Properties

Label 5445.2.a.bn.1.4
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.27648.1
Defining polynomial: \(x^{4} - 6 x^{2} + 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.33441\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.33441 q^{2} +3.44949 q^{4} +1.00000 q^{5} -1.04930 q^{7} +3.38371 q^{8} +O(q^{10})\) \(q+2.33441 q^{2} +3.44949 q^{4} +1.00000 q^{5} -1.04930 q^{7} +3.38371 q^{8} +2.33441 q^{10} +5.71812 q^{13} -2.44949 q^{14} +1.00000 q^{16} -1.04930 q^{17} +4.66883 q^{19} +3.44949 q^{20} +4.89898 q^{23} +1.00000 q^{25} +13.3485 q^{26} -3.61953 q^{28} +2.57024 q^{29} -2.00000 q^{31} -4.43300 q^{32} -2.44949 q^{34} -1.04930 q^{35} -6.89898 q^{37} +10.8990 q^{38} +3.38371 q^{40} +6.76742 q^{41} +1.04930 q^{43} +11.4362 q^{46} -9.79796 q^{47} -5.89898 q^{49} +2.33441 q^{50} +19.7246 q^{52} +10.8990 q^{53} -3.55051 q^{56} +6.00000 q^{58} +10.8990 q^{59} -7.23907 q^{61} -4.66883 q^{62} -12.3485 q^{64} +5.71812 q^{65} +8.89898 q^{67} -3.61953 q^{68} -2.44949 q^{70} +1.10102 q^{71} +1.52094 q^{73} -16.1051 q^{74} +16.1051 q^{76} +9.80930 q^{79} +1.00000 q^{80} +15.7980 q^{82} -3.61953 q^{83} -1.04930 q^{85} +2.44949 q^{86} +3.79796 q^{89} -6.00000 q^{91} +16.8990 q^{92} -22.8725 q^{94} +4.66883 q^{95} +16.6969 q^{97} -13.7707 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + 4q^{5} + O(q^{10}) \) \( 4q + 4q^{4} + 4q^{5} + 4q^{16} + 4q^{20} + 4q^{25} + 24q^{26} - 8q^{31} - 8q^{37} + 24q^{38} - 4q^{49} + 24q^{53} - 24q^{56} + 24q^{58} + 24q^{59} - 20q^{64} + 16q^{67} + 24q^{71} + 4q^{80} + 24q^{82} - 24q^{89} - 24q^{91} + 48q^{92} + 8q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.33441 1.65068 0.825340 0.564636i \(-0.190983\pi\)
0.825340 + 0.564636i \(0.190983\pi\)
\(3\) 0 0
\(4\) 3.44949 1.72474
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.04930 −0.396596 −0.198298 0.980142i \(-0.563541\pi\)
−0.198298 + 0.980142i \(0.563541\pi\)
\(8\) 3.38371 1.19632
\(9\) 0 0
\(10\) 2.33441 0.738207
\(11\) 0 0
\(12\) 0 0
\(13\) 5.71812 1.58592 0.792961 0.609272i \(-0.208538\pi\)
0.792961 + 0.609272i \(0.208538\pi\)
\(14\) −2.44949 −0.654654
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.04930 −0.254491 −0.127246 0.991871i \(-0.540614\pi\)
−0.127246 + 0.991871i \(0.540614\pi\)
\(18\) 0 0
\(19\) 4.66883 1.07110 0.535551 0.844503i \(-0.320104\pi\)
0.535551 + 0.844503i \(0.320104\pi\)
\(20\) 3.44949 0.771329
\(21\) 0 0
\(22\) 0 0
\(23\) 4.89898 1.02151 0.510754 0.859727i \(-0.329366\pi\)
0.510754 + 0.859727i \(0.329366\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 13.3485 2.61785
\(27\) 0 0
\(28\) −3.61953 −0.684027
\(29\) 2.57024 0.477281 0.238641 0.971108i \(-0.423298\pi\)
0.238641 + 0.971108i \(0.423298\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −4.43300 −0.783652
\(33\) 0 0
\(34\) −2.44949 −0.420084
\(35\) −1.04930 −0.177363
\(36\) 0 0
\(37\) −6.89898 −1.13419 −0.567093 0.823654i \(-0.691932\pi\)
−0.567093 + 0.823654i \(0.691932\pi\)
\(38\) 10.8990 1.76805
\(39\) 0 0
\(40\) 3.38371 0.535011
\(41\) 6.76742 1.05689 0.528447 0.848967i \(-0.322775\pi\)
0.528447 + 0.848967i \(0.322775\pi\)
\(42\) 0 0
\(43\) 1.04930 0.160016 0.0800080 0.996794i \(-0.474505\pi\)
0.0800080 + 0.996794i \(0.474505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 11.4362 1.68618
\(47\) −9.79796 −1.42918 −0.714590 0.699544i \(-0.753387\pi\)
−0.714590 + 0.699544i \(0.753387\pi\)
\(48\) 0 0
\(49\) −5.89898 −0.842711
\(50\) 2.33441 0.330136
\(51\) 0 0
\(52\) 19.7246 2.73531
\(53\) 10.8990 1.49709 0.748545 0.663084i \(-0.230753\pi\)
0.748545 + 0.663084i \(0.230753\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.55051 −0.474457
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 10.8990 1.41893 0.709463 0.704743i \(-0.248937\pi\)
0.709463 + 0.704743i \(0.248937\pi\)
\(60\) 0 0
\(61\) −7.23907 −0.926867 −0.463434 0.886132i \(-0.653383\pi\)
−0.463434 + 0.886132i \(0.653383\pi\)
\(62\) −4.66883 −0.592942
\(63\) 0 0
\(64\) −12.3485 −1.54356
\(65\) 5.71812 0.709246
\(66\) 0 0
\(67\) 8.89898 1.08718 0.543592 0.839350i \(-0.317064\pi\)
0.543592 + 0.839350i \(0.317064\pi\)
\(68\) −3.61953 −0.438933
\(69\) 0 0
\(70\) −2.44949 −0.292770
\(71\) 1.10102 0.130667 0.0653335 0.997863i \(-0.479189\pi\)
0.0653335 + 0.997863i \(0.479189\pi\)
\(72\) 0 0
\(73\) 1.52094 0.178013 0.0890064 0.996031i \(-0.471631\pi\)
0.0890064 + 0.996031i \(0.471631\pi\)
\(74\) −16.1051 −1.87218
\(75\) 0 0
\(76\) 16.1051 1.84738
\(77\) 0 0
\(78\) 0 0
\(79\) 9.80930 1.10363 0.551816 0.833966i \(-0.313935\pi\)
0.551816 + 0.833966i \(0.313935\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 15.7980 1.74459
\(83\) −3.61953 −0.397295 −0.198648 0.980071i \(-0.563655\pi\)
−0.198648 + 0.980071i \(0.563655\pi\)
\(84\) 0 0
\(85\) −1.04930 −0.113812
\(86\) 2.44949 0.264135
\(87\) 0 0
\(88\) 0 0
\(89\) 3.79796 0.402583 0.201291 0.979531i \(-0.435486\pi\)
0.201291 + 0.979531i \(0.435486\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 16.8990 1.76184
\(93\) 0 0
\(94\) −22.8725 −2.35912
\(95\) 4.66883 0.479012
\(96\) 0 0
\(97\) 16.6969 1.69532 0.847659 0.530542i \(-0.178012\pi\)
0.847659 + 0.530542i \(0.178012\pi\)
\(98\) −13.7707 −1.39105
\(99\) 0 0
\(100\) 3.44949 0.344949
\(101\) −18.2037 −1.81133 −0.905666 0.423991i \(-0.860629\pi\)
−0.905666 + 0.423991i \(0.860629\pi\)
\(102\) 0 0
\(103\) 12.8990 1.27097 0.635487 0.772111i \(-0.280799\pi\)
0.635487 + 0.772111i \(0.280799\pi\)
\(104\) 19.3485 1.89727
\(105\) 0 0
\(106\) 25.4427 2.47122
\(107\) −7.81671 −0.755670 −0.377835 0.925873i \(-0.623331\pi\)
−0.377835 + 0.925873i \(0.623331\pi\)
\(108\) 0 0
\(109\) −13.5348 −1.29640 −0.648201 0.761469i \(-0.724478\pi\)
−0.648201 + 0.761469i \(0.724478\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.04930 −0.0991491
\(113\) 1.10102 0.103575 0.0517876 0.998658i \(-0.483508\pi\)
0.0517876 + 0.998658i \(0.483508\pi\)
\(114\) 0 0
\(115\) 4.89898 0.456832
\(116\) 8.86601 0.823188
\(117\) 0 0
\(118\) 25.4427 2.34219
\(119\) 1.10102 0.100930
\(120\) 0 0
\(121\) 0 0
\(122\) −16.8990 −1.52996
\(123\) 0 0
\(124\) −6.89898 −0.619547
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −19.7246 −1.75028 −0.875138 0.483873i \(-0.839230\pi\)
−0.875138 + 0.483873i \(0.839230\pi\)
\(128\) −19.9604 −1.76427
\(129\) 0 0
\(130\) 13.3485 1.17074
\(131\) 15.6334 1.36590 0.682949 0.730466i \(-0.260697\pi\)
0.682949 + 0.730466i \(0.260697\pi\)
\(132\) 0 0
\(133\) −4.89898 −0.424795
\(134\) 20.7739 1.79459
\(135\) 0 0
\(136\) −3.55051 −0.304454
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 10.9646 0.930005 0.465003 0.885309i \(-0.346053\pi\)
0.465003 + 0.885309i \(0.346053\pi\)
\(140\) −3.61953 −0.305906
\(141\) 0 0
\(142\) 2.57024 0.215690
\(143\) 0 0
\(144\) 0 0
\(145\) 2.57024 0.213447
\(146\) 3.55051 0.293842
\(147\) 0 0
\(148\) −23.7980 −1.95618
\(149\) 0.471647 0.0386389 0.0193194 0.999813i \(-0.493850\pi\)
0.0193194 + 0.999813i \(0.493850\pi\)
\(150\) 0 0
\(151\) 0.471647 0.0383821 0.0191911 0.999816i \(-0.493891\pi\)
0.0191911 + 0.999816i \(0.493891\pi\)
\(152\) 15.7980 1.28138
\(153\) 0 0
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 7.79796 0.622345 0.311172 0.950353i \(-0.399278\pi\)
0.311172 + 0.950353i \(0.399278\pi\)
\(158\) 22.8990 1.82174
\(159\) 0 0
\(160\) −4.43300 −0.350460
\(161\) −5.14048 −0.405126
\(162\) 0 0
\(163\) −6.69694 −0.524545 −0.262272 0.964994i \(-0.584472\pi\)
−0.262272 + 0.964994i \(0.584472\pi\)
\(164\) 23.3441 1.82287
\(165\) 0 0
\(166\) −8.44949 −0.655808
\(167\) 12.0139 0.929663 0.464832 0.885399i \(-0.346115\pi\)
0.464832 + 0.885399i \(0.346115\pi\)
\(168\) 0 0
\(169\) 19.6969 1.51515
\(170\) −2.44949 −0.187867
\(171\) 0 0
\(172\) 3.61953 0.275987
\(173\) 13.4288 1.02098 0.510488 0.859885i \(-0.329465\pi\)
0.510488 + 0.859885i \(0.329465\pi\)
\(174\) 0 0
\(175\) −1.04930 −0.0793193
\(176\) 0 0
\(177\) 0 0
\(178\) 8.86601 0.664536
\(179\) 21.7980 1.62926 0.814628 0.579984i \(-0.196941\pi\)
0.814628 + 0.579984i \(0.196941\pi\)
\(180\) 0 0
\(181\) 3.10102 0.230497 0.115249 0.993337i \(-0.463234\pi\)
0.115249 + 0.993337i \(0.463234\pi\)
\(182\) −14.0065 −1.03823
\(183\) 0 0
\(184\) 16.5767 1.22205
\(185\) −6.89898 −0.507223
\(186\) 0 0
\(187\) 0 0
\(188\) −33.7980 −2.46497
\(189\) 0 0
\(190\) 10.8990 0.790695
\(191\) −19.5959 −1.41791 −0.708955 0.705253i \(-0.750833\pi\)
−0.708955 + 0.705253i \(0.750833\pi\)
\(192\) 0 0
\(193\) −20.1963 −1.45376 −0.726879 0.686765i \(-0.759030\pi\)
−0.726879 + 0.686765i \(0.759030\pi\)
\(194\) 38.9776 2.79843
\(195\) 0 0
\(196\) −20.3485 −1.45346
\(197\) −19.7246 −1.40532 −0.702660 0.711526i \(-0.748005\pi\)
−0.702660 + 0.711526i \(0.748005\pi\)
\(198\) 0 0
\(199\) −17.7980 −1.26166 −0.630832 0.775920i \(-0.717286\pi\)
−0.630832 + 0.775920i \(0.717286\pi\)
\(200\) 3.38371 0.239264
\(201\) 0 0
\(202\) −42.4949 −2.98993
\(203\) −2.69694 −0.189288
\(204\) 0 0
\(205\) 6.76742 0.472657
\(206\) 30.1116 2.09797
\(207\) 0 0
\(208\) 5.71812 0.396481
\(209\) 0 0
\(210\) 0 0
\(211\) 14.0065 0.964246 0.482123 0.876103i \(-0.339866\pi\)
0.482123 + 0.876103i \(0.339866\pi\)
\(212\) 37.5959 2.58210
\(213\) 0 0
\(214\) −18.2474 −1.24737
\(215\) 1.04930 0.0715613
\(216\) 0 0
\(217\) 2.09859 0.142462
\(218\) −31.5959 −2.13995
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −1.79796 −0.120400 −0.0602001 0.998186i \(-0.519174\pi\)
−0.0602001 + 0.998186i \(0.519174\pi\)
\(224\) 4.65153 0.310793
\(225\) 0 0
\(226\) 2.57024 0.170970
\(227\) 1.52094 0.100949 0.0504743 0.998725i \(-0.483927\pi\)
0.0504743 + 0.998725i \(0.483927\pi\)
\(228\) 0 0
\(229\) −21.5959 −1.42710 −0.713549 0.700605i \(-0.752913\pi\)
−0.713549 + 0.700605i \(0.752913\pi\)
\(230\) 11.4362 0.754084
\(231\) 0 0
\(232\) 8.69694 0.570982
\(233\) −10.3870 −0.680472 −0.340236 0.940340i \(-0.610507\pi\)
−0.340236 + 0.940340i \(0.610507\pi\)
\(234\) 0 0
\(235\) −9.79796 −0.639148
\(236\) 37.5959 2.44729
\(237\) 0 0
\(238\) 2.57024 0.166604
\(239\) −25.9144 −1.67626 −0.838131 0.545469i \(-0.816352\pi\)
−0.838131 + 0.545469i \(0.816352\pi\)
\(240\) 0 0
\(241\) −16.5767 −1.06780 −0.533900 0.845547i \(-0.679274\pi\)
−0.533900 + 0.845547i \(0.679274\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −24.9711 −1.59861
\(245\) −5.89898 −0.376872
\(246\) 0 0
\(247\) 26.6969 1.69869
\(248\) −6.76742 −0.429732
\(249\) 0 0
\(250\) 2.33441 0.147641
\(251\) 13.1010 0.826929 0.413465 0.910520i \(-0.364319\pi\)
0.413465 + 0.910520i \(0.364319\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −46.0454 −2.88915
\(255\) 0 0
\(256\) −21.8990 −1.36869
\(257\) −25.5959 −1.59663 −0.798315 0.602240i \(-0.794275\pi\)
−0.798315 + 0.602240i \(0.794275\pi\)
\(258\) 0 0
\(259\) 7.23907 0.449814
\(260\) 19.7246 1.22327
\(261\) 0 0
\(262\) 36.4949 2.25466
\(263\) −20.1963 −1.24535 −0.622677 0.782479i \(-0.713955\pi\)
−0.622677 + 0.782479i \(0.713955\pi\)
\(264\) 0 0
\(265\) 10.8990 0.669519
\(266\) −11.4362 −0.701201
\(267\) 0 0
\(268\) 30.6969 1.87511
\(269\) −7.59592 −0.463131 −0.231566 0.972819i \(-0.574385\pi\)
−0.231566 + 0.972819i \(0.574385\pi\)
\(270\) 0 0
\(271\) −22.4008 −1.36075 −0.680377 0.732862i \(-0.738184\pi\)
−0.680377 + 0.732862i \(0.738184\pi\)
\(272\) −1.04930 −0.0636229
\(273\) 0 0
\(274\) 42.0195 2.53849
\(275\) 0 0
\(276\) 0 0
\(277\) 26.4920 1.59175 0.795876 0.605460i \(-0.207011\pi\)
0.795876 + 0.605460i \(0.207011\pi\)
\(278\) 25.5959 1.53514
\(279\) 0 0
\(280\) −3.55051 −0.212184
\(281\) −6.76742 −0.403710 −0.201855 0.979415i \(-0.564697\pi\)
−0.201855 + 0.979415i \(0.564697\pi\)
\(282\) 0 0
\(283\) −29.0623 −1.72757 −0.863786 0.503859i \(-0.831913\pi\)
−0.863786 + 0.503859i \(0.831913\pi\)
\(284\) 3.79796 0.225367
\(285\) 0 0
\(286\) 0 0
\(287\) −7.10102 −0.419160
\(288\) 0 0
\(289\) −15.8990 −0.935234
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) 5.24648 0.307027
\(293\) −29.0623 −1.69784 −0.848918 0.528525i \(-0.822745\pi\)
−0.848918 + 0.528525i \(0.822745\pi\)
\(294\) 0 0
\(295\) 10.8990 0.634563
\(296\) −23.3441 −1.35685
\(297\) 0 0
\(298\) 1.10102 0.0637804
\(299\) 28.0130 1.62003
\(300\) 0 0
\(301\) −1.10102 −0.0634618
\(302\) 1.10102 0.0633566
\(303\) 0 0
\(304\) 4.66883 0.267776
\(305\) −7.23907 −0.414508
\(306\) 0 0
\(307\) −15.5274 −0.886197 −0.443099 0.896473i \(-0.646121\pi\)
−0.443099 + 0.896473i \(0.646121\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.66883 −0.265172
\(311\) 22.8990 1.29848 0.649241 0.760583i \(-0.275087\pi\)
0.649241 + 0.760583i \(0.275087\pi\)
\(312\) 0 0
\(313\) −21.1010 −1.19270 −0.596350 0.802724i \(-0.703383\pi\)
−0.596350 + 0.802724i \(0.703383\pi\)
\(314\) 18.2037 1.02729
\(315\) 0 0
\(316\) 33.8371 1.90349
\(317\) −8.69694 −0.488469 −0.244234 0.969716i \(-0.578537\pi\)
−0.244234 + 0.969716i \(0.578537\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −12.3485 −0.690300
\(321\) 0 0
\(322\) −12.0000 −0.668734
\(323\) −4.89898 −0.272587
\(324\) 0 0
\(325\) 5.71812 0.317184
\(326\) −15.6334 −0.865856
\(327\) 0 0
\(328\) 22.8990 1.26438
\(329\) 10.2810 0.566807
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) −12.4855 −0.685233
\(333\) 0 0
\(334\) 28.0454 1.53458
\(335\) 8.89898 0.486203
\(336\) 0 0
\(337\) −10.8586 −0.591506 −0.295753 0.955265i \(-0.595570\pi\)
−0.295753 + 0.955265i \(0.595570\pi\)
\(338\) 45.9808 2.50103
\(339\) 0 0
\(340\) −3.61953 −0.196297
\(341\) 0 0
\(342\) 0 0
\(343\) 13.5348 0.730813
\(344\) 3.55051 0.191431
\(345\) 0 0
\(346\) 31.3485 1.68530
\(347\) 12.9572 0.695578 0.347789 0.937573i \(-0.386932\pi\)
0.347789 + 0.937573i \(0.386932\pi\)
\(348\) 0 0
\(349\) −7.23907 −0.387498 −0.193749 0.981051i \(-0.562065\pi\)
−0.193749 + 0.981051i \(0.562065\pi\)
\(350\) −2.44949 −0.130931
\(351\) 0 0
\(352\) 0 0
\(353\) −10.8990 −0.580094 −0.290047 0.957012i \(-0.593671\pi\)
−0.290047 + 0.957012i \(0.593671\pi\)
\(354\) 0 0
\(355\) 1.10102 0.0584361
\(356\) 13.1010 0.694353
\(357\) 0 0
\(358\) 50.8855 2.68938
\(359\) 31.0549 1.63901 0.819506 0.573070i \(-0.194248\pi\)
0.819506 + 0.573070i \(0.194248\pi\)
\(360\) 0 0
\(361\) 2.79796 0.147261
\(362\) 7.23907 0.380477
\(363\) 0 0
\(364\) −20.6969 −1.08481
\(365\) 1.52094 0.0796098
\(366\) 0 0
\(367\) −23.5959 −1.23170 −0.615848 0.787865i \(-0.711187\pi\)
−0.615848 + 0.787865i \(0.711187\pi\)
\(368\) 4.89898 0.255377
\(369\) 0 0
\(370\) −16.1051 −0.837263
\(371\) −11.4362 −0.593740
\(372\) 0 0
\(373\) 9.91530 0.513395 0.256698 0.966492i \(-0.417366\pi\)
0.256698 + 0.966492i \(0.417366\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −33.1534 −1.70976
\(377\) 14.6969 0.756931
\(378\) 0 0
\(379\) −23.5959 −1.21204 −0.606020 0.795449i \(-0.707235\pi\)
−0.606020 + 0.795449i \(0.707235\pi\)
\(380\) 16.1051 0.826173
\(381\) 0 0
\(382\) −45.7450 −2.34052
\(383\) 16.8990 0.863498 0.431749 0.901994i \(-0.357897\pi\)
0.431749 + 0.901994i \(0.357897\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −47.1464 −2.39969
\(387\) 0 0
\(388\) 57.5959 2.92399
\(389\) −31.5959 −1.60198 −0.800988 0.598680i \(-0.795692\pi\)
−0.800988 + 0.598680i \(0.795692\pi\)
\(390\) 0 0
\(391\) −5.14048 −0.259965
\(392\) −19.9604 −1.00815
\(393\) 0 0
\(394\) −46.0454 −2.31973
\(395\) 9.80930 0.493560
\(396\) 0 0
\(397\) 28.6969 1.44026 0.720129 0.693840i \(-0.244083\pi\)
0.720129 + 0.693840i \(0.244083\pi\)
\(398\) −41.5478 −2.08260
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 8.20204 0.409590 0.204795 0.978805i \(-0.434347\pi\)
0.204795 + 0.978805i \(0.434347\pi\)
\(402\) 0 0
\(403\) −11.4362 −0.569680
\(404\) −62.7934 −3.12409
\(405\) 0 0
\(406\) −6.29577 −0.312454
\(407\) 0 0
\(408\) 0 0
\(409\) 17.7320 0.876792 0.438396 0.898782i \(-0.355547\pi\)
0.438396 + 0.898782i \(0.355547\pi\)
\(410\) 15.7980 0.780206
\(411\) 0 0
\(412\) 44.4949 2.19211
\(413\) −11.4362 −0.562741
\(414\) 0 0
\(415\) −3.61953 −0.177676
\(416\) −25.3485 −1.24281
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −5.30306 −0.258455 −0.129228 0.991615i \(-0.541250\pi\)
−0.129228 + 0.991615i \(0.541250\pi\)
\(422\) 32.6969 1.59166
\(423\) 0 0
\(424\) 36.8790 1.79100
\(425\) −1.04930 −0.0508983
\(426\) 0 0
\(427\) 7.59592 0.367592
\(428\) −26.9637 −1.30334
\(429\) 0 0
\(430\) 2.44949 0.118125
\(431\) 4.19718 0.202171 0.101086 0.994878i \(-0.467768\pi\)
0.101086 + 0.994878i \(0.467768\pi\)
\(432\) 0 0
\(433\) −12.6969 −0.610176 −0.305088 0.952324i \(-0.598686\pi\)
−0.305088 + 0.952324i \(0.598686\pi\)
\(434\) 4.89898 0.235159
\(435\) 0 0
\(436\) −46.6883 −2.23596
\(437\) 22.8725 1.09414
\(438\) 0 0
\(439\) −39.9209 −1.90532 −0.952659 0.304039i \(-0.901665\pi\)
−0.952659 + 0.304039i \(0.901665\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −14.0065 −0.666221
\(443\) −2.69694 −0.128135 −0.0640677 0.997946i \(-0.520407\pi\)
−0.0640677 + 0.997946i \(0.520407\pi\)
\(444\) 0 0
\(445\) 3.79796 0.180041
\(446\) −4.19718 −0.198742
\(447\) 0 0
\(448\) 12.9572 0.612170
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 3.79796 0.178641
\(453\) 0 0
\(454\) 3.55051 0.166634
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) 27.4353 1.28337 0.641685 0.766968i \(-0.278236\pi\)
0.641685 + 0.766968i \(0.278236\pi\)
\(458\) −50.4138 −2.35568
\(459\) 0 0
\(460\) 16.8990 0.787919
\(461\) −13.0632 −0.608413 −0.304207 0.952606i \(-0.598391\pi\)
−0.304207 + 0.952606i \(0.598391\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 2.57024 0.119320
\(465\) 0 0
\(466\) −24.2474 −1.12324
\(467\) −4.89898 −0.226698 −0.113349 0.993555i \(-0.536158\pi\)
−0.113349 + 0.993555i \(0.536158\pi\)
\(468\) 0 0
\(469\) −9.33766 −0.431173
\(470\) −22.8725 −1.05503
\(471\) 0 0
\(472\) 36.8790 1.69749
\(473\) 0 0
\(474\) 0 0
\(475\) 4.66883 0.214221
\(476\) 3.79796 0.174079
\(477\) 0 0
\(478\) −60.4949 −2.76697
\(479\) 29.1683 1.33273 0.666366 0.745625i \(-0.267849\pi\)
0.666366 + 0.745625i \(0.267849\pi\)
\(480\) 0 0
\(481\) −39.4492 −1.79873
\(482\) −38.6969 −1.76260
\(483\) 0 0
\(484\) 0 0
\(485\) 16.6969 0.758169
\(486\) 0 0
\(487\) 8.89898 0.403251 0.201626 0.979463i \(-0.435378\pi\)
0.201626 + 0.979463i \(0.435378\pi\)
\(488\) −24.4949 −1.10883
\(489\) 0 0
\(490\) −13.7707 −0.622095
\(491\) −37.3506 −1.68561 −0.842805 0.538219i \(-0.819097\pi\)
−0.842805 + 0.538219i \(0.819097\pi\)
\(492\) 0 0
\(493\) −2.69694 −0.121464
\(494\) 62.3217 2.80399
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −1.15530 −0.0518221
\(498\) 0 0
\(499\) 13.7980 0.617681 0.308841 0.951114i \(-0.400059\pi\)
0.308841 + 0.951114i \(0.400059\pi\)
\(500\) 3.44949 0.154266
\(501\) 0 0
\(502\) 30.5832 1.36500
\(503\) 24.3934 1.08765 0.543825 0.839199i \(-0.316976\pi\)
0.543825 + 0.839199i \(0.316976\pi\)
\(504\) 0 0
\(505\) −18.2037 −0.810053
\(506\) 0 0
\(507\) 0 0
\(508\) −68.0398 −3.01878
\(509\) −9.79796 −0.434287 −0.217143 0.976140i \(-0.569674\pi\)
−0.217143 + 0.976140i \(0.569674\pi\)
\(510\) 0 0
\(511\) −1.59592 −0.0705993
\(512\) −11.2004 −0.494993
\(513\) 0 0
\(514\) −59.7515 −2.63552
\(515\) 12.8990 0.568397
\(516\) 0 0
\(517\) 0 0
\(518\) 16.8990 0.742499
\(519\) 0 0
\(520\) 19.3485 0.848487
\(521\) 23.3939 1.02490 0.512452 0.858716i \(-0.328737\pi\)
0.512452 + 0.858716i \(0.328737\pi\)
\(522\) 0 0
\(523\) −11.5422 −0.504707 −0.252354 0.967635i \(-0.581205\pi\)
−0.252354 + 0.967635i \(0.581205\pi\)
\(524\) 53.9274 2.35583
\(525\) 0 0
\(526\) −47.1464 −2.05568
\(527\) 2.09859 0.0914160
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 25.4427 1.10516
\(531\) 0 0
\(532\) −16.8990 −0.732664
\(533\) 38.6969 1.67615
\(534\) 0 0
\(535\) −7.81671 −0.337946
\(536\) 30.1116 1.30062
\(537\) 0 0
\(538\) −17.7320 −0.764482
\(539\) 0 0
\(540\) 0 0
\(541\) −31.2669 −1.34427 −0.672134 0.740430i \(-0.734622\pi\)
−0.672134 + 0.740430i \(0.734622\pi\)
\(542\) −52.2929 −2.24617
\(543\) 0 0
\(544\) 4.65153 0.199433
\(545\) −13.5348 −0.579769
\(546\) 0 0
\(547\) −32.1042 −1.37267 −0.686337 0.727283i \(-0.740783\pi\)
−0.686337 + 0.727283i \(0.740783\pi\)
\(548\) 62.0908 2.65239
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) −10.2929 −0.437697
\(554\) 61.8434 2.62747
\(555\) 0 0
\(556\) 37.8223 1.60402
\(557\) 5.24648 0.222300 0.111150 0.993804i \(-0.464547\pi\)
0.111150 + 0.993804i \(0.464547\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) −1.04930 −0.0443408
\(561\) 0 0
\(562\) −15.7980 −0.666397
\(563\) 6.87342 0.289680 0.144840 0.989455i \(-0.453733\pi\)
0.144840 + 0.989455i \(0.453733\pi\)
\(564\) 0 0
\(565\) 1.10102 0.0463203
\(566\) −67.8434 −2.85167
\(567\) 0 0
\(568\) 3.72553 0.156320
\(569\) −5.61212 −0.235272 −0.117636 0.993057i \(-0.537532\pi\)
−0.117636 + 0.993057i \(0.537532\pi\)
\(570\) 0 0
\(571\) −10.9646 −0.458854 −0.229427 0.973326i \(-0.573685\pi\)
−0.229427 + 0.973326i \(0.573685\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −16.5767 −0.691899
\(575\) 4.89898 0.204302
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) −37.1148 −1.54377
\(579\) 0 0
\(580\) 8.86601 0.368141
\(581\) 3.79796 0.157566
\(582\) 0 0
\(583\) 0 0
\(584\) 5.14643 0.212961
\(585\) 0 0
\(586\) −67.8434 −2.80258
\(587\) 14.6969 0.606608 0.303304 0.952894i \(-0.401910\pi\)
0.303304 + 0.952894i \(0.401910\pi\)
\(588\) 0 0
\(589\) −9.33766 −0.384751
\(590\) 25.4427 1.04746
\(591\) 0 0
\(592\) −6.89898 −0.283546
\(593\) −17.6260 −0.723814 −0.361907 0.932214i \(-0.617874\pi\)
−0.361907 + 0.932214i \(0.617874\pi\)
\(594\) 0 0
\(595\) 1.10102 0.0451374
\(596\) 1.62694 0.0666422
\(597\) 0 0
\(598\) 65.3939 2.67415
\(599\) 31.5959 1.29097 0.645487 0.763771i \(-0.276654\pi\)
0.645487 + 0.763771i \(0.276654\pi\)
\(600\) 0 0
\(601\) −3.04189 −0.124081 −0.0620405 0.998074i \(-0.519761\pi\)
−0.0620405 + 0.998074i \(0.519761\pi\)
\(602\) −2.57024 −0.104755
\(603\) 0 0
\(604\) 1.62694 0.0661994
\(605\) 0 0
\(606\) 0 0
\(607\) −15.5274 −0.630239 −0.315119 0.949052i \(-0.602045\pi\)
−0.315119 + 0.949052i \(0.602045\pi\)
\(608\) −20.6969 −0.839372
\(609\) 0 0
\(610\) −16.8990 −0.684220
\(611\) −56.0259 −2.26657
\(612\) 0 0
\(613\) −29.5339 −1.19286 −0.596432 0.802664i \(-0.703415\pi\)
−0.596432 + 0.802664i \(0.703415\pi\)
\(614\) −36.2474 −1.46283
\(615\) 0 0
\(616\) 0 0
\(617\) 13.1010 0.527427 0.263714 0.964601i \(-0.415053\pi\)
0.263714 + 0.964601i \(0.415053\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) −6.89898 −0.277070
\(621\) 0 0
\(622\) 53.4557 2.14338
\(623\) −3.98518 −0.159663
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −49.2585 −1.96877
\(627\) 0 0
\(628\) 26.8990 1.07339
\(629\) 7.23907 0.288640
\(630\) 0 0
\(631\) 21.3939 0.851677 0.425838 0.904799i \(-0.359979\pi\)
0.425838 + 0.904799i \(0.359979\pi\)
\(632\) 33.1918 1.32030
\(633\) 0 0
\(634\) −20.3023 −0.806306
\(635\) −19.7246 −0.782747
\(636\) 0 0
\(637\) −33.7311 −1.33647
\(638\) 0 0
\(639\) 0 0
\(640\) −19.9604 −0.789005
\(641\) −37.5959 −1.48495 −0.742475 0.669874i \(-0.766348\pi\)
−0.742475 + 0.669874i \(0.766348\pi\)
\(642\) 0 0
\(643\) 8.49490 0.335006 0.167503 0.985872i \(-0.446430\pi\)
0.167503 + 0.985872i \(0.446430\pi\)
\(644\) −17.7320 −0.698739
\(645\) 0 0
\(646\) −11.4362 −0.449953
\(647\) 40.8990 1.60790 0.803952 0.594694i \(-0.202727\pi\)
0.803952 + 0.594694i \(0.202727\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 13.3485 0.523570
\(651\) 0 0
\(652\) −23.1010 −0.904706
\(653\) 1.59592 0.0624531 0.0312265 0.999512i \(-0.490059\pi\)
0.0312265 + 0.999512i \(0.490059\pi\)
\(654\) 0 0
\(655\) 15.6334 0.610849
\(656\) 6.76742 0.264223
\(657\) 0 0
\(658\) 24.0000 0.935617
\(659\) 36.4073 1.41823 0.709114 0.705094i \(-0.249095\pi\)
0.709114 + 0.705094i \(0.249095\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) −23.3441 −0.907296
\(663\) 0 0
\(664\) −12.2474 −0.475293
\(665\) −4.89898 −0.189974
\(666\) 0 0
\(667\) 12.5915 0.487546
\(668\) 41.4418 1.60343
\(669\) 0 0
\(670\) 20.7739 0.802566
\(671\) 0 0
\(672\) 0 0
\(673\) 2.46424 0.0949894 0.0474947 0.998871i \(-0.484876\pi\)
0.0474947 + 0.998871i \(0.484876\pi\)
\(674\) −25.3485 −0.976387
\(675\) 0 0
\(676\) 67.9444 2.61325
\(677\) 30.2176 1.16136 0.580678 0.814134i \(-0.302788\pi\)
0.580678 + 0.814134i \(0.302788\pi\)
\(678\) 0 0
\(679\) −17.5200 −0.672357
\(680\) −3.55051 −0.136156
\(681\) 0 0
\(682\) 0 0
\(683\) 31.5959 1.20898 0.604492 0.796611i \(-0.293376\pi\)
0.604492 + 0.796611i \(0.293376\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 31.5959 1.20634
\(687\) 0 0
\(688\) 1.04930 0.0400040
\(689\) 62.3217 2.37427
\(690\) 0 0
\(691\) −18.2020 −0.692438 −0.346219 0.938154i \(-0.612535\pi\)
−0.346219 + 0.938154i \(0.612535\pi\)
\(692\) 46.3226 1.76092
\(693\) 0 0
\(694\) 30.2474 1.14818
\(695\) 10.9646 0.415911
\(696\) 0 0
\(697\) −7.10102 −0.268970
\(698\) −16.8990 −0.639636
\(699\) 0 0
\(700\) −3.61953 −0.136805
\(701\) −10.0213 −0.378499 −0.189250 0.981929i \(-0.560606\pi\)
−0.189250 + 0.981929i \(0.560606\pi\)
\(702\) 0 0
\(703\) −32.2102 −1.21483
\(704\) 0 0
\(705\) 0 0
\(706\) −25.4427 −0.957550
\(707\) 19.1010 0.718368
\(708\) 0 0
\(709\) 6.69694 0.251509 0.125754 0.992061i \(-0.459865\pi\)
0.125754 + 0.992061i \(0.459865\pi\)
\(710\) 2.57024 0.0964593
\(711\) 0 0
\(712\) 12.8512 0.481619
\(713\) −9.79796 −0.366936
\(714\) 0 0
\(715\) 0 0
\(716\) 75.1918 2.81005
\(717\) 0 0
\(718\) 72.4949 2.70549
\(719\) −14.2020 −0.529647 −0.264823 0.964297i \(-0.585314\pi\)
−0.264823 + 0.964297i \(0.585314\pi\)
\(720\) 0 0
\(721\) −13.5348 −0.504064
\(722\) 6.53160 0.243081
\(723\) 0 0
\(724\) 10.6969 0.397549
\(725\) 2.57024 0.0954562
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −20.3023 −0.752452
\(729\) 0 0
\(730\) 3.55051 0.131410
\(731\) −1.10102 −0.0407227
\(732\) 0 0
\(733\) −32.5758 −1.20321 −0.601607 0.798792i \(-0.705473\pi\)
−0.601607 + 0.798792i \(0.705473\pi\)
\(734\) −55.0826 −2.03314
\(735\) 0 0
\(736\) −21.7172 −0.800507
\(737\) 0 0
\(738\) 0 0
\(739\) −7.71071 −0.283643 −0.141822 0.989892i \(-0.545296\pi\)
−0.141822 + 0.989892i \(0.545296\pi\)
\(740\) −23.7980 −0.874830
\(741\) 0 0
\(742\) −26.6969 −0.980075
\(743\) 34.8864 1.27986 0.639929 0.768434i \(-0.278964\pi\)
0.639929 + 0.768434i \(0.278964\pi\)
\(744\) 0 0
\(745\) 0.471647 0.0172798
\(746\) 23.1464 0.847451
\(747\) 0 0
\(748\) 0 0
\(749\) 8.20204 0.299696
\(750\) 0 0
\(751\) 31.1918 1.13821 0.569103 0.822266i \(-0.307291\pi\)
0.569103 + 0.822266i \(0.307291\pi\)
\(752\) −9.79796 −0.357295
\(753\) 0 0
\(754\) 34.3087 1.24945
\(755\) 0.471647 0.0171650
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) −55.0826 −2.00069
\(759\) 0 0
\(760\) 15.7980 0.573052
\(761\) 22.4008 0.812030 0.406015 0.913866i \(-0.366918\pi\)
0.406015 + 0.913866i \(0.366918\pi\)
\(762\) 0 0
\(763\) 14.2020 0.514148
\(764\) −67.5959 −2.44553
\(765\) 0 0
\(766\) 39.4492 1.42536
\(767\) 62.3217 2.25031
\(768\) 0 0
\(769\) 11.4362 0.412402 0.206201 0.978510i \(-0.433890\pi\)
0.206201 + 0.978510i \(0.433890\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −69.6668 −2.50736
\(773\) −20.2020 −0.726617 −0.363308 0.931669i \(-0.618353\pi\)
−0.363308 + 0.931669i \(0.618353\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 56.4976 2.02815
\(777\) 0 0
\(778\) −73.7580 −2.64435
\(779\) 31.5959 1.13204
\(780\) 0 0
\(781\) 0 0
\(782\) −12.0000 −0.429119
\(783\) 0 0
\(784\) −5.89898 −0.210678
\(785\) 7.79796 0.278321
\(786\) 0 0
\(787\) −44.6957 −1.59323 −0.796615 0.604487i \(-0.793378\pi\)
−0.796615 + 0.604487i \(0.793378\pi\)
\(788\) −68.0398 −2.42382
\(789\) 0 0
\(790\) 22.8990 0.814709
\(791\) −1.15530 −0.0410776
\(792\) 0 0
\(793\) −41.3939 −1.46994
\(794\) 66.9905 2.37741
\(795\) 0 0
\(796\) −61.3939 −2.17605
\(797\) 1.10102 0.0390001 0.0195001 0.999810i \(-0.493793\pi\)
0.0195001 + 0.999810i \(0.493793\pi\)
\(798\) 0 0
\(799\) 10.2810 0.363714
\(800\) −4.43300 −0.156730
\(801\) 0 0
\(802\) 19.1470 0.676103
\(803\) 0 0
\(804\) 0 0
\(805\) −5.14048 −0.181178
\(806\) −26.6969 −0.940360
\(807\) 0 0
\(808\) −61.5959 −2.16694
\(809\) 31.5265 1.10841 0.554206 0.832379i \(-0.313022\pi\)
0.554206 + 0.832379i \(0.313022\pi\)
\(810\) 0 0
\(811\) −16.1051 −0.565526 −0.282763 0.959190i \(-0.591251\pi\)
−0.282763 + 0.959190i \(0.591251\pi\)
\(812\) −9.30306 −0.326473
\(813\) 0 0
\(814\) 0 0
\(815\) −6.69694 −0.234584
\(816\) 0 0
\(817\) 4.89898 0.171394
\(818\) 41.3939 1.44730
\(819\) 0 0
\(820\) 23.3441 0.815213
\(821\) −39.9209 −1.39325 −0.696624 0.717437i \(-0.745315\pi\)
−0.696624 + 0.717437i \(0.745315\pi\)
\(822\) 0 0
\(823\) 39.1010 1.36298 0.681488 0.731829i \(-0.261333\pi\)
0.681488 + 0.731829i \(0.261333\pi\)
\(824\) 43.6464 1.52049
\(825\) 0 0
\(826\) −26.6969 −0.928905
\(827\) −43.0688 −1.49765 −0.748824 0.662769i \(-0.769381\pi\)
−0.748824 + 0.662769i \(0.769381\pi\)
\(828\) 0 0
\(829\) −42.6969 −1.48293 −0.741463 0.670994i \(-0.765868\pi\)
−0.741463 + 0.670994i \(0.765868\pi\)
\(830\) −8.44949 −0.293286
\(831\) 0 0
\(832\) −70.6101 −2.44796
\(833\) 6.18977 0.214463
\(834\) 0 0
\(835\) 12.0139 0.415758
\(836\) 0 0
\(837\) 0 0
\(838\) 28.0130 0.967692
\(839\) −30.4949 −1.05280 −0.526400 0.850237i \(-0.676459\pi\)
−0.526400 + 0.850237i \(0.676459\pi\)
\(840\) 0 0
\(841\) −22.3939 −0.772203
\(842\) −12.3795 −0.426627
\(843\) 0 0
\(844\) 48.3152 1.66308
\(845\) 19.6969 0.677595
\(846\) 0 0
\(847\) 0 0
\(848\) 10.8990 0.374272
\(849\) 0 0
\(850\) −2.44949 −0.0840168
\(851\) −33.7980 −1.15858
\(852\) 0 0
\(853\) 15.9991 0.547798 0.273899 0.961758i \(-0.411687\pi\)
0.273899 + 0.961758i \(0.411687\pi\)
\(854\) 17.7320 0.606777
\(855\) 0 0
\(856\) −26.4495 −0.904025
\(857\) −18.5693 −0.634316 −0.317158 0.948373i \(-0.602728\pi\)
−0.317158 + 0.948373i \(0.602728\pi\)
\(858\) 0 0
\(859\) −34.0000 −1.16007 −0.580033 0.814593i \(-0.696960\pi\)
−0.580033 + 0.814593i \(0.696960\pi\)
\(860\) 3.61953 0.123425
\(861\) 0 0
\(862\) 9.79796 0.333720
\(863\) −21.3031 −0.725165 −0.362582 0.931952i \(-0.618105\pi\)
−0.362582 + 0.931952i \(0.618105\pi\)
\(864\) 0 0
\(865\) 13.4288 0.456594
\(866\) −29.6399 −1.00721
\(867\) 0 0
\(868\) 7.23907 0.245710
\(869\) 0 0
\(870\) 0 0
\(871\) 50.8855 1.72419
\(872\) −45.7980 −1.55091
\(873\) 0 0
\(874\) 53.3939 1.80607
\(875\) −1.04930 −0.0354727
\(876\) 0 0
\(877\) 16.2111 0.547409 0.273705 0.961814i \(-0.411751\pi\)
0.273705 + 0.961814i \(0.411751\pi\)
\(878\) −93.1918 −3.14507
\(879\) 0 0
\(880\) 0 0
\(881\) 21.7980 0.734392 0.367196 0.930144i \(-0.380318\pi\)
0.367196 + 0.930144i \(0.380318\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −20.6969 −0.696113
\(885\) 0 0
\(886\) −6.29577 −0.211511
\(887\) 29.5339 0.991652 0.495826 0.868422i \(-0.334865\pi\)
0.495826 + 0.868422i \(0.334865\pi\)
\(888\) 0 0
\(889\) 20.6969 0.694153
\(890\) 8.86601 0.297189
\(891\) 0 0
\(892\) −6.20204 −0.207660
\(893\) −45.7450 −1.53080
\(894\) 0 0
\(895\) 21.7980 0.728625
\(896\) 20.9444 0.699703
\(897\) 0 0
\(898\) 42.0195 1.40221
\(899\) −5.14048 −0.171444
\(900\) 0 0
\(901\) −11.4362 −0.380997
\(902\) 0 0
\(903\) 0 0
\(904\) 3.72553 0.123909
\(905\) 3.10102 0.103081
\(906\) 0 0
\(907\) −9.39388 −0.311919 −0.155959 0.987763i \(-0.549847\pi\)
−0.155959 + 0.987763i \(0.549847\pi\)
\(908\) 5.24648 0.174110
\(909\) 0 0
\(910\) −14.0065 −0.464310
\(911\) 14.2020 0.470535 0.235267 0.971931i \(-0.424403\pi\)
0.235267 + 0.971931i \(0.424403\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 64.0454 2.11843
\(915\) 0 0
\(916\) −74.4949 −2.46138
\(917\) −16.4041 −0.541711
\(918\) 0 0
\(919\) 19.3590 0.638593 0.319297 0.947655i \(-0.396553\pi\)
0.319297 + 0.947655i \(0.396553\pi\)
\(920\) 16.5767 0.546518
\(921\) 0 0
\(922\) −30.4949 −1.00430
\(923\) 6.29577 0.207228
\(924\) 0 0
\(925\) −6.89898 −0.226837
\(926\) 18.6753 0.613709
\(927\) 0 0
\(928\) −11.3939 −0.374022
\(929\) 53.3939 1.75180 0.875898 0.482496i \(-0.160270\pi\)
0.875898 + 0.482496i \(0.160270\pi\)
\(930\) 0 0
\(931\) −27.5413 −0.902630
\(932\) −35.8297 −1.17364
\(933\) 0 0
\(934\) −11.4362 −0.374205
\(935\) 0 0
\(936\) 0 0
\(937\) 32.7878 1.07113 0.535565 0.844494i \(-0.320099\pi\)
0.535565 + 0.844494i \(0.320099\pi\)
\(938\) −21.7980 −0.711729
\(939\) 0 0
\(940\) −33.7980 −1.10237
\(941\) −42.9628 −1.40055 −0.700273 0.713875i \(-0.746938\pi\)
−0.700273 + 0.713875i \(0.746938\pi\)
\(942\) 0 0
\(943\) 33.1534 1.07962
\(944\) 10.8990 0.354732
\(945\) 0 0
\(946\) 0 0
\(947\) 48.0000 1.55979 0.779895 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) 0 0
\(949\) 8.69694 0.282315
\(950\) 10.8990 0.353610
\(951\) 0 0
\(952\) 3.72553 0.120745
\(953\) 0.106000 0.00343369 0.00171684 0.999999i \(-0.499454\pi\)
0.00171684 + 0.999999i \(0.499454\pi\)
\(954\) 0 0
\(955\) −19.5959 −0.634109
\(956\) −89.3914 −2.89112
\(957\) 0 0
\(958\) 68.0908 2.19991
\(959\) −18.8873 −0.609903
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −92.0908 −2.96913
\(963\) 0 0
\(964\) −57.1812 −1.84168
\(965\) −20.1963 −0.650140
\(966\) 0 0
\(967\) 32.1042 1.03240 0.516200 0.856468i \(-0.327346\pi\)
0.516200 + 0.856468i \(0.327346\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 38.9776 1.25149
\(971\) 29.3939 0.943294 0.471647 0.881787i \(-0.343660\pi\)
0.471647 + 0.881787i \(0.343660\pi\)
\(972\) 0 0
\(973\) −11.5051 −0.368837
\(974\) 20.7739 0.665639
\(975\) 0 0
\(976\) −7.23907 −0.231717
\(977\) 33.1918 1.06190 0.530950 0.847403i \(-0.321835\pi\)
0.530950 + 0.847403i \(0.321835\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −20.3485 −0.650008
\(981\) 0 0
\(982\) −87.1918 −2.78240
\(983\) 2.69694 0.0860190 0.0430095 0.999075i \(-0.486305\pi\)
0.0430095 + 0.999075i \(0.486305\pi\)
\(984\) 0 0
\(985\) −19.7246 −0.628478
\(986\) −6.29577 −0.200498
\(987\) 0 0
\(988\) 92.0908 2.92980
\(989\) 5.14048 0.163458
\(990\) 0 0
\(991\) 25.1918 0.800245 0.400123 0.916462i \(-0.368968\pi\)
0.400123 + 0.916462i \(0.368968\pi\)
\(992\) 8.86601 0.281496
\(993\) 0 0
\(994\) −2.69694 −0.0855417
\(995\) −17.7980 −0.564233
\(996\) 0 0
\(997\) −13.1692 −0.417072 −0.208536 0.978015i \(-0.566870\pi\)
−0.208536 + 0.978015i \(0.566870\pi\)
\(998\) 32.2102 1.01959
\(999\) 0 0
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 5445.2.a.bn.1.4 yes 4
3.2 odd 2 5445.2.a.bm.1.1 4
11.10 odd 2 inner 5445.2.a.bn.1.1 yes 4
33.32 even 2 5445.2.a.bm.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5445.2.a.bm.1.1 4 3.2 odd 2
5445.2.a.bm.1.4 yes 4 33.32 even 2
5445.2.a.bn.1.1 yes 4 11.10 odd 2 inner
5445.2.a.bn.1.4 yes 4 1.1 even 1 trivial