi\)
0.390519 + 0.920595i \(0.372296\pi\)
\(678\) 9774.00 0.553640
\(679\) 0 0
\(680\) 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
\(686\) 0 0
\(687\) 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 \(-0.862595\pi\)
−0.908268 + 0.418388i \(0.862595\pi\)
\(692\) −426.000 −0.0234019
\(693\) 0 0
\(694\) 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
\(700\) 0 0
\(701\) −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
\(707\) 0 0
\(708\) −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
\(714\) 0 0
\(715\) 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 \(-0.859207\pi\)
−0.903763 + 0.428033i \(0.859207\pi\)
\(720\) −3195.00 −0.165376
\(721\) 0 0
\(722\) 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 \(-0.753539\pi\)
−0.714925 + 0.699201i \(0.753539\pi\)
\(728\) 0 0
\(729\) 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 \(-0.649846\pi\)
−0.453560 + 0.891226i \(0.649846\pi\)
\(734\) 10524.0 0.529221
\(735\) 0 0
\(736\) 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
\(742\) 0 0
\(743\) −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
\(749\) 0 0
\(750\) 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
\(756\) 0 0
\(757\) −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 \(-0.692521\pi\)
−0.568615 + 0.822604i \(0.692521\pi\)
\(762\) 11196.0 0.532268
\(763\) 0 0
\(764\) 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 \(-0.686808\pi\)
−0.553763 + 0.832675i \(0.686808\pi\)
\(770\) 0 0
\(771\) −7578.00 −0.353975
\(772\) 2666.00 0.124289
\(773\) −11538.0 −0.536860 −0.268430 0.963299i \(-0.586505\pi\)
−0.268430 + 0.963299i \(0.586505\pi\)
\(774\) 2484.00 0.115356
\(775\) −5000.00 −0.231749
\(776\) −6006.00 −0.277839
\(777\) 0 0
\(778\) −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
\(784\) 0 0
\(785\) −11890.0 −0.540602
\(786\) −20952.0 −0.950805
\(787\) 14884.0 0.674152 0.337076 0.941478i \(-0.390562\pi\)
0.337076 + 0.941478i \(0.390562\pi\)
\(788\) −2718.00 −0.122874
\(789\) 16344.0 0.737467
\(790\) −7800.00 −0.351280
\(791\) 0 0
\(792\) 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 \(-0.418957\pi\)
0.251864 + 0.967763i \(0.418957\pi\)
\(798\) 0 0
\(799\) −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
\(805\) 0 0
\(806\) 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 \(-0.123684\pi\)
0.925453 + 0.378862i \(0.123684\pi\)
\(812\) 0 0
\(813\) 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
\(819\) 0 0
\(820\) −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
\(826\) 0 0
\(827\) 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 \(-0.608438\pi\)
−0.334117 + 0.942532i \(0.608438\pi\)
\(830\) −2340.00 −0.0978585
\(831\) 11886.0 0.496174
\(832\) −32042.0 −1.33516
\(833\) 0 0
\(834\) −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 \(-0.591637\pi\)
−0.283927 + 0.958846i \(0.591637\pi\)
\(840\) 0 0
\(841\) −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
\(847\) 0 0
\(848\) −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 \(-0.311113\pi\)
0.559189 + 0.829040i \(0.311113\pi\)
\(854\) 0 0
\(855\) 5580.00 0.223195
\(856\) 29484.0 1.17727
\(857\) 7314.00 0.291530 0.145765 0.989319i \(-0.453436\pi\)
0.145765 + 0.989319i \(0.453436\pi\)
\(858\) 15984.0 0.635996
\(859\) 28780.0 1.14314 0.571572 0.820552i \(-0.306334\pi\)
0.571572 + 0.820552i \(0.306334\pi\)
\(860\) 460.000 0.0182394
\(861\) 0 0
\(862\) 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
\(868\) 0 0
\(869\) 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
\(875\) 0 0
\(876\) 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 \(-0.483889\pi\)
0.0505936 + 0.998719i \(0.483889\pi\)
\(882\) 0 0
\(883\) 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 \(-0.192496\pi\)
0.822648 + 0.568550i \(0.192496\pi\)
\(888\) 4410.00 0.166655
\(889\) 0 0
\(890\) −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
\(896\) 0 0
\(897\) 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
\(903\) 0 0
\(904\) −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
\(910\) 0 0
\(911\) −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
\(917\) 0 0
\(918\) −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
\(924\) 0 0
\(925\) −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 \(-0.677355\pi\)
−0.528792 + 0.848751i \(0.677355\pi\)
\(930\) −9000.00 −0.317335
\(931\) 0 0
\(932\) −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 \(-0.787080\pi\)
−0.784499 + 0.620130i \(0.787080\pi\)
\(938\) 0 0
\(939\) 22026.0 0.765486
\(940\) 120.000 0.00416380
\(941\) −6090.00 −0.210976 −0.105488 0.994421i \(-0.533640\pi\)
−0.105488 + 0.994421i \(0.533640\pi\)
\(942\) −21402.0 −0.740249
\(943\) 39600.0 1.36750
\(944\) 1704.00 0.0587505
\(945\) 0 0
\(946\) −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
\(952\) 0 0
\(953\) 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
\(959\) 0 0
\(960\) 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
\(966\) 0 0
\(967\) −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 \(-0.632565\pi\)
−0.404532 + 0.914524i \(0.632565\pi\)
\(972\) 243.000 0.00801875
\(973\) 0 0
\(974\) −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
\(980\) 0 0
\(981\) −13266.0 −0.431754
\(982\) −56736.0 −1.84371
\(983\) −6960.00 −0.225829 −0.112914 0.993605i \(-0.536019\pi\)
−0.112914 + 0.993605i \(0.536019\pi\)
\(984\) 20790.0 0.673538
\(985\) −13590.0 −0.439608
\(986\) 12636.0 0.408126
\(987\) 0 0
\(988\) −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
\(994\) 0 0
\(995\) 19160.0 0.610465
\(996\) −468.000 −0.0148887
\(997\) −9938.00 −0.315687 −0.157843 0.987464i \(-0.550454\pi\)
−0.157843 + 0.987464i \(0.550454\pi\)
\(998\) 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 735.4.a.i.1.1 1
3.2 odd 2 2205.4.a.c.1.1 1
7.6 odd 2 15.4.a.b.1.1 1
21.20 even 2 45.4.a.b.1.1 1
28.27 even 2 240.4.a.f.1.1 1
35.13 even 4 75.4.b.a.49.1 2
35.27 even 4 75.4.b.a.49.2 2
35.34 odd 2 75.4.a.a.1.1 1
56.13 odd 2 960.4.a.bi.1.1 1
56.27 even 2 960.4.a.l.1.1 1
63.13 odd 6 405.4.e.d.136.1 2
63.20 even 6 405.4.e.k.271.1 2
63.34 odd 6 405.4.e.d.271.1 2
63.41 even 6 405.4.e.k.136.1 2
77.76 even 2 1815.4.a.a.1.1 1
84.83 odd 2 720.4.a.r.1.1 1
105.62 odd 4 225.4.b.d.199.1 2
105.83 odd 4 225.4.b.d.199.2 2
105.104 even 2 225.4.a.g.1.1 1
140.27 odd 4 1200.4.f.m.49.1 2
140.83 odd 4 1200.4.f.m.49.2 2
140.139 even 2 1200.4.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 7.6 odd 2
45.4.a.b.1.1 1 21.20 even 2
75.4.a.a.1.1 1 35.34 odd 2
75.4.b.a.49.1 2 35.13 even 4
75.4.b.a.49.2 2 35.27 even 4
225.4.a.g.1.1 1 105.104 even 2
225.4.b.d.199.1 2 105.62 odd 4
225.4.b.d.199.2 2 105.83 odd 4
240.4.a.f.1.1 1 28.27 even 2
405.4.e.d.136.1 2 63.13 odd 6
405.4.e.d.271.1 2 63.34 odd 6
405.4.e.k.136.1 2 63.41 even 6
405.4.e.k.271.1 2 63.20 even 6
720.4.a.r.1.1 1 84.83 odd 2
735.4.a.i.1.1 1 1.1 even 1 trivial
960.4.a.l.1.1 1 56.27 even 2
960.4.a.bi.1.1 1 56.13 odd 2
1200.4.a.o.1.1 1 140.139 even 2
1200.4.f.m.49.1 2 140.27 odd 4
1200.4.f.m.49.2 2 140.83 odd 4
1815.4.a.a.1.1 1 77.76 even 2
2205.4.a.c.1.1 1 3.2 odd 2