Properties

Label 150.4.c.d.49.1
Level $150$
Weight $4$
Character 150.49
Analytic conductor $8.850$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 150.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.85028650086\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 150.49
Dual form 150.4.c.d.49.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} +6.00000 q^{6} -16.0000i q^{7} +8.00000i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} +6.00000 q^{6} -16.0000i q^{7} +8.00000i q^{8} -9.00000 q^{9} +12.0000 q^{11} -12.0000i q^{12} -38.0000i q^{13} -32.0000 q^{14} +16.0000 q^{16} -126.000i q^{17} +18.0000i q^{18} -20.0000 q^{19} +48.0000 q^{21} -24.0000i q^{22} -168.000i q^{23} -24.0000 q^{24} -76.0000 q^{26} -27.0000i q^{27} +64.0000i q^{28} -30.0000 q^{29} -88.0000 q^{31} -32.0000i q^{32} +36.0000i q^{33} -252.000 q^{34} +36.0000 q^{36} +254.000i q^{37} +40.0000i q^{38} +114.000 q^{39} +42.0000 q^{41} -96.0000i q^{42} +52.0000i q^{43} -48.0000 q^{44} -336.000 q^{46} -96.0000i q^{47} +48.0000i q^{48} +87.0000 q^{49} +378.000 q^{51} +152.000i q^{52} -198.000i q^{53} -54.0000 q^{54} +128.000 q^{56} -60.0000i q^{57} +60.0000i q^{58} +660.000 q^{59} -538.000 q^{61} +176.000i q^{62} +144.000i q^{63} -64.0000 q^{64} +72.0000 q^{66} +884.000i q^{67} +504.000i q^{68} +504.000 q^{69} +792.000 q^{71} -72.0000i q^{72} -218.000i q^{73} +508.000 q^{74} +80.0000 q^{76} -192.000i q^{77} -228.000i q^{78} +520.000 q^{79} +81.0000 q^{81} -84.0000i q^{82} +492.000i q^{83} -192.000 q^{84} +104.000 q^{86} -90.0000i q^{87} +96.0000i q^{88} -810.000 q^{89} -608.000 q^{91} +672.000i q^{92} -264.000i q^{93} -192.000 q^{94} +96.0000 q^{96} +1154.00i q^{97} -174.000i q^{98} -108.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} + 12q^{6} - 18q^{9} + O(q^{10}) \) \( 2q - 8q^{4} + 12q^{6} - 18q^{9} + 24q^{11} - 64q^{14} + 32q^{16} - 40q^{19} + 96q^{21} - 48q^{24} - 152q^{26} - 60q^{29} - 176q^{31} - 504q^{34} + 72q^{36} + 228q^{39} + 84q^{41} - 96q^{44} - 672q^{46} + 174q^{49} + 756q^{51} - 108q^{54} + 256q^{56} + 1320q^{59} - 1076q^{61} - 128q^{64} + 144q^{66} + 1008q^{69} + 1584q^{71} + 1016q^{74} + 160q^{76} + 1040q^{79} + 162q^{81} - 384q^{84} + 208q^{86} - 1620q^{89} - 1216q^{91} - 384q^{94} + 192q^{96} - 216q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) 3.00000i 0.577350i
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 6.00000 0.408248
\(7\) − 16.0000i − 0.863919i −0.901893 0.431959i \(-0.857822\pi\)
0.901893 0.431959i \(-0.142178\pi\)
\(8\) 8.00000i 0.353553i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 12.0000 0.328921 0.164461 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(12\) − 12.0000i − 0.288675i
\(13\) − 38.0000i − 0.810716i −0.914158 0.405358i \(-0.867147\pi\)
0.914158 0.405358i \(-0.132853\pi\)
\(14\) −32.0000 −0.610883
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 126.000i − 1.79762i −0.438342 0.898808i \(-0.644434\pi\)
0.438342 0.898808i \(-0.355566\pi\)
\(18\) 18.0000i 0.235702i
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 0 0
\(21\) 48.0000 0.498784
\(22\) − 24.0000i − 0.232583i
\(23\) − 168.000i − 1.52306i −0.648129 0.761531i \(-0.724448\pi\)
0.648129 0.761531i \(-0.275552\pi\)
\(24\) −24.0000 −0.204124
\(25\) 0 0
\(26\) −76.0000 −0.573263
\(27\) − 27.0000i − 0.192450i
\(28\) 64.0000i 0.431959i
\(29\) −30.0000 −0.192099 −0.0960493 0.995377i \(-0.530621\pi\)
−0.0960493 + 0.995377i \(0.530621\pi\)
\(30\) 0 0
\(31\) −88.0000 −0.509847 −0.254924 0.966961i \(-0.582050\pi\)
−0.254924 + 0.966961i \(0.582050\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 36.0000i 0.189903i
\(34\) −252.000 −1.27111
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 254.000i 1.12858i 0.825578 + 0.564288i \(0.190849\pi\)
−0.825578 + 0.564288i \(0.809151\pi\)
\(38\) 40.0000i 0.170759i
\(39\) 114.000 0.468067
\(40\) 0 0
\(41\) 42.0000 0.159983 0.0799914 0.996796i \(-0.474511\pi\)
0.0799914 + 0.996796i \(0.474511\pi\)
\(42\) − 96.0000i − 0.352693i
\(43\) 52.0000i 0.184417i 0.995740 + 0.0922084i \(0.0293926\pi\)
−0.995740 + 0.0922084i \(0.970607\pi\)
\(44\) −48.0000 −0.164461
\(45\) 0 0
\(46\) −336.000 −1.07697
\(47\) − 96.0000i − 0.297937i −0.988842 0.148969i \(-0.952405\pi\)
0.988842 0.148969i \(-0.0475953\pi\)
\(48\) 48.0000i 0.144338i
\(49\) 87.0000 0.253644
\(50\) 0 0
\(51\) 378.000 1.03785
\(52\) 152.000i 0.405358i
\(53\) − 198.000i − 0.513158i −0.966523 0.256579i \(-0.917405\pi\)
0.966523 0.256579i \(-0.0825954\pi\)
\(54\) −54.0000 −0.136083
\(55\) 0 0
\(56\) 128.000 0.305441
\(57\) − 60.0000i − 0.139424i
\(58\) 60.0000i 0.135834i
\(59\) 660.000 1.45635 0.728175 0.685391i \(-0.240369\pi\)
0.728175 + 0.685391i \(0.240369\pi\)
\(60\) 0 0
\(61\) −538.000 −1.12924 −0.564622 0.825350i \(-0.690978\pi\)
−0.564622 + 0.825350i \(0.690978\pi\)
\(62\) 176.000i 0.360516i
\(63\) 144.000i 0.287973i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 72.0000 0.134282
\(67\) 884.000i 1.61191i 0.591979 + 0.805954i \(0.298347\pi\)
−0.591979 + 0.805954i \(0.701653\pi\)
\(68\) 504.000i 0.898808i
\(69\) 504.000 0.879340
\(70\) 0 0
\(71\) 792.000 1.32385 0.661923 0.749572i \(-0.269740\pi\)
0.661923 + 0.749572i \(0.269740\pi\)
\(72\) − 72.0000i − 0.117851i
\(73\) − 218.000i − 0.349520i −0.984611 0.174760i \(-0.944085\pi\)
0.984611 0.174760i \(-0.0559150\pi\)
\(74\) 508.000 0.798024
\(75\) 0 0
\(76\) 80.0000 0.120745
\(77\) − 192.000i − 0.284161i
\(78\) − 228.000i − 0.330973i
\(79\) 520.000 0.740564 0.370282 0.928919i \(-0.379261\pi\)
0.370282 + 0.928919i \(0.379261\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 84.0000i − 0.113125i
\(83\) 492.000i 0.650651i 0.945602 + 0.325325i \(0.105474\pi\)
−0.945602 + 0.325325i \(0.894526\pi\)
\(84\) −192.000 −0.249392
\(85\) 0 0
\(86\) 104.000 0.130402
\(87\) − 90.0000i − 0.110908i
\(88\) 96.0000i 0.116291i
\(89\) −810.000 −0.964717 −0.482359 0.875974i \(-0.660220\pi\)
−0.482359 + 0.875974i \(0.660220\pi\)
\(90\) 0 0
\(91\) −608.000 −0.700393
\(92\) 672.000i 0.761531i
\(93\) − 264.000i − 0.294360i
\(94\) −192.000 −0.210673
\(95\) 0 0
\(96\) 96.0000 0.102062
\(97\) 1154.00i 1.20795i 0.797004 + 0.603974i \(0.206417\pi\)
−0.797004 + 0.603974i \(0.793583\pi\)
\(98\) − 174.000i − 0.179354i
\(99\) −108.000 −0.109640
\(100\) 0 0
\(101\) −618.000 −0.608845 −0.304422 0.952537i \(-0.598463\pi\)
−0.304422 + 0.952537i \(0.598463\pi\)
\(102\) − 756.000i − 0.733874i
\(103\) − 128.000i − 0.122449i −0.998124 0.0612243i \(-0.980499\pi\)
0.998124 0.0612243i \(-0.0195005\pi\)
\(104\) 304.000 0.286631
\(105\) 0 0
\(106\) −396.000 −0.362858
\(107\) − 1476.00i − 1.33355i −0.745257 0.666777i \(-0.767673\pi\)
0.745257 0.666777i \(-0.232327\pi\)
\(108\) 108.000i 0.0962250i
\(109\) −1190.00 −1.04570 −0.522850 0.852425i \(-0.675131\pi\)
−0.522850 + 0.852425i \(0.675131\pi\)
\(110\) 0 0
\(111\) −762.000 −0.651584
\(112\) − 256.000i − 0.215980i
\(113\) 462.000i 0.384613i 0.981335 + 0.192307i \(0.0615968\pi\)
−0.981335 + 0.192307i \(0.938403\pi\)
\(114\) −120.000 −0.0985880
\(115\) 0 0
\(116\) 120.000 0.0960493
\(117\) 342.000i 0.270239i
\(118\) − 1320.00i − 1.02980i
\(119\) −2016.00 −1.55300
\(120\) 0 0
\(121\) −1187.00 −0.891811
\(122\) 1076.00i 0.798496i
\(123\) 126.000i 0.0923662i
\(124\) 352.000 0.254924
\(125\) 0 0
\(126\) 288.000 0.203628
\(127\) − 2536.00i − 1.77192i −0.463763 0.885959i \(-0.653501\pi\)
0.463763 0.885959i \(-0.346499\pi\)
\(128\) 128.000i 0.0883883i
\(129\) −156.000 −0.106473
\(130\) 0 0
\(131\) 2292.00 1.52865 0.764324 0.644832i \(-0.223073\pi\)
0.764324 + 0.644832i \(0.223073\pi\)
\(132\) − 144.000i − 0.0949514i
\(133\) 320.000i 0.208628i
\(134\) 1768.00 1.13979
\(135\) 0 0
\(136\) 1008.00 0.635554
\(137\) − 726.000i − 0.452747i −0.974041 0.226374i \(-0.927313\pi\)
0.974041 0.226374i \(-0.0726870\pi\)
\(138\) − 1008.00i − 0.621787i
\(139\) −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
\(142\) − 1584.00i − 0.936101i
\(143\) − 456.000i − 0.266662i
\(144\) −144.000 −0.0833333
\(145\) 0 0
\(146\) −436.000 −0.247148
\(147\) 261.000i 0.146442i
\(148\) − 1016.00i − 0.564288i
\(149\) −1590.00 −0.874214 −0.437107 0.899410i \(-0.643997\pi\)
−0.437107 + 0.899410i \(0.643997\pi\)
\(150\) 0 0
\(151\) 2432.00 1.31068 0.655342 0.755332i \(-0.272524\pi\)
0.655342 + 0.755332i \(0.272524\pi\)
\(152\) − 160.000i − 0.0853797i
\(153\) 1134.00i 0.599206i
\(154\) −384.000 −0.200932
\(155\) 0 0
\(156\) −456.000 −0.234033
\(157\) 614.000i 0.312118i 0.987748 + 0.156059i \(0.0498790\pi\)
−0.987748 + 0.156059i \(0.950121\pi\)
\(158\) − 1040.00i − 0.523658i
\(159\) 594.000 0.296272
\(160\) 0 0
\(161\) −2688.00 −1.31580
\(162\) − 162.000i − 0.0785674i
\(163\) 1852.00i 0.889938i 0.895546 + 0.444969i \(0.146785\pi\)
−0.895546 + 0.444969i \(0.853215\pi\)
\(164\) −168.000 −0.0799914
\(165\) 0 0
\(166\) 984.000 0.460080
\(167\) − 2136.00i − 0.989752i −0.868964 0.494876i \(-0.835213\pi\)
0.868964 0.494876i \(-0.164787\pi\)
\(168\) 384.000i 0.176347i
\(169\) 753.000 0.342740
\(170\) 0 0
\(171\) 180.000 0.0804967
\(172\) − 208.000i − 0.0922084i
\(173\) − 1758.00i − 0.772591i −0.922375 0.386296i \(-0.873754\pi\)
0.922375 0.386296i \(-0.126246\pi\)
\(174\) −180.000 −0.0784239
\(175\) 0 0
\(176\) 192.000 0.0822304
\(177\) 1980.00i 0.840824i
\(178\) 1620.00i 0.682158i
\(179\) 540.000 0.225483 0.112742 0.993624i \(-0.464037\pi\)
0.112742 + 0.993624i \(0.464037\pi\)
\(180\) 0 0
\(181\) 1982.00 0.813928 0.406964 0.913444i \(-0.366588\pi\)
0.406964 + 0.913444i \(0.366588\pi\)
\(182\) 1216.00i 0.495252i
\(183\) − 1614.00i − 0.651969i
\(184\) 1344.00 0.538484
\(185\) 0 0
\(186\) −528.000 −0.208144
\(187\) − 1512.00i − 0.591275i
\(188\) 384.000i 0.148969i
\(189\) −432.000 −0.166261
\(190\) 0 0
\(191\) −2688.00 −1.01831 −0.509154 0.860675i \(-0.670042\pi\)
−0.509154 + 0.860675i \(0.670042\pi\)
\(192\) − 192.000i − 0.0721688i
\(193\) 2302.00i 0.858557i 0.903172 + 0.429279i \(0.141232\pi\)
−0.903172 + 0.429279i \(0.858768\pi\)
\(194\) 2308.00 0.854148
\(195\) 0 0
\(196\) −348.000 −0.126822
\(197\) 4374.00i 1.58190i 0.611880 + 0.790951i \(0.290414\pi\)
−0.611880 + 0.790951i \(0.709586\pi\)
\(198\) 216.000i 0.0775275i
\(199\) 1600.00 0.569955 0.284977 0.958534i \(-0.408014\pi\)
0.284977 + 0.958534i \(0.408014\pi\)
\(200\) 0 0
\(201\) −2652.00 −0.930635
\(202\) 1236.00i 0.430518i
\(203\) 480.000i 0.165958i
\(204\) −1512.00 −0.518927
\(205\) 0 0
\(206\) −256.000 −0.0865843
\(207\) 1512.00i 0.507687i
\(208\) − 608.000i − 0.202679i
\(209\) −240.000 −0.0794313
\(210\) 0 0
\(211\) 3332.00 1.08713 0.543565 0.839367i \(-0.317074\pi\)
0.543565 + 0.839367i \(0.317074\pi\)
\(212\) 792.000i 0.256579i
\(213\) 2376.00i 0.764323i
\(214\) −2952.00 −0.942965
\(215\) 0 0
\(216\) 216.000 0.0680414
\(217\) 1408.00i 0.440467i
\(218\) 2380.00i 0.739422i
\(219\) 654.000 0.201796
\(220\) 0 0
\(221\) −4788.00 −1.45736
\(222\) 1524.00i 0.460740i
\(223\) − 2648.00i − 0.795171i −0.917565 0.397586i \(-0.869848\pi\)
0.917565 0.397586i \(-0.130152\pi\)
\(224\) −512.000 −0.152721
\(225\) 0 0
\(226\) 924.000 0.271963
\(227\) 2244.00i 0.656121i 0.944657 + 0.328061i \(0.106395\pi\)
−0.944657 + 0.328061i \(0.893605\pi\)
\(228\) 240.000i 0.0697122i
\(229\) 5650.00 1.63040 0.815202 0.579177i \(-0.196626\pi\)
0.815202 + 0.579177i \(0.196626\pi\)
\(230\) 0 0
\(231\) 576.000 0.164061
\(232\) − 240.000i − 0.0679171i
\(233\) − 4698.00i − 1.32093i −0.750858 0.660464i \(-0.770360\pi\)
0.750858 0.660464i \(-0.229640\pi\)
\(234\) 684.000 0.191088
\(235\) 0 0
\(236\) −2640.00 −0.728175
\(237\) 1560.00i 0.427565i
\(238\) 4032.00i 1.09813i
\(239\) 1200.00 0.324776 0.162388 0.986727i \(-0.448080\pi\)
0.162388 + 0.986727i \(0.448080\pi\)
\(240\) 0 0
\(241\) −718.000 −0.191911 −0.0959553 0.995386i \(-0.530591\pi\)
−0.0959553 + 0.995386i \(0.530591\pi\)
\(242\) 2374.00i 0.630605i
\(243\) 243.000i 0.0641500i
\(244\) 2152.00 0.564622
\(245\) 0 0
\(246\) 252.000 0.0653127
\(247\) 760.000i 0.195780i
\(248\) − 704.000i − 0.180258i
\(249\) −1476.00 −0.375653
\(250\) 0 0
\(251\) 6012.00 1.51185 0.755924 0.654659i \(-0.227188\pi\)
0.755924 + 0.654659i \(0.227188\pi\)
\(252\) − 576.000i − 0.143986i
\(253\) − 2016.00i − 0.500968i
\(254\) −5072.00 −1.25294
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 2046.00i − 0.496599i −0.968683 0.248300i \(-0.920128\pi\)
0.968683 0.248300i \(-0.0798717\pi\)
\(258\) 312.000i 0.0752879i
\(259\) 4064.00 0.974999
\(260\) 0 0
\(261\) 270.000 0.0640329
\(262\) − 4584.00i − 1.08092i
\(263\) 6072.00i 1.42363i 0.702365 + 0.711817i \(0.252127\pi\)
−0.702365 + 0.711817i \(0.747873\pi\)
\(264\) −288.000 −0.0671408
\(265\) 0 0
\(266\) 640.000 0.147522
\(267\) − 2430.00i − 0.556980i
\(268\) − 3536.00i − 0.805954i
\(269\) 6930.00 1.57074 0.785371 0.619025i \(-0.212472\pi\)
0.785371 + 0.619025i \(0.212472\pi\)
\(270\) 0 0
\(271\) 1352.00 0.303056 0.151528 0.988453i \(-0.451581\pi\)
0.151528 + 0.988453i \(0.451581\pi\)
\(272\) − 2016.00i − 0.449404i
\(273\) − 1824.00i − 0.404372i
\(274\) −1452.00 −0.320141
\(275\) 0 0
\(276\) −2016.00 −0.439670
\(277\) − 1186.00i − 0.257256i −0.991693 0.128628i \(-0.958943\pi\)
0.991693 0.128628i \(-0.0410573\pi\)
\(278\) 760.000i 0.163963i
\(279\) 792.000 0.169949
\(280\) 0 0
\(281\) 2442.00 0.518425 0.259213 0.965820i \(-0.416537\pi\)
0.259213 + 0.965820i \(0.416537\pi\)
\(282\) − 576.000i − 0.121632i
\(283\) − 2828.00i − 0.594018i −0.954875 0.297009i \(-0.904011\pi\)
0.954875 0.297009i \(-0.0959892\pi\)
\(284\) −3168.00 −0.661923
\(285\) 0 0
\(286\) −912.000 −0.188558
\(287\) − 672.000i − 0.138212i
\(288\) 288.000i 0.0589256i
\(289\) −10963.0 −2.23143
\(290\) 0 0
\(291\) −3462.00 −0.697409
\(292\) 872.000i 0.174760i
\(293\) − 4758.00i − 0.948687i −0.880340 0.474344i \(-0.842685\pi\)
0.880340 0.474344i \(-0.157315\pi\)
\(294\) 522.000 0.103550
\(295\) 0 0
\(296\) −2032.00 −0.399012
\(297\) − 324.000i − 0.0633010i
\(298\) 3180.00i 0.618163i
\(299\) −6384.00 −1.23477
\(300\) 0 0
\(301\) 832.000 0.159321
\(302\) − 4864.00i − 0.926794i
\(303\) − 1854.00i − 0.351517i
\(304\) −320.000 −0.0603726
\(305\) 0 0
\(306\) 2268.00 0.423702
\(307\) − 8476.00i − 1.57574i −0.615844 0.787868i \(-0.711185\pi\)
0.615844 0.787868i \(-0.288815\pi\)
\(308\) 768.000i 0.142081i
\(309\) 384.000 0.0706958
\(310\) 0 0
\(311\) 4632.00 0.844555 0.422278 0.906467i \(-0.361231\pi\)
0.422278 + 0.906467i \(0.361231\pi\)
\(312\) 912.000i 0.165487i
\(313\) 4822.00i 0.870785i 0.900241 + 0.435392i \(0.143390\pi\)
−0.900241 + 0.435392i \(0.856610\pi\)
\(314\) 1228.00 0.220701
\(315\) 0 0
\(316\) −2080.00 −0.370282
\(317\) − 3426.00i − 0.607014i −0.952829 0.303507i \(-0.901842\pi\)
0.952829 0.303507i \(-0.0981575\pi\)
\(318\) − 1188.00i − 0.209496i
\(319\) −360.000 −0.0631854
\(320\) 0 0
\(321\) 4428.00 0.769928
\(322\) 5376.00i 0.930412i
\(323\) 2520.00i 0.434107i
\(324\) −324.000 −0.0555556
\(325\) 0 0
\(326\) 3704.00 0.629281
\(327\) − 3570.00i − 0.603735i
\(328\) 336.000i 0.0565625i
\(329\) −1536.00 −0.257393
\(330\) 0 0
\(331\) −2788.00 −0.462968 −0.231484 0.972839i \(-0.574358\pi\)
−0.231484 + 0.972839i \(0.574358\pi\)
\(332\) − 1968.00i − 0.325325i
\(333\) − 2286.00i − 0.376192i
\(334\) −4272.00 −0.699861
\(335\) 0 0
\(336\) 768.000 0.124696
\(337\) 434.000i 0.0701528i 0.999385 + 0.0350764i \(0.0111675\pi\)
−0.999385 + 0.0350764i \(0.988833\pi\)
\(338\) − 1506.00i − 0.242354i
\(339\) −1386.00 −0.222057
\(340\) 0 0
\(341\) −1056.00 −0.167700
\(342\) − 360.000i − 0.0569198i
\(343\) − 6880.00i − 1.08305i
\(344\) −416.000 −0.0652012
\(345\) 0 0
\(346\) −3516.00 −0.546304
\(347\) 6684.00i 1.03405i 0.855970 + 0.517026i \(0.172961\pi\)
−0.855970 + 0.517026i \(0.827039\pi\)
\(348\) 360.000i 0.0554541i
\(349\) −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
\(352\) − 384.000i − 0.0581456i
\(353\) 7422.00i 1.11907i 0.828805 + 0.559537i \(0.189021\pi\)
−0.828805 + 0.559537i \(0.810979\pi\)
\(354\) 3960.00 0.594553
\(355\) 0 0
\(356\) 3240.00 0.482359
\(357\) − 6048.00i − 0.896622i
\(358\) − 1080.00i − 0.159441i
\(359\) 10440.0 1.53482 0.767412 0.641154i \(-0.221544\pi\)
0.767412 + 0.641154i \(0.221544\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) − 3964.00i − 0.575534i
\(363\) − 3561.00i − 0.514887i
\(364\) 2432.00 0.350196
\(365\) 0 0
\(366\) −3228.00 −0.461012
\(367\) 10424.0i 1.48264i 0.671153 + 0.741319i \(0.265800\pi\)
−0.671153 + 0.741319i \(0.734200\pi\)
\(368\) − 2688.00i − 0.380765i
\(369\) −378.000 −0.0533276
\(370\) 0 0
\(371\) −3168.00 −0.443327
\(372\) 1056.00i 0.147180i
\(373\) − 3278.00i − 0.455036i −0.973774 0.227518i \(-0.926939\pi\)
0.973774 0.227518i \(-0.0730610\pi\)
\(374\) −3024.00 −0.418094
\(375\) 0 0
\(376\) 768.000 0.105337
\(377\) 1140.00i 0.155737i
\(378\) 864.000i 0.117564i
\(379\) −6140.00 −0.832165 −0.416083 0.909327i \(-0.636597\pi\)
−0.416083 + 0.909327i \(0.636597\pi\)
\(380\) 0 0
\(381\) 7608.00 1.02302
\(382\) 5376.00i 0.720053i
\(383\) 3072.00i 0.409848i 0.978778 + 0.204924i \(0.0656948\pi\)
−0.978778 + 0.204924i \(0.934305\pi\)
\(384\) −384.000 −0.0510310
\(385\) 0 0
\(386\) 4604.00 0.607092
\(387\) − 468.000i − 0.0614723i
\(388\) − 4616.00i − 0.603974i
\(389\) −6150.00 −0.801587 −0.400794 0.916168i \(-0.631266\pi\)
−0.400794 + 0.916168i \(0.631266\pi\)
\(390\) 0 0
\(391\) −21168.0 −2.73788
\(392\) 696.000i 0.0896768i
\(393\) 6876.00i 0.882566i
\(394\) 8748.00 1.11857
\(395\) 0 0
\(396\) 432.000 0.0548202
\(397\) − 106.000i − 0.0134005i −0.999978 0.00670024i \(-0.997867\pi\)
0.999978 0.00670024i \(-0.00213277\pi\)
\(398\) − 3200.00i − 0.403019i
\(399\) −960.000 −0.120451
\(400\) 0 0
\(401\) −1758.00 −0.218929 −0.109464 0.993991i \(-0.534914\pi\)
−0.109464 + 0.993991i \(0.534914\pi\)
\(402\) 5304.00i 0.658058i
\(403\) 3344.00i 0.413341i
\(404\) 2472.00 0.304422
\(405\) 0 0
\(406\) 960.000 0.117350
\(407\) 3048.00i 0.371213i
\(408\) 3024.00i 0.366937i
\(409\) 3670.00 0.443691 0.221846 0.975082i \(-0.428792\pi\)
0.221846 + 0.975082i \(0.428792\pi\)
\(410\) 0 0
\(411\) 2178.00 0.261394
\(412\) 512.000i 0.0612243i
\(413\) − 10560.0i − 1.25817i
\(414\) 3024.00 0.358989
\(415\) 0 0
\(416\) −1216.00 −0.143316
\(417\) − 1140.00i − 0.133875i
\(418\) 480.000i 0.0561664i
\(419\) 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 \(0.337069\pi\)
\(422\) − 6664.00i − 0.768717i
\(423\) 864.000i 0.0993123i
\(424\) 1584.00 0.181429
\(425\) 0 0
\(426\) 4752.00 0.540458
\(427\) 8608.00i 0.975575i
\(428\) 5904.00i 0.666777i
\(429\) 1368.00 0.153957
\(430\) 0 0
\(431\) 9792.00 1.09435 0.547174 0.837019i \(-0.315704\pi\)
0.547174 + 0.837019i \(0.315704\pi\)
\(432\) − 432.000i − 0.0481125i
\(433\) 7342.00i 0.814859i 0.913237 + 0.407430i \(0.133575\pi\)
−0.913237 + 0.407430i \(0.866425\pi\)
\(434\) 2816.00 0.311457
\(435\) 0 0
\(436\) 4760.00 0.522850
\(437\) 3360.00i 0.367805i
\(438\) − 1308.00i − 0.142691i
\(439\) −10640.0 −1.15676 −0.578382 0.815766i \(-0.696316\pi\)
−0.578382 + 0.815766i \(0.696316\pi\)
\(440\) 0 0
\(441\) −783.000 −0.0845481
\(442\) 9576.00i 1.03051i
\(443\) 17412.0i 1.86742i 0.358024 + 0.933712i \(0.383451\pi\)
−0.358024 + 0.933712i \(0.616549\pi\)
\(444\) 3048.00 0.325792
\(445\) 0 0
\(446\) −5296.00 −0.562271
\(447\) − 4770.00i − 0.504728i
\(448\) 1024.00i 0.107990i
\(449\) 1710.00 0.179732 0.0898662 0.995954i \(-0.471356\pi\)
0.0898662 + 0.995954i \(0.471356\pi\)
\(450\) 0 0
\(451\) 504.000 0.0526218
\(452\) − 1848.00i − 0.192307i
\(453\) 7296.00i 0.756724i
\(454\) 4488.00 0.463948
\(455\) 0 0
\(456\) 480.000 0.0492940
\(457\) − 646.000i − 0.0661239i −0.999453 0.0330619i \(-0.989474\pi\)
0.999453 0.0330619i \(-0.0105259\pi\)
\(458\) − 11300.0i − 1.15287i
\(459\) −3402.00 −0.345952
\(460\) 0 0
\(461\) −6018.00 −0.607996 −0.303998 0.952673i \(-0.598322\pi\)
−0.303998 + 0.952673i \(0.598322\pi\)
\(462\) − 1152.00i − 0.116008i
\(463\) 6712.00i 0.673722i 0.941554 + 0.336861i \(0.109365\pi\)
−0.941554 + 0.336861i \(0.890635\pi\)
\(464\) −480.000 −0.0480247
\(465\) 0 0
\(466\) −9396.00 −0.934037
\(467\) 5364.00i 0.531512i 0.964040 + 0.265756i \(0.0856216\pi\)
−0.964040 + 0.265756i \(0.914378\pi\)
\(468\) − 1368.00i − 0.135119i
\(469\) 14144.0 1.39256
\(470\) 0 0
\(471\) −1842.00 −0.180201
\(472\) 5280.00i 0.514898i
\(473\) 624.000i 0.0606587i
\(474\) 3120.00 0.302334
\(475\) 0 0
\(476\) 8064.00 0.776498
\(477\) 1782.00i 0.171053i
\(478\) − 2400.00i − 0.229652i
\(479\) −9840.00 −0.938624 −0.469312 0.883032i \(-0.655498\pi\)
−0.469312 + 0.883032i \(0.655498\pi\)
\(480\) 0 0
\(481\) 9652.00 0.914955
\(482\) 1436.00i 0.135701i
\(483\) − 8064.00i − 0.759678i
\(484\) 4748.00 0.445905
\(485\) 0 0
\(486\) 486.000 0.0453609
\(487\) 1424.00i 0.132500i 0.997803 + 0.0662501i \(0.0211035\pi\)
−0.997803 + 0.0662501i \(0.978896\pi\)
\(488\) − 4304.00i − 0.399248i
\(489\) −5556.00 −0.513806
\(490\) 0 0
\(491\) −4548.00 −0.418021 −0.209011 0.977913i \(-0.567024\pi\)
−0.209011 + 0.977913i \(0.567024\pi\)
\(492\) − 504.000i − 0.0461831i
\(493\) 3780.00i 0.345320i
\(494\) 1520.00 0.138437
\(495\) 0 0
\(496\) −1408.00 −0.127462
\(497\) − 12672.0i − 1.14370i
\(498\) 2952.00i 0.265627i
\(499\) −6500.00 −0.583126 −0.291563 0.956552i \(-0.594175\pi\)
−0.291563 + 0.956552i \(0.594175\pi\)
\(500\) 0 0
\(501\) 6408.00 0.571434
\(502\) − 12024.0i − 1.06904i
\(503\) − 12168.0i − 1.07862i −0.842108 0.539308i \(-0.818686\pi\)
0.842108 0.539308i \(-0.181314\pi\)
\(504\) −1152.00 −0.101814
\(505\) 0 0
\(506\) −4032.00 −0.354238
\(507\) 2259.00i 0.197881i
\(508\) 10144.0i 0.885959i
\(509\) 21090.0 1.83654 0.918269 0.395957i \(-0.129587\pi\)
0.918269 + 0.395957i \(0.129587\pi\)
\(510\) 0 0
\(511\) −3488.00 −0.301957
\(512\) − 512.000i − 0.0441942i
\(513\) 540.000i 0.0464748i
\(514\) −4092.00 −0.351149
\(515\) 0 0
\(516\) 624.000 0.0532366
\(517\) − 1152.00i − 0.0979979i
\(518\) − 8128.00i − 0.689428i
\(519\) 5274.00 0.446056
\(520\) 0 0
\(521\) −5238.00 −0.440462 −0.220231 0.975448i \(-0.570681\pi\)
−0.220231 + 0.975448i \(0.570681\pi\)
\(522\) − 540.000i − 0.0452781i
\(523\) − 8588.00i − 0.718025i −0.933333 0.359012i \(-0.883114\pi\)
0.933333 0.359012i \(-0.116886\pi\)
\(524\) −9168.00 −0.764324
\(525\) 0 0
\(526\) 12144.0 1.00666
\(527\) 11088.0i 0.916510i
\(528\) 576.000i 0.0474757i
\(529\) −16057.0 −1.31972
\(530\) 0 0
\(531\) −5940.00 −0.485450
\(532\) − 1280.00i − 0.104314i
\(533\) − 1596.00i − 0.129701i
\(534\) −4860.00 −0.393844
\(535\) 0 0
\(536\) −7072.00 −0.569895
\(537\) 1620.00i 0.130183i
\(538\) − 13860.0i − 1.11068i
\(539\) 1044.00 0.0834291
\(540\) 0 0
\(541\) 3062.00 0.243338 0.121669 0.992571i \(-0.461175\pi\)
0.121669 + 0.992571i \(0.461175\pi\)
\(542\) − 2704.00i − 0.214293i
\(543\) 5946.00i 0.469921i
\(544\) −4032.00 −0.317777
\(545\) 0 0
\(546\) −3648.00 −0.285934
\(547\) − 8476.00i − 0.662537i −0.943537 0.331268i \(-0.892523\pi\)
0.943537 0.331268i \(-0.107477\pi\)
\(548\) 2904.00i 0.226374i
\(549\) 4842.00 0.376414
\(550\) 0 0
\(551\) 600.000 0.0463899
\(552\) 4032.00i 0.310894i
\(553\) − 8320.00i − 0.639787i
\(554\) −2372.00 −0.181907
\(555\) 0 0
\(556\) 1520.00 0.115939
\(557\) − 12546.0i − 0.954383i −0.878799 0.477191i \(-0.841655\pi\)
0.878799 0.477191i \(-0.158345\pi\)
\(558\) − 1584.00i − 0.120172i
\(559\) 1976.00 0.149510
\(560\) 0 0
\(561\) 4536.00 0.341373
\(562\) − 4884.00i − 0.366582i
\(563\) 12.0000i 0 0.000898294i 1.00000 0.000449147i \(0.000142968\pi\)
−1.00000 0.000449147i \(0.999857\pi\)
\(564\) −1152.00 −0.0860070
\(565\) 0 0
\(566\) −5656.00 −0.420034
\(567\) − 1296.00i − 0.0959910i
\(568\) 6336.00i 0.468050i
\(569\) −19290.0 −1.42123 −0.710614 0.703582i \(-0.751583\pi\)
−0.710614 + 0.703582i \(0.751583\pi\)
\(570\) 0 0
\(571\) −12148.0 −0.890329 −0.445165 0.895449i \(-0.646855\pi\)
−0.445165 + 0.895449i \(0.646855\pi\)
\(572\) 1824.00i 0.133331i
\(573\) − 8064.00i − 0.587920i
\(574\) −1344.00 −0.0977308
\(575\) 0 0
\(576\) 576.000 0.0416667
\(577\) − 10366.0i − 0.747907i −0.927447 0.373953i \(-0.878002\pi\)
0.927447 0.373953i \(-0.121998\pi\)
\(578\) 21926.0i 1.57786i
\(579\) −6906.00 −0.495688
\(580\) 0 0
\(581\) 7872.00 0.562109
\(582\) 6924.00i 0.493143i
\(583\) − 2376.00i − 0.168789i
\(584\) 1744.00 0.123574
\(585\) 0 0
\(586\) −9516.00 −0.670823
\(587\) 7644.00i 0.537482i 0.963213 + 0.268741i \(0.0866075\pi\)
−0.963213 + 0.268741i \(0.913393\pi\)
\(588\) − 1044.00i − 0.0732208i
\(589\) 1760.00 0.123123
\(590\) 0 0
\(591\) −13122.0 −0.913311
\(592\) 4064.00i 0.282144i
\(593\) − 8658.00i − 0.599564i −0.954008 0.299782i \(-0.903086\pi\)
0.954008 0.299782i \(-0.0969139\pi\)
\(594\) −648.000 −0.0447605
\(595\) 0 0
\(596\) 6360.00 0.437107
\(597\) 4800.00i 0.329064i
\(598\) 12768.0i 0.873114i
\(599\) −25800.0 −1.75987 −0.879933 0.475098i \(-0.842413\pi\)
−0.879933 + 0.475098i \(0.842413\pi\)
\(600\) 0 0
\(601\) 16202.0 1.09966 0.549828 0.835278i \(-0.314693\pi\)
0.549828 + 0.835278i \(0.314693\pi\)
\(602\) − 1664.00i − 0.112657i
\(603\) − 7956.00i − 0.537302i
\(604\) −9728.00 −0.655342
\(605\) 0 0
\(606\) −3708.00 −0.248560
\(607\) − 24136.0i − 1.61392i −0.590605 0.806960i \(-0.701111\pi\)
0.590605 0.806960i \(-0.298889\pi\)
\(608\) 640.000i 0.0426898i
\(609\) −1440.00 −0.0958157
\(610\) 0 0
\(611\) −3648.00 −0.241542
\(612\) − 4536.00i − 0.299603i
\(613\) 4642.00i 0.305854i 0.988237 + 0.152927i \(0.0488700\pi\)
−0.988237 + 0.152927i \(0.951130\pi\)
\(614\) −16952.0 −1.11421
\(615\) 0 0
\(616\) 1536.00 0.100466
\(617\) − 6726.00i − 0.438863i −0.975628 0.219432i \(-0.929580\pi\)
0.975628 0.219432i \(-0.0704203\pi\)
\(618\) − 768.000i − 0.0499895i
\(619\) 21220.0 1.37787 0.688937 0.724821i \(-0.258078\pi\)
0.688937 + 0.724821i \(0.258078\pi\)
\(620\) 0 0
\(621\) −4536.00 −0.293113
\(622\) − 9264.00i − 0.597191i
\(623\) 12960.0i 0.833437i
\(624\) 1824.00 0.117017
\(625\) 0 0
\(626\) 9644.00 0.615738
\(627\) − 720.000i − 0.0458597i
\(628\) − 2456.00i − 0.156059i
\(629\) 32004.0 2.02875
\(630\) 0 0
\(631\) 29792.0 1.87956 0.939779 0.341783i \(-0.111031\pi\)
0.939779 + 0.341783i \(0.111031\pi\)
\(632\) 4160.00i 0.261829i
\(633\) 9996.00i 0.627655i
\(634\) −6852.00 −0.429223
\(635\) 0 0
\(636\) −2376.00 −0.148136
\(637\) − 3306.00i − 0.205633i
\(638\) 720.000i 0.0446788i
\(639\) −7128.00 −0.441282
\(640\) 0 0
\(641\) −10158.0 −0.625923 −0.312962 0.949766i \(-0.601321\pi\)
−0.312962 + 0.949766i \(0.601321\pi\)
\(642\) − 8856.00i − 0.544421i
\(643\) − 29828.0i − 1.82940i −0.404138 0.914698i \(-0.632429\pi\)
0.404138 0.914698i \(-0.367571\pi\)
\(644\) 10752.0 0.657901
\(645\) 0 0
\(646\) 5040.00 0.306960
\(647\) 1944.00i 0.118124i 0.998254 + 0.0590622i \(0.0188110\pi\)
−0.998254 + 0.0590622i \(0.981189\pi\)
\(648\) 648.000i 0.0392837i
\(649\) 7920.00 0.479025
\(650\) 0 0
\(651\) −4224.00 −0.254304
\(652\) − 7408.00i − 0.444969i
\(653\) − 26718.0i − 1.60116i −0.599227 0.800579i \(-0.704525\pi\)
0.599227 0.800579i \(-0.295475\pi\)
\(654\) −7140.00 −0.426905
\(655\) 0 0
\(656\) 672.000 0.0399957
\(657\) 1962.00i 0.116507i
\(658\) 3072.00i 0.182005i
\(659\) −4260.00 −0.251815 −0.125907 0.992042i \(-0.540184\pi\)
−0.125907 + 0.992042i \(0.540184\pi\)
\(660\) 0 0
\(661\) 22862.0 1.34528 0.672639 0.739971i \(-0.265161\pi\)
0.672639 + 0.739971i \(0.265161\pi\)
\(662\) 5576.00i 0.327368i
\(663\) − 14364.0i − 0.841405i
\(664\) −3936.00 −0.230040
\(665\) 0 0
\(666\) −4572.00 −0.266008
\(667\) 5040.00i 0.292578i
\(668\) 8544.00i 0.494876i
\(669\) 7944.00 0.459092
\(670\) 0 0
\(671\) −6456.00 −0.371432
\(672\) − 1536.00i − 0.0881733i
\(673\) 32542.0i 1.86390i 0.362592 + 0.931948i \(0.381892\pi\)
−0.362592 + 0.931948i \(0.618108\pi\)
\(674\) 868.000 0.0496055
\(675\) 0 0
\(676\) −3012.00 −0.171370
\(677\) 14214.0i 0.806925i 0.914996 + 0.403463i \(0.132193\pi\)
−0.914996 + 0.403463i \(0.867807\pi\)
\(678\) 2772.00i 0.157018i
\(679\) 18464.0 1.04357
\(680\) 0 0
\(681\) −6732.00 −0.378812
\(682\) 2112.00i 0.118582i
\(683\) 7092.00i 0.397317i 0.980069 + 0.198659i \(0.0636585\pi\)
−0.980069 + 0.198659i \(0.936341\pi\)
\(684\) −720.000 −0.0402484
\(685\) 0 0
\(686\) −13760.0 −0.765830
\(687\) 16950.0i 0.941314i
\(688\) 832.000i 0.0461042i
\(689\) −7524.00 −0.416026
\(690\) 0 0
\(691\) −13228.0 −0.728244 −0.364122 0.931351i \(-0.618631\pi\)
−0.364122 + 0.931351i \(0.618631\pi\)
\(692\) 7032.00i 0.386296i
\(693\) 1728.00i 0.0947205i
\(694\) 13368.0 0.731185
\(695\) 0 0
\(696\) 720.000 0.0392120
\(697\) − 5292.00i − 0.287588i
\(698\) 5260.00i 0.285235i
\(699\) 14094.0 0.762638
\(700\) 0 0
\(701\) 28062.0 1.51196 0.755982 0.654592i \(-0.227160\pi\)
0.755982 + 0.654592i \(0.227160\pi\)
\(702\) 2052.00i 0.110324i
\(703\) − 5080.00i − 0.272540i
\(704\) −768.000 −0.0411152
\(705\) 0 0
\(706\) 14844.0 0.791305
\(707\) 9888.00i 0.525992i
\(708\) − 7920.00i − 0.420412i
\(709\) 27250.0 1.44343 0.721717 0.692188i \(-0.243353\pi\)
0.721717 + 0.692188i \(0.243353\pi\)
\(710\) 0 0
\(711\) −4680.00 −0.246855
\(712\) − 6480.00i − 0.341079i
\(713\) 14784.0i 0.776529i
\(714\) −12096.0 −0.634008
\(715\) 0 0
\(716\) −2160.00 −0.112742
\(717\) 3600.00i 0.187510i
\(718\) − 20880.0i − 1.08529i
\(719\) 14400.0 0.746912 0.373456 0.927648i \(-0.378173\pi\)
0.373456 + 0.927648i \(0.378173\pi\)
\(720\) 0 0
\(721\) −2048.00 −0.105786
\(722\) 12918.0i 0.665870i
\(723\) − 2154.00i − 0.110800i
\(724\) −7928.00 −0.406964
\(725\) 0 0
\(726\) −7122.00 −0.364080
\(727\) 17984.0i 0.917455i 0.888577 + 0.458727i \(0.151695\pi\)
−0.888577 + 0.458727i \(0.848305\pi\)
\(728\) − 4864.00i − 0.247626i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 6552.00 0.331511
\(732\) 6456.00i 0.325984i
\(733\) − 16598.0i − 0.836373i −0.908361 0.418186i \(-0.862666\pi\)
0.908361 0.418186i \(-0.137334\pi\)
\(734\) 20848.0 1.04838
\(735\) 0 0
\(736\) −5376.00 −0.269242
\(737\) 10608.0i 0.530191i
\(738\) 756.000i 0.0377083i
\(739\) −1460.00 −0.0726752 −0.0363376 0.999340i \(-0.511569\pi\)
−0.0363376 + 0.999340i \(0.511569\pi\)
\(740\) 0 0
\(741\) −2280.00 −0.113034
\(742\) 6336.00i 0.313480i
\(743\) 30072.0i 1.48484i 0.669936 + 0.742419i \(0.266322\pi\)
−0.669936 + 0.742419i \(0.733678\pi\)
\(744\) 2112.00 0.104072
\(745\) 0 0
\(746\) −6556.00 −0.321759
\(747\) − 4428.00i − 0.216884i
\(748\) 6048.00i 0.295637i
\(749\) −23616.0 −1.15208
\(750\) 0 0
\(751\) −18088.0 −0.878882 −0.439441 0.898271i \(-0.644823\pi\)
−0.439441 + 0.898271i \(0.644823\pi\)
\(752\) − 1536.00i − 0.0744843i
\(753\) 18036.0i 0.872866i
\(754\) 2280.00 0.110123
\(755\) 0 0
\(756\) 1728.00 0.0831306
\(757\) 24734.0i 1.18755i 0.804633 + 0.593773i \(0.202362\pi\)
−0.804633 + 0.593773i \(0.797638\pi\)
\(758\) 12280.0i 0.588430i
\(759\) 6048.00 0.289234
\(760\) 0 0
\(761\) −22278.0 −1.06120 −0.530602 0.847621i \(-0.678034\pi\)
−0.530602 + 0.847621i \(0.678034\pi\)
\(762\) − 15216.0i − 0.723383i
\(763\) 19040.0i 0.903400i
\(764\) 10752.0 0.509154
\(765\) 0 0
\(766\) 6144.00 0.289806
\(767\) − 25080.0i − 1.18069i
\(768\) 768.000i 0.0360844i
\(769\) −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
\(772\) − 9208.00i − 0.429279i
\(773\) − 29718.0i − 1.38277i −0.722486 0.691386i \(-0.757001\pi\)
0.722486 0.691386i \(-0.242999\pi\)
\(774\) −936.000 −0.0434675
\(775\) 0 0
\(776\) −9232.00 −0.427074
\(777\) 12192.0i 0.562916i
\(778\) 12300.0i 0.566808i
\(779\) −840.000 −0.0386343
\(780\) 0 0
\(781\) 9504.00 0.435442
\(782\) 42336.0i 1.93597i
\(783\) 810.000i 0.0369694i
\(784\) 1392.00 0.0634111
\(785\) 0 0
\(786\) 13752.0 0.624068
\(787\) 9524.00i 0.431377i 0.976462 + 0.215689i \(0.0691996\pi\)
−0.976462 + 0.215689i \(0.930800\pi\)
\(788\) − 17496.0i − 0.790951i
\(789\) −18216.0 −0.821935
\(790\) 0 0
\(791\) 7392.00 0.332275
\(792\) − 864.000i − 0.0387638i
\(793\) 20444.0i 0.915495i
\(794\) −212.000 −0.00947556
\(795\) 0 0
\(796\) −6400.00 −0.284977
\(797\) − 33906.0i − 1.50692i −0.657496 0.753458i \(-0.728384\pi\)
0.657496 0.753458i \(-0.271616\pi\)
\(798\) 1920.00i 0.0851720i
\(799\) −12096.0 −0.535577
\(800\) 0 0
\(801\) 7290.00 0.321572
\(802\) 3516.00i 0.154806i
\(803\) − 2616.00i − 0.114965i
\(804\) 10608.0 0.465318
\(805\) 0 0
\(806\) 6688.00 0.292276
\(807\) 20790.0i 0.906868i
\(808\) − 4944.00i − 0.215259i
\(809\) 630.000 0.0273790 0.0136895 0.999906i \(-0.495642\pi\)
0.0136895 + 0.999906i \(0.495642\pi\)
\(810\) 0 0
\(811\) −20788.0 −0.900081 −0.450040 0.893008i \(-0.648590\pi\)
−0.450040 + 0.893008i \(0.648590\pi\)
\(812\) − 1920.00i − 0.0829788i
\(813\) 4056.00i 0.174969i
\(814\) 6096.00 0.262487
\(815\) 0 0
\(816\) 6048.00 0.259464
\(817\) − 1040.00i − 0.0445349i
\(818\) − 7340.00i − 0.313737i
\(819\) 5472.00 0.233464
\(820\) 0 0
\(821\) −43098.0 −1.83207 −0.916036 0.401097i \(-0.868629\pi\)
−0.916036 + 0.401097i \(0.868629\pi\)
\(822\) − 4356.00i − 0.184833i
\(823\) 14272.0i 0.604484i 0.953231 + 0.302242i \(0.0977351\pi\)
−0.953231 + 0.302242i \(0.902265\pi\)
\(824\) 1024.00 0.0432921
\(825\) 0 0
\(826\) −21120.0 −0.889660
\(827\) 13644.0i 0.573698i 0.957976 + 0.286849i \(0.0926078\pi\)
−0.957976 + 0.286849i \(0.907392\pi\)
\(828\) − 6048.00i − 0.253844i
\(829\) 2410.00 0.100968 0.0504842 0.998725i \(-0.483924\pi\)
0.0504842 + 0.998725i \(0.483924\pi\)
\(830\) 0 0
\(831\) 3558.00 0.148527
\(832\) 2432.00i 0.101339i
\(833\) − 10962.0i − 0.455955i
\(834\) −2280.00 −0.0946642
\(835\) 0 0
\(836\) 960.000 0.0397157
\(837\) 2376.00i 0.0981202i
\(838\) − 19320.0i − 0.796418i
\(839\) −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
\(842\) − 16924.0i − 0.692684i
\(843\) 7326.00i 0.299313i
\(844\) −13328.0 −0.543565
\(845\) 0 0
\(846\) 1728.00 0.0702244
\(847\) 18992.0i 0.770452i
\(848\) − 3168.00i − 0.128290i
\(849\) 8484.00 0.342957
\(850\) 0 0
\(851\) 42672.0 1.71889
\(852\) − 9504.00i − 0.382162i
\(853\) − 32078.0i − 1.28761i −0.765190 0.643804i \(-0.777355\pi\)
0.765190 0.643804i \(-0.222645\pi\)
\(854\) 17216.0 0.689835
\(855\) 0 0
\(856\) 11808.0 0.471483
\(857\) − 14406.0i − 0.574212i −0.957899 0.287106i \(-0.907307\pi\)
0.957899 0.287106i \(-0.0926932\pi\)
\(858\) − 2736.00i − 0.108864i
\(859\) −30620.0 −1.21623 −0.608115 0.793849i \(-0.708074\pi\)
−0.608115 + 0.793849i \(0.708074\pi\)
\(860\) 0 0
\(861\) 2016.00 0.0797969
\(862\) − 19584.0i − 0.773821i
\(863\) − 17568.0i − 0.692957i −0.938058 0.346478i \(-0.887377\pi\)
0.938058 0.346478i \(-0.112623\pi\)
\(864\) −864.000 −0.0340207
\(865\) 0 0
\(866\) 14684.0 0.576192
\(867\) − 32889.0i − 1.28831i
\(868\) − 5632.00i − 0.220233i
\(869\) 6240.00 0.243587
\(870\) 0 0
\(871\) 33592.0 1.30680
\(872\) − 9520.00i − 0.369711i
\(873\) − 10386.0i − 0.402649i
\(874\) 6720.00 0.260077
\(875\) 0 0
\(876\) −2616.00 −0.100898
\(877\) − 21706.0i − 0.835758i −0.908503 0.417879i \(-0.862774\pi\)
0.908503 0.417879i \(-0.137226\pi\)
\(878\) 21280.0i 0.817956i
\(879\) 14274.0 0.547725
\(880\) 0 0
\(881\) −14958.0 −0.572018 −0.286009 0.958227i \(-0.592329\pi\)
−0.286009 + 0.958227i \(0.592329\pi\)
\(882\) 1566.00i 0.0597845i
\(883\) 32812.0i 1.25052i 0.780415 + 0.625261i \(0.215008\pi\)
−0.780415 + 0.625261i \(0.784992\pi\)
\(884\) 19152.0 0.728678
\(885\) 0 0
\(886\) 34824.0 1.32047
\(887\) − 38856.0i − 1.47086i −0.677598 0.735432i \(-0.736979\pi\)
0.677598 0.735432i \(-0.263021\pi\)
\(888\) − 6096.00i − 0.230370i
\(889\) −40576.0 −1.53079
\(890\) 0 0
\(891\) 972.000 0.0365468
\(892\) 10592.0i 0.397586i
\(893\) 1920.00i 0.0719489i
\(894\) −9540.00 −0.356896
\(895\) 0 0
\(896\) 2048.00 0.0763604
\(897\) − 19152.0i − 0.712895i
\(898\) − 3420.00i − 0.127090i
\(899\) 2640.00 0.0979410
\(900\) 0 0
\(901\) −24948.0 −0.922462
\(902\) − 1008.00i − 0.0372092i
\(903\) 2496.00i 0.0919841i
\(904\) −3696.00 −0.135981
\(905\) 0 0
\(906\) 14592.0 0.535085
\(907\) − 28276.0i − 1.03516i −0.855635 0.517579i \(-0.826833\pi\)
0.855635 0.517579i \(-0.173167\pi\)
\(908\) − 8976.00i − 0.328061i
\(909\) 5562.00 0.202948
\(910\) 0 0
\(911\) 8112.00 0.295019 0.147510 0.989061i \(-0.452874\pi\)
0.147510 + 0.989061i \(0.452874\pi\)
\(912\) − 960.000i − 0.0348561i
\(913\) 5904.00i 0.214013i
\(914\) −1292.00 −0.0467566
\(915\) 0 0
\(916\) −22600.0 −0.815202
\(917\) − 36672.0i − 1.32063i
\(918\) 6804.00i 0.244625i
\(919\) 26080.0 0.936126 0.468063 0.883695i \(-0.344952\pi\)
0.468063 + 0.883695i \(0.344952\pi\)
\(920\) 0 0
\(921\) 25428.0 0.909751
\(922\) 12036.0i 0.429918i
\(923\) − 30096.0i − 1.07326i
\(924\) −2304.00 −0.0820303
\(925\) 0 0
\(926\) 13424.0 0.476393
\(927\) 1152.00i 0.0408162i
\(928\) 960.000i 0.0339586i
\(929\) −49170.0 −1.73651 −0.868254 0.496120i \(-0.834757\pi\)
−0.868254 + 0.496120i \(0.834757\pi\)
\(930\) 0 0
\(931\) −1740.00 −0.0612526
\(932\) 18792.0i 0.660464i
\(933\) 13896.0i 0.487604i
\(934\) 10728.0 0.375836
\(935\) 0 0
\(936\) −2736.00 −0.0955438
\(937\) 48314.0i 1.68447i 0.539110 + 0.842236i \(0.318761\pi\)
−0.539110 + 0.842236i \(0.681239\pi\)
\(938\) − 28288.0i − 0.984687i
\(939\) −14466.0 −0.502748
\(940\) 0 0
\(941\) 34782.0 1.20495 0.602477 0.798137i \(-0.294181\pi\)
0.602477 + 0.798137i \(0.294181\pi\)
\(942\) 3684.00i 0.127422i
\(943\) − 7056.00i − 0.243664i
\(944\) 10560.0 0.364088
\(945\) 0 0
\(946\) 1248.00 0.0428922
\(947\) − 25116.0i − 0.861838i −0.902391 0.430919i \(-0.858190\pi\)
0.902391 0.430919i \(-0.141810\pi\)
\(948\) − 6240.00i − 0.213782i
\(949\) −8284.00 −0.283361
\(950\) 0 0
\(951\) 10278.0 0.350460
\(952\) − 16128.0i − 0.549067i
\(953\) 15462.0i 0.525565i 0.964855 + 0.262782i \(0.0846401\pi\)
−0.964855 + 0.262782i \(0.915360\pi\)
\(954\) 3564.00 0.120953
\(955\) 0 0
\(956\) −4800.00 −0.162388
\(957\) − 1080.00i − 0.0364801i
\(958\) 19680.0i 0.663708i
\(959\) −11616.0 −0.391137
\(960\) 0 0
\(961\) −22047.0 −0.740056
\(962\) − 19304.0i − 0.646971i
\(963\) 13284.0i 0.444518i
\(964\) 2872.00 0.0959553
\(965\) 0 0
\(966\) −16128.0 −0.537174
\(967\) − 736.000i − 0.0244759i −0.999925 0.0122379i \(-0.996104\pi\)
0.999925 0.0122379i \(-0.00389555\pi\)
\(968\) − 9496.00i − 0.315303i
\(969\) −7560.00 −0.250632
\(970\) 0 0
\(971\) −29268.0 −0.967307 −0.483653 0.875260i \(-0.660690\pi\)
−0.483653 + 0.875260i \(0.660690\pi\)
\(972\) − 972.000i − 0.0320750i
\(973\) 6080.00i 0.200325i
\(974\) 2848.00 0.0936918
\(975\) 0 0
\(976\) −8608.00 −0.282311
\(977\) 16674.0i 0.546007i 0.962013 + 0.273003i \(0.0880170\pi\)
−0.962013 + 0.273003i \(0.911983\pi\)
\(978\) 11112.0i 0.363316i
\(979\) −9720.00 −0.317316
\(980\) 0 0
\(981\) 10710.0 0.348567
\(982\) 9096.00i 0.295586i
\(983\) 31272.0i 1.01467i 0.861749 + 0.507336i \(0.169370\pi\)
−0.861749 + 0.507336i \(0.830630\pi\)
\(984\) −1008.00 −0.0326564
\(985\) 0 0
\(986\) 7560.00 0.244178
\(987\) − 4608.00i − 0.148606i
\(988\) − 3040.00i − 0.0978900i
\(989\) 8736.00 0.280878
\(990\) 0 0
\(991\) −15928.0 −0.510565 −0.255282 0.966867i \(-0.582168\pi\)
−0.255282 + 0.966867i \(0.582168\pi\)
\(992\) 2816.00i 0.0901291i
\(993\) − 8364.00i − 0.267295i
\(994\) −25344.0 −0.808715
\(995\) 0 0
\(996\) 5904.00 0.187827
\(997\) 42014.0i 1.33460i 0.744789 + 0.667300i \(0.232550\pi\)
−0.744789 + 0.667300i \(0.767450\pi\)
\(998\) 13000.0i 0.412332i
\(999\) 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.c.d.49.1 2
3.2 odd 2 450.4.c.e.199.2 2
4.3 odd 2 1200.4.f.j.49.1 2
5.2 odd 4 150.4.a.i.1.1 1
5.3 odd 4 6.4.a.a.1.1 1
5.4 even 2 inner 150.4.c.d.49.2 2
15.2 even 4 450.4.a.h.1.1 1
15.8 even 4 18.4.a.a.1.1 1
15.14 odd 2 450.4.c.e.199.1 2
20.3 even 4 48.4.a.c.1.1 1
20.7 even 4 1200.4.a.b.1.1 1
20.19 odd 2 1200.4.f.j.49.2 2
35.3 even 12 294.4.e.g.79.1 2
35.13 even 4 294.4.a.e.1.1 1
35.18 odd 12 294.4.e.h.79.1 2
35.23 odd 12 294.4.e.h.67.1 2
35.33 even 12 294.4.e.g.67.1 2
40.3 even 4 192.4.a.c.1.1 1
40.13 odd 4 192.4.a.i.1.1 1
45.13 odd 12 162.4.c.f.55.1 2
45.23 even 12 162.4.c.c.55.1 2
45.38 even 12 162.4.c.c.109.1 2
45.43 odd 12 162.4.c.f.109.1 2
55.43 even 4 726.4.a.f.1.1 1
60.23 odd 4 144.4.a.c.1.1 1
65.8 even 4 1014.4.b.d.337.1 2
65.18 even 4 1014.4.b.d.337.2 2
65.38 odd 4 1014.4.a.g.1.1 1
80.3 even 4 768.4.d.c.385.2 2
80.13 odd 4 768.4.d.n.385.1 2
80.43 even 4 768.4.d.c.385.1 2
80.53 odd 4 768.4.d.n.385.2 2
85.33 odd 4 1734.4.a.d.1.1 1
95.18 even 4 2166.4.a.i.1.1 1
105.23 even 12 882.4.g.i.361.1 2
105.38 odd 12 882.4.g.f.667.1 2
105.53 even 12 882.4.g.i.667.1 2
105.68 odd 12 882.4.g.f.361.1 2
105.83 odd 4 882.4.a.n.1.1 1
120.53 even 4 576.4.a.q.1.1 1
120.83 odd 4 576.4.a.r.1.1 1
140.83 odd 4 2352.4.a.e.1.1 1
165.98 odd 4 2178.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 1 5.3 odd 4
18.4.a.a.1.1 1 15.8 even 4
48.4.a.c.1.1 1 20.3 even 4
144.4.a.c.1.1 1 60.23 odd 4
150.4.a.i.1.1 1 5.2 odd 4
150.4.c.d.49.1 2 1.1 even 1 trivial
150.4.c.d.49.2 2 5.4 even 2 inner
162.4.c.c.55.1 2 45.23 even 12
162.4.c.c.109.1 2 45.38 even 12
162.4.c.f.55.1 2 45.13 odd 12
162.4.c.f.109.1 2 45.43 odd 12
192.4.a.c.1.1 1 40.3 even 4
192.4.a.i.1.1 1 40.13 odd 4
294.4.a.e.1.1 1 35.13 even 4
294.4.e.g.67.1 2 35.33 even 12
294.4.e.g.79.1 2 35.3 even 12
294.4.e.h.67.1 2 35.23 odd 12
294.4.e.h.79.1 2 35.18 odd 12
450.4.a.h.1.1 1 15.2 even 4
450.4.c.e.199.1 2 15.14 odd 2
450.4.c.e.199.2 2 3.2 odd 2
576.4.a.q.1.1 1 120.53 even 4
576.4.a.r.1.1 1 120.83 odd 4
726.4.a.f.1.1 1 55.43 even 4
768.4.d.c.385.1 2 80.43 even 4
768.4.d.c.385.2 2 80.3 even 4
768.4.d.n.385.1 2 80.13 odd 4
768.4.d.n.385.2 2 80.53 odd 4
882.4.a.n.1.1 1 105.83 odd 4
882.4.g.f.361.1 2 105.68 odd 12
882.4.g.f.667.1 2 105.38 odd 12
882.4.g.i.361.1 2 105.23 even 12
882.4.g.i.667.1 2 105.53 even 12
1014.4.a.g.1.1 1 65.38 odd 4
1014.4.b.d.337.1 2 65.8 even 4
1014.4.b.d.337.2 2 65.18 even 4
1200.4.a.b.1.1 1 20.7 even 4
1200.4.f.j.49.1 2 4.3 odd 2
1200.4.f.j.49.2 2 20.19 odd 2
1734.4.a.d.1.1 1 85.33 odd 4
2166.4.a.i.1.1 1 95.18 even 4
2178.4.a.e.1.1 1 165.98 odd 4
2352.4.a.e.1.1 1 140.83 odd 4