Properties

Label 294.4.a.e.1.1
Level $294$
Weight $4$
Character 294.1
Self dual yes
Analytic conductor $17.347$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3465615417\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 294.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} -6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} -6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} +12.0000 q^{10} +12.0000 q^{11} +12.0000 q^{12} -38.0000 q^{13} -18.0000 q^{15} +16.0000 q^{16} +126.000 q^{17} -18.0000 q^{18} -20.0000 q^{19} -24.0000 q^{20} -24.0000 q^{22} +168.000 q^{23} -24.0000 q^{24} -89.0000 q^{25} +76.0000 q^{26} +27.0000 q^{27} +30.0000 q^{29} +36.0000 q^{30} +88.0000 q^{31} -32.0000 q^{32} +36.0000 q^{33} -252.000 q^{34} +36.0000 q^{36} +254.000 q^{37} +40.0000 q^{38} -114.000 q^{39} +48.0000 q^{40} -42.0000 q^{41} -52.0000 q^{43} +48.0000 q^{44} -54.0000 q^{45} -336.000 q^{46} +96.0000 q^{47} +48.0000 q^{48} +178.000 q^{50} +378.000 q^{51} -152.000 q^{52} +198.000 q^{53} -54.0000 q^{54} -72.0000 q^{55} -60.0000 q^{57} -60.0000 q^{58} +660.000 q^{59} -72.0000 q^{60} +538.000 q^{61} -176.000 q^{62} +64.0000 q^{64} +228.000 q^{65} -72.0000 q^{66} +884.000 q^{67} +504.000 q^{68} +504.000 q^{69} +792.000 q^{71} -72.0000 q^{72} -218.000 q^{73} -508.000 q^{74} -267.000 q^{75} -80.0000 q^{76} +228.000 q^{78} -520.000 q^{79} -96.0000 q^{80} +81.0000 q^{81} +84.0000 q^{82} +492.000 q^{83} -756.000 q^{85} +104.000 q^{86} +90.0000 q^{87} -96.0000 q^{88} -810.000 q^{89} +108.000 q^{90} +672.000 q^{92} +264.000 q^{93} -192.000 q^{94} +120.000 q^{95} -96.0000 q^{96} -1154.00 q^{97} +108.000 q^{99} +O(q^{100})\)

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
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 12.0000 0.379473
\(11\) 12.0000 0.328921 0.164461 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(12\) 12.0000 0.288675
\(13\) −38.0000 −0.810716 −0.405358 0.914158i \(-0.632853\pi\)
−0.405358 + 0.914158i \(0.632853\pi\)
\(14\) 0 0
\(15\) −18.0000 −0.309839
\(16\) 16.0000 0.250000
\(17\) 126.000 1.79762 0.898808 0.438342i \(-0.144434\pi\)
0.898808 + 0.438342i \(0.144434\pi\)
\(18\) −18.0000 −0.235702
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) −24.0000 −0.268328
\(21\) 0 0
\(22\) −24.0000 −0.232583
\(23\) 168.000 1.52306 0.761531 0.648129i \(-0.224448\pi\)
0.761531 + 0.648129i \(0.224448\pi\)
\(24\) −24.0000 −0.204124
\(25\) −89.0000 −0.712000
\(26\) 76.0000 0.573263
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 30.0000 0.192099 0.0960493 0.995377i \(-0.469379\pi\)
0.0960493 + 0.995377i \(0.469379\pi\)
\(30\) 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
\(34\) −252.000 −1.27111
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 254.000 1.12858 0.564288 0.825578i \(-0.309151\pi\)
0.564288 + 0.825578i \(0.309151\pi\)
\(38\) 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 \(0.525489\pi\)
\(42\) 0 0
\(43\) −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 \(-0.452405\pi\)
0.148969 + 0.988842i \(0.452405\pi\)
\(48\) 48.0000 0.144338
\(49\) 0 0
\(50\) 178.000 0.503460
\(51\) 378.000 1.03785
\(52\) −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
\(56\) 0 0
\(57\) −60.0000 −0.139424
\(58\) −60.0000 −0.135834
\(59\) 660.000 1.45635 0.728175 0.685391i \(-0.240369\pi\)
0.728175 + 0.685391i \(0.240369\pi\)
\(60\) −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
\(63\) 0 0
\(64\) 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 \(0.201653\pi\)
\(68\) 504.000 0.898808
\(69\) 504.000 0.879340
\(70\) 0 0
\(71\) 792.000 1.32385 0.661923 0.749572i \(-0.269740\pi\)
0.661923 + 0.749572i \(0.269740\pi\)
\(72\) −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
\(77\) 0 0
\(78\) 228.000 0.330973
\(79\) −520.000 −0.740564 −0.370282 0.928919i \(-0.620739\pi\)
−0.370282 + 0.928919i \(0.620739\pi\)
\(80\) −96.0000 −0.134164
\(81\) 81.0000 0.111111
\(82\) 84.0000 0.113125
\(83\) 492.000 0.650651 0.325325 0.945602i \(-0.394526\pi\)
0.325325 + 0.945602i \(0.394526\pi\)
\(84\) 0 0
\(85\) −756.000 −0.964703
\(86\) 104.000 0.130402
\(87\) 90.0000 0.110908
\(88\) −96.0000 −0.116291
\(89\) −810.000 −0.964717 −0.482359 0.875974i \(-0.660220\pi\)
−0.482359 + 0.875974i \(0.660220\pi\)
\(90\) 108.000 0.126491
\(91\) 0 0
\(92\) 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 \(0.706417\pi\)
\(98\) 0 0
\(99\) 108.000 0.109640
\(100\) −356.000 −0.356000
\(101\) 618.000 0.608845 0.304422 0.952537i \(-0.401537\pi\)
0.304422 + 0.952537i \(0.401537\pi\)
\(102\) −756.000 −0.733874
\(103\) −128.000 −0.122449 −0.0612243 0.998124i \(-0.519501\pi\)
−0.0612243 + 0.998124i \(0.519501\pi\)
\(104\) 304.000 0.286631
\(105\) 0 0
\(106\) −396.000 −0.362858
\(107\) −1476.00 −1.33355 −0.666777 0.745257i \(-0.732327\pi\)
−0.666777 + 0.745257i \(0.732327\pi\)
\(108\) 108.000 0.0962250
\(109\) 1190.00 1.04570 0.522850 0.852425i \(-0.324869\pi\)
0.522850 + 0.852425i \(0.324869\pi\)
\(110\) 144.000 0.124817
\(111\) 762.000 0.651584
\(112\) 0 0
\(113\) −462.000 −0.384613 −0.192307 0.981335i \(-0.561597\pi\)
−0.192307 + 0.981335i \(0.561597\pi\)
\(114\) 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
\(126\) 0 0
\(127\) −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
\(133\) 0 0
\(134\) −1768.00 −1.13979
\(135\) −162.000 −0.103280
\(136\) −1008.00 −0.635554
\(137\) −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 \(0.536988\pi\)
\(140\) 0 0
\(141\) 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
\(147\) 0 0
\(148\) 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
\(153\) 1134.00 0.599206
\(154\) 0 0
\(155\) −528.000 −0.273613
\(156\) −456.000 −0.234033
\(157\) −614.000 −0.312118 −0.156059 0.987748i \(-0.549879\pi\)
−0.156059 + 0.987748i \(0.549879\pi\)
\(158\) 1040.00 0.523658
\(159\) 594.000 0.296272
\(160\) 192.000 0.0948683
\(161\) 0 0
\(162\) −162.000 −0.0785674
\(163\) −1852.00 −0.889938 −0.444969 0.895546i \(-0.646785\pi\)
−0.444969 + 0.895546i \(0.646785\pi\)
\(164\) −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 \(0.335213\pi\)
\(168\) 0 0
\(169\) −753.000 −0.342740
\(170\) 1512.00 0.682148
\(171\) −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
\(175\) 0 0
\(176\) 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 \(0.633412\pi\)
\(182\) 0 0
\(183\) 1614.00 0.651969
\(184\) −1344.00 −0.538484
\(185\) −1524.00 −0.605658
\(186\) −528.000 −0.208144
\(187\) 1512.00 0.591275
\(188\) 384.000 0.148969
\(189\) 0 0
\(190\) −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 \(-0.641232\pi\)
−0.429279 + 0.903172i \(0.641232\pi\)
\(194\) 2308.00 0.854148
\(195\) 684.000 0.251191
\(196\) 0 0
\(197\) 4374.00 1.58190 0.790951 0.611880i \(-0.209586\pi\)
0.790951 + 0.611880i \(0.209586\pi\)
\(198\) −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
\(203\) 0 0
\(204\) 1512.00 0.518927
\(205\) 252.000 0.0858558
\(206\) 256.000 0.0865843
\(207\) 1512.00 0.507687
\(208\) −608.000 −0.202679
\(209\) −240.000 −0.0794313
\(210\) 0 0
\(211\) 3332.00 1.08713 0.543565 0.839367i \(-0.317074\pi\)
0.543565 + 0.839367i \(0.317074\pi\)
\(212\) 792.000 0.256579
\(213\) 2376.00 0.764323
\(214\) 2952.00 0.942965
\(215\) 312.000 0.0989685
\(216\) −216.000 −0.0680414
\(217\) 0 0
\(218\) −2380.00 −0.739422
\(219\) −654.000 −0.201796
\(220\) −288.000 −0.0882589
\(221\) −4788.00 −1.45736
\(222\) −1524.00 −0.460740
\(223\) −2648.00 −0.795171 −0.397586 0.917565i \(-0.630152\pi\)
−0.397586 + 0.917565i \(0.630152\pi\)
\(224\) 0 0
\(225\) −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 \(-0.196626\pi\)
0.815202 + 0.579177i \(0.196626\pi\)
\(230\) 2016.00 0.577961
\(231\) 0 0
\(232\) −240.000 −0.0679171
\(233\) 4698.00 1.32093 0.660464 0.750858i \(-0.270360\pi\)
0.660464 + 0.750858i \(0.270360\pi\)
\(234\) 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
\(245\) 0 0
\(246\) 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 \(-0.772812\pi\)
−0.755924 + 0.654659i \(0.772812\pi\)
\(252\) 0 0
\(253\) 2016.00 0.500968
\(254\) 5072.00 1.25294
\(255\) −2268.00 −0.556971
\(256\) 256.000 0.0625000
\(257\) 2046.00 0.496599 0.248300 0.968683i \(-0.420128\pi\)
0.248300 + 0.968683i \(0.420128\pi\)
\(258\) 312.000 0.0752879
\(259\) 0 0
\(260\) 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
\(266\) 0 0
\(267\) −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 \(-0.548419\pi\)
−0.151528 + 0.988453i \(0.548419\pi\)
\(272\) 2016.00 0.449404
\(273\) 0 0
\(274\) 1452.00 0.320141
\(275\) −1068.00 −0.234192
\(276\) 2016.00 0.439670
\(277\) −1186.00 −0.257256 −0.128628 0.991693i \(-0.541057\pi\)
−0.128628 + 0.991693i \(0.541057\pi\)
\(278\) 760.000 0.163963
\(279\) 792.000 0.169949
\(280\) 0 0
\(281\) 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
\(287\) 0 0
\(288\) −288.000 −0.0589256
\(289\) 10963.0 2.23143
\(290\) 360.000 0.0728963
\(291\) −3462.00 −0.697409
\(292\) −872.000 −0.174760
\(293\) −4758.00 −0.948687 −0.474344 0.880340i \(-0.657315\pi\)
−0.474344 + 0.880340i \(0.657315\pi\)
\(294\) 0 0
\(295\) −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
\(301\) 0 0
\(302\) −4864.00 −0.926794
\(303\) 1854.00 0.351517
\(304\) −320.000 −0.0603726
\(305\) −3228.00 −0.606016
\(306\) −2268.00 −0.423702
\(307\) 8476.00 1.57574 0.787868 0.615844i \(-0.211185\pi\)
0.787868 + 0.615844i \(0.211185\pi\)
\(308\) 0 0
\(309\) −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
\(315\) 0 0
\(316\) −2080.00 −0.370282
\(317\) −3426.00 −0.607014 −0.303507 0.952829i \(-0.598158\pi\)
−0.303507 + 0.952829i \(0.598158\pi\)
\(318\) −1188.00 −0.209496
\(319\) 360.000 0.0631854
\(320\) −384.000 −0.0670820
\(321\) −4428.00 −0.769928
\(322\) 0 0
\(323\) −2520.00 −0.434107
\(324\) 324.000 0.0555556
\(325\) 3382.00 0.577230
\(326\) 3704.00 0.629281
\(327\) 3570.00 0.603735
\(328\) 336.000 0.0565625
\(329\) 0 0
\(330\) 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
\(336\) 0 0
\(337\) 434.000 0.0701528 0.0350764 0.999385i \(-0.488833\pi\)
0.0350764 + 0.999385i \(0.488833\pi\)
\(338\) 1506.00 0.242354
\(339\) −1386.00 −0.222057
\(340\) −3024.00 −0.482351
\(341\) 1056.00 0.167700
\(342\) 360.000 0.0569198
\(343\) 0 0
\(344\) 416.000 0.0652012
\(345\) −3024.00 −0.471903
\(346\) 3516.00 0.546304
\(347\) 6684.00 1.03405 0.517026 0.855970i \(-0.327039\pi\)
0.517026 + 0.855970i \(0.327039\pi\)
\(348\) 360.000 0.0554541
\(349\) −2630.00 −0.403383 −0.201692 0.979449i \(-0.564644\pi\)
−0.201692 + 0.979449i \(0.564644\pi\)
\(350\) 0 0
\(351\) −1026.00 −0.156022
\(352\) −384.000 −0.0581456
\(353\) 7422.00 1.11907 0.559537 0.828805i \(-0.310979\pi\)
0.559537 + 0.828805i \(0.310979\pi\)
\(354\) −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
\(364\) 0 0
\(365\) 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
\(371\) 0 0
\(372\) 1056.00 0.147180
\(373\) 3278.00 0.455036 0.227518 0.973774i \(-0.426939\pi\)
0.227518 + 0.973774i \(0.426939\pi\)
\(374\) −3024.00 −0.418094
\(375\) 3852.00 0.530444
\(376\) −768.000 −0.105337
\(377\) −1140.00 −0.155737
\(378\) 0 0
\(379\) 6140.00 0.832165 0.416083 0.909327i \(-0.363403\pi\)
0.416083 + 0.909327i \(0.363403\pi\)
\(380\) 480.000 0.0647986
\(381\) −7608.00 −1.02302
\(382\) 5376.00 0.720053
\(383\) 3072.00 0.409848 0.204924 0.978778i \(-0.434305\pi\)
0.204924 + 0.978778i \(0.434305\pi\)
\(384\) −384.000 −0.0510310
\(385\) 0 0
\(386\) 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
\(391\) 21168.0 2.73788
\(392\) 0 0
\(393\) −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
\(399\) 0 0
\(400\) −1424.00 −0.178000
\(401\) −1758.00 −0.218929 −0.109464 0.993991i \(-0.534914\pi\)
−0.109464 + 0.993991i \(0.534914\pi\)
\(402\) −5304.00 −0.658058
\(403\) −3344.00 −0.413341
\(404\) 2472.00 0.304422
\(405\) −486.000 −0.0596285
\(406\) 0 0
\(407\) 3048.00 0.371213
\(408\) −3024.00 −0.366937
\(409\) 3670.00 0.443691 0.221846 0.975082i \(-0.428792\pi\)
0.221846 + 0.975082i \(0.428792\pi\)
\(410\) −504.000 −0.0607092
\(411\) −2178.00 −0.261394
\(412\) −512.000 −0.0612243
\(413\) 0 0
\(414\) −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 \(0.309588\pi\)
\(420\) 0 0
\(421\) 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
\(425\) −11214.0 −1.27990
\(426\) −4752.00 −0.540458
\(427\) 0 0
\(428\) −5904.00 −0.666777
\(429\) −1368.00 −0.153957
\(430\) −624.000 −0.0699813
\(431\) 9792.00 1.09435 0.547174 0.837019i \(-0.315704\pi\)
0.547174 + 0.837019i \(0.315704\pi\)
\(432\) 432.000 0.0481125
\(433\) 7342.00 0.814859 0.407430 0.913237i \(-0.366425\pi\)
0.407430 + 0.913237i \(0.366425\pi\)
\(434\) 0 0
\(435\) −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
\(441\) 0 0
\(442\) 9576.00 1.03051
\(443\) −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
\(448\) 0 0
\(449\) −1710.00 −0.179732 −0.0898662 0.995954i \(-0.528644\pi\)
−0.0898662 + 0.995954i \(0.528644\pi\)
\(450\) 1602.00 0.167820
\(451\) −504.000 −0.0526218
\(452\) −1848.00 −0.192307
\(453\) 7296.00 0.756724
\(454\) 4488.00 0.463948
\(455\) 0 0
\(456\) 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
\(459\) 3402.00 0.345952
\(460\) −4032.00 −0.408680
\(461\) 6018.00 0.607996 0.303998 0.952673i \(-0.401678\pi\)
0.303998 + 0.952673i \(0.401678\pi\)
\(462\) 0 0
\(463\) −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 \(-0.585622\pi\)
−0.265756 + 0.964040i \(0.585622\pi\)
\(468\) −1368.00 −0.135119
\(469\) 0 0
\(470\) 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
\(483\) 0 0
\(484\) −4748.00 −0.445905
\(485\) 6924.00 0.648253
\(486\) −486.000 −0.0453609
\(487\) 1424.00 0.132500 0.0662501 0.997803i \(-0.478896\pi\)
0.0662501 + 0.997803i \(0.478896\pi\)
\(488\) −4304.00 −0.399248
\(489\) −5556.00 −0.513806
\(490\) 0 0
\(491\) −4548.00 −0.418021 −0.209011 0.977913i \(-0.567024\pi\)
−0.209011 + 0.977913i \(0.567024\pi\)
\(492\) −504.000 −0.0461831
\(493\) 3780.00 0.345320
\(494\) −1520.00 −0.138437
\(495\) −648.000 −0.0588393
\(496\) 1408.00 0.127462
\(497\) 0 0
\(498\) −2952.00 −0.265627
\(499\) 6500.00 0.583126 0.291563 0.956552i \(-0.405825\pi\)
0.291563 + 0.956552i \(0.405825\pi\)
\(500\) 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 \(0.681314\pi\)
\(504\) 0 0
\(505\) −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 \(-0.129587\pi\)
0.918269 + 0.395957i \(0.129587\pi\)
\(510\) 4536.00 0.393838
\(511\) 0 0
\(512\) −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
\(518\) 0 0
\(519\) −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 \(-0.616886\pi\)
−0.359012 + 0.933333i \(0.616886\pi\)
\(524\) −9168.00 −0.764324
\(525\) 0 0
\(526\) 12144.0 1.00666
\(527\) 11088.0 0.916510
\(528\) 576.000 0.0474757
\(529\) 16057.0 1.31972
\(530\) 2376.00 0.194730
\(531\) 5940.00 0.485450
\(532\) 0 0
\(533\) 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
\(539\) 0 0
\(540\) −648.000 −0.0516398
\(541\) 3062.00 0.243338 0.121669 0.992571i \(-0.461175\pi\)
0.121669 + 0.992571i \(0.461175\pi\)
\(542\) 2704.00 0.214293
\(543\) −5946.00 −0.469921
\(544\) −4032.00 −0.317777
\(545\) −7140.00 −0.561182
\(546\) 0 0
\(547\) −8476.00 −0.662537 −0.331268 0.943537i \(-0.607477\pi\)
−0.331268 + 0.943537i \(0.607477\pi\)
\(548\) −2904.00 −0.226374
\(549\) 4842.00 0.376414
\(550\) 2136.00 0.165599
\(551\) −600.000 −0.0463899
\(552\) −4032.00 −0.310894
\(553\) 0 0
\(554\) 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
\(560\) 0 0
\(561\) 4536.00 0.341373
\(562\) −4884.00 −0.366582
\(563\) 12.0000 0.000898294 0 0.000449147 1.00000i \(-0.499857\pi\)
0.000449147 1.00000i \(0.499857\pi\)
\(564\) 1152.00 0.0860070
\(565\) 2772.00 0.206405
\(566\) 5656.00 0.420034
\(567\) 0 0
\(568\) −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 \(-0.646855\pi\)
−0.445165 + 0.895449i \(0.646855\pi\)
\(572\) −1824.00 −0.133331
\(573\) −8064.00 −0.587920
\(574\) 0 0
\(575\) −14952.0 −1.08442
\(576\) 576.000 0.0416667
\(577\) 10366.0 0.747907 0.373953 0.927447i \(-0.378002\pi\)
0.373953 + 0.927447i \(0.378002\pi\)
\(578\) −21926.0 −1.57786
\(579\) −6906.00 −0.495688
\(580\) −720.000 −0.0515455
\(581\) 0 0
\(582\) 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 \(-0.586607\pi\)
−0.268741 + 0.963213i \(0.586607\pi\)
\(588\) 0 0
\(589\) −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 \(-0.596914\pi\)
−0.299782 + 0.954008i \(0.596914\pi\)
\(594\) −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 \(0.685307\pi\)
\(602\) 0 0
\(603\) 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
\(609\) 0 0
\(610\) 6456.00 0.428518
\(611\) −3648.00 −0.241542
\(612\) 4536.00 0.299603
\(613\) −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
\(616\) 0 0
\(617\) −6726.00 −0.438863 −0.219432 0.975628i \(-0.570420\pi\)
−0.219432 + 0.975628i \(0.570420\pi\)
\(618\) 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
\(623\) 0 0
\(624\) −1824.00 −0.117017
\(625\) 3421.00 0.218944
\(626\) −9644.00 −0.615738
\(627\) −720.000 −0.0458597
\(628\) −2456.00 −0.156059
\(629\) 32004.0 2.02875
\(630\) 0 0
\(631\) 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
\(637\) 0 0
\(638\) −720.000 −0.0446788
\(639\) 7128.00 0.441282
\(640\) 768.000 0.0474342
\(641\) −10158.0 −0.625923 −0.312962 0.949766i \(-0.601321\pi\)
−0.312962 + 0.949766i \(0.601321\pi\)
\(642\) 8856.00 0.544421
\(643\) −29828.0 −1.82940 −0.914698 0.404138i \(-0.867571\pi\)
−0.914698 + 0.404138i \(0.867571\pi\)
\(644\) 0 0
\(645\) 936.000 0.0571395
\(646\) 5040.00 0.306960
\(647\) −1944.00 −0.118124 −0.0590622 0.998254i \(-0.518811\pi\)
−0.0590622 + 0.998254i \(0.518811\pi\)
\(648\) −648.000 −0.0392837
\(649\) 7920.00 0.479025
\(650\) −6764.00 −0.408163
\(651\) 0 0
\(652\) −7408.00 −0.444969
\(653\) 26718.0 1.60116 0.800579 0.599227i \(-0.204525\pi\)
0.800579 + 0.599227i \(0.204525\pi\)
\(654\) −7140.00 −0.426905
\(655\) 13752.0 0.820359
\(656\) −672.000 −0.0399957
\(657\) −1962.00 −0.116507
\(658\) 0 0
\(659\) 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 \(-0.734839\pi\)
−0.672639 + 0.739971i \(0.734839\pi\)
\(662\) 5576.00 0.327368
\(663\) −14364.0 −0.841405
\(664\) −3936.00 −0.230040
\(665\) 0 0
\(666\) −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
\(672\) 0 0
\(673\) −32542.0 −1.86390 −0.931948 0.362592i \(-0.881892\pi\)
−0.931948 + 0.362592i \(0.881892\pi\)
\(674\) −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
\(679\) 0 0
\(680\) 6048.00 0.341074
\(681\) −6732.00 −0.378812
\(682\) −2112.00 −0.118582
\(683\) −7092.00 −0.397317 −0.198659 0.980069i \(-0.563659\pi\)
−0.198659 + 0.980069i \(0.563659\pi\)
\(684\) −720.000 −0.0402484
\(685\) 4356.00 0.242970
\(686\) 0 0
\(687\) 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
\(693\) 0 0
\(694\) −13368.0 −0.731185
\(695\) 2280.00 0.124439
\(696\) −720.000 −0.0392120
\(697\) −5292.00 −0.287588
\(698\) 5260.00 0.285235
\(699\) 14094.0 0.762638
\(700\) 0 0
\(701\) 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
\(707\) 0 0
\(708\) 7920.00 0.420412
\(709\) −27250.0 −1.44343 −0.721717 0.692188i \(-0.756647\pi\)
−0.721717 + 0.692188i \(0.756647\pi\)
\(710\) 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
\(721\) 0 0
\(722\) 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 \(-0.651695\pi\)
−0.458727 + 0.888577i \(0.651695\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −2616.00 −0.132634
\(731\) −6552.00 −0.331511
\(732\) 6456.00 0.325984
\(733\) −16598.0 −0.836373 −0.418186 0.908361i \(-0.637334\pi\)
−0.418186 + 0.908361i \(0.637334\pi\)
\(734\) 20848.0 1.04838
\(735\) 0 0
\(736\) −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
\(742\) 0 0
\(743\) −30072.0 −1.48484 −0.742419 0.669936i \(-0.766322\pi\)
−0.742419 + 0.669936i \(0.766322\pi\)
\(744\) −2112.00 −0.104072
\(745\) −9540.00 −0.469152
\(746\) −6556.00 −0.321759
\(747\) 4428.00 0.216884
\(748\) 6048.00 0.295637
\(749\) 0 0
\(750\) −7704.00 −0.375080
\(751\) −18088.0 −0.878882 −0.439441 0.898271i \(-0.644823\pi\)
−0.439441 + 0.898271i \(0.644823\pi\)
\(752\) 1536.00 0.0744843
\(753\) −18036.0 −0.872866
\(754\) 2280.00 0.110123
\(755\) −14592.0 −0.703387
\(756\) 0 0
\(757\) 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 \(-0.321966\pi\)
0.530602 + 0.847621i \(0.321966\pi\)
\(762\) 15216.0 0.723383
\(763\) 0 0
\(764\) −10752.0 −0.509154
\(765\) −6804.00 −0.321568
\(766\) −6144.00 −0.289806
\(767\) −25080.0 −1.18069
\(768\) 768.000 0.0360844
\(769\) −16130.0 −0.756388 −0.378194 0.925726i \(-0.623455\pi\)
−0.378194 + 0.925726i \(0.623455\pi\)
\(770\) 0 0
\(771\) 6138.00 0.286712
\(772\) −9208.00 −0.429279
\(773\) −29718.0 −1.38277 −0.691386 0.722486i \(-0.742999\pi\)
−0.691386 + 0.722486i \(0.742999\pi\)
\(774\) 936.000 0.0434675
\(775\) −7832.00 −0.363011
\(776\) 9232.00 0.427074
\(777\) 0 0
\(778\) −12300.0 −0.566808
\(779\) 840.000 0.0386343
\(780\) 2736.00 0.125596
\(781\) 9504.00 0.435442
\(782\) −42336.0 −1.93597
\(783\) 810.000 0.0369694
\(784\) 0 0
\(785\) 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
\(791\) 0 0
\(792\) −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 \(-0.228384\pi\)
0.753458 + 0.657496i \(0.228384\pi\)
\(798\) 0 0
\(799\) 12096.0 0.535577
\(800\) 2848.00 0.125865
\(801\) −7290.00 −0.321572
\(802\) 3516.00 0.154806
\(803\) −2616.00 −0.114965
\(804\) 10608.0 0.465318
\(805\) 0 0
\(806\) 6688.00 0.292276
\(807\) 20790.0 0.906868
\(808\) −4944.00 −0.215259
\(809\) −630.000 −0.0273790 −0.0136895 0.999906i \(-0.504358\pi\)
−0.0136895 + 0.999906i \(0.504358\pi\)
\(810\) 972.000 0.0421637
\(811\) 20788.0 0.900081 0.450040 0.893008i \(-0.351410\pi\)
0.450040 + 0.893008i \(0.351410\pi\)
\(812\) 0 0
\(813\) −4056.00 −0.174969
\(814\) −6096.00 −0.262487
\(815\) 11112.0 0.477591
\(816\) 6048.00 0.259464
\(817\) 1040.00 0.0445349
\(818\) −7340.00 −0.313737
\(819\) 0 0
\(820\) 1008.00 0.0429279
\(821\) −43098.0 −1.83207 −0.916036 0.401097i \(-0.868629\pi\)
−0.916036 + 0.401097i \(0.868629\pi\)
\(822\) 4356.00 0.184833
\(823\) −14272.0 −0.604484 −0.302242 0.953231i \(-0.597735\pi\)
−0.302242 + 0.953231i \(0.597735\pi\)
\(824\) 1024.00 0.0432921
\(825\) −3204.00 −0.135211
\(826\) 0 0
\(827\) 13644.0 0.573698 0.286849 0.957976i \(-0.407392\pi\)
0.286849 + 0.957976i \(0.407392\pi\)
\(828\) 6048.00 0.253844
\(829\) 2410.00 0.100968 0.0504842 0.998725i \(-0.483924\pi\)
0.0504842 + 0.998725i \(0.483924\pi\)
\(830\) 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 \(0.658096\pi\)
\(840\) 0 0
\(841\) −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
\(847\) 0 0
\(848\) 3168.00 0.128290
\(849\) −8484.00 −0.342957
\(850\) 22428.0 0.905028
\(851\) 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 \(0.722645\pi\)
\(854\) 0 0
\(855\) 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 \(-0.708074\pi\)
−0.608115 + 0.793849i \(0.708074\pi\)
\(860\) 1248.00 0.0494842
\(861\) 0 0
\(862\) −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
\(867\) 32889.0 1.28831
\(868\) 0 0
\(869\) −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
\(875\) 0 0
\(876\) −2616.00 −0.100898
\(877\) −21706.0 −0.835758 −0.417879 0.908503i \(-0.637226\pi\)
−0.417879 + 0.908503i \(0.637226\pi\)
\(878\) 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 \(0.407671\pi\)
\(882\) 0 0
\(883\) −32812.0 −1.25052 −0.625261 0.780415i \(-0.715008\pi\)
−0.625261 + 0.780415i \(0.715008\pi\)
\(884\) −19152.0 −0.728678
\(885\) −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
\(889\) 0 0
\(890\) −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
\(896\) 0 0
\(897\) −19152.0 −0.712895
\(898\) 3420.00 0.127090
\(899\) 2640.00 0.0979410
\(900\) −3204.00 −0.118667
\(901\) 24948.0 0.922462
\(902\) 1008.00 0.0372092
\(903\) 0 0
\(904\) 3696.00 0.135981
\(905\) 11892.0 0.436799
\(906\) −14592.0 −0.535085
\(907\) −28276.0 −1.03516 −0.517579 0.855635i \(-0.673167\pi\)
−0.517579 + 0.855635i \(0.673167\pi\)
\(908\) −8976.00 −0.328061
\(909\) 5562.00 0.202948
\(910\) 0 0
\(911\) 8112.00 0.295019 0.147510 0.989061i \(-0.452874\pi\)
0.147510 + 0.989061i \(0.452874\pi\)
\(912\) −960.000 −0.0348561
\(913\) 5904.00 0.214013
\(914\) 1292.00 0.0467566
\(915\) −9684.00 −0.349883
\(916\) 22600.0 0.815202
\(917\) 0 0
\(918\) −6804.00 −0.244625
\(919\) −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
\(924\) 0 0
\(925\) −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
\(931\) 0 0
\(932\) 18792.0 0.660464
\(933\) −13896.0 −0.487604
\(934\) 10728.0 0.375836
\(935\) −9072.00 −0.317311
\(936\) 2736.00 0.0955438
\(937\) −48314.0 −1.68447 −0.842236 0.539110i \(-0.818761\pi\)
−0.842236 + 0.539110i \(0.818761\pi\)
\(938\) 0 0
\(939\) 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
\(945\) 0 0
\(946\) 1248.00 0.0428922
\(947\) −25116.0 −0.861838 −0.430919 0.902391i \(-0.641810\pi\)
−0.430919 + 0.902391i \(0.641810\pi\)
\(948\) −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
\(959\) 0 0
\(960\) −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
\(966\) 0 0
\(967\) −736.000 −0.0244759 −0.0122379 0.999925i \(-0.503896\pi\)
−0.0122379 + 0.999925i \(0.503896\pi\)
\(968\) 9496.00 0.315303
\(969\) −7560.00 −0.250632
\(970\) −13848.0 −0.458384
\(971\) 29268.0 0.967307 0.483653 0.875260i \(-0.339310\pi\)
0.483653 + 0.875260i \(0.339310\pi\)
\(972\) 972.000 0.0320750
\(973\) 0 0
\(974\) −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
\(980\) 0 0
\(981\) 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
\(986\) −7560.00 −0.244178
\(987\) 0 0
\(988\) 3040.00 0.0978900
\(989\) −8736.00 −0.280878
\(990\) 1296.00 0.0416056
\(991\) −15928.0 −0.510565 −0.255282 0.966867i \(-0.582168\pi\)
−0.255282 + 0.966867i \(0.582168\pi\)
\(992\) −2816.00 −0.0901291
\(993\) −8364.00 −0.267295
\(994\) 0 0
\(995\) −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 294.4.a.e.1.1 1
3.2 odd 2 882.4.a.n.1.1 1
4.3 odd 2 2352.4.a.e.1.1 1
7.2 even 3 294.4.e.g.67.1 2
7.3 odd 6 294.4.e.h.79.1 2
7.4 even 3 294.4.e.g.79.1 2
7.5 odd 6 294.4.e.h.67.1 2
7.6 odd 2 6.4.a.a.1.1 1
21.2 odd 6 882.4.g.f.361.1 2
21.5 even 6 882.4.g.i.361.1 2
21.11 odd 6 882.4.g.f.667.1 2
21.17 even 6 882.4.g.i.667.1 2
21.20 even 2 18.4.a.a.1.1 1
28.27 even 2 48.4.a.c.1.1 1
35.13 even 4 150.4.c.d.49.2 2
35.27 even 4 150.4.c.d.49.1 2
35.34 odd 2 150.4.a.i.1.1 1
56.13 odd 2 192.4.a.i.1.1 1
56.27 even 2 192.4.a.c.1.1 1
63.13 odd 6 162.4.c.f.55.1 2
63.20 even 6 162.4.c.c.109.1 2
63.34 odd 6 162.4.c.f.109.1 2
63.41 even 6 162.4.c.c.55.1 2
77.76 even 2 726.4.a.f.1.1 1
84.83 odd 2 144.4.a.c.1.1 1
91.34 even 4 1014.4.b.d.337.1 2
91.83 even 4 1014.4.b.d.337.2 2
91.90 odd 2 1014.4.a.g.1.1 1
105.62 odd 4 450.4.c.e.199.2 2
105.83 odd 4 450.4.c.e.199.1 2
105.104 even 2 450.4.a.h.1.1 1
112.13 odd 4 768.4.d.n.385.1 2
112.27 even 4 768.4.d.c.385.1 2
112.69 odd 4 768.4.d.n.385.2 2
112.83 even 4 768.4.d.c.385.2 2
119.118 odd 2 1734.4.a.d.1.1 1
133.132 even 2 2166.4.a.i.1.1 1
140.27 odd 4 1200.4.f.j.49.1 2
140.83 odd 4 1200.4.f.j.49.2 2
140.139 even 2 1200.4.a.b.1.1 1
168.83 odd 2 576.4.a.r.1.1 1
168.125 even 2 576.4.a.q.1.1 1
231.230 odd 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 7.6 odd 2
18.4.a.a.1.1 1 21.20 even 2
48.4.a.c.1.1 1 28.27 even 2
144.4.a.c.1.1 1 84.83 odd 2
150.4.a.i.1.1 1 35.34 odd 2
150.4.c.d.49.1 2 35.27 even 4
150.4.c.d.49.2 2 35.13 even 4
162.4.c.c.55.1 2 63.41 even 6
162.4.c.c.109.1 2 63.20 even 6
162.4.c.f.55.1 2 63.13 odd 6
162.4.c.f.109.1 2 63.34 odd 6
192.4.a.c.1.1 1 56.27 even 2
192.4.a.i.1.1 1 56.13 odd 2
294.4.a.e.1.1 1 1.1 even 1 trivial
294.4.e.g.67.1 2 7.2 even 3
294.4.e.g.79.1 2 7.4 even 3
294.4.e.h.67.1 2 7.5 odd 6
294.4.e.h.79.1 2 7.3 odd 6
450.4.a.h.1.1 1 105.104 even 2
450.4.c.e.199.1 2 105.83 odd 4
450.4.c.e.199.2 2 105.62 odd 4
576.4.a.q.1.1 1 168.125 even 2
576.4.a.r.1.1 1 168.83 odd 2
726.4.a.f.1.1 1 77.76 even 2
768.4.d.c.385.1 2 112.27 even 4
768.4.d.c.385.2 2 112.83 even 4
768.4.d.n.385.1 2 112.13 odd 4
768.4.d.n.385.2 2 112.69 odd 4
882.4.a.n.1.1 1 3.2 odd 2
882.4.g.f.361.1 2 21.2 odd 6
882.4.g.f.667.1 2 21.11 odd 6
882.4.g.i.361.1 2 21.5 even 6
882.4.g.i.667.1 2 21.17 even 6
1014.4.a.g.1.1 1 91.90 odd 2
1014.4.b.d.337.1 2 91.34 even 4
1014.4.b.d.337.2 2 91.83 even 4
1200.4.a.b.1.1 1 140.139 even 2
1200.4.f.j.49.1 2 140.27 odd 4
1200.4.f.j.49.2 2 140.83 odd 4
1734.4.a.d.1.1 1 119.118 odd 2
2166.4.a.i.1.1 1 133.132 even 2
2178.4.a.e.1.1 1 231.230 odd 2
2352.4.a.e.1.1 1 4.3 odd 2