Properties

Label 1734.4.a.d.1.1
Level $1734$
Weight $4$
Character 1734.1
Self dual yes
Analytic conductor $102.309$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1734.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(102.309311950\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1734.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} -6.00000 q^{6} +16.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} -6.00000 q^{6} +16.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +12.0000 q^{10} -12.0000 q^{11} +12.0000 q^{12} +38.0000 q^{13} -32.0000 q^{14} -18.0000 q^{15} +16.0000 q^{16} -18.0000 q^{18} +20.0000 q^{19} -24.0000 q^{20} +48.0000 q^{21} +24.0000 q^{22} -168.000 q^{23} -24.0000 q^{24} -89.0000 q^{25} -76.0000 q^{26} +27.0000 q^{27} +64.0000 q^{28} -30.0000 q^{29} +36.0000 q^{30} +88.0000 q^{31} -32.0000 q^{32} -36.0000 q^{33} -96.0000 q^{35} +36.0000 q^{36} -254.000 q^{37} -40.0000 q^{38} +114.000 q^{39} +48.0000 q^{40} -42.0000 q^{41} -96.0000 q^{42} -52.0000 q^{43} -48.0000 q^{44} -54.0000 q^{45} +336.000 q^{46} -96.0000 q^{47} +48.0000 q^{48} -87.0000 q^{49} +178.000 q^{50} +152.000 q^{52} +198.000 q^{53} -54.0000 q^{54} +72.0000 q^{55} -128.000 q^{56} +60.0000 q^{57} +60.0000 q^{58} -660.000 q^{59} -72.0000 q^{60} +538.000 q^{61} -176.000 q^{62} +144.000 q^{63} +64.0000 q^{64} -228.000 q^{65} +72.0000 q^{66} +884.000 q^{67} -504.000 q^{69} +192.000 q^{70} -792.000 q^{71} -72.0000 q^{72} -218.000 q^{73} +508.000 q^{74} -267.000 q^{75} +80.0000 q^{76} -192.000 q^{77} -228.000 q^{78} +520.000 q^{79} -96.0000 q^{80} +81.0000 q^{81} +84.0000 q^{82} -492.000 q^{83} +192.000 q^{84} +104.000 q^{86} -90.0000 q^{87} +96.0000 q^{88} +810.000 q^{89} +108.000 q^{90} +608.000 q^{91} -672.000 q^{92} +264.000 q^{93} +192.000 q^{94} -120.000 q^{95} -96.0000 q^{96} -1154.00 q^{97} +174.000 q^{98} -108.000 q^{99} +O(q^{100})\)

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
\(5\) −6.00000 −0.536656 −0.268328 0.963328i \(-0.586471\pi\)
−0.268328 + 0.963328i \(0.586471\pi\)
\(6\) −6.00000 −0.408248
\(7\) 16.0000 0.863919 0.431959 0.901893i \(-0.357822\pi\)
0.431959 + 0.901893i \(0.357822\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 12.0000 0.379473
\(11\) −12.0000 −0.328921 −0.164461 0.986384i \(-0.552588\pi\)
−0.164461 + 0.986384i \(0.552588\pi\)
\(12\) 12.0000 0.288675
\(13\) 38.0000 0.810716 0.405358 0.914158i \(-0.367147\pi\)
0.405358 + 0.914158i \(0.367147\pi\)
\(14\) −32.0000 −0.610883
\(15\) −18.0000 −0.309839
\(16\) 16.0000 0.250000
\(17\) 0 0
\(18\) −18.0000 −0.235702
\(19\) 20.0000 0.241490 0.120745 0.992684i \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) −24.0000 −0.268328
\(21\) 48.0000 0.498784
\(22\) 24.0000 0.232583
\(23\) −168.000 −1.52306 −0.761531 0.648129i \(-0.775552\pi\)
−0.761531 + 0.648129i \(0.775552\pi\)
\(24\) −24.0000 −0.204124
\(25\) −89.0000 −0.712000
\(26\) −76.0000 −0.573263
\(27\) 27.0000 0.192450
\(28\) 64.0000 0.431959
\(29\) −30.0000 −0.192099 −0.0960493 0.995377i \(-0.530621\pi\)
−0.0960493 + 0.995377i \(0.530621\pi\)
\(30\) 36.0000 0.219089
\(31\) 88.0000 0.509847 0.254924 0.966961i \(-0.417950\pi\)
0.254924 + 0.966961i \(0.417950\pi\)
\(32\) −32.0000 −0.176777
\(33\) −36.0000 −0.189903
\(34\) 0 0
\(35\) −96.0000 −0.463627
\(36\) 36.0000 0.166667
\(37\) −254.000 −1.12858 −0.564288 0.825578i \(-0.690849\pi\)
−0.564288 + 0.825578i \(0.690849\pi\)
\(38\) −40.0000 −0.170759
\(39\) 114.000 0.468067
\(40\) 48.0000 0.189737
\(41\) −42.0000 −0.159983 −0.0799914 0.996796i \(-0.525489\pi\)
−0.0799914 + 0.996796i \(0.525489\pi\)
\(42\) −96.0000 −0.352693
\(43\) −52.0000 −0.184417 −0.0922084 0.995740i \(-0.529393\pi\)
−0.0922084 + 0.995740i \(0.529393\pi\)
\(44\) −48.0000 −0.164461
\(45\) −54.0000 −0.178885
\(46\) 336.000 1.07697
\(47\) −96.0000 −0.297937 −0.148969 0.988842i \(-0.547595\pi\)
−0.148969 + 0.988842i \(0.547595\pi\)
\(48\) 48.0000 0.144338
\(49\) −87.0000 −0.253644
\(50\) 178.000 0.503460
\(51\) 0 0
\(52\) 152.000 0.405358
\(53\) 198.000 0.513158 0.256579 0.966523i \(-0.417405\pi\)
0.256579 + 0.966523i \(0.417405\pi\)
\(54\) −54.0000 −0.136083
\(55\) 72.0000 0.176518
\(56\) −128.000 −0.305441
\(57\) 60.0000 0.139424
\(58\) 60.0000 0.135834
\(59\) −660.000 −1.45635 −0.728175 0.685391i \(-0.759631\pi\)
−0.728175 + 0.685391i \(0.759631\pi\)
\(60\) −72.0000 −0.154919
\(61\) 538.000 1.12924 0.564622 0.825350i \(-0.309022\pi\)
0.564622 + 0.825350i \(0.309022\pi\)
\(62\) −176.000 −0.360516
\(63\) 144.000 0.287973
\(64\) 64.0000 0.125000
\(65\) −228.000 −0.435076
\(66\) 72.0000 0.134282
\(67\) 884.000 1.61191 0.805954 0.591979i \(-0.201653\pi\)
0.805954 + 0.591979i \(0.201653\pi\)
\(68\) 0 0
\(69\) −504.000 −0.879340
\(70\) 192.000 0.327834
\(71\) −792.000 −1.32385 −0.661923 0.749572i \(-0.730260\pi\)
−0.661923 + 0.749572i \(0.730260\pi\)
\(72\) −72.0000 −0.117851
\(73\) −218.000 −0.349520 −0.174760 0.984611i \(-0.555915\pi\)
−0.174760 + 0.984611i \(0.555915\pi\)
\(74\) 508.000 0.798024
\(75\) −267.000 −0.411073
\(76\) 80.0000 0.120745
\(77\) −192.000 −0.284161
\(78\) −228.000 −0.330973
\(79\) 520.000 0.740564 0.370282 0.928919i \(-0.379261\pi\)
0.370282 + 0.928919i \(0.379261\pi\)
\(80\) −96.0000 −0.134164
\(81\) 81.0000 0.111111
\(82\) 84.0000 0.113125
\(83\) −492.000 −0.650651 −0.325325 0.945602i \(-0.605474\pi\)
−0.325325 + 0.945602i \(0.605474\pi\)
\(84\) 192.000 0.249392
\(85\) 0 0
\(86\) 104.000 0.130402
\(87\) −90.0000 −0.110908
\(88\) 96.0000 0.116291
\(89\) 810.000 0.964717 0.482359 0.875974i \(-0.339780\pi\)
0.482359 + 0.875974i \(0.339780\pi\)
\(90\) 108.000 0.126491
\(91\) 608.000 0.700393
\(92\) −672.000 −0.761531
\(93\) 264.000 0.294360
\(94\) 192.000 0.210673
\(95\) −120.000 −0.129597
\(96\) −96.0000 −0.102062
\(97\) −1154.00 −1.20795 −0.603974 0.797004i \(-0.706417\pi\)
−0.603974 + 0.797004i \(0.706417\pi\)
\(98\) 174.000 0.179354
\(99\) −108.000 −0.109640
\(100\) −356.000 −0.356000
\(101\) −618.000 −0.608845 −0.304422 0.952537i \(-0.598463\pi\)
−0.304422 + 0.952537i \(0.598463\pi\)
\(102\) 0 0
\(103\) 128.000 0.122449 0.0612243 0.998124i \(-0.480499\pi\)
0.0612243 + 0.998124i \(0.480499\pi\)
\(104\) −304.000 −0.286631
\(105\) −288.000 −0.267675
\(106\) −396.000 −0.362858
\(107\) 1476.00 1.33355 0.666777 0.745257i \(-0.267673\pi\)
0.666777 + 0.745257i \(0.267673\pi\)
\(108\) 108.000 0.0962250
\(109\) −1190.00 −1.04570 −0.522850 0.852425i \(-0.675131\pi\)
−0.522850 + 0.852425i \(0.675131\pi\)
\(110\) −144.000 −0.124817
\(111\) −762.000 −0.651584
\(112\) 256.000 0.215980
\(113\) 462.000 0.384613 0.192307 0.981335i \(-0.438403\pi\)
0.192307 + 0.981335i \(0.438403\pi\)
\(114\) −120.000 −0.0985880
\(115\) 1008.00 0.817361
\(116\) −120.000 −0.0960493
\(117\) 342.000 0.270239
\(118\) 1320.00 1.02980
\(119\) 0 0
\(120\) 144.000 0.109545
\(121\) −1187.00 −0.891811
\(122\) −1076.00 −0.798496
\(123\) −126.000 −0.0923662
\(124\) 352.000 0.254924
\(125\) 1284.00 0.918756
\(126\) −288.000 −0.203628
\(127\) −2536.00 −1.77192 −0.885959 0.463763i \(-0.846499\pi\)
−0.885959 + 0.463763i \(0.846499\pi\)
\(128\) −128.000 −0.0883883
\(129\) −156.000 −0.106473
\(130\) 456.000 0.307645
\(131\) −2292.00 −1.52865 −0.764324 0.644832i \(-0.776927\pi\)
−0.764324 + 0.644832i \(0.776927\pi\)
\(132\) −144.000 −0.0949514
\(133\) 320.000 0.208628
\(134\) −1768.00 −1.13979
\(135\) −162.000 −0.103280
\(136\) 0 0
\(137\) −726.000 −0.452747 −0.226374 0.974041i \(-0.572687\pi\)
−0.226374 + 0.974041i \(0.572687\pi\)
\(138\) 1008.00 0.621787
\(139\) −380.000 −0.231879 −0.115939 0.993256i \(-0.536988\pi\)
−0.115939 + 0.993256i \(0.536988\pi\)
\(140\) −384.000 −0.231814
\(141\) −288.000 −0.172014
\(142\) 1584.00 0.936101
\(143\) −456.000 −0.266662
\(144\) 144.000 0.0833333
\(145\) 180.000 0.103091
\(146\) 436.000 0.247148
\(147\) −261.000 −0.146442
\(148\) −1016.00 −0.564288
\(149\) 1590.00 0.874214 0.437107 0.899410i \(-0.356003\pi\)
0.437107 + 0.899410i \(0.356003\pi\)
\(150\) 534.000 0.290673
\(151\) 2432.00 1.31068 0.655342 0.755332i \(-0.272524\pi\)
0.655342 + 0.755332i \(0.272524\pi\)
\(152\) −160.000 −0.0853797
\(153\) 0 0
\(154\) 384.000 0.200932
\(155\) −528.000 −0.273613
\(156\) 456.000 0.234033
\(157\) 614.000 0.312118 0.156059 0.987748i \(-0.450121\pi\)
0.156059 + 0.987748i \(0.450121\pi\)
\(158\) −1040.00 −0.523658
\(159\) 594.000 0.296272
\(160\) 192.000 0.0948683
\(161\) −2688.00 −1.31580
\(162\) −162.000 −0.0785674
\(163\) 1852.00 0.889938 0.444969 0.895546i \(-0.353215\pi\)
0.444969 + 0.895546i \(0.353215\pi\)
\(164\) −168.000 −0.0799914
\(165\) 216.000 0.101913
\(166\) 984.000 0.460080
\(167\) 2136.00 0.989752 0.494876 0.868964i \(-0.335213\pi\)
0.494876 + 0.868964i \(0.335213\pi\)
\(168\) −384.000 −0.176347
\(169\) −753.000 −0.342740
\(170\) 0 0
\(171\) 180.000 0.0804967
\(172\) −208.000 −0.0922084
\(173\) −1758.00 −0.772591 −0.386296 0.922375i \(-0.626246\pi\)
−0.386296 + 0.922375i \(0.626246\pi\)
\(174\) 180.000 0.0784239
\(175\) −1424.00 −0.615110
\(176\) −192.000 −0.0822304
\(177\) −1980.00 −0.840824
\(178\) −1620.00 −0.682158
\(179\) −540.000 −0.225483 −0.112742 0.993624i \(-0.535963\pi\)
−0.112742 + 0.993624i \(0.535963\pi\)
\(180\) −216.000 −0.0894427
\(181\) −1982.00 −0.813928 −0.406964 0.913444i \(-0.633412\pi\)
−0.406964 + 0.913444i \(0.633412\pi\)
\(182\) −1216.00 −0.495252
\(183\) 1614.00 0.651969
\(184\) 1344.00 0.538484
\(185\) 1524.00 0.605658
\(186\) −528.000 −0.208144
\(187\) 0 0
\(188\) −384.000 −0.148969
\(189\) 432.000 0.166261
\(190\) 240.000 0.0916391
\(191\) −2688.00 −1.01831 −0.509154 0.860675i \(-0.670042\pi\)
−0.509154 + 0.860675i \(0.670042\pi\)
\(192\) 192.000 0.0721688
\(193\) 2302.00 0.858557 0.429279 0.903172i \(-0.358768\pi\)
0.429279 + 0.903172i \(0.358768\pi\)
\(194\) 2308.00 0.854148
\(195\) −684.000 −0.251191
\(196\) −348.000 −0.126822
\(197\) −4374.00 −1.58190 −0.790951 0.611880i \(-0.790414\pi\)
−0.790951 + 0.611880i \(0.790414\pi\)
\(198\) 216.000 0.0775275
\(199\) 1600.00 0.569955 0.284977 0.958534i \(-0.408014\pi\)
0.284977 + 0.958534i \(0.408014\pi\)
\(200\) 712.000 0.251730
\(201\) 2652.00 0.930635
\(202\) 1236.00 0.430518
\(203\) −480.000 −0.165958
\(204\) 0 0
\(205\) 252.000 0.0858558
\(206\) −256.000 −0.0865843
\(207\) −1512.00 −0.507687
\(208\) 608.000 0.202679
\(209\) −240.000 −0.0794313
\(210\) 576.000 0.189275
\(211\) −3332.00 −1.08713 −0.543565 0.839367i \(-0.682926\pi\)
−0.543565 + 0.839367i \(0.682926\pi\)
\(212\) 792.000 0.256579
\(213\) −2376.00 −0.764323
\(214\) −2952.00 −0.942965
\(215\) 312.000 0.0989685
\(216\) −216.000 −0.0680414
\(217\) 1408.00 0.440467
\(218\) 2380.00 0.739422
\(219\) −654.000 −0.201796
\(220\) 288.000 0.0882589
\(221\) 0 0
\(222\) 1524.00 0.460740
\(223\) 2648.00 0.795171 0.397586 0.917565i \(-0.369848\pi\)
0.397586 + 0.917565i \(0.369848\pi\)
\(224\) −512.000 −0.152721
\(225\) −801.000 −0.237333
\(226\) −924.000 −0.271963
\(227\) −2244.00 −0.656121 −0.328061 0.944657i \(-0.606395\pi\)
−0.328061 + 0.944657i \(0.606395\pi\)
\(228\) 240.000 0.0697122
\(229\) −5650.00 −1.63040 −0.815202 0.579177i \(-0.803374\pi\)
−0.815202 + 0.579177i \(0.803374\pi\)
\(230\) −2016.00 −0.577961
\(231\) −576.000 −0.164061
\(232\) 240.000 0.0679171
\(233\) −4698.00 −1.32093 −0.660464 0.750858i \(-0.729640\pi\)
−0.660464 + 0.750858i \(0.729640\pi\)
\(234\) −684.000 −0.191088
\(235\) 576.000 0.159890
\(236\) −2640.00 −0.728175
\(237\) 1560.00 0.427565
\(238\) 0 0
\(239\) −1200.00 −0.324776 −0.162388 0.986727i \(-0.551920\pi\)
−0.162388 + 0.986727i \(0.551920\pi\)
\(240\) −288.000 −0.0774597
\(241\) 718.000 0.191911 0.0959553 0.995386i \(-0.469409\pi\)
0.0959553 + 0.995386i \(0.469409\pi\)
\(242\) 2374.00 0.630605
\(243\) 243.000 0.0641500
\(244\) 2152.00 0.564622
\(245\) 522.000 0.136120
\(246\) 252.000 0.0653127
\(247\) 760.000 0.195780
\(248\) −704.000 −0.180258
\(249\) −1476.00 −0.375653
\(250\) −2568.00 −0.649658
\(251\) 6012.00 1.51185 0.755924 0.654659i \(-0.227188\pi\)
0.755924 + 0.654659i \(0.227188\pi\)
\(252\) 576.000 0.143986
\(253\) 2016.00 0.500968
\(254\) 5072.00 1.25294
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −2046.00 −0.496599 −0.248300 0.968683i \(-0.579872\pi\)
−0.248300 + 0.968683i \(0.579872\pi\)
\(258\) 312.000 0.0752879
\(259\) −4064.00 −0.974999
\(260\) −912.000 −0.217538
\(261\) −270.000 −0.0640329
\(262\) 4584.00 1.08092
\(263\) −6072.00 −1.42363 −0.711817 0.702365i \(-0.752127\pi\)
−0.711817 + 0.702365i \(0.752127\pi\)
\(264\) 288.000 0.0671408
\(265\) −1188.00 −0.275390
\(266\) −640.000 −0.147522
\(267\) 2430.00 0.556980
\(268\) 3536.00 0.805954
\(269\) 6930.00 1.57074 0.785371 0.619025i \(-0.212472\pi\)
0.785371 + 0.619025i \(0.212472\pi\)
\(270\) 324.000 0.0730297
\(271\) 1352.00 0.303056 0.151528 0.988453i \(-0.451581\pi\)
0.151528 + 0.988453i \(0.451581\pi\)
\(272\) 0 0
\(273\) 1824.00 0.404372
\(274\) 1452.00 0.320141
\(275\) 1068.00 0.234192
\(276\) −2016.00 −0.439670
\(277\) 1186.00 0.257256 0.128628 0.991693i \(-0.458943\pi\)
0.128628 + 0.991693i \(0.458943\pi\)
\(278\) 760.000 0.163963
\(279\) 792.000 0.169949
\(280\) 768.000 0.163917
\(281\) 2442.00 0.518425 0.259213 0.965820i \(-0.416537\pi\)
0.259213 + 0.965820i \(0.416537\pi\)
\(282\) 576.000 0.121632
\(283\) −2828.00 −0.594018 −0.297009 0.954875i \(-0.595989\pi\)
−0.297009 + 0.954875i \(0.595989\pi\)
\(284\) −3168.00 −0.661923
\(285\) −360.000 −0.0748230
\(286\) 912.000 0.188558
\(287\) −672.000 −0.138212
\(288\) −288.000 −0.0589256
\(289\) 0 0
\(290\) −360.000 −0.0728963
\(291\) −3462.00 −0.697409
\(292\) −872.000 −0.174760
\(293\) 4758.00 0.948687 0.474344 0.880340i \(-0.342685\pi\)
0.474344 + 0.880340i \(0.342685\pi\)
\(294\) 522.000 0.103550
\(295\) 3960.00 0.781560
\(296\) 2032.00 0.399012
\(297\) −324.000 −0.0633010
\(298\) −3180.00 −0.618163
\(299\) −6384.00 −1.23477
\(300\) −1068.00 −0.205537
\(301\) −832.000 −0.159321
\(302\) −4864.00 −0.926794
\(303\) −1854.00 −0.351517
\(304\) 320.000 0.0603726
\(305\) −3228.00 −0.606016
\(306\) 0 0
\(307\) −8476.00 −1.57574 −0.787868 0.615844i \(-0.788815\pi\)
−0.787868 + 0.615844i \(0.788815\pi\)
\(308\) −768.000 −0.142081
\(309\) 384.000 0.0706958
\(310\) 1056.00 0.193473
\(311\) −4632.00 −0.844555 −0.422278 0.906467i \(-0.638769\pi\)
−0.422278 + 0.906467i \(0.638769\pi\)
\(312\) −912.000 −0.165487
\(313\) 4822.00 0.870785 0.435392 0.900241i \(-0.356610\pi\)
0.435392 + 0.900241i \(0.356610\pi\)
\(314\) −1228.00 −0.220701
\(315\) −864.000 −0.154542
\(316\) 2080.00 0.370282
\(317\) 3426.00 0.607014 0.303507 0.952829i \(-0.401842\pi\)
0.303507 + 0.952829i \(0.401842\pi\)
\(318\) −1188.00 −0.209496
\(319\) 360.000 0.0631854
\(320\) −384.000 −0.0670820
\(321\) 4428.00 0.769928
\(322\) 5376.00 0.930412
\(323\) 0 0
\(324\) 324.000 0.0555556
\(325\) −3382.00 −0.577230
\(326\) −3704.00 −0.629281
\(327\) −3570.00 −0.603735
\(328\) 336.000 0.0565625
\(329\) −1536.00 −0.257393
\(330\) −432.000 −0.0720631
\(331\) −2788.00 −0.462968 −0.231484 0.972839i \(-0.574358\pi\)
−0.231484 + 0.972839i \(0.574358\pi\)
\(332\) −1968.00 −0.325325
\(333\) −2286.00 −0.376192
\(334\) −4272.00 −0.699861
\(335\) −5304.00 −0.865040
\(336\) 768.000 0.124696
\(337\) −434.000 −0.0701528 −0.0350764 0.999385i \(-0.511167\pi\)
−0.0350764 + 0.999385i \(0.511167\pi\)
\(338\) 1506.00 0.242354
\(339\) 1386.00 0.222057
\(340\) 0 0
\(341\) −1056.00 −0.167700
\(342\) −360.000 −0.0569198
\(343\) −6880.00 −1.08305
\(344\) 416.000 0.0652012
\(345\) 3024.00 0.471903
\(346\) 3516.00 0.546304
\(347\) −6684.00 −1.03405 −0.517026 0.855970i \(-0.672961\pi\)
−0.517026 + 0.855970i \(0.672961\pi\)
\(348\) −360.000 −0.0554541
\(349\) 2630.00 0.403383 0.201692 0.979449i \(-0.435356\pi\)
0.201692 + 0.979449i \(0.435356\pi\)
\(350\) 2848.00 0.434949
\(351\) 1026.00 0.156022
\(352\) 384.000 0.0581456
\(353\) −7422.00 −1.11907 −0.559537 0.828805i \(-0.689021\pi\)
−0.559537 + 0.828805i \(0.689021\pi\)
\(354\) 3960.00 0.594553
\(355\) 4752.00 0.710451
\(356\) 3240.00 0.482359
\(357\) 0 0
\(358\) 1080.00 0.159441
\(359\) −10440.0 −1.53482 −0.767412 0.641154i \(-0.778456\pi\)
−0.767412 + 0.641154i \(0.778456\pi\)
\(360\) 432.000 0.0632456
\(361\) −6459.00 −0.941682
\(362\) 3964.00 0.575534
\(363\) −3561.00 −0.514887
\(364\) 2432.00 0.350196
\(365\) 1308.00 0.187572
\(366\) −3228.00 −0.461012
\(367\) −10424.0 −1.48264 −0.741319 0.671153i \(-0.765800\pi\)
−0.741319 + 0.671153i \(0.765800\pi\)
\(368\) −2688.00 −0.380765
\(369\) −378.000 −0.0533276
\(370\) −3048.00 −0.428265
\(371\) 3168.00 0.443327
\(372\) 1056.00 0.147180
\(373\) 3278.00 0.455036 0.227518 0.973774i \(-0.426939\pi\)
0.227518 + 0.973774i \(0.426939\pi\)
\(374\) 0 0
\(375\) 3852.00 0.530444
\(376\) 768.000 0.105337
\(377\) −1140.00 −0.155737
\(378\) −864.000 −0.117564
\(379\) −6140.00 −0.832165 −0.416083 0.909327i \(-0.636597\pi\)
−0.416083 + 0.909327i \(0.636597\pi\)
\(380\) −480.000 −0.0647986
\(381\) −7608.00 −1.02302
\(382\) 5376.00 0.720053
\(383\) −3072.00 −0.409848 −0.204924 0.978778i \(-0.565695\pi\)
−0.204924 + 0.978778i \(0.565695\pi\)
\(384\) −384.000 −0.0510310
\(385\) 1152.00 0.152497
\(386\) −4604.00 −0.607092
\(387\) −468.000 −0.0614723
\(388\) −4616.00 −0.603974
\(389\) 6150.00 0.801587 0.400794 0.916168i \(-0.368734\pi\)
0.400794 + 0.916168i \(0.368734\pi\)
\(390\) 1368.00 0.177619
\(391\) 0 0
\(392\) 696.000 0.0896768
\(393\) −6876.00 −0.882566
\(394\) 8748.00 1.11857
\(395\) −3120.00 −0.397428
\(396\) −432.000 −0.0548202
\(397\) 106.000 0.0134005 0.00670024 0.999978i \(-0.497867\pi\)
0.00670024 + 0.999978i \(0.497867\pi\)
\(398\) −3200.00 −0.403019
\(399\) 960.000 0.120451
\(400\) −1424.00 −0.178000
\(401\) 1758.00 0.218929 0.109464 0.993991i \(-0.465086\pi\)
0.109464 + 0.993991i \(0.465086\pi\)
\(402\) −5304.00 −0.658058
\(403\) 3344.00 0.413341
\(404\) −2472.00 −0.304422
\(405\) −486.000 −0.0596285
\(406\) 960.000 0.117350
\(407\) 3048.00 0.371213
\(408\) 0 0
\(409\) −3670.00 −0.443691 −0.221846 0.975082i \(-0.571208\pi\)
−0.221846 + 0.975082i \(0.571208\pi\)
\(410\) −504.000 −0.0607092
\(411\) −2178.00 −0.261394
\(412\) 512.000 0.0612243
\(413\) −10560.0 −1.25817
\(414\) 3024.00 0.358989
\(415\) 2952.00 0.349176
\(416\) −1216.00 −0.143316
\(417\) −1140.00 −0.133875
\(418\) 480.000 0.0561664
\(419\) 9660.00 1.12631 0.563153 0.826353i \(-0.309588\pi\)
0.563153 + 0.826353i \(0.309588\pi\)
\(420\) −1152.00 −0.133838
\(421\) 8462.00 0.979602 0.489801 0.871834i \(-0.337069\pi\)
0.489801 + 0.871834i \(0.337069\pi\)
\(422\) 6664.00 0.768717
\(423\) −864.000 −0.0993123
\(424\) −1584.00 −0.181429
\(425\) 0 0
\(426\) 4752.00 0.540458
\(427\) 8608.00 0.975575
\(428\) 5904.00 0.666777
\(429\) −1368.00 −0.153957
\(430\) −624.000 −0.0699813
\(431\) −9792.00 −1.09435 −0.547174 0.837019i \(-0.684296\pi\)
−0.547174 + 0.837019i \(0.684296\pi\)
\(432\) 432.000 0.0481125
\(433\) −7342.00 −0.814859 −0.407430 0.913237i \(-0.633575\pi\)
−0.407430 + 0.913237i \(0.633575\pi\)
\(434\) −2816.00 −0.311457
\(435\) 540.000 0.0595196
\(436\) −4760.00 −0.522850
\(437\) −3360.00 −0.367805
\(438\) 1308.00 0.142691
\(439\) −10640.0 −1.15676 −0.578382 0.815766i \(-0.696316\pi\)
−0.578382 + 0.815766i \(0.696316\pi\)
\(440\) −576.000 −0.0624085
\(441\) −783.000 −0.0845481
\(442\) 0 0
\(443\) −17412.0 −1.86742 −0.933712 0.358024i \(-0.883451\pi\)
−0.933712 + 0.358024i \(0.883451\pi\)
\(444\) −3048.00 −0.325792
\(445\) −4860.00 −0.517722
\(446\) −5296.00 −0.562271
\(447\) 4770.00 0.504728
\(448\) 1024.00 0.107990
\(449\) 1710.00 0.179732 0.0898662 0.995954i \(-0.471356\pi\)
0.0898662 + 0.995954i \(0.471356\pi\)
\(450\) 1602.00 0.167820
\(451\) 504.000 0.0526218
\(452\) 1848.00 0.192307
\(453\) 7296.00 0.756724
\(454\) 4488.00 0.463948
\(455\) −3648.00 −0.375870
\(456\) −480.000 −0.0492940
\(457\) −646.000 −0.0661239 −0.0330619 0.999453i \(-0.510526\pi\)
−0.0330619 + 0.999453i \(0.510526\pi\)
\(458\) 11300.0 1.15287
\(459\) 0 0
\(460\) 4032.00 0.408680
\(461\) −6018.00 −0.607996 −0.303998 0.952673i \(-0.598322\pi\)
−0.303998 + 0.952673i \(0.598322\pi\)
\(462\) 1152.00 0.116008
\(463\) −6712.00 −0.673722 −0.336861 0.941554i \(-0.609365\pi\)
−0.336861 + 0.941554i \(0.609365\pi\)
\(464\) −480.000 −0.0480247
\(465\) −1584.00 −0.157970
\(466\) 9396.00 0.934037
\(467\) 5364.00 0.531512 0.265756 0.964040i \(-0.414378\pi\)
0.265756 + 0.964040i \(0.414378\pi\)
\(468\) 1368.00 0.135119
\(469\) 14144.0 1.39256
\(470\) −1152.00 −0.113059
\(471\) 1842.00 0.180201
\(472\) 5280.00 0.514898
\(473\) 624.000 0.0606587
\(474\) −3120.00 −0.302334
\(475\) −1780.00 −0.171941
\(476\) 0 0
\(477\) 1782.00 0.171053
\(478\) 2400.00 0.229652
\(479\) −9840.00 −0.938624 −0.469312 0.883032i \(-0.655498\pi\)
−0.469312 + 0.883032i \(0.655498\pi\)
\(480\) 576.000 0.0547723
\(481\) −9652.00 −0.914955
\(482\) −1436.00 −0.135701
\(483\) −8064.00 −0.759678
\(484\) −4748.00 −0.445905
\(485\) 6924.00 0.648253
\(486\) −486.000 −0.0453609
\(487\) −1424.00 −0.132500 −0.0662501 0.997803i \(-0.521104\pi\)
−0.0662501 + 0.997803i \(0.521104\pi\)
\(488\) −4304.00 −0.399248
\(489\) 5556.00 0.513806
\(490\) −1044.00 −0.0962513
\(491\) −4548.00 −0.418021 −0.209011 0.977913i \(-0.567024\pi\)
−0.209011 + 0.977913i \(0.567024\pi\)
\(492\) −504.000 −0.0461831
\(493\) 0 0
\(494\) −1520.00 −0.138437
\(495\) 648.000 0.0588393
\(496\) 1408.00 0.127462
\(497\) −12672.0 −1.14370
\(498\) 2952.00 0.265627
\(499\) −6500.00 −0.583126 −0.291563 0.956552i \(-0.594175\pi\)
−0.291563 + 0.956552i \(0.594175\pi\)
\(500\) 5136.00 0.459378
\(501\) 6408.00 0.571434
\(502\) −12024.0 −1.06904
\(503\) −12168.0 −1.07862 −0.539308 0.842108i \(-0.681314\pi\)
−0.539308 + 0.842108i \(0.681314\pi\)
\(504\) −1152.00 −0.101814
\(505\) 3708.00 0.326740
\(506\) −4032.00 −0.354238
\(507\) −2259.00 −0.197881
\(508\) −10144.0 −0.885959
\(509\) −21090.0 −1.83654 −0.918269 0.395957i \(-0.870413\pi\)
−0.918269 + 0.395957i \(0.870413\pi\)
\(510\) 0 0
\(511\) −3488.00 −0.301957
\(512\) −512.000 −0.0441942
\(513\) 540.000 0.0464748
\(514\) 4092.00 0.351149
\(515\) −768.000 −0.0657129
\(516\) −624.000 −0.0532366
\(517\) 1152.00 0.0979979
\(518\) 8128.00 0.689428
\(519\) −5274.00 −0.446056
\(520\) 1824.00 0.153822
\(521\) 5238.00 0.440462 0.220231 0.975448i \(-0.429319\pi\)
0.220231 + 0.975448i \(0.429319\pi\)
\(522\) 540.000 0.0452781
\(523\) 8588.00 0.718025 0.359012 0.933333i \(-0.383114\pi\)
0.359012 + 0.933333i \(0.383114\pi\)
\(524\) −9168.00 −0.764324
\(525\) −4272.00 −0.355134
\(526\) 12144.0 1.00666
\(527\) 0 0
\(528\) −576.000 −0.0474757
\(529\) 16057.0 1.31972
\(530\) 2376.00 0.194730
\(531\) −5940.00 −0.485450
\(532\) 1280.00 0.104314
\(533\) −1596.00 −0.129701
\(534\) −4860.00 −0.393844
\(535\) −8856.00 −0.715660
\(536\) −7072.00 −0.569895
\(537\) −1620.00 −0.130183
\(538\) −13860.0 −1.11068
\(539\) 1044.00 0.0834291
\(540\) −648.000 −0.0516398
\(541\) −3062.00 −0.243338 −0.121669 0.992571i \(-0.538825\pi\)
−0.121669 + 0.992571i \(0.538825\pi\)
\(542\) −2704.00 −0.214293
\(543\) −5946.00 −0.469921
\(544\) 0 0
\(545\) 7140.00 0.561182
\(546\) −3648.00 −0.285934
\(547\) 8476.00 0.662537 0.331268 0.943537i \(-0.392523\pi\)
0.331268 + 0.943537i \(0.392523\pi\)
\(548\) −2904.00 −0.226374
\(549\) 4842.00 0.376414
\(550\) −2136.00 −0.165599
\(551\) −600.000 −0.0463899
\(552\) 4032.00 0.310894
\(553\) 8320.00 0.639787
\(554\) −2372.00 −0.181907
\(555\) 4572.00 0.349677
\(556\) −1520.00 −0.115939
\(557\) −12546.0 −0.954383 −0.477191 0.878799i \(-0.658345\pi\)
−0.477191 + 0.878799i \(0.658345\pi\)
\(558\) −1584.00 −0.120172
\(559\) −1976.00 −0.149510
\(560\) −1536.00 −0.115907
\(561\) 0 0
\(562\) −4884.00 −0.366582
\(563\) −12.0000 −0.000898294 0 −0.000449147 1.00000i \(-0.500143\pi\)
−0.000449147 1.00000i \(0.500143\pi\)
\(564\) −1152.00 −0.0860070
\(565\) −2772.00 −0.206405
\(566\) 5656.00 0.420034
\(567\) 1296.00 0.0959910
\(568\) 6336.00 0.468050
\(569\) 19290.0 1.42123 0.710614 0.703582i \(-0.248417\pi\)
0.710614 + 0.703582i \(0.248417\pi\)
\(570\) 720.000 0.0529079
\(571\) 12148.0 0.890329 0.445165 0.895449i \(-0.353145\pi\)
0.445165 + 0.895449i \(0.353145\pi\)
\(572\) −1824.00 −0.133331
\(573\) −8064.00 −0.587920
\(574\) 1344.00 0.0977308
\(575\) 14952.0 1.08442
\(576\) 576.000 0.0416667
\(577\) −10366.0 −0.747907 −0.373953 0.927447i \(-0.621998\pi\)
−0.373953 + 0.927447i \(0.621998\pi\)
\(578\) 0 0
\(579\) 6906.00 0.495688
\(580\) 720.000 0.0515455
\(581\) −7872.00 −0.562109
\(582\) 6924.00 0.493143
\(583\) −2376.00 −0.168789
\(584\) 1744.00 0.123574
\(585\) −2052.00 −0.145025
\(586\) −9516.00 −0.670823
\(587\) 7644.00 0.537482 0.268741 0.963213i \(-0.413393\pi\)
0.268741 + 0.963213i \(0.413393\pi\)
\(588\) −1044.00 −0.0732208
\(589\) 1760.00 0.123123
\(590\) −7920.00 −0.552646
\(591\) −13122.0 −0.913311
\(592\) −4064.00 −0.282144
\(593\) 8658.00 0.599564 0.299782 0.954008i \(-0.403086\pi\)
0.299782 + 0.954008i \(0.403086\pi\)
\(594\) 648.000 0.0447605
\(595\) 0 0
\(596\) 6360.00 0.437107
\(597\) 4800.00 0.329064
\(598\) 12768.0 0.873114
\(599\) 25800.0 1.75987 0.879933 0.475098i \(-0.157587\pi\)
0.879933 + 0.475098i \(0.157587\pi\)
\(600\) 2136.00 0.145336
\(601\) −16202.0 −1.09966 −0.549828 0.835278i \(-0.685307\pi\)
−0.549828 + 0.835278i \(0.685307\pi\)
\(602\) 1664.00 0.112657
\(603\) 7956.00 0.537302
\(604\) 9728.00 0.655342
\(605\) 7122.00 0.478596
\(606\) 3708.00 0.248560
\(607\) 24136.0 1.61392 0.806960 0.590605i \(-0.201111\pi\)
0.806960 + 0.590605i \(0.201111\pi\)
\(608\) −640.000 −0.0426898
\(609\) −1440.00 −0.0958157
\(610\) 6456.00 0.428518
\(611\) −3648.00 −0.241542
\(612\) 0 0
\(613\) −4642.00 −0.305854 −0.152927 0.988237i \(-0.548870\pi\)
−0.152927 + 0.988237i \(0.548870\pi\)
\(614\) 16952.0 1.11421
\(615\) 756.000 0.0495689
\(616\) 1536.00 0.100466
\(617\) 6726.00 0.438863 0.219432 0.975628i \(-0.429580\pi\)
0.219432 + 0.975628i \(0.429580\pi\)
\(618\) −768.000 −0.0499895
\(619\) 21220.0 1.37787 0.688937 0.724821i \(-0.258078\pi\)
0.688937 + 0.724821i \(0.258078\pi\)
\(620\) −2112.00 −0.136806
\(621\) −4536.00 −0.293113
\(622\) 9264.00 0.597191
\(623\) 12960.0 0.833437
\(624\) 1824.00 0.117017
\(625\) 3421.00 0.218944
\(626\) −9644.00 −0.615738
\(627\) −720.000 −0.0458597
\(628\) 2456.00 0.156059
\(629\) 0 0
\(630\) 1728.00 0.109278
\(631\) 29792.0 1.87956 0.939779 0.341783i \(-0.111031\pi\)
0.939779 + 0.341783i \(0.111031\pi\)
\(632\) −4160.00 −0.261829
\(633\) −9996.00 −0.627655
\(634\) −6852.00 −0.429223
\(635\) 15216.0 0.950911
\(636\) 2376.00 0.148136
\(637\) −3306.00 −0.205633
\(638\) −720.000 −0.0446788
\(639\) −7128.00 −0.441282
\(640\) 768.000 0.0474342
\(641\) 10158.0 0.625923 0.312962 0.949766i \(-0.398679\pi\)
0.312962 + 0.949766i \(0.398679\pi\)
\(642\) −8856.00 −0.544421
\(643\) −29828.0 −1.82940 −0.914698 0.404138i \(-0.867571\pi\)
−0.914698 + 0.404138i \(0.867571\pi\)
\(644\) −10752.0 −0.657901
\(645\) 936.000 0.0571395
\(646\) 0 0
\(647\) 1944.00 0.118124 0.0590622 0.998254i \(-0.481189\pi\)
0.0590622 + 0.998254i \(0.481189\pi\)
\(648\) −648.000 −0.0392837
\(649\) 7920.00 0.479025
\(650\) 6764.00 0.408163
\(651\) 4224.00 0.254304
\(652\) 7408.00 0.444969
\(653\) −26718.0 −1.60116 −0.800579 0.599227i \(-0.795475\pi\)
−0.800579 + 0.599227i \(0.795475\pi\)
\(654\) 7140.00 0.426905
\(655\) 13752.0 0.820359
\(656\) −672.000 −0.0399957
\(657\) −1962.00 −0.116507
\(658\) 3072.00 0.182005
\(659\) 4260.00 0.251815 0.125907 0.992042i \(-0.459816\pi\)
0.125907 + 0.992042i \(0.459816\pi\)
\(660\) 864.000 0.0509563
\(661\) 22862.0 1.34528 0.672639 0.739971i \(-0.265161\pi\)
0.672639 + 0.739971i \(0.265161\pi\)
\(662\) 5576.00 0.327368
\(663\) 0 0
\(664\) 3936.00 0.230040
\(665\) −1920.00 −0.111962
\(666\) 4572.00 0.266008
\(667\) 5040.00 0.292578
\(668\) 8544.00 0.494876
\(669\) 7944.00 0.459092
\(670\) 10608.0 0.611676
\(671\) −6456.00 −0.371432
\(672\) −1536.00 −0.0881733
\(673\) 32542.0 1.86390 0.931948 0.362592i \(-0.118108\pi\)
0.931948 + 0.362592i \(0.118108\pi\)
\(674\) 868.000 0.0496055
\(675\) −2403.00 −0.137024
\(676\) −3012.00 −0.171370
\(677\) −14214.0 −0.806925 −0.403463 0.914996i \(-0.632193\pi\)
−0.403463 + 0.914996i \(0.632193\pi\)
\(678\) −2772.00 −0.157018
\(679\) −18464.0 −1.04357
\(680\) 0 0
\(681\) −6732.00 −0.378812
\(682\) 2112.00 0.118582
\(683\) 7092.00 0.397317 0.198659 0.980069i \(-0.436341\pi\)
0.198659 + 0.980069i \(0.436341\pi\)
\(684\) 720.000 0.0402484
\(685\) 4356.00 0.242970
\(686\) 13760.0 0.765830
\(687\) −16950.0 −0.941314
\(688\) −832.000 −0.0461042
\(689\) 7524.00 0.416026
\(690\) −6048.00 −0.333686
\(691\) 13228.0 0.728244 0.364122 0.931351i \(-0.381369\pi\)
0.364122 + 0.931351i \(0.381369\pi\)
\(692\) −7032.00 −0.386296
\(693\) −1728.00 −0.0947205
\(694\) 13368.0 0.731185
\(695\) 2280.00 0.124439
\(696\) 720.000 0.0392120
\(697\) 0 0
\(698\) −5260.00 −0.285235
\(699\) −14094.0 −0.762638
\(700\) −5696.00 −0.307555
\(701\) 28062.0 1.51196 0.755982 0.654592i \(-0.227160\pi\)
0.755982 + 0.654592i \(0.227160\pi\)
\(702\) −2052.00 −0.110324
\(703\) −5080.00 −0.272540
\(704\) −768.000 −0.0411152
\(705\) 1728.00 0.0923124
\(706\) 14844.0 0.791305
\(707\) −9888.00 −0.525992
\(708\) −7920.00 −0.420412
\(709\) 27250.0 1.44343 0.721717 0.692188i \(-0.243353\pi\)
0.721717 + 0.692188i \(0.243353\pi\)
\(710\) −9504.00 −0.502364
\(711\) 4680.00 0.246855
\(712\) −6480.00 −0.341079
\(713\) −14784.0 −0.776529
\(714\) 0 0
\(715\) 2736.00 0.143106
\(716\) −2160.00 −0.112742
\(717\) −3600.00 −0.187510
\(718\) 20880.0 1.08529
\(719\) 14400.0 0.746912 0.373456 0.927648i \(-0.378173\pi\)
0.373456 + 0.927648i \(0.378173\pi\)
\(720\) −864.000 −0.0447214
\(721\) 2048.00 0.105786
\(722\) 12918.0 0.665870
\(723\) 2154.00 0.110800
\(724\) −7928.00 −0.406964
\(725\) 2670.00 0.136774
\(726\) 7122.00 0.364080
\(727\) 17984.0 0.917455 0.458727 0.888577i \(-0.348305\pi\)
0.458727 + 0.888577i \(0.348305\pi\)
\(728\) −4864.00 −0.247626
\(729\) 729.000 0.0370370
\(730\) −2616.00 −0.132634
\(731\) 0 0
\(732\) 6456.00 0.325984
\(733\) 16598.0 0.836373 0.418186 0.908361i \(-0.362666\pi\)
0.418186 + 0.908361i \(0.362666\pi\)
\(734\) 20848.0 1.04838
\(735\) 1566.00 0.0785888
\(736\) 5376.00 0.269242
\(737\) −10608.0 −0.530191
\(738\) 756.000 0.0377083
\(739\) 1460.00 0.0726752 0.0363376 0.999340i \(-0.488431\pi\)
0.0363376 + 0.999340i \(0.488431\pi\)
\(740\) 6096.00 0.302829
\(741\) 2280.00 0.113034
\(742\) −6336.00 −0.313480
\(743\) 30072.0 1.48484 0.742419 0.669936i \(-0.233678\pi\)
0.742419 + 0.669936i \(0.233678\pi\)
\(744\) −2112.00 −0.104072
\(745\) −9540.00 −0.469152
\(746\) −6556.00 −0.321759
\(747\) −4428.00 −0.216884
\(748\) 0 0
\(749\) 23616.0 1.15208
\(750\) −7704.00 −0.375080
\(751\) 18088.0 0.878882 0.439441 0.898271i \(-0.355177\pi\)
0.439441 + 0.898271i \(0.355177\pi\)
\(752\) −1536.00 −0.0744843
\(753\) 18036.0 0.872866
\(754\) 2280.00 0.110123
\(755\) −14592.0 −0.703387
\(756\) 1728.00 0.0831306
\(757\) 24734.0 1.18755 0.593773 0.804633i \(-0.297638\pi\)
0.593773 + 0.804633i \(0.297638\pi\)
\(758\) 12280.0 0.588430
\(759\) 6048.00 0.289234
\(760\) 960.000 0.0458196
\(761\) −22278.0 −1.06120 −0.530602 0.847621i \(-0.678034\pi\)
−0.530602 + 0.847621i \(0.678034\pi\)
\(762\) 15216.0 0.723383
\(763\) −19040.0 −0.903400
\(764\) −10752.0 −0.509154
\(765\) 0 0
\(766\) 6144.00 0.289806
\(767\) −25080.0 −1.18069
\(768\) 768.000 0.0360844
\(769\) 16130.0 0.756388 0.378194 0.925726i \(-0.376545\pi\)
0.378194 + 0.925726i \(0.376545\pi\)
\(770\) −2304.00 −0.107832
\(771\) −6138.00 −0.286712
\(772\) 9208.00 0.429279
\(773\) 29718.0 1.38277 0.691386 0.722486i \(-0.257001\pi\)
0.691386 + 0.722486i \(0.257001\pi\)
\(774\) 936.000 0.0434675
\(775\) −7832.00 −0.363011
\(776\) 9232.00 0.427074
\(777\) −12192.0 −0.562916
\(778\) −12300.0 −0.566808
\(779\) −840.000 −0.0386343
\(780\) −2736.00 −0.125596
\(781\) 9504.00 0.435442
\(782\) 0 0
\(783\) −810.000 −0.0369694
\(784\) −1392.00 −0.0634111
\(785\) −3684.00 −0.167500
\(786\) 13752.0 0.624068
\(787\) −9524.00 −0.431377 −0.215689 0.976462i \(-0.569200\pi\)
−0.215689 + 0.976462i \(0.569200\pi\)
\(788\) −17496.0 −0.790951
\(789\) −18216.0 −0.821935
\(790\) 6240.00 0.281024
\(791\) 7392.00 0.332275
\(792\) 864.000 0.0387638
\(793\) 20444.0 0.915495
\(794\) −212.000 −0.00947556
\(795\) −3564.00 −0.158996
\(796\) 6400.00 0.284977
\(797\) −33906.0 −1.50692 −0.753458 0.657496i \(-0.771616\pi\)
−0.753458 + 0.657496i \(0.771616\pi\)
\(798\) −1920.00 −0.0851720
\(799\) 0 0
\(800\) 2848.00 0.125865
\(801\) 7290.00 0.321572
\(802\) −3516.00 −0.154806
\(803\) 2616.00 0.114965
\(804\) 10608.0 0.465318
\(805\) 16128.0 0.706133
\(806\) −6688.00 −0.292276
\(807\) 20790.0 0.906868
\(808\) 4944.00 0.215259
\(809\) 630.000 0.0273790 0.0136895 0.999906i \(-0.495642\pi\)
0.0136895 + 0.999906i \(0.495642\pi\)
\(810\) 972.000 0.0421637
\(811\) 20788.0 0.900081 0.450040 0.893008i \(-0.351410\pi\)
0.450040 + 0.893008i \(0.351410\pi\)
\(812\) −1920.00 −0.0829788
\(813\) 4056.00 0.174969
\(814\) −6096.00 −0.262487
\(815\) −11112.0 −0.477591
\(816\) 0 0
\(817\) −1040.00 −0.0445349
\(818\) 7340.00 0.313737
\(819\) 5472.00 0.233464
\(820\) 1008.00 0.0429279
\(821\) 43098.0 1.83207 0.916036 0.401097i \(-0.131371\pi\)
0.916036 + 0.401097i \(0.131371\pi\)
\(822\) 4356.00 0.184833
\(823\) 14272.0 0.604484 0.302242 0.953231i \(-0.402265\pi\)
0.302242 + 0.953231i \(0.402265\pi\)
\(824\) −1024.00 −0.0432921
\(825\) 3204.00 0.135211
\(826\) 21120.0 0.889660
\(827\) −13644.0 −0.573698 −0.286849 0.957976i \(-0.592608\pi\)
−0.286849 + 0.957976i \(0.592608\pi\)
\(828\) −6048.00 −0.253844
\(829\) −2410.00 −0.100968 −0.0504842 0.998725i \(-0.516076\pi\)
−0.0504842 + 0.998725i \(0.516076\pi\)
\(830\) −5904.00 −0.246905
\(831\) 3558.00 0.148527
\(832\) 2432.00 0.101339
\(833\) 0 0
\(834\) 2280.00 0.0946642
\(835\) −12816.0 −0.531157
\(836\) −960.000 −0.0397157
\(837\) 2376.00 0.0981202
\(838\) −19320.0 −0.796418
\(839\) −23160.0 −0.953006 −0.476503 0.879173i \(-0.658096\pi\)
−0.476503 + 0.879173i \(0.658096\pi\)
\(840\) 2304.00 0.0946376
\(841\) −23489.0 −0.963098
\(842\) −16924.0 −0.692684
\(843\) 7326.00 0.299313
\(844\) −13328.0 −0.543565
\(845\) 4518.00 0.183934
\(846\) 1728.00 0.0702244
\(847\) −18992.0 −0.770452
\(848\) 3168.00 0.128290
\(849\) −8484.00 −0.342957
\(850\) 0 0
\(851\) 42672.0 1.71889
\(852\) −9504.00 −0.382162
\(853\) −32078.0 −1.28761 −0.643804 0.765190i \(-0.722645\pi\)
−0.643804 + 0.765190i \(0.722645\pi\)
\(854\) −17216.0 −0.689835
\(855\) −1080.00 −0.0431991
\(856\) −11808.0 −0.471483
\(857\) 14406.0 0.574212 0.287106 0.957899i \(-0.407307\pi\)
0.287106 + 0.957899i \(0.407307\pi\)
\(858\) 2736.00 0.108864
\(859\) 30620.0 1.21623 0.608115 0.793849i \(-0.291926\pi\)
0.608115 + 0.793849i \(0.291926\pi\)
\(860\) 1248.00 0.0494842
\(861\) −2016.00 −0.0797969
\(862\) 19584.0 0.773821
\(863\) 17568.0 0.692957 0.346478 0.938058i \(-0.387377\pi\)
0.346478 + 0.938058i \(0.387377\pi\)
\(864\) −864.000 −0.0340207
\(865\) 10548.0 0.414616
\(866\) 14684.0 0.576192
\(867\) 0 0
\(868\) 5632.00 0.220233
\(869\) −6240.00 −0.243587
\(870\) −1080.00 −0.0420867
\(871\) 33592.0 1.30680
\(872\) 9520.00 0.369711
\(873\) −10386.0 −0.402649
\(874\) 6720.00 0.260077
\(875\) 20544.0 0.793730
\(876\) −2616.00 −0.100898
\(877\) 21706.0 0.835758 0.417879 0.908503i \(-0.362774\pi\)
0.417879 + 0.908503i \(0.362774\pi\)
\(878\) 21280.0 0.817956
\(879\) 14274.0 0.547725
\(880\) 1152.00 0.0441294
\(881\) 14958.0 0.572018 0.286009 0.958227i \(-0.407671\pi\)
0.286009 + 0.958227i \(0.407671\pi\)
\(882\) 1566.00 0.0597845
\(883\) −32812.0 −1.25052 −0.625261 0.780415i \(-0.715008\pi\)
−0.625261 + 0.780415i \(0.715008\pi\)
\(884\) 0 0
\(885\) 11880.0 0.451234
\(886\) 34824.0 1.32047
\(887\) 38856.0 1.47086 0.735432 0.677598i \(-0.236979\pi\)
0.735432 + 0.677598i \(0.236979\pi\)
\(888\) 6096.00 0.230370
\(889\) −40576.0 −1.53079
\(890\) 9720.00 0.366084
\(891\) −972.000 −0.0365468
\(892\) 10592.0 0.397586
\(893\) −1920.00 −0.0719489
\(894\) −9540.00 −0.356896
\(895\) 3240.00 0.121007
\(896\) −2048.00 −0.0763604
\(897\) −19152.0 −0.712895
\(898\) −3420.00 −0.127090
\(899\) −2640.00 −0.0979410
\(900\) −3204.00 −0.118667
\(901\) 0 0
\(902\) −1008.00 −0.0372092
\(903\) −2496.00 −0.0919841
\(904\) −3696.00 −0.135981
\(905\) 11892.0 0.436799
\(906\) −14592.0 −0.535085
\(907\) 28276.0 1.03516 0.517579 0.855635i \(-0.326833\pi\)
0.517579 + 0.855635i \(0.326833\pi\)
\(908\) −8976.00 −0.328061
\(909\) −5562.00 −0.202948
\(910\) 7296.00 0.265780
\(911\) −8112.00 −0.295019 −0.147510 0.989061i \(-0.547126\pi\)
−0.147510 + 0.989061i \(0.547126\pi\)
\(912\) 960.000 0.0348561
\(913\) 5904.00 0.214013
\(914\) 1292.00 0.0467566
\(915\) −9684.00 −0.349883
\(916\) −22600.0 −0.815202
\(917\) −36672.0 −1.32063
\(918\) 0 0
\(919\) −26080.0 −0.936126 −0.468063 0.883695i \(-0.655048\pi\)
−0.468063 + 0.883695i \(0.655048\pi\)
\(920\) −8064.00 −0.288981
\(921\) −25428.0 −0.909751
\(922\) 12036.0 0.429918
\(923\) −30096.0 −1.07326
\(924\) −2304.00 −0.0820303
\(925\) 22606.0 0.803547
\(926\) 13424.0 0.476393
\(927\) 1152.00 0.0408162
\(928\) 960.000 0.0339586
\(929\) −49170.0 −1.73651 −0.868254 0.496120i \(-0.834757\pi\)
−0.868254 + 0.496120i \(0.834757\pi\)
\(930\) 3168.00 0.111702
\(931\) −1740.00 −0.0612526
\(932\) −18792.0 −0.660464
\(933\) −13896.0 −0.487604
\(934\) −10728.0 −0.375836
\(935\) 0 0
\(936\) −2736.00 −0.0955438
\(937\) 48314.0 1.68447 0.842236 0.539110i \(-0.181239\pi\)
0.842236 + 0.539110i \(0.181239\pi\)
\(938\) −28288.0 −0.984687
\(939\) 14466.0 0.502748
\(940\) 2304.00 0.0799449
\(941\) −34782.0 −1.20495 −0.602477 0.798137i \(-0.705819\pi\)
−0.602477 + 0.798137i \(0.705819\pi\)
\(942\) −3684.00 −0.127422
\(943\) 7056.00 0.243664
\(944\) −10560.0 −0.364088
\(945\) −2592.00 −0.0892251
\(946\) −1248.00 −0.0428922
\(947\) 25116.0 0.861838 0.430919 0.902391i \(-0.358190\pi\)
0.430919 + 0.902391i \(0.358190\pi\)
\(948\) 6240.00 0.213782
\(949\) −8284.00 −0.283361
\(950\) 3560.00 0.121581
\(951\) 10278.0 0.350460
\(952\) 0 0
\(953\) −15462.0 −0.525565 −0.262782 0.964855i \(-0.584640\pi\)
−0.262782 + 0.964855i \(0.584640\pi\)
\(954\) −3564.00 −0.120953
\(955\) 16128.0 0.546481
\(956\) −4800.00 −0.162388
\(957\) 1080.00 0.0364801
\(958\) 19680.0 0.663708
\(959\) −11616.0 −0.391137
\(960\) −1152.00 −0.0387298
\(961\) −22047.0 −0.740056
\(962\) 19304.0 0.646971
\(963\) 13284.0 0.444518
\(964\) 2872.00 0.0959553
\(965\) −13812.0 −0.460750
\(966\) 16128.0 0.537174
\(967\) −736.000 −0.0244759 −0.0122379 0.999925i \(-0.503896\pi\)
−0.0122379 + 0.999925i \(0.503896\pi\)
\(968\) 9496.00 0.315303
\(969\) 0 0
\(970\) −13848.0 −0.458384
\(971\) −29268.0 −0.967307 −0.483653 0.875260i \(-0.660690\pi\)
−0.483653 + 0.875260i \(0.660690\pi\)
\(972\) 972.000 0.0320750
\(973\) −6080.00 −0.200325
\(974\) 2848.00 0.0936918
\(975\) −10146.0 −0.333264
\(976\) 8608.00 0.282311
\(977\) 16674.0 0.546007 0.273003 0.962013i \(-0.411983\pi\)
0.273003 + 0.962013i \(0.411983\pi\)
\(978\) −11112.0 −0.363316
\(979\) −9720.00 −0.317316
\(980\) 2088.00 0.0680599
\(981\) −10710.0 −0.348567
\(982\) 9096.00 0.295586
\(983\) 31272.0 1.01467 0.507336 0.861749i \(-0.330630\pi\)
0.507336 + 0.861749i \(0.330630\pi\)
\(984\) 1008.00 0.0326564
\(985\) 26244.0 0.848937
\(986\) 0 0
\(987\) −4608.00 −0.148606
\(988\) 3040.00 0.0978900
\(989\) 8736.00 0.280878
\(990\) −1296.00 −0.0416056
\(991\) 15928.0 0.510565 0.255282 0.966867i \(-0.417832\pi\)
0.255282 + 0.966867i \(0.417832\pi\)
\(992\) −2816.00 −0.0901291
\(993\) −8364.00 −0.267295
\(994\) 25344.0 0.808715
\(995\) −9600.00 −0.305870
\(996\) −5904.00 −0.187827
\(997\) −42014.0 −1.33460 −0.667300 0.744789i \(-0.732550\pi\)
−0.667300 + 0.744789i \(0.732550\pi\)
\(998\) 13000.0 0.412332
\(999\) −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 1734.4.a.d.1.1 1
17.16 even 2 6.4.a.a.1.1 1
51.50 odd 2 18.4.a.a.1.1 1
68.67 odd 2 48.4.a.c.1.1 1
85.33 odd 4 150.4.c.d.49.2 2
85.67 odd 4 150.4.c.d.49.1 2
85.84 even 2 150.4.a.i.1.1 1
119.16 even 6 294.4.e.h.67.1 2
119.33 odd 6 294.4.e.g.67.1 2
119.67 even 6 294.4.e.h.79.1 2
119.101 odd 6 294.4.e.g.79.1 2
119.118 odd 2 294.4.a.e.1.1 1
136.67 odd 2 192.4.a.c.1.1 1
136.101 even 2 192.4.a.i.1.1 1
153.16 even 6 162.4.c.f.109.1 2
153.50 odd 6 162.4.c.c.55.1 2
153.67 even 6 162.4.c.f.55.1 2
153.101 odd 6 162.4.c.c.109.1 2
187.186 odd 2 726.4.a.f.1.1 1
204.203 even 2 144.4.a.c.1.1 1
221.135 odd 4 1014.4.b.d.337.2 2
221.203 odd 4 1014.4.b.d.337.1 2
221.220 even 2 1014.4.a.g.1.1 1
255.152 even 4 450.4.c.e.199.2 2
255.203 even 4 450.4.c.e.199.1 2
255.254 odd 2 450.4.a.h.1.1 1
272.67 odd 4 768.4.d.c.385.2 2
272.101 even 4 768.4.d.n.385.2 2
272.203 odd 4 768.4.d.c.385.1 2
272.237 even 4 768.4.d.n.385.1 2
323.322 odd 2 2166.4.a.i.1.1 1
340.67 even 4 1200.4.f.j.49.1 2
340.203 even 4 1200.4.f.j.49.2 2
340.339 odd 2 1200.4.a.b.1.1 1
357.101 even 6 882.4.g.f.667.1 2
357.152 even 6 882.4.g.f.361.1 2
357.254 odd 6 882.4.g.i.361.1 2
357.305 odd 6 882.4.g.i.667.1 2
357.356 even 2 882.4.a.n.1.1 1
408.101 odd 2 576.4.a.q.1.1 1
408.203 even 2 576.4.a.r.1.1 1
476.475 even 2 2352.4.a.e.1.1 1
561.560 even 2 2178.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 1 17.16 even 2
18.4.a.a.1.1 1 51.50 odd 2
48.4.a.c.1.1 1 68.67 odd 2
144.4.a.c.1.1 1 204.203 even 2
150.4.a.i.1.1 1 85.84 even 2
150.4.c.d.49.1 2 85.67 odd 4
150.4.c.d.49.2 2 85.33 odd 4
162.4.c.c.55.1 2 153.50 odd 6
162.4.c.c.109.1 2 153.101 odd 6
162.4.c.f.55.1 2 153.67 even 6
162.4.c.f.109.1 2 153.16 even 6
192.4.a.c.1.1 1 136.67 odd 2
192.4.a.i.1.1 1 136.101 even 2
294.4.a.e.1.1 1 119.118 odd 2
294.4.e.g.67.1 2 119.33 odd 6
294.4.e.g.79.1 2 119.101 odd 6
294.4.e.h.67.1 2 119.16 even 6
294.4.e.h.79.1 2 119.67 even 6
450.4.a.h.1.1 1 255.254 odd 2
450.4.c.e.199.1 2 255.203 even 4
450.4.c.e.199.2 2 255.152 even 4
576.4.a.q.1.1 1 408.101 odd 2
576.4.a.r.1.1 1 408.203 even 2
726.4.a.f.1.1 1 187.186 odd 2
768.4.d.c.385.1 2 272.203 odd 4
768.4.d.c.385.2 2 272.67 odd 4
768.4.d.n.385.1 2 272.237 even 4
768.4.d.n.385.2 2 272.101 even 4
882.4.a.n.1.1 1 357.356 even 2
882.4.g.f.361.1 2 357.152 even 6
882.4.g.f.667.1 2 357.101 even 6
882.4.g.i.361.1 2 357.254 odd 6
882.4.g.i.667.1 2 357.305 odd 6
1014.4.a.g.1.1 1 221.220 even 2
1014.4.b.d.337.1 2 221.203 odd 4
1014.4.b.d.337.2 2 221.135 odd 4
1200.4.a.b.1.1 1 340.339 odd 2
1200.4.f.j.49.1 2 340.67 even 4
1200.4.f.j.49.2 2 340.203 even 4
1734.4.a.d.1.1 1 1.1 even 1 trivial
2166.4.a.i.1.1 1 323.322 odd 2
2178.4.a.e.1.1 1 561.560 even 2
2352.4.a.e.1.1 1 476.475 even 2