Properties

Label 3822.2.a.bs.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.41421 q^{10} +0.414214 q^{11} +1.00000 q^{12} +1.00000 q^{13} -1.41421 q^{15} +1.00000 q^{16} +5.24264 q^{17} +1.00000 q^{18} +3.00000 q^{19} -1.41421 q^{20} +0.414214 q^{22} -9.07107 q^{23} +1.00000 q^{24} -3.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +5.00000 q^{29} -1.41421 q^{30} +6.82843 q^{31} +1.00000 q^{32} +0.414214 q^{33} +5.24264 q^{34} +1.00000 q^{36} -5.07107 q^{37} +3.00000 q^{38} +1.00000 q^{39} -1.41421 q^{40} +4.58579 q^{41} +4.82843 q^{43} +0.414214 q^{44} -1.41421 q^{45} -9.07107 q^{46} +9.00000 q^{47} +1.00000 q^{48} -3.00000 q^{50} +5.24264 q^{51} +1.00000 q^{52} +12.3137 q^{53} +1.00000 q^{54} -0.585786 q^{55} +3.00000 q^{57} +5.00000 q^{58} +7.58579 q^{59} -1.41421 q^{60} -5.24264 q^{61} +6.82843 q^{62} +1.00000 q^{64} -1.41421 q^{65} +0.414214 q^{66} -14.6569 q^{67} +5.24264 q^{68} -9.07107 q^{69} -1.00000 q^{71} +1.00000 q^{72} -1.07107 q^{73} -5.07107 q^{74} -3.00000 q^{75} +3.00000 q^{76} +1.00000 q^{78} +13.3137 q^{79} -1.41421 q^{80} +1.00000 q^{81} +4.58579 q^{82} -7.65685 q^{83} -7.41421 q^{85} +4.82843 q^{86} +5.00000 q^{87} +0.414214 q^{88} -6.24264 q^{89} -1.41421 q^{90} -9.07107 q^{92} +6.82843 q^{93} +9.00000 q^{94} -4.24264 q^{95} +1.00000 q^{96} +7.89949 q^{97} +0.414214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 6 q^{19} - 2 q^{22} - 4 q^{23} + 2 q^{24} - 6 q^{25} + 2 q^{26} + 2 q^{27} + 10 q^{29} + 8 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{34} + 2 q^{36} + 4 q^{37} + 6 q^{38} + 2 q^{39} + 12 q^{41} + 4 q^{43} - 2 q^{44} - 4 q^{46} + 18 q^{47} + 2 q^{48} - 6 q^{50} + 2 q^{51} + 2 q^{52} + 2 q^{53} + 2 q^{54} - 4 q^{55} + 6 q^{57} + 10 q^{58} + 18 q^{59} - 2 q^{61} + 8 q^{62} + 2 q^{64} - 2 q^{66} - 18 q^{67} + 2 q^{68} - 4 q^{69} - 2 q^{71} + 2 q^{72} + 12 q^{73} + 4 q^{74} - 6 q^{75} + 6 q^{76} + 2 q^{78} + 4 q^{79} + 2 q^{81} + 12 q^{82} - 4 q^{83} - 12 q^{85} + 4 q^{86} + 10 q^{87} - 2 q^{88} - 4 q^{89} - 4 q^{92} + 8 q^{93} + 18 q^{94} + 2 q^{96} - 4 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.41421 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.41421 −0.447214
\(11\) 0.414214 0.124890 0.0624450 0.998048i \(-0.480110\pi\)
0.0624450 + 0.998048i \(0.480110\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 1.00000 0.250000
\(17\) 5.24264 1.27153 0.635764 0.771884i \(-0.280685\pi\)
0.635764 + 0.771884i \(0.280685\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −1.41421 −0.316228
\(21\) 0 0
\(22\) 0.414214 0.0883106
\(23\) −9.07107 −1.89145 −0.945724 0.324970i \(-0.894646\pi\)
−0.945724 + 0.324970i \(0.894646\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.00000 −0.600000
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) −1.41421 −0.258199
\(31\) 6.82843 1.22642 0.613211 0.789919i \(-0.289878\pi\)
0.613211 + 0.789919i \(0.289878\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.414214 0.0721053
\(34\) 5.24264 0.899105
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −5.07107 −0.833678 −0.416839 0.908980i \(-0.636862\pi\)
−0.416839 + 0.908980i \(0.636862\pi\)
\(38\) 3.00000 0.486664
\(39\) 1.00000 0.160128
\(40\) −1.41421 −0.223607
\(41\) 4.58579 0.716180 0.358090 0.933687i \(-0.383428\pi\)
0.358090 + 0.933687i \(0.383428\pi\)
\(42\) 0 0
\(43\) 4.82843 0.736328 0.368164 0.929761i \(-0.379986\pi\)
0.368164 + 0.929761i \(0.379986\pi\)
\(44\) 0.414214 0.0624450
\(45\) −1.41421 −0.210819
\(46\) −9.07107 −1.33746
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −3.00000 −0.424264
\(51\) 5.24264 0.734117
\(52\) 1.00000 0.138675
\(53\) 12.3137 1.69142 0.845709 0.533644i \(-0.179178\pi\)
0.845709 + 0.533644i \(0.179178\pi\)
\(54\) 1.00000 0.136083
\(55\) −0.585786 −0.0789874
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 5.00000 0.656532
\(59\) 7.58579 0.987585 0.493793 0.869580i \(-0.335610\pi\)
0.493793 + 0.869580i \(0.335610\pi\)
\(60\) −1.41421 −0.182574
\(61\) −5.24264 −0.671251 −0.335626 0.941995i \(-0.608948\pi\)
−0.335626 + 0.941995i \(0.608948\pi\)
\(62\) 6.82843 0.867211
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.41421 −0.175412
\(66\) 0.414214 0.0509862
\(67\) −14.6569 −1.79062 −0.895310 0.445444i \(-0.853046\pi\)
−0.895310 + 0.445444i \(0.853046\pi\)
\(68\) 5.24264 0.635764
\(69\) −9.07107 −1.09203
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.07107 −0.125359 −0.0626795 0.998034i \(-0.519965\pi\)
−0.0626795 + 0.998034i \(0.519965\pi\)
\(74\) −5.07107 −0.589500
\(75\) −3.00000 −0.346410
\(76\) 3.00000 0.344124
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 13.3137 1.49791 0.748955 0.662621i \(-0.230556\pi\)
0.748955 + 0.662621i \(0.230556\pi\)
\(80\) −1.41421 −0.158114
\(81\) 1.00000 0.111111
\(82\) 4.58579 0.506415
\(83\) −7.65685 −0.840449 −0.420224 0.907420i \(-0.638049\pi\)
−0.420224 + 0.907420i \(0.638049\pi\)
\(84\) 0 0
\(85\) −7.41421 −0.804184
\(86\) 4.82843 0.520663
\(87\) 5.00000 0.536056
\(88\) 0.414214 0.0441553
\(89\) −6.24264 −0.661719 −0.330859 0.943680i \(-0.607339\pi\)
−0.330859 + 0.943680i \(0.607339\pi\)
\(90\) −1.41421 −0.149071
\(91\) 0 0
\(92\) −9.07107 −0.945724
\(93\) 6.82843 0.708075
\(94\) 9.00000 0.928279
\(95\) −4.24264 −0.435286
\(96\) 1.00000 0.102062
\(97\) 7.89949 0.802072 0.401036 0.916062i \(-0.368650\pi\)
0.401036 + 0.916062i \(0.368650\pi\)
\(98\) 0 0
\(99\) 0.414214 0.0416300
\(100\) −3.00000 −0.300000
\(101\) −1.17157 −0.116576 −0.0582879 0.998300i \(-0.518564\pi\)
−0.0582879 + 0.998300i \(0.518564\pi\)
\(102\) 5.24264 0.519099
\(103\) 6.48528 0.639014 0.319507 0.947584i \(-0.396483\pi\)
0.319507 + 0.947584i \(0.396483\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 12.3137 1.19601
\(107\) 13.6569 1.32026 0.660129 0.751152i \(-0.270502\pi\)
0.660129 + 0.751152i \(0.270502\pi\)
\(108\) 1.00000 0.0962250
\(109\) 1.65685 0.158698 0.0793489 0.996847i \(-0.474716\pi\)
0.0793489 + 0.996847i \(0.474716\pi\)
\(110\) −0.585786 −0.0558525
\(111\) −5.07107 −0.481324
\(112\) 0 0
\(113\) −6.07107 −0.571118 −0.285559 0.958361i \(-0.592179\pi\)
−0.285559 + 0.958361i \(0.592179\pi\)
\(114\) 3.00000 0.280976
\(115\) 12.8284 1.19626
\(116\) 5.00000 0.464238
\(117\) 1.00000 0.0924500
\(118\) 7.58579 0.698328
\(119\) 0 0
\(120\) −1.41421 −0.129099
\(121\) −10.8284 −0.984402
\(122\) −5.24264 −0.474646
\(123\) 4.58579 0.413486
\(124\) 6.82843 0.613211
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) 9.89949 0.878438 0.439219 0.898380i \(-0.355255\pi\)
0.439219 + 0.898380i \(0.355255\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.82843 0.425119
\(130\) −1.41421 −0.124035
\(131\) 9.89949 0.864923 0.432461 0.901652i \(-0.357645\pi\)
0.432461 + 0.901652i \(0.357645\pi\)
\(132\) 0.414214 0.0360527
\(133\) 0 0
\(134\) −14.6569 −1.26616
\(135\) −1.41421 −0.121716
\(136\) 5.24264 0.449553
\(137\) −13.0711 −1.11674 −0.558368 0.829593i \(-0.688572\pi\)
−0.558368 + 0.829593i \(0.688572\pi\)
\(138\) −9.07107 −0.772181
\(139\) −21.2132 −1.79928 −0.899640 0.436632i \(-0.856171\pi\)
−0.899640 + 0.436632i \(0.856171\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) −1.00000 −0.0839181
\(143\) 0.414214 0.0346383
\(144\) 1.00000 0.0833333
\(145\) −7.07107 −0.587220
\(146\) −1.07107 −0.0886422
\(147\) 0 0
\(148\) −5.07107 −0.416839
\(149\) 15.8995 1.30254 0.651269 0.758847i \(-0.274237\pi\)
0.651269 + 0.758847i \(0.274237\pi\)
\(150\) −3.00000 −0.244949
\(151\) −7.58579 −0.617323 −0.308661 0.951172i \(-0.599881\pi\)
−0.308661 + 0.951172i \(0.599881\pi\)
\(152\) 3.00000 0.243332
\(153\) 5.24264 0.423842
\(154\) 0 0
\(155\) −9.65685 −0.775657
\(156\) 1.00000 0.0800641
\(157\) 7.92893 0.632798 0.316399 0.948626i \(-0.397526\pi\)
0.316399 + 0.948626i \(0.397526\pi\)
\(158\) 13.3137 1.05918
\(159\) 12.3137 0.976541
\(160\) −1.41421 −0.111803
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 17.1421 1.34268 0.671338 0.741151i \(-0.265720\pi\)
0.671338 + 0.741151i \(0.265720\pi\)
\(164\) 4.58579 0.358090
\(165\) −0.585786 −0.0456034
\(166\) −7.65685 −0.594287
\(167\) 12.3137 0.952863 0.476432 0.879211i \(-0.341930\pi\)
0.476432 + 0.879211i \(0.341930\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −7.41421 −0.568644
\(171\) 3.00000 0.229416
\(172\) 4.82843 0.368164
\(173\) 20.3137 1.54442 0.772211 0.635366i \(-0.219151\pi\)
0.772211 + 0.635366i \(0.219151\pi\)
\(174\) 5.00000 0.379049
\(175\) 0 0
\(176\) 0.414214 0.0312225
\(177\) 7.58579 0.570183
\(178\) −6.24264 −0.467906
\(179\) −12.3431 −0.922570 −0.461285 0.887252i \(-0.652611\pi\)
−0.461285 + 0.887252i \(0.652611\pi\)
\(180\) −1.41421 −0.105409
\(181\) −11.5858 −0.861165 −0.430582 0.902551i \(-0.641692\pi\)
−0.430582 + 0.902551i \(0.641692\pi\)
\(182\) 0 0
\(183\) −5.24264 −0.387547
\(184\) −9.07107 −0.668728
\(185\) 7.17157 0.527265
\(186\) 6.82843 0.500685
\(187\) 2.17157 0.158801
\(188\) 9.00000 0.656392
\(189\) 0 0
\(190\) −4.24264 −0.307794
\(191\) 2.34315 0.169544 0.0847720 0.996400i \(-0.472984\pi\)
0.0847720 + 0.996400i \(0.472984\pi\)
\(192\) 1.00000 0.0721688
\(193\) −15.7574 −1.13424 −0.567120 0.823635i \(-0.691942\pi\)
−0.567120 + 0.823635i \(0.691942\pi\)
\(194\) 7.89949 0.567151
\(195\) −1.41421 −0.101274
\(196\) 0 0
\(197\) 9.55635 0.680862 0.340431 0.940270i \(-0.389427\pi\)
0.340431 + 0.940270i \(0.389427\pi\)
\(198\) 0.414214 0.0294369
\(199\) 10.5858 0.750407 0.375203 0.926943i \(-0.377573\pi\)
0.375203 + 0.926943i \(0.377573\pi\)
\(200\) −3.00000 −0.212132
\(201\) −14.6569 −1.03381
\(202\) −1.17157 −0.0824316
\(203\) 0 0
\(204\) 5.24264 0.367058
\(205\) −6.48528 −0.452952
\(206\) 6.48528 0.451851
\(207\) −9.07107 −0.630483
\(208\) 1.00000 0.0693375
\(209\) 1.24264 0.0859553
\(210\) 0 0
\(211\) −28.3848 −1.95409 −0.977044 0.213036i \(-0.931665\pi\)
−0.977044 + 0.213036i \(0.931665\pi\)
\(212\) 12.3137 0.845709
\(213\) −1.00000 −0.0685189
\(214\) 13.6569 0.933563
\(215\) −6.82843 −0.465695
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 1.65685 0.112216
\(219\) −1.07107 −0.0723761
\(220\) −0.585786 −0.0394937
\(221\) 5.24264 0.352658
\(222\) −5.07107 −0.340348
\(223\) 21.3848 1.43203 0.716015 0.698085i \(-0.245964\pi\)
0.716015 + 0.698085i \(0.245964\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) −6.07107 −0.403841
\(227\) −26.1421 −1.73511 −0.867557 0.497337i \(-0.834311\pi\)
−0.867557 + 0.497337i \(0.834311\pi\)
\(228\) 3.00000 0.198680
\(229\) −7.17157 −0.473911 −0.236955 0.971521i \(-0.576150\pi\)
−0.236955 + 0.971521i \(0.576150\pi\)
\(230\) 12.8284 0.845881
\(231\) 0 0
\(232\) 5.00000 0.328266
\(233\) −3.72792 −0.244224 −0.122112 0.992516i \(-0.538967\pi\)
−0.122112 + 0.992516i \(0.538967\pi\)
\(234\) 1.00000 0.0653720
\(235\) −12.7279 −0.830278
\(236\) 7.58579 0.493793
\(237\) 13.3137 0.864818
\(238\) 0 0
\(239\) −2.51472 −0.162664 −0.0813318 0.996687i \(-0.525917\pi\)
−0.0813318 + 0.996687i \(0.525917\pi\)
\(240\) −1.41421 −0.0912871
\(241\) −16.4853 −1.06191 −0.530955 0.847400i \(-0.678167\pi\)
−0.530955 + 0.847400i \(0.678167\pi\)
\(242\) −10.8284 −0.696078
\(243\) 1.00000 0.0641500
\(244\) −5.24264 −0.335626
\(245\) 0 0
\(246\) 4.58579 0.292379
\(247\) 3.00000 0.190885
\(248\) 6.82843 0.433606
\(249\) −7.65685 −0.485233
\(250\) 11.3137 0.715542
\(251\) 12.8284 0.809723 0.404862 0.914378i \(-0.367320\pi\)
0.404862 + 0.914378i \(0.367320\pi\)
\(252\) 0 0
\(253\) −3.75736 −0.236223
\(254\) 9.89949 0.621150
\(255\) −7.41421 −0.464296
\(256\) 1.00000 0.0625000
\(257\) −21.7990 −1.35978 −0.679892 0.733312i \(-0.737973\pi\)
−0.679892 + 0.733312i \(0.737973\pi\)
\(258\) 4.82843 0.300605
\(259\) 0 0
\(260\) −1.41421 −0.0877058
\(261\) 5.00000 0.309492
\(262\) 9.89949 0.611593
\(263\) −7.55635 −0.465944 −0.232972 0.972483i \(-0.574845\pi\)
−0.232972 + 0.972483i \(0.574845\pi\)
\(264\) 0.414214 0.0254931
\(265\) −17.4142 −1.06975
\(266\) 0 0
\(267\) −6.24264 −0.382043
\(268\) −14.6569 −0.895310
\(269\) −3.68629 −0.224757 −0.112379 0.993665i \(-0.535847\pi\)
−0.112379 + 0.993665i \(0.535847\pi\)
\(270\) −1.41421 −0.0860663
\(271\) −15.3848 −0.934559 −0.467279 0.884110i \(-0.654766\pi\)
−0.467279 + 0.884110i \(0.654766\pi\)
\(272\) 5.24264 0.317882
\(273\) 0 0
\(274\) −13.0711 −0.789652
\(275\) −1.24264 −0.0749341
\(276\) −9.07107 −0.546014
\(277\) −16.4142 −0.986235 −0.493117 0.869963i \(-0.664143\pi\)
−0.493117 + 0.869963i \(0.664143\pi\)
\(278\) −21.2132 −1.27228
\(279\) 6.82843 0.408807
\(280\) 0 0
\(281\) 7.31371 0.436299 0.218150 0.975915i \(-0.429998\pi\)
0.218150 + 0.975915i \(0.429998\pi\)
\(282\) 9.00000 0.535942
\(283\) 17.0711 1.01477 0.507385 0.861720i \(-0.330612\pi\)
0.507385 + 0.861720i \(0.330612\pi\)
\(284\) −1.00000 −0.0593391
\(285\) −4.24264 −0.251312
\(286\) 0.414214 0.0244930
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 10.4853 0.616781
\(290\) −7.07107 −0.415227
\(291\) 7.89949 0.463077
\(292\) −1.07107 −0.0626795
\(293\) −18.7279 −1.09410 −0.547048 0.837101i \(-0.684249\pi\)
−0.547048 + 0.837101i \(0.684249\pi\)
\(294\) 0 0
\(295\) −10.7279 −0.624604
\(296\) −5.07107 −0.294750
\(297\) 0.414214 0.0240351
\(298\) 15.8995 0.921033
\(299\) −9.07107 −0.524593
\(300\) −3.00000 −0.173205
\(301\) 0 0
\(302\) −7.58579 −0.436513
\(303\) −1.17157 −0.0673051
\(304\) 3.00000 0.172062
\(305\) 7.41421 0.424537
\(306\) 5.24264 0.299702
\(307\) −19.6274 −1.12020 −0.560098 0.828426i \(-0.689237\pi\)
−0.560098 + 0.828426i \(0.689237\pi\)
\(308\) 0 0
\(309\) 6.48528 0.368935
\(310\) −9.65685 −0.548472
\(311\) −6.58579 −0.373446 −0.186723 0.982413i \(-0.559787\pi\)
−0.186723 + 0.982413i \(0.559787\pi\)
\(312\) 1.00000 0.0566139
\(313\) 18.1421 1.02545 0.512727 0.858552i \(-0.328635\pi\)
0.512727 + 0.858552i \(0.328635\pi\)
\(314\) 7.92893 0.447456
\(315\) 0 0
\(316\) 13.3137 0.748955
\(317\) 0.828427 0.0465291 0.0232646 0.999729i \(-0.492594\pi\)
0.0232646 + 0.999729i \(0.492594\pi\)
\(318\) 12.3137 0.690518
\(319\) 2.07107 0.115958
\(320\) −1.41421 −0.0790569
\(321\) 13.6569 0.762251
\(322\) 0 0
\(323\) 15.7279 0.875125
\(324\) 1.00000 0.0555556
\(325\) −3.00000 −0.166410
\(326\) 17.1421 0.949415
\(327\) 1.65685 0.0916242
\(328\) 4.58579 0.253208
\(329\) 0 0
\(330\) −0.585786 −0.0322465
\(331\) −22.4853 −1.23590 −0.617951 0.786216i \(-0.712037\pi\)
−0.617951 + 0.786216i \(0.712037\pi\)
\(332\) −7.65685 −0.420224
\(333\) −5.07107 −0.277893
\(334\) 12.3137 0.673776
\(335\) 20.7279 1.13249
\(336\) 0 0
\(337\) 28.6569 1.56104 0.780519 0.625132i \(-0.214955\pi\)
0.780519 + 0.625132i \(0.214955\pi\)
\(338\) 1.00000 0.0543928
\(339\) −6.07107 −0.329735
\(340\) −7.41421 −0.402092
\(341\) 2.82843 0.153168
\(342\) 3.00000 0.162221
\(343\) 0 0
\(344\) 4.82843 0.260331
\(345\) 12.8284 0.690659
\(346\) 20.3137 1.09207
\(347\) 5.75736 0.309071 0.154536 0.987987i \(-0.450612\pi\)
0.154536 + 0.987987i \(0.450612\pi\)
\(348\) 5.00000 0.268028
\(349\) 5.41421 0.289816 0.144908 0.989445i \(-0.453711\pi\)
0.144908 + 0.989445i \(0.453711\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0.414214 0.0220777
\(353\) 12.6274 0.672090 0.336045 0.941846i \(-0.390911\pi\)
0.336045 + 0.941846i \(0.390911\pi\)
\(354\) 7.58579 0.403180
\(355\) 1.41421 0.0750587
\(356\) −6.24264 −0.330859
\(357\) 0 0
\(358\) −12.3431 −0.652356
\(359\) −15.4558 −0.815728 −0.407864 0.913043i \(-0.633726\pi\)
−0.407864 + 0.913043i \(0.633726\pi\)
\(360\) −1.41421 −0.0745356
\(361\) −10.0000 −0.526316
\(362\) −11.5858 −0.608935
\(363\) −10.8284 −0.568345
\(364\) 0 0
\(365\) 1.51472 0.0792840
\(366\) −5.24264 −0.274037
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) −9.07107 −0.472862
\(369\) 4.58579 0.238727
\(370\) 7.17157 0.372832
\(371\) 0 0
\(372\) 6.82843 0.354037
\(373\) −11.1005 −0.574762 −0.287381 0.957816i \(-0.592785\pi\)
−0.287381 + 0.957816i \(0.592785\pi\)
\(374\) 2.17157 0.112289
\(375\) 11.3137 0.584237
\(376\) 9.00000 0.464140
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) −33.6569 −1.72884 −0.864418 0.502773i \(-0.832313\pi\)
−0.864418 + 0.502773i \(0.832313\pi\)
\(380\) −4.24264 −0.217643
\(381\) 9.89949 0.507166
\(382\) 2.34315 0.119886
\(383\) 6.48528 0.331382 0.165691 0.986178i \(-0.447015\pi\)
0.165691 + 0.986178i \(0.447015\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −15.7574 −0.802028
\(387\) 4.82843 0.245443
\(388\) 7.89949 0.401036
\(389\) 23.0000 1.16615 0.583073 0.812420i \(-0.301850\pi\)
0.583073 + 0.812420i \(0.301850\pi\)
\(390\) −1.41421 −0.0716115
\(391\) −47.5563 −2.40503
\(392\) 0 0
\(393\) 9.89949 0.499363
\(394\) 9.55635 0.481442
\(395\) −18.8284 −0.947361
\(396\) 0.414214 0.0208150
\(397\) −19.7574 −0.991593 −0.495797 0.868439i \(-0.665124\pi\)
−0.495797 + 0.868439i \(0.665124\pi\)
\(398\) 10.5858 0.530618
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) 19.0711 0.952364 0.476182 0.879347i \(-0.342020\pi\)
0.476182 + 0.879347i \(0.342020\pi\)
\(402\) −14.6569 −0.731017
\(403\) 6.82843 0.340148
\(404\) −1.17157 −0.0582879
\(405\) −1.41421 −0.0702728
\(406\) 0 0
\(407\) −2.10051 −0.104118
\(408\) 5.24264 0.259549
\(409\) −2.58579 −0.127859 −0.0639295 0.997954i \(-0.520363\pi\)
−0.0639295 + 0.997954i \(0.520363\pi\)
\(410\) −6.48528 −0.320285
\(411\) −13.0711 −0.644748
\(412\) 6.48528 0.319507
\(413\) 0 0
\(414\) −9.07107 −0.445819
\(415\) 10.8284 0.531547
\(416\) 1.00000 0.0490290
\(417\) −21.2132 −1.03882
\(418\) 1.24264 0.0607795
\(419\) −27.4558 −1.34131 −0.670653 0.741771i \(-0.733986\pi\)
−0.670653 + 0.741771i \(0.733986\pi\)
\(420\) 0 0
\(421\) 10.5858 0.515920 0.257960 0.966156i \(-0.416950\pi\)
0.257960 + 0.966156i \(0.416950\pi\)
\(422\) −28.3848 −1.38175
\(423\) 9.00000 0.437595
\(424\) 12.3137 0.598007
\(425\) −15.7279 −0.762916
\(426\) −1.00000 −0.0484502
\(427\) 0 0
\(428\) 13.6569 0.660129
\(429\) 0.414214 0.0199984
\(430\) −6.82843 −0.329296
\(431\) 14.4853 0.697731 0.348866 0.937173i \(-0.386567\pi\)
0.348866 + 0.937173i \(0.386567\pi\)
\(432\) 1.00000 0.0481125
\(433\) −19.0000 −0.913082 −0.456541 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) 0 0
\(435\) −7.07107 −0.339032
\(436\) 1.65685 0.0793489
\(437\) −27.2132 −1.30178
\(438\) −1.07107 −0.0511776
\(439\) −10.5858 −0.505232 −0.252616 0.967567i \(-0.581291\pi\)
−0.252616 + 0.967567i \(0.581291\pi\)
\(440\) −0.585786 −0.0279263
\(441\) 0 0
\(442\) 5.24264 0.249367
\(443\) 22.7279 1.07984 0.539918 0.841718i \(-0.318455\pi\)
0.539918 + 0.841718i \(0.318455\pi\)
\(444\) −5.07107 −0.240662
\(445\) 8.82843 0.418508
\(446\) 21.3848 1.01260
\(447\) 15.8995 0.752020
\(448\) 0 0
\(449\) −33.9411 −1.60178 −0.800890 0.598811i \(-0.795640\pi\)
−0.800890 + 0.598811i \(0.795640\pi\)
\(450\) −3.00000 −0.141421
\(451\) 1.89949 0.0894437
\(452\) −6.07107 −0.285559
\(453\) −7.58579 −0.356411
\(454\) −26.1421 −1.22691
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) −15.2132 −0.711644 −0.355822 0.934554i \(-0.615799\pi\)
−0.355822 + 0.934554i \(0.615799\pi\)
\(458\) −7.17157 −0.335106
\(459\) 5.24264 0.244706
\(460\) 12.8284 0.598128
\(461\) −33.4558 −1.55819 −0.779097 0.626903i \(-0.784322\pi\)
−0.779097 + 0.626903i \(0.784322\pi\)
\(462\) 0 0
\(463\) −37.9411 −1.76327 −0.881637 0.471929i \(-0.843558\pi\)
−0.881637 + 0.471929i \(0.843558\pi\)
\(464\) 5.00000 0.232119
\(465\) −9.65685 −0.447826
\(466\) −3.72792 −0.172693
\(467\) −0.242641 −0.0112281 −0.00561404 0.999984i \(-0.501787\pi\)
−0.00561404 + 0.999984i \(0.501787\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) −12.7279 −0.587095
\(471\) 7.92893 0.365346
\(472\) 7.58579 0.349164
\(473\) 2.00000 0.0919601
\(474\) 13.3137 0.611519
\(475\) −9.00000 −0.412948
\(476\) 0 0
\(477\) 12.3137 0.563806
\(478\) −2.51472 −0.115021
\(479\) −31.1421 −1.42292 −0.711460 0.702726i \(-0.751966\pi\)
−0.711460 + 0.702726i \(0.751966\pi\)
\(480\) −1.41421 −0.0645497
\(481\) −5.07107 −0.231221
\(482\) −16.4853 −0.750884
\(483\) 0 0
\(484\) −10.8284 −0.492201
\(485\) −11.1716 −0.507275
\(486\) 1.00000 0.0453609
\(487\) −23.7279 −1.07521 −0.537607 0.843195i \(-0.680672\pi\)
−0.537607 + 0.843195i \(0.680672\pi\)
\(488\) −5.24264 −0.237323
\(489\) 17.1421 0.775194
\(490\) 0 0
\(491\) −26.2843 −1.18619 −0.593096 0.805132i \(-0.702095\pi\)
−0.593096 + 0.805132i \(0.702095\pi\)
\(492\) 4.58579 0.206743
\(493\) 26.2132 1.18058
\(494\) 3.00000 0.134976
\(495\) −0.585786 −0.0263291
\(496\) 6.82843 0.306605
\(497\) 0 0
\(498\) −7.65685 −0.343112
\(499\) −30.0000 −1.34298 −0.671492 0.741012i \(-0.734346\pi\)
−0.671492 + 0.741012i \(0.734346\pi\)
\(500\) 11.3137 0.505964
\(501\) 12.3137 0.550136
\(502\) 12.8284 0.572561
\(503\) 35.7990 1.59620 0.798099 0.602526i \(-0.205839\pi\)
0.798099 + 0.602526i \(0.205839\pi\)
\(504\) 0 0
\(505\) 1.65685 0.0737290
\(506\) −3.75736 −0.167035
\(507\) 1.00000 0.0444116
\(508\) 9.89949 0.439219
\(509\) 12.9706 0.574910 0.287455 0.957794i \(-0.407191\pi\)
0.287455 + 0.957794i \(0.407191\pi\)
\(510\) −7.41421 −0.328307
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 3.00000 0.132453
\(514\) −21.7990 −0.961512
\(515\) −9.17157 −0.404148
\(516\) 4.82843 0.212560
\(517\) 3.72792 0.163954
\(518\) 0 0
\(519\) 20.3137 0.891673
\(520\) −1.41421 −0.0620174
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 5.00000 0.218844
\(523\) −3.85786 −0.168693 −0.0843463 0.996437i \(-0.526880\pi\)
−0.0843463 + 0.996437i \(0.526880\pi\)
\(524\) 9.89949 0.432461
\(525\) 0 0
\(526\) −7.55635 −0.329472
\(527\) 35.7990 1.55943
\(528\) 0.414214 0.0180263
\(529\) 59.2843 2.57758
\(530\) −17.4142 −0.756425
\(531\) 7.58579 0.329195
\(532\) 0 0
\(533\) 4.58579 0.198632
\(534\) −6.24264 −0.270145
\(535\) −19.3137 −0.835004
\(536\) −14.6569 −0.633080
\(537\) −12.3431 −0.532646
\(538\) −3.68629 −0.158927
\(539\) 0 0
\(540\) −1.41421 −0.0608581
\(541\) 12.8701 0.553327 0.276663 0.960967i \(-0.410771\pi\)
0.276663 + 0.960967i \(0.410771\pi\)
\(542\) −15.3848 −0.660833
\(543\) −11.5858 −0.497194
\(544\) 5.24264 0.224776
\(545\) −2.34315 −0.100369
\(546\) 0 0
\(547\) −5.75736 −0.246167 −0.123083 0.992396i \(-0.539278\pi\)
−0.123083 + 0.992396i \(0.539278\pi\)
\(548\) −13.0711 −0.558368
\(549\) −5.24264 −0.223750
\(550\) −1.24264 −0.0529864
\(551\) 15.0000 0.639021
\(552\) −9.07107 −0.386090
\(553\) 0 0
\(554\) −16.4142 −0.697373
\(555\) 7.17157 0.304416
\(556\) −21.2132 −0.899640
\(557\) −10.9706 −0.464838 −0.232419 0.972616i \(-0.574664\pi\)
−0.232419 + 0.972616i \(0.574664\pi\)
\(558\) 6.82843 0.289070
\(559\) 4.82843 0.204221
\(560\) 0 0
\(561\) 2.17157 0.0916839
\(562\) 7.31371 0.308510
\(563\) −0.485281 −0.0204522 −0.0102261 0.999948i \(-0.503255\pi\)
−0.0102261 + 0.999948i \(0.503255\pi\)
\(564\) 9.00000 0.378968
\(565\) 8.58579 0.361207
\(566\) 17.0711 0.717551
\(567\) 0 0
\(568\) −1.00000 −0.0419591
\(569\) −25.5858 −1.07261 −0.536306 0.844024i \(-0.680181\pi\)
−0.536306 + 0.844024i \(0.680181\pi\)
\(570\) −4.24264 −0.177705
\(571\) −1.65685 −0.0693372 −0.0346686 0.999399i \(-0.511038\pi\)
−0.0346686 + 0.999399i \(0.511038\pi\)
\(572\) 0.414214 0.0173191
\(573\) 2.34315 0.0978863
\(574\) 0 0
\(575\) 27.2132 1.13487
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 10.4853 0.436130
\(579\) −15.7574 −0.654854
\(580\) −7.07107 −0.293610
\(581\) 0 0
\(582\) 7.89949 0.327445
\(583\) 5.10051 0.211241
\(584\) −1.07107 −0.0443211
\(585\) −1.41421 −0.0584705
\(586\) −18.7279 −0.773643
\(587\) 11.2426 0.464033 0.232017 0.972712i \(-0.425468\pi\)
0.232017 + 0.972712i \(0.425468\pi\)
\(588\) 0 0
\(589\) 20.4853 0.844081
\(590\) −10.7279 −0.441662
\(591\) 9.55635 0.393096
\(592\) −5.07107 −0.208420
\(593\) −28.5858 −1.17388 −0.586939 0.809631i \(-0.699667\pi\)
−0.586939 + 0.809631i \(0.699667\pi\)
\(594\) 0.414214 0.0169954
\(595\) 0 0
\(596\) 15.8995 0.651269
\(597\) 10.5858 0.433247
\(598\) −9.07107 −0.370944
\(599\) −14.8701 −0.607574 −0.303787 0.952740i \(-0.598251\pi\)
−0.303787 + 0.952740i \(0.598251\pi\)
\(600\) −3.00000 −0.122474
\(601\) −17.8284 −0.727237 −0.363618 0.931548i \(-0.618459\pi\)
−0.363618 + 0.931548i \(0.618459\pi\)
\(602\) 0 0
\(603\) −14.6569 −0.596873
\(604\) −7.58579 −0.308661
\(605\) 15.3137 0.622591
\(606\) −1.17157 −0.0475919
\(607\) 27.0711 1.09878 0.549390 0.835566i \(-0.314860\pi\)
0.549390 + 0.835566i \(0.314860\pi\)
\(608\) 3.00000 0.121666
\(609\) 0 0
\(610\) 7.41421 0.300193
\(611\) 9.00000 0.364101
\(612\) 5.24264 0.211921
\(613\) −35.7990 −1.44591 −0.722954 0.690896i \(-0.757216\pi\)
−0.722954 + 0.690896i \(0.757216\pi\)
\(614\) −19.6274 −0.792098
\(615\) −6.48528 −0.261512
\(616\) 0 0
\(617\) −6.48528 −0.261088 −0.130544 0.991443i \(-0.541672\pi\)
−0.130544 + 0.991443i \(0.541672\pi\)
\(618\) 6.48528 0.260876
\(619\) 43.4558 1.74664 0.873319 0.487149i \(-0.161963\pi\)
0.873319 + 0.487149i \(0.161963\pi\)
\(620\) −9.65685 −0.387829
\(621\) −9.07107 −0.364009
\(622\) −6.58579 −0.264066
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) −1.00000 −0.0400000
\(626\) 18.1421 0.725106
\(627\) 1.24264 0.0496263
\(628\) 7.92893 0.316399
\(629\) −26.5858 −1.06004
\(630\) 0 0
\(631\) 36.6274 1.45811 0.729057 0.684453i \(-0.239959\pi\)
0.729057 + 0.684453i \(0.239959\pi\)
\(632\) 13.3137 0.529591
\(633\) −28.3848 −1.12819
\(634\) 0.828427 0.0329010
\(635\) −14.0000 −0.555573
\(636\) 12.3137 0.488270
\(637\) 0 0
\(638\) 2.07107 0.0819944
\(639\) −1.00000 −0.0395594
\(640\) −1.41421 −0.0559017
\(641\) 33.1716 1.31020 0.655099 0.755543i \(-0.272627\pi\)
0.655099 + 0.755543i \(0.272627\pi\)
\(642\) 13.6569 0.538993
\(643\) 36.3137 1.43207 0.716036 0.698063i \(-0.245954\pi\)
0.716036 + 0.698063i \(0.245954\pi\)
\(644\) 0 0
\(645\) −6.82843 −0.268869
\(646\) 15.7279 0.618807
\(647\) 28.6274 1.12546 0.562730 0.826641i \(-0.309751\pi\)
0.562730 + 0.826641i \(0.309751\pi\)
\(648\) 1.00000 0.0392837
\(649\) 3.14214 0.123340
\(650\) −3.00000 −0.117670
\(651\) 0 0
\(652\) 17.1421 0.671338
\(653\) 15.1127 0.591406 0.295703 0.955280i \(-0.404446\pi\)
0.295703 + 0.955280i \(0.404446\pi\)
\(654\) 1.65685 0.0647881
\(655\) −14.0000 −0.547025
\(656\) 4.58579 0.179045
\(657\) −1.07107 −0.0417863
\(658\) 0 0
\(659\) 30.7279 1.19699 0.598495 0.801127i \(-0.295766\pi\)
0.598495 + 0.801127i \(0.295766\pi\)
\(660\) −0.585786 −0.0228017
\(661\) −13.5563 −0.527281 −0.263640 0.964621i \(-0.584923\pi\)
−0.263640 + 0.964621i \(0.584923\pi\)
\(662\) −22.4853 −0.873915
\(663\) 5.24264 0.203607
\(664\) −7.65685 −0.297144
\(665\) 0 0
\(666\) −5.07107 −0.196500
\(667\) −45.3553 −1.75617
\(668\) 12.3137 0.476432
\(669\) 21.3848 0.826783
\(670\) 20.7279 0.800789
\(671\) −2.17157 −0.0838326
\(672\) 0 0
\(673\) −1.17157 −0.0451608 −0.0225804 0.999745i \(-0.507188\pi\)
−0.0225804 + 0.999745i \(0.507188\pi\)
\(674\) 28.6569 1.10382
\(675\) −3.00000 −0.115470
\(676\) 1.00000 0.0384615
\(677\) 28.1716 1.08272 0.541361 0.840790i \(-0.317909\pi\)
0.541361 + 0.840790i \(0.317909\pi\)
\(678\) −6.07107 −0.233158
\(679\) 0 0
\(680\) −7.41421 −0.284322
\(681\) −26.1421 −1.00177
\(682\) 2.82843 0.108306
\(683\) 5.45584 0.208762 0.104381 0.994537i \(-0.466714\pi\)
0.104381 + 0.994537i \(0.466714\pi\)
\(684\) 3.00000 0.114708
\(685\) 18.4853 0.706286
\(686\) 0 0
\(687\) −7.17157 −0.273613
\(688\) 4.82843 0.184082
\(689\) 12.3137 0.469115
\(690\) 12.8284 0.488370
\(691\) 11.1421 0.423867 0.211933 0.977284i \(-0.432024\pi\)
0.211933 + 0.977284i \(0.432024\pi\)
\(692\) 20.3137 0.772211
\(693\) 0 0
\(694\) 5.75736 0.218546
\(695\) 30.0000 1.13796
\(696\) 5.00000 0.189525
\(697\) 24.0416 0.910642
\(698\) 5.41421 0.204931
\(699\) −3.72792 −0.141003
\(700\) 0 0
\(701\) 48.4853 1.83126 0.915632 0.402018i \(-0.131691\pi\)
0.915632 + 0.402018i \(0.131691\pi\)
\(702\) 1.00000 0.0377426
\(703\) −15.2132 −0.573777
\(704\) 0.414214 0.0156113
\(705\) −12.7279 −0.479361
\(706\) 12.6274 0.475239
\(707\) 0 0
\(708\) 7.58579 0.285091
\(709\) −32.0416 −1.20335 −0.601674 0.798741i \(-0.705499\pi\)
−0.601674 + 0.798741i \(0.705499\pi\)
\(710\) 1.41421 0.0530745
\(711\) 13.3137 0.499303
\(712\) −6.24264 −0.233953
\(713\) −61.9411 −2.31971
\(714\) 0 0
\(715\) −0.585786 −0.0219072
\(716\) −12.3431 −0.461285
\(717\) −2.51472 −0.0939139
\(718\) −15.4558 −0.576807
\(719\) −22.2843 −0.831063 −0.415532 0.909579i \(-0.636404\pi\)
−0.415532 + 0.909579i \(0.636404\pi\)
\(720\) −1.41421 −0.0527046
\(721\) 0 0
\(722\) −10.0000 −0.372161
\(723\) −16.4853 −0.613094
\(724\) −11.5858 −0.430582
\(725\) −15.0000 −0.557086
\(726\) −10.8284 −0.401881
\(727\) −38.8284 −1.44007 −0.720033 0.693939i \(-0.755874\pi\)
−0.720033 + 0.693939i \(0.755874\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 1.51472 0.0560623
\(731\) 25.3137 0.936261
\(732\) −5.24264 −0.193774
\(733\) −29.0122 −1.07159 −0.535795 0.844348i \(-0.679988\pi\)
−0.535795 + 0.844348i \(0.679988\pi\)
\(734\) 14.0000 0.516749
\(735\) 0 0
\(736\) −9.07107 −0.334364
\(737\) −6.07107 −0.223631
\(738\) 4.58579 0.168805
\(739\) 6.97056 0.256416 0.128208 0.991747i \(-0.459077\pi\)
0.128208 + 0.991747i \(0.459077\pi\)
\(740\) 7.17157 0.263632
\(741\) 3.00000 0.110208
\(742\) 0 0
\(743\) 11.4853 0.421354 0.210677 0.977556i \(-0.432433\pi\)
0.210677 + 0.977556i \(0.432433\pi\)
\(744\) 6.82843 0.250342
\(745\) −22.4853 −0.823797
\(746\) −11.1005 −0.406418
\(747\) −7.65685 −0.280150
\(748\) 2.17157 0.0794006
\(749\) 0 0
\(750\) 11.3137 0.413118
\(751\) −34.3431 −1.25320 −0.626600 0.779341i \(-0.715554\pi\)
−0.626600 + 0.779341i \(0.715554\pi\)
\(752\) 9.00000 0.328196
\(753\) 12.8284 0.467494
\(754\) 5.00000 0.182089
\(755\) 10.7279 0.390429
\(756\) 0 0
\(757\) 23.4437 0.852074 0.426037 0.904706i \(-0.359909\pi\)
0.426037 + 0.904706i \(0.359909\pi\)
\(758\) −33.6569 −1.22247
\(759\) −3.75736 −0.136384
\(760\) −4.24264 −0.153897
\(761\) −17.8579 −0.647347 −0.323674 0.946169i \(-0.604918\pi\)
−0.323674 + 0.946169i \(0.604918\pi\)
\(762\) 9.89949 0.358621
\(763\) 0 0
\(764\) 2.34315 0.0847720
\(765\) −7.41421 −0.268061
\(766\) 6.48528 0.234323
\(767\) 7.58579 0.273907
\(768\) 1.00000 0.0360844
\(769\) −6.92893 −0.249864 −0.124932 0.992165i \(-0.539871\pi\)
−0.124932 + 0.992165i \(0.539871\pi\)
\(770\) 0 0
\(771\) −21.7990 −0.785071
\(772\) −15.7574 −0.567120
\(773\) −29.5980 −1.06457 −0.532283 0.846567i \(-0.678666\pi\)
−0.532283 + 0.846567i \(0.678666\pi\)
\(774\) 4.82843 0.173554
\(775\) −20.4853 −0.735853
\(776\) 7.89949 0.283575
\(777\) 0 0
\(778\) 23.0000 0.824590
\(779\) 13.7574 0.492909
\(780\) −1.41421 −0.0506370
\(781\) −0.414214 −0.0148217
\(782\) −47.5563 −1.70061
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) −11.2132 −0.400216
\(786\) 9.89949 0.353103
\(787\) −8.17157 −0.291285 −0.145643 0.989337i \(-0.546525\pi\)
−0.145643 + 0.989337i \(0.546525\pi\)
\(788\) 9.55635 0.340431
\(789\) −7.55635 −0.269013
\(790\) −18.8284 −0.669885
\(791\) 0 0
\(792\) 0.414214 0.0147184
\(793\) −5.24264 −0.186172
\(794\) −19.7574 −0.701162
\(795\) −17.4142 −0.617619
\(796\) 10.5858 0.375203
\(797\) −36.9706 −1.30956 −0.654782 0.755818i \(-0.727240\pi\)
−0.654782 + 0.755818i \(0.727240\pi\)
\(798\) 0 0
\(799\) 47.1838 1.66924
\(800\) −3.00000 −0.106066
\(801\) −6.24264 −0.220573
\(802\) 19.0711 0.673423
\(803\) −0.443651 −0.0156561
\(804\) −14.6569 −0.516907
\(805\) 0 0
\(806\) 6.82843 0.240521
\(807\) −3.68629 −0.129764
\(808\) −1.17157 −0.0412158
\(809\) 35.0416 1.23200 0.615999 0.787747i \(-0.288753\pi\)
0.615999 + 0.787747i \(0.288753\pi\)
\(810\) −1.41421 −0.0496904
\(811\) 26.7696 0.940006 0.470003 0.882665i \(-0.344253\pi\)
0.470003 + 0.882665i \(0.344253\pi\)
\(812\) 0 0
\(813\) −15.3848 −0.539568
\(814\) −2.10051 −0.0736227
\(815\) −24.2426 −0.849183
\(816\) 5.24264 0.183529
\(817\) 14.4853 0.506776
\(818\) −2.58579 −0.0904099
\(819\) 0 0
\(820\) −6.48528 −0.226476
\(821\) −9.61522 −0.335574 −0.167787 0.985823i \(-0.553662\pi\)
−0.167787 + 0.985823i \(0.553662\pi\)
\(822\) −13.0711 −0.455906
\(823\) −8.10051 −0.282366 −0.141183 0.989984i \(-0.545091\pi\)
−0.141183 + 0.989984i \(0.545091\pi\)
\(824\) 6.48528 0.225925
\(825\) −1.24264 −0.0432632
\(826\) 0 0
\(827\) 13.5269 0.470377 0.235188 0.971950i \(-0.424429\pi\)
0.235188 + 0.971950i \(0.424429\pi\)
\(828\) −9.07107 −0.315241
\(829\) 18.6985 0.649425 0.324713 0.945813i \(-0.394732\pi\)
0.324713 + 0.945813i \(0.394732\pi\)
\(830\) 10.8284 0.375860
\(831\) −16.4142 −0.569403
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −21.2132 −0.734553
\(835\) −17.4142 −0.602644
\(836\) 1.24264 0.0429776
\(837\) 6.82843 0.236025
\(838\) −27.4558 −0.948446
\(839\) 2.51472 0.0868177 0.0434089 0.999057i \(-0.486178\pi\)
0.0434089 + 0.999057i \(0.486178\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 10.5858 0.364810
\(843\) 7.31371 0.251898
\(844\) −28.3848 −0.977044
\(845\) −1.41421 −0.0486504
\(846\) 9.00000 0.309426
\(847\) 0 0
\(848\) 12.3137 0.422854
\(849\) 17.0711 0.585878
\(850\) −15.7279 −0.539463
\(851\) 46.0000 1.57686
\(852\) −1.00000 −0.0342594
\(853\) −17.6985 −0.605985 −0.302992 0.952993i \(-0.597986\pi\)
−0.302992 + 0.952993i \(0.597986\pi\)
\(854\) 0 0
\(855\) −4.24264 −0.145095
\(856\) 13.6569 0.466782
\(857\) −51.3848 −1.75527 −0.877635 0.479329i \(-0.840880\pi\)
−0.877635 + 0.479329i \(0.840880\pi\)
\(858\) 0.414214 0.0141410
\(859\) 0.201010 0.00685838 0.00342919 0.999994i \(-0.498908\pi\)
0.00342919 + 0.999994i \(0.498908\pi\)
\(860\) −6.82843 −0.232847
\(861\) 0 0
\(862\) 14.4853 0.493371
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 1.00000 0.0340207
\(865\) −28.7279 −0.976779
\(866\) −19.0000 −0.645646
\(867\) 10.4853 0.356099
\(868\) 0 0
\(869\) 5.51472 0.187074
\(870\) −7.07107 −0.239732
\(871\) −14.6569 −0.496629
\(872\) 1.65685 0.0561082
\(873\) 7.89949 0.267357
\(874\) −27.2132 −0.920500
\(875\) 0 0
\(876\) −1.07107 −0.0361880
\(877\) 23.4142 0.790642 0.395321 0.918543i \(-0.370633\pi\)
0.395321 + 0.918543i \(0.370633\pi\)
\(878\) −10.5858 −0.357253
\(879\) −18.7279 −0.631677
\(880\) −0.585786 −0.0197469
\(881\) −41.1127 −1.38512 −0.692561 0.721359i \(-0.743518\pi\)
−0.692561 + 0.721359i \(0.743518\pi\)
\(882\) 0 0
\(883\) −0.928932 −0.0312611 −0.0156305 0.999878i \(-0.504976\pi\)
−0.0156305 + 0.999878i \(0.504976\pi\)
\(884\) 5.24264 0.176329
\(885\) −10.7279 −0.360615
\(886\) 22.7279 0.763559
\(887\) 41.3137 1.38718 0.693589 0.720371i \(-0.256028\pi\)
0.693589 + 0.720371i \(0.256028\pi\)
\(888\) −5.07107 −0.170174
\(889\) 0 0
\(890\) 8.82843 0.295930
\(891\) 0.414214 0.0138767
\(892\) 21.3848 0.716015
\(893\) 27.0000 0.903521
\(894\) 15.8995 0.531759
\(895\) 17.4558 0.583485
\(896\) 0 0
\(897\) −9.07107 −0.302874
\(898\) −33.9411 −1.13263
\(899\) 34.1421 1.13870
\(900\) −3.00000 −0.100000
\(901\) 64.5563 2.15068
\(902\) 1.89949 0.0632463
\(903\) 0 0
\(904\) −6.07107 −0.201921
\(905\) 16.3848 0.544648
\(906\) −7.58579 −0.252021
\(907\) −32.1005 −1.06588 −0.532940 0.846153i \(-0.678913\pi\)
−0.532940 + 0.846153i \(0.678913\pi\)
\(908\) −26.1421 −0.867557
\(909\) −1.17157 −0.0388586
\(910\) 0 0
\(911\) 54.0000 1.78910 0.894550 0.446968i \(-0.147496\pi\)
0.894550 + 0.446968i \(0.147496\pi\)
\(912\) 3.00000 0.0993399
\(913\) −3.17157 −0.104964
\(914\) −15.2132 −0.503208
\(915\) 7.41421 0.245106
\(916\) −7.17157 −0.236955
\(917\) 0 0
\(918\) 5.24264 0.173033
\(919\) −25.7990 −0.851030 −0.425515 0.904951i \(-0.639907\pi\)
−0.425515 + 0.904951i \(0.639907\pi\)
\(920\) 12.8284 0.422941
\(921\) −19.6274 −0.646745
\(922\) −33.4558 −1.10181
\(923\) −1.00000 −0.0329154
\(924\) 0 0
\(925\) 15.2132 0.500207
\(926\) −37.9411 −1.24682
\(927\) 6.48528 0.213005
\(928\) 5.00000 0.164133
\(929\) 18.2843 0.599887 0.299944 0.953957i \(-0.403032\pi\)
0.299944 + 0.953957i \(0.403032\pi\)
\(930\) −9.65685 −0.316661
\(931\) 0 0
\(932\) −3.72792 −0.122112
\(933\) −6.58579 −0.215609
\(934\) −0.242641 −0.00793945
\(935\) −3.07107 −0.100435
\(936\) 1.00000 0.0326860
\(937\) 40.5980 1.32628 0.663139 0.748496i \(-0.269224\pi\)
0.663139 + 0.748496i \(0.269224\pi\)
\(938\) 0 0
\(939\) 18.1421 0.592046
\(940\) −12.7279 −0.415139
\(941\) −20.9706 −0.683621 −0.341810 0.939769i \(-0.611040\pi\)
−0.341810 + 0.939769i \(0.611040\pi\)
\(942\) 7.92893 0.258339
\(943\) −41.5980 −1.35462
\(944\) 7.58579 0.246896
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) 17.2426 0.560311 0.280155 0.959955i \(-0.409614\pi\)
0.280155 + 0.959955i \(0.409614\pi\)
\(948\) 13.3137 0.432409
\(949\) −1.07107 −0.0347683
\(950\) −9.00000 −0.291999
\(951\) 0.828427 0.0268636
\(952\) 0 0
\(953\) −10.2132 −0.330838 −0.165419 0.986223i \(-0.552898\pi\)
−0.165419 + 0.986223i \(0.552898\pi\)
\(954\) 12.3137 0.398671
\(955\) −3.31371 −0.107229
\(956\) −2.51472 −0.0813318
\(957\) 2.07107 0.0669481
\(958\) −31.1421 −1.00616
\(959\) 0 0
\(960\) −1.41421 −0.0456435
\(961\) 15.6274 0.504110
\(962\) −5.07107 −0.163498
\(963\) 13.6569 0.440086
\(964\) −16.4853 −0.530955
\(965\) 22.2843 0.717356
\(966\) 0 0
\(967\) 9.38478 0.301794 0.150897 0.988549i \(-0.451784\pi\)
0.150897 + 0.988549i \(0.451784\pi\)
\(968\) −10.8284 −0.348039
\(969\) 15.7279 0.505254
\(970\) −11.1716 −0.358698
\(971\) 57.5563 1.84707 0.923536 0.383513i \(-0.125286\pi\)
0.923536 + 0.383513i \(0.125286\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −23.7279 −0.760292
\(975\) −3.00000 −0.0960769
\(976\) −5.24264 −0.167813
\(977\) 2.48528 0.0795112 0.0397556 0.999209i \(-0.487342\pi\)
0.0397556 + 0.999209i \(0.487342\pi\)
\(978\) 17.1421 0.548145
\(979\) −2.58579 −0.0826421
\(980\) 0 0
\(981\) 1.65685 0.0528993
\(982\) −26.2843 −0.838765
\(983\) −22.1716 −0.707163 −0.353582 0.935404i \(-0.615036\pi\)
−0.353582 + 0.935404i \(0.615036\pi\)
\(984\) 4.58579 0.146190
\(985\) −13.5147 −0.430615
\(986\) 26.2132 0.834798
\(987\) 0 0
\(988\) 3.00000 0.0954427
\(989\) −43.7990 −1.39273
\(990\) −0.585786 −0.0186175
\(991\) −7.31371 −0.232328 −0.116164 0.993230i \(-0.537060\pi\)
−0.116164 + 0.993230i \(0.537060\pi\)
\(992\) 6.82843 0.216803
\(993\) −22.4853 −0.713549
\(994\) 0 0
\(995\) −14.9706 −0.474599
\(996\) −7.65685 −0.242617
\(997\) 6.69848 0.212143 0.106072 0.994358i \(-0.466173\pi\)
0.106072 + 0.994358i \(0.466173\pi\)
\(998\) −30.0000 −0.949633
\(999\) −5.07107 −0.160441
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 3822.2.a.bs.1.1 2
7.2 even 3 546.2.i.h.235.2 yes 4
7.4 even 3 546.2.i.h.79.2 4
7.6 odd 2 3822.2.a.bp.1.2 2
21.2 odd 6 1638.2.j.n.235.1 4
21.11 odd 6 1638.2.j.n.1171.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.h.79.2 4 7.4 even 3
546.2.i.h.235.2 yes 4 7.2 even 3
1638.2.j.n.235.1 4 21.2 odd 6
1638.2.j.n.1171.1 4 21.11 odd 6
3822.2.a.bp.1.2 2 7.6 odd 2
3822.2.a.bs.1.1 2 1.1 even 1 trivial