Properties

Label 150.4.a.i.1.1
Level $150$
Weight $4$
Character 150.1
Self dual yes
Analytic conductor $8.850$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.85028650086\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 150.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +16.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +16.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +12.0000 q^{11} +12.0000 q^{12} -38.0000 q^{13} +32.0000 q^{14} +16.0000 q^{16} +126.000 q^{17} +18.0000 q^{18} +20.0000 q^{19} +48.0000 q^{21} +24.0000 q^{22} -168.000 q^{23} +24.0000 q^{24} -76.0000 q^{26} +27.0000 q^{27} +64.0000 q^{28} +30.0000 q^{29} -88.0000 q^{31} +32.0000 q^{32} +36.0000 q^{33} +252.000 q^{34} +36.0000 q^{36} -254.000 q^{37} +40.0000 q^{38} -114.000 q^{39} +42.0000 q^{41} +96.0000 q^{42} +52.0000 q^{43} +48.0000 q^{44} -336.000 q^{46} +96.0000 q^{47} +48.0000 q^{48} -87.0000 q^{49} +378.000 q^{51} -152.000 q^{52} -198.000 q^{53} +54.0000 q^{54} +128.000 q^{56} +60.0000 q^{57} +60.0000 q^{58} -660.000 q^{59} -538.000 q^{61} -176.000 q^{62} +144.000 q^{63} +64.0000 q^{64} +72.0000 q^{66} -884.000 q^{67} +504.000 q^{68} -504.000 q^{69} +792.000 q^{71} +72.0000 q^{72} -218.000 q^{73} -508.000 q^{74} +80.0000 q^{76} +192.000 q^{77} -228.000 q^{78} -520.000 q^{79} +81.0000 q^{81} +84.0000 q^{82} +492.000 q^{83} +192.000 q^{84} +104.000 q^{86} +90.0000 q^{87} +96.0000 q^{88} +810.000 q^{89} -608.000 q^{91} -672.000 q^{92} -264.000 q^{93} +192.000 q^{94} +96.0000 q^{96} -1154.00 q^{97} -174.000 q^{98} +108.000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 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
\(10\) 0 0
\(11\) 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 \(0.632853\pi\)
\(14\) 32.0000 0.610883
\(15\) 0 0
\(16\) 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 \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 0 0
\(21\) 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
\(25\) 0 0
\(26\) −76.0000 −0.573263
\(27\) 27.0000 0.192450
\(28\) 64.0000 0.431959
\(29\) 30.0000 0.192099 0.0960493 0.995377i \(-0.469379\pi\)
0.0960493 + 0.995377i \(0.469379\pi\)
\(30\) 0 0
\(31\) −88.0000 −0.509847 −0.254924 0.966961i \(-0.582050\pi\)
−0.254924 + 0.966961i \(0.582050\pi\)
\(32\) 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 \(-0.690849\pi\)
−0.564288 + 0.825578i \(0.690849\pi\)
\(38\) 40.0000 0.170759
\(39\) −114.000 −0.468067
\(40\) 0 0
\(41\) 42.0000 0.159983 0.0799914 0.996796i \(-0.474511\pi\)
0.0799914 + 0.996796i \(0.474511\pi\)
\(42\) 96.0000 0.352693
\(43\) 52.0000 0.184417 0.0922084 0.995740i \(-0.470607\pi\)
0.0922084 + 0.995740i \(0.470607\pi\)
\(44\) 48.0000 0.164461
\(45\) 0 0
\(46\) −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
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) 378.000 1.03785
\(52\) −152.000 −0.405358
\(53\) −198.000 −0.513158 −0.256579 0.966523i \(-0.582595\pi\)
−0.256579 + 0.966523i \(0.582595\pi\)
\(54\) 54.0000 0.136083
\(55\) 0 0
\(56\) 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 \(0.759631\pi\)
\(60\) 0 0
\(61\) −538.000 −1.12924 −0.564622 0.825350i \(-0.690978\pi\)
−0.564622 + 0.825350i \(0.690978\pi\)
\(62\) −176.000 −0.360516
\(63\) 144.000 0.287973
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 72.0000 0.134282
\(67\) −884.000 −1.61191 −0.805954 0.591979i \(-0.798347\pi\)
−0.805954 + 0.591979i \(0.798347\pi\)
\(68\) 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
\(75\) 0 0
\(76\) 80.0000 0.120745
\(77\) 192.000 0.284161
\(78\) −228.000 −0.330973
\(79\) −520.000 −0.740564 −0.370282 0.928919i \(-0.620739\pi\)
−0.370282 + 0.928919i \(0.620739\pi\)
\(80\) 0 0
\(81\) 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 \(0.394526\pi\)
\(84\) 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 \(0.339780\pi\)
\(90\) 0 0
\(91\) −608.000 −0.700393
\(92\) −672.000 −0.761531
\(93\) −264.000 −0.294360
\(94\) 192.000 0.210673
\(95\) 0 0
\(96\) 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
\(100\) 0 0
\(101\) −618.000 −0.608845 −0.304422 0.952537i \(-0.598463\pi\)
−0.304422 + 0.952537i \(0.598463\pi\)
\(102\) 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 \(-0.267673\pi\)
0.666777 + 0.745257i \(0.267673\pi\)
\(108\) 108.000 0.0962250
\(109\) 1190.00 1.04570 0.522850 0.852425i \(-0.324869\pi\)
0.522850 + 0.852425i \(0.324869\pi\)
\(110\) 0 0
\(111\) −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
\(115\) 0 0
\(116\) 120.000 0.0960493
\(117\) −342.000 −0.270239
\(118\) −1320.00 −1.02980
\(119\) 2016.00 1.55300
\(120\) 0 0
\(121\) −1187.00 −0.891811
\(122\) −1076.00 −0.798496
\(123\) 126.000 0.0923662
\(124\) −352.000 −0.254924
\(125\) 0 0
\(126\) 288.000 0.203628
\(127\) 2536.00 1.77192 0.885959 0.463763i \(-0.153501\pi\)
0.885959 + 0.463763i \(0.153501\pi\)
\(128\) 128.000 0.0883883
\(129\) 156.000 0.106473
\(130\) 0 0
\(131\) 2292.00 1.52865 0.764324 0.644832i \(-0.223073\pi\)
0.764324 + 0.644832i \(0.223073\pi\)
\(132\) 144.000 0.0949514
\(133\) 320.000 0.208628
\(134\) −1768.00 −1.13979
\(135\) 0 0
\(136\) 1008.00 0.635554
\(137\) 726.000 0.452747 0.226374 0.974041i \(-0.427313\pi\)
0.226374 + 0.974041i \(0.427313\pi\)
\(138\) −1008.00 −0.621787
\(139\) 380.000 0.231879 0.115939 0.993256i \(-0.463012\pi\)
0.115939 + 0.993256i \(0.463012\pi\)
\(140\) 0 0
\(141\) 288.000 0.172014
\(142\) 1584.00 0.936101
\(143\) −456.000 −0.266662
\(144\) 144.000 0.0833333
\(145\) 0 0
\(146\) −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 \(0.356003\pi\)
\(150\) 0 0
\(151\) 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
\(154\) 384.000 0.200932
\(155\) 0 0
\(156\) −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
\(160\) 0 0
\(161\) −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
\(165\) 0 0
\(166\) 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
\(175\) 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 \(0.535963\pi\)
\(180\) 0 0
\(181\) 1982.00 0.813928 0.406964 0.913444i \(-0.366588\pi\)
0.406964 + 0.913444i \(0.366588\pi\)
\(182\) −1216.00 −0.495252
\(183\) −1614.00 −0.651969
\(184\) −1344.00 −0.538484
\(185\) 0 0
\(186\) −528.000 −0.208144
\(187\) 1512.00 0.591275
\(188\) 384.000 0.148969
\(189\) 432.000 0.166261
\(190\) 0 0
\(191\) −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
\(195\) 0 0
\(196\) −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 \(-0.591986\pi\)
−0.284977 + 0.958534i \(0.591986\pi\)
\(200\) 0 0
\(201\) −2652.00 −0.930635
\(202\) −1236.00 −0.430518
\(203\) 480.000 0.165958
\(204\) 1512.00 0.518927
\(205\) 0 0
\(206\) −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
\(215\) 0 0
\(216\) 216.000 0.0680414
\(217\) −1408.00 −0.440467
\(218\) 2380.00 0.739422
\(219\) −654.000 −0.201796
\(220\) 0 0
\(221\) −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 \(0.630152\pi\)
\(224\) 512.000 0.152721
\(225\) 0 0
\(226\) 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 \(0.803374\pi\)
\(230\) 0 0
\(231\) 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
\(235\) 0 0
\(236\) −2640.00 −0.728175
\(237\) −1560.00 −0.427565
\(238\) 4032.00 1.09813
\(239\) −1200.00 −0.324776 −0.162388 0.986727i \(-0.551920\pi\)
−0.162388 + 0.986727i \(0.551920\pi\)
\(240\) 0 0
\(241\) −718.000 −0.191911 −0.0959553 0.995386i \(-0.530591\pi\)
−0.0959553 + 0.995386i \(0.530591\pi\)
\(242\) −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
\(250\) 0 0
\(251\) 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 \(-0.420128\pi\)
0.248300 + 0.968683i \(0.420128\pi\)
\(258\) 312.000 0.0752879
\(259\) −4064.00 −0.974999
\(260\) 0 0
\(261\) 270.000 0.0640329
\(262\) 4584.00 1.08092
\(263\) 6072.00 1.42363 0.711817 0.702365i \(-0.247873\pi\)
0.711817 + 0.702365i \(0.247873\pi\)
\(264\) 288.000 0.0671408
\(265\) 0 0
\(266\) 640.000 0.147522
\(267\) 2430.00 0.556980
\(268\) −3536.00 −0.805954
\(269\) −6930.00 −1.57074 −0.785371 0.619025i \(-0.787528\pi\)
−0.785371 + 0.619025i \(0.787528\pi\)
\(270\) 0 0
\(271\) 1352.00 0.303056 0.151528 0.988453i \(-0.451581\pi\)
0.151528 + 0.988453i \(0.451581\pi\)
\(272\) 2016.00 0.449404
\(273\) −1824.00 −0.404372
\(274\) 1452.00 0.320141
\(275\) 0 0
\(276\) −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
\(280\) 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
\(285\) 0 0
\(286\) −912.000 −0.188558
\(287\) 672.000 0.138212
\(288\) 288.000 0.0589256
\(289\) 10963.0 2.23143
\(290\) 0 0
\(291\) −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 \(0.657315\pi\)
\(294\) −522.000 −0.103550
\(295\) 0 0
\(296\) −2032.00 −0.399012
\(297\) 324.000 0.0633010
\(298\) 3180.00 0.618163
\(299\) 6384.00 1.23477
\(300\) 0 0
\(301\) 832.000 0.159321
\(302\) 4864.00 0.926794
\(303\) −1854.00 −0.351517
\(304\) 320.000 0.0603726
\(305\) 0 0
\(306\) 2268.00 0.423702
\(307\) 8476.00 1.57574 0.787868 0.615844i \(-0.211185\pi\)
0.787868 + 0.615844i \(0.211185\pi\)
\(308\) 768.000 0.142081
\(309\) −384.000 −0.0706958
\(310\) 0 0
\(311\) 4632.00 0.844555 0.422278 0.906467i \(-0.361231\pi\)
0.422278 + 0.906467i \(0.361231\pi\)
\(312\) −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 \(-0.401842\pi\)
0.303507 + 0.952829i \(0.401842\pi\)
\(318\) −1188.00 −0.209496
\(319\) 360.000 0.0631854
\(320\) 0 0
\(321\) 4428.00 0.769928
\(322\) −5376.00 −0.930412
\(323\) 2520.00 0.434107
\(324\) 324.000 0.0555556
\(325\) 0 0
\(326\) 3704.00 0.629281
\(327\) 3570.00 0.603735
\(328\) 336.000 0.0565625
\(329\) 1536.00 0.257393
\(330\) 0 0
\(331\) −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
\(335\) 0 0
\(336\) 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
\(345\) 0 0
\(346\) −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 \(0.435356\pi\)
\(350\) 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
\(355\) 0 0
\(356\) 3240.00 0.482359
\(357\) 6048.00 0.896622
\(358\) −1080.00 −0.159441
\(359\) −10440.0 −1.53482 −0.767412 0.641154i \(-0.778456\pi\)
−0.767412 + 0.641154i \(0.778456\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 3964.00 0.575534
\(363\) −3561.00 −0.514887
\(364\) −2432.00 −0.350196
\(365\) 0 0
\(366\) −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
\(370\) 0 0
\(371\) −3168.00 −0.443327
\(372\) −1056.00 −0.147180
\(373\) −3278.00 −0.455036 −0.227518 0.973774i \(-0.573061\pi\)
−0.227518 + 0.973774i \(0.573061\pi\)
\(374\) 3024.00 0.418094
\(375\) 0 0
\(376\) 768.000 0.105337
\(377\) −1140.00 −0.155737
\(378\) 864.000 0.117564
\(379\) 6140.00 0.832165 0.416083 0.909327i \(-0.363403\pi\)
0.416083 + 0.909327i \(0.363403\pi\)
\(380\) 0 0
\(381\) 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 \(0.368734\pi\)
\(390\) 0 0
\(391\) −21168.0 −2.73788
\(392\) −696.000 −0.0896768
\(393\) 6876.00 0.882566
\(394\) −8748.00 −1.11857
\(395\) 0 0
\(396\) 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
\(400\) 0 0
\(401\) −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
\(405\) 0 0
\(406\) 960.000 0.117350
\(407\) −3048.00 −0.371213
\(408\) 3024.00 0.366937
\(409\) −3670.00 −0.443691 −0.221846 0.975082i \(-0.571208\pi\)
−0.221846 + 0.975082i \(0.571208\pi\)
\(410\) 0 0
\(411\) 2178.00 0.261394
\(412\) −512.000 −0.0612243
\(413\) −10560.0 −1.25817
\(414\) −3024.00 −0.358989
\(415\) 0 0
\(416\) −1216.00 −0.143316
\(417\) 1140.00 0.133875
\(418\) 480.000 0.0561664
\(419\) −9660.00 −1.12631 −0.563153 0.826353i \(-0.690412\pi\)
−0.563153 + 0.826353i \(0.690412\pi\)
\(420\) 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
\(425\) 0 0
\(426\) 4752.00 0.540458
\(427\) −8608.00 −0.975575
\(428\) 5904.00 0.666777
\(429\) −1368.00 −0.153957
\(430\) 0 0
\(431\) 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 \(0.366425\pi\)
\(434\) −2816.00 −0.311457
\(435\) 0 0
\(436\) 4760.00 0.522850
\(437\) −3360.00 −0.367805
\(438\) −1308.00 −0.142691
\(439\) 10640.0 1.15676 0.578382 0.815766i \(-0.303684\pi\)
0.578382 + 0.815766i \(0.303684\pi\)
\(440\) 0 0
\(441\) −783.000 −0.0845481
\(442\) −9576.00 −1.03051
\(443\) 17412.0 1.86742 0.933712 0.358024i \(-0.116549\pi\)
0.933712 + 0.358024i \(0.116549\pi\)
\(444\) −3048.00 −0.325792
\(445\) 0 0
\(446\) −5296.00 −0.562271
\(447\) 4770.00 0.504728
\(448\) 1024.00 0.107990
\(449\) −1710.00 −0.179732 −0.0898662 0.995954i \(-0.528644\pi\)
−0.0898662 + 0.995954i \(0.528644\pi\)
\(450\) 0 0
\(451\) 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 \(-0.489474\pi\)
0.0330619 + 0.999453i \(0.489474\pi\)
\(458\) −11300.0 −1.15287
\(459\) 3402.00 0.345952
\(460\) 0 0
\(461\) −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 \(-0.390635\pi\)
0.336861 + 0.941554i \(0.390635\pi\)
\(464\) 480.000 0.0480247
\(465\) 0 0
\(466\) −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
\(469\) −14144.0 −1.39256
\(470\) 0 0
\(471\) −1842.00 −0.180201
\(472\) −5280.00 −0.514898
\(473\) 624.000 0.0606587
\(474\) −3120.00 −0.302334
\(475\) 0 0
\(476\) 8064.00 0.776498
\(477\) −1782.00 −0.171053
\(478\) −2400.00 −0.229652
\(479\) 9840.00 0.938624 0.469312 0.883032i \(-0.344502\pi\)
0.469312 + 0.883032i \(0.344502\pi\)
\(480\) 0 0
\(481\) 9652.00 0.914955
\(482\) −1436.00 −0.135701
\(483\) −8064.00 −0.759678
\(484\) −4748.00 −0.445905
\(485\) 0 0
\(486\) 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
\(490\) 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
\(495\) 0 0
\(496\) −1408.00 −0.127462
\(497\) 12672.0 1.14370
\(498\) 2952.00 0.265627
\(499\) 6500.00 0.583126 0.291563 0.956552i \(-0.405825\pi\)
0.291563 + 0.956552i \(0.405825\pi\)
\(500\) 0 0
\(501\) 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
\(505\) 0 0
\(506\) −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
\(515\) 0 0
\(516\) 624.000 0.0532366
\(517\) 1152.00 0.0979979
\(518\) −8128.00 −0.689428
\(519\) −5274.00 −0.446056
\(520\) 0 0
\(521\) −5238.00 −0.440462 −0.220231 0.975448i \(-0.570681\pi\)
−0.220231 + 0.975448i \(0.570681\pi\)
\(522\) 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
\(530\) 0 0
\(531\) −5940.00 −0.485450
\(532\) 1280.00 0.104314
\(533\) −1596.00 −0.129701
\(534\) 4860.00 0.393844
\(535\) 0 0
\(536\) −7072.00 −0.569895
\(537\) −1620.00 −0.130183
\(538\) −13860.0 −1.11068
\(539\) −1044.00 −0.0834291
\(540\) 0 0
\(541\) 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
\(545\) 0 0
\(546\) −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
\(550\) 0 0
\(551\) 600.000 0.0463899
\(552\) −4032.00 −0.310894
\(553\) −8320.00 −0.639787
\(554\) 2372.00 0.181907
\(555\) 0 0
\(556\) 1520.00 0.115939
\(557\) 12546.0 0.954383 0.477191 0.878799i \(-0.341655\pi\)
0.477191 + 0.878799i \(0.341655\pi\)
\(558\) −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
\(565\) 0 0
\(566\) −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 \(0.248417\pi\)
\(570\) 0 0
\(571\) −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
\(574\) 1344.00 0.0977308
\(575\) 0 0
\(576\) 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
\(580\) 0 0
\(581\) 7872.00 0.562109
\(582\) −6924.00 −0.493143
\(583\) −2376.00 −0.168789
\(584\) −1744.00 −0.123574
\(585\) 0 0
\(586\) −9516.00 −0.670823
\(587\) −7644.00 −0.537482 −0.268741 0.963213i \(-0.586607\pi\)
−0.268741 + 0.963213i \(0.586607\pi\)
\(588\) −1044.00 −0.0732208
\(589\) −1760.00 −0.123123
\(590\) 0 0
\(591\) −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 \(0.157587\pi\)
\(600\) 0 0
\(601\) 16202.0 1.09966 0.549828 0.835278i \(-0.314693\pi\)
0.549828 + 0.835278i \(0.314693\pi\)
\(602\) 1664.00 0.112657
\(603\) −7956.00 −0.537302
\(604\) 9728.00 0.655342
\(605\) 0 0
\(606\) −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
\(610\) 0 0
\(611\) −3648.00 −0.241542
\(612\) 4536.00 0.299603
\(613\) 4642.00 0.305854 0.152927 0.988237i \(-0.451130\pi\)
0.152927 + 0.988237i \(0.451130\pi\)
\(614\) 16952.0 1.11421
\(615\) 0 0
\(616\) 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 \(-0.741922\pi\)
−0.688937 + 0.724821i \(0.741922\pi\)
\(620\) 0 0
\(621\) −4536.00 −0.293113
\(622\) 9264.00 0.597191
\(623\) 12960.0 0.833437
\(624\) −1824.00 −0.117017
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) −2376.00 −0.148136
\(637\) 3306.00 0.205633
\(638\) 720.000 0.0446788
\(639\) 7128.00 0.441282
\(640\) 0 0
\(641\) −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 \(0.867571\pi\)
\(644\) −10752.0 −0.657901
\(645\) 0 0
\(646\) 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
\(650\) 0 0
\(651\) −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
\(655\) 0 0
\(656\) 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 \(0.459816\pi\)
\(660\) 0 0
\(661\) 22862.0 1.34528 0.672639 0.739971i \(-0.265161\pi\)
0.672639 + 0.739971i \(0.265161\pi\)
\(662\) −5576.00 −0.327368
\(663\) −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
\(670\) 0 0
\(671\) −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
\(675\) 0 0
\(676\) −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
\(685\) 0 0
\(686\) −13760.0 −0.765830
\(687\) −16950.0 −0.941314
\(688\) 832.000 0.0461042
\(689\) 7524.00 0.416026
\(690\) 0 0
\(691\) −13228.0 −0.728244 −0.364122 0.931351i \(-0.618631\pi\)
−0.364122 + 0.931351i \(0.618631\pi\)
\(692\) −7032.00 −0.386296
\(693\) 1728.00 0.0947205
\(694\) −13368.0 −0.731185
\(695\) 0 0
\(696\) 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
\(705\) 0 0
\(706\) 14844.0 0.791305
\(707\) −9888.00 −0.525992
\(708\) −7920.00 −0.420412
\(709\) −27250.0 −1.44343 −0.721717 0.692188i \(-0.756647\pi\)
−0.721717 + 0.692188i \(0.756647\pi\)
\(710\) 0 0
\(711\) −4680.00 −0.246855
\(712\) 6480.00 0.341079
\(713\) 14784.0 0.776529
\(714\) 12096.0 0.634008
\(715\) 0 0
\(716\) −2160.00 −0.112742
\(717\) −3600.00 −0.187510
\(718\) −20880.0 −1.08529
\(719\) −14400.0 −0.746912 −0.373456 0.927648i \(-0.621827\pi\)
−0.373456 + 0.927648i \(0.621827\pi\)
\(720\) 0 0
\(721\) −2048.00 −0.105786
\(722\) −12918.0 −0.665870
\(723\) −2154.00 −0.110800
\(724\) 7928.00 0.406964
\(725\) 0 0
\(726\) −7122.00 −0.364080
\(727\) −17984.0 −0.917455 −0.458727 0.888577i \(-0.651695\pi\)
−0.458727 + 0.888577i \(0.651695\pi\)
\(728\) −4864.00 −0.247626
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 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 \(0.488431\pi\)
\(740\) 0 0
\(741\) −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
\(745\) 0 0
\(746\) −6556.00 −0.321759
\(747\) 4428.00 0.216884
\(748\) 6048.00 0.295637
\(749\) 23616.0 1.15208
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) 1728.00 0.0831306
\(757\) −24734.0 −1.18755 −0.593773 0.804633i \(-0.702362\pi\)
−0.593773 + 0.804633i \(0.702362\pi\)
\(758\) 12280.0 0.588430
\(759\) −6048.00 −0.289234
\(760\) 0 0
\(761\) −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 \(0.376545\pi\)
\(770\) 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
\(775\) 0 0
\(776\) −9232.00 −0.427074
\(777\) −12192.0 −0.562916
\(778\) 12300.0 0.566808
\(779\) 840.000 0.0386343
\(780\) 0 0
\(781\) 9504.00 0.435442
\(782\) −42336.0 −1.93597
\(783\) 810.000 0.0369694
\(784\) −1392.00 −0.0634111
\(785\) 0 0
\(786\) 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
\(790\) 0 0
\(791\) 7392.00 0.332275
\(792\) 864.000 0.0387638
\(793\) 20444.0 0.915495
\(794\) 212.000 0.00947556
\(795\) 0 0
\(796\) −6400.00 −0.284977
\(797\) 33906.0 1.50692 0.753458 0.657496i \(-0.228384\pi\)
0.753458 + 0.657496i \(0.228384\pi\)
\(798\) 1920.00 0.0851720
\(799\) 12096.0 0.535577
\(800\) 0 0
\(801\) 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 \(0.504358\pi\)
\(810\) 0 0
\(811\) −20788.0 −0.900081 −0.450040 0.893008i \(-0.648590\pi\)
−0.450040 + 0.893008i \(0.648590\pi\)
\(812\) 1920.00 0.0829788
\(813\) 4056.00 0.174969
\(814\) −6096.00 −0.262487
\(815\) 0 0
\(816\) 6048.00 0.259464
\(817\) 1040.00 0.0445349
\(818\) −7340.00 −0.313737
\(819\) −5472.00 −0.233464
\(820\) 0 0
\(821\) −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 \(-0.402265\pi\)
0.302242 + 0.953231i \(0.402265\pi\)
\(824\) −1024.00 −0.0432921
\(825\) 0 0
\(826\) −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 \(0.516076\pi\)
\(830\) 0 0
\(831\) 3558.00 0.148527
\(832\) −2432.00 −0.101339
\(833\) −10962.0 −0.455955
\(834\) 2280.00 0.0946642
\(835\) 0 0
\(836\) 960.000 0.0397157
\(837\) −2376.00 −0.0981202
\(838\) −19320.0 −0.796418
\(839\) 23160.0 0.953006 0.476503 0.879173i \(-0.341904\pi\)
0.476503 + 0.879173i \(0.341904\pi\)
\(840\) 0 0
\(841\) −23489.0 −0.963098
\(842\) 16924.0 0.692684
\(843\) 7326.00 0.299313
\(844\) 13328.0 0.543565
\(845\) 0 0
\(846\) 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
\(855\) 0 0
\(856\) 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 \(0.291926\pi\)
\(860\) 0 0
\(861\) 2016.00 0.0797969
\(862\) 19584.0 0.773821
\(863\) −17568.0 −0.692957 −0.346478 0.938058i \(-0.612623\pi\)
−0.346478 + 0.938058i \(0.612623\pi\)
\(864\) 864.000 0.0340207
\(865\) 0 0
\(866\) 14684.0 0.576192
\(867\) 32889.0 1.28831
\(868\) −5632.00 −0.220233
\(869\) −6240.00 −0.243587
\(870\) 0 0
\(871\) 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 \(-0.362774\pi\)
0.417879 + 0.908503i \(0.362774\pi\)
\(878\) 21280.0 0.817956
\(879\) −14274.0 −0.547725
\(880\) 0 0
\(881\) −14958.0 −0.572018 −0.286009 0.958227i \(-0.592329\pi\)
−0.286009 + 0.958227i \(0.592329\pi\)
\(882\) −1566.00 −0.0597845
\(883\) 32812.0 1.25052 0.625261 0.780415i \(-0.284992\pi\)
0.625261 + 0.780415i \(0.284992\pi\)
\(884\) −19152.0 −0.728678
\(885\) 0 0
\(886\) 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
\(890\) 0 0
\(891\) 972.000 0.0365468
\(892\) −10592.0 −0.397586
\(893\) 1920.00 0.0719489
\(894\) 9540.00 0.356896
\(895\) 0 0
\(896\) 2048.00 0.0763604
\(897\) 19152.0 0.712895
\(898\) −3420.00 −0.127090
\(899\) −2640.00 −0.0979410
\(900\) 0 0
\(901\) −24948.0 −0.922462
\(902\) 1008.00 0.0372092
\(903\) 2496.00 0.0919841
\(904\) 3696.00 0.135981
\(905\) 0 0
\(906\) 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
\(910\) 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
\(915\) 0 0
\(916\) −22600.0 −0.815202
\(917\) 36672.0 1.32063
\(918\) 6804.00 0.244625
\(919\) −26080.0 −0.936126 −0.468063 0.883695i \(-0.655048\pi\)
−0.468063 + 0.883695i \(0.655048\pi\)
\(920\) 0 0
\(921\) 25428.0 0.909751
\(922\) −12036.0 −0.429918
\(923\) −30096.0 −1.07326
\(924\) 2304.00 0.0820303
\(925\) 0 0
\(926\) 13424.0 0.476393
\(927\) −1152.00 −0.0408162
\(928\) 960.000 0.0339586
\(929\) 49170.0 1.73651 0.868254 0.496120i \(-0.165243\pi\)
0.868254 + 0.496120i \(0.165243\pi\)
\(930\) 0 0
\(931\) −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 \(-0.818761\pi\)
−0.842236 + 0.539110i \(0.818761\pi\)
\(938\) −28288.0 −0.984687
\(939\) 14466.0 0.502748
\(940\) 0 0
\(941\) 34782.0 1.20495 0.602477 0.798137i \(-0.294181\pi\)
0.602477 + 0.798137i \(0.294181\pi\)
\(942\) −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 \(-0.358190\pi\)
0.430919 + 0.902391i \(0.358190\pi\)
\(948\) −6240.00 −0.213782
\(949\) 8284.00 0.283361
\(950\) 0 0
\(951\) 10278.0 0.350460
\(952\) 16128.0 0.549067
\(953\) 15462.0 0.525565 0.262782 0.964855i \(-0.415360\pi\)
0.262782 + 0.964855i \(0.415360\pi\)
\(954\) −3564.00 −0.120953
\(955\) 0 0
\(956\) −4800.00 −0.162388
\(957\) 1080.00 0.0364801
\(958\) 19680.0 0.663708
\(959\) 11616.0 0.391137
\(960\) 0 0
\(961\) −22047.0 −0.740056
\(962\) 19304.0 0.646971
\(963\) 13284.0 0.444518
\(964\) −2872.00 −0.0959553
\(965\) 0 0
\(966\) −16128.0 −0.537174
\(967\) 736.000 0.0244759 0.0122379 0.999925i \(-0.496104\pi\)
0.0122379 + 0.999925i \(0.496104\pi\)
\(968\) −9496.00 −0.315303
\(969\) 7560.00 0.250632
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) −8608.00 −0.282311
\(977\) −16674.0 −0.546007 −0.273003 0.962013i \(-0.588017\pi\)
−0.273003 + 0.962013i \(0.588017\pi\)
\(978\) 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
\(985\) 0 0
\(986\) 7560.00 0.244178
\(987\) 4608.00 0.148606
\(988\) −3040.00 −0.0978900
\(989\) −8736.00 −0.280878
\(990\) 0 0
\(991\) −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
\(994\) 25344.0 0.808715
\(995\) 0 0
\(996\) 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 150.4.a.i.1.1 1
3.2 odd 2 450.4.a.h.1.1 1
4.3 odd 2 1200.4.a.b.1.1 1
5.2 odd 4 150.4.c.d.49.2 2
5.3 odd 4 150.4.c.d.49.1 2
5.4 even 2 6.4.a.a.1.1 1
15.2 even 4 450.4.c.e.199.1 2
15.8 even 4 450.4.c.e.199.2 2
15.14 odd 2 18.4.a.a.1.1 1
20.3 even 4 1200.4.f.j.49.1 2
20.7 even 4 1200.4.f.j.49.2 2
20.19 odd 2 48.4.a.c.1.1 1
35.4 even 6 294.4.e.h.79.1 2
35.9 even 6 294.4.e.h.67.1 2
35.19 odd 6 294.4.e.g.67.1 2
35.24 odd 6 294.4.e.g.79.1 2
35.34 odd 2 294.4.a.e.1.1 1
40.19 odd 2 192.4.a.c.1.1 1
40.29 even 2 192.4.a.i.1.1 1
45.4 even 6 162.4.c.f.55.1 2
45.14 odd 6 162.4.c.c.55.1 2
45.29 odd 6 162.4.c.c.109.1 2
45.34 even 6 162.4.c.f.109.1 2
55.54 odd 2 726.4.a.f.1.1 1
60.59 even 2 144.4.a.c.1.1 1
65.34 odd 4 1014.4.b.d.337.1 2
65.44 odd 4 1014.4.b.d.337.2 2
65.64 even 2 1014.4.a.g.1.1 1
80.19 odd 4 768.4.d.c.385.2 2
80.29 even 4 768.4.d.n.385.1 2
80.59 odd 4 768.4.d.c.385.1 2
80.69 even 4 768.4.d.n.385.2 2
85.84 even 2 1734.4.a.d.1.1 1
95.94 odd 2 2166.4.a.i.1.1 1
105.44 odd 6 882.4.g.i.361.1 2
105.59 even 6 882.4.g.f.667.1 2
105.74 odd 6 882.4.g.i.667.1 2
105.89 even 6 882.4.g.f.361.1 2
105.104 even 2 882.4.a.n.1.1 1
120.29 odd 2 576.4.a.q.1.1 1
120.59 even 2 576.4.a.r.1.1 1
140.139 even 2 2352.4.a.e.1.1 1
165.164 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 5.4 even 2
18.4.a.a.1.1 1 15.14 odd 2
48.4.a.c.1.1 1 20.19 odd 2
144.4.a.c.1.1 1 60.59 even 2
150.4.a.i.1.1 1 1.1 even 1 trivial
150.4.c.d.49.1 2 5.3 odd 4
150.4.c.d.49.2 2 5.2 odd 4
162.4.c.c.55.1 2 45.14 odd 6
162.4.c.c.109.1 2 45.29 odd 6
162.4.c.f.55.1 2 45.4 even 6
162.4.c.f.109.1 2 45.34 even 6
192.4.a.c.1.1 1 40.19 odd 2
192.4.a.i.1.1 1 40.29 even 2
294.4.a.e.1.1 1 35.34 odd 2
294.4.e.g.67.1 2 35.19 odd 6
294.4.e.g.79.1 2 35.24 odd 6
294.4.e.h.67.1 2 35.9 even 6
294.4.e.h.79.1 2 35.4 even 6
450.4.a.h.1.1 1 3.2 odd 2
450.4.c.e.199.1 2 15.2 even 4
450.4.c.e.199.2 2 15.8 even 4
576.4.a.q.1.1 1 120.29 odd 2
576.4.a.r.1.1 1 120.59 even 2
726.4.a.f.1.1 1 55.54 odd 2
768.4.d.c.385.1 2 80.59 odd 4
768.4.d.c.385.2 2 80.19 odd 4
768.4.d.n.385.1 2 80.29 even 4
768.4.d.n.385.2 2 80.69 even 4
882.4.a.n.1.1 1 105.104 even 2
882.4.g.f.361.1 2 105.89 even 6
882.4.g.f.667.1 2 105.59 even 6
882.4.g.i.361.1 2 105.44 odd 6
882.4.g.i.667.1 2 105.74 odd 6
1014.4.a.g.1.1 1 65.64 even 2
1014.4.b.d.337.1 2 65.34 odd 4
1014.4.b.d.337.2 2 65.44 odd 4
1200.4.a.b.1.1 1 4.3 odd 2
1200.4.f.j.49.1 2 20.3 even 4
1200.4.f.j.49.2 2 20.7 even 4
1734.4.a.d.1.1 1 85.84 even 2
2166.4.a.i.1.1 1 95.94 odd 2
2178.4.a.e.1.1 1 165.164 even 2
2352.4.a.e.1.1 1 140.139 even 2