Properties

Label 2541.2.a.z.1.1
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 2541.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.79129 q^{2} -1.00000 q^{3} +1.20871 q^{4} +3.00000 q^{5} +1.79129 q^{6} -1.00000 q^{7} +1.41742 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.79129 q^{2} -1.00000 q^{3} +1.20871 q^{4} +3.00000 q^{5} +1.79129 q^{6} -1.00000 q^{7} +1.41742 q^{8} +1.00000 q^{9} -5.37386 q^{10} -1.20871 q^{12} -1.00000 q^{13} +1.79129 q^{14} -3.00000 q^{15} -4.95644 q^{16} -7.58258 q^{17} -1.79129 q^{18} +6.58258 q^{19} +3.62614 q^{20} +1.00000 q^{21} -5.58258 q^{23} -1.41742 q^{24} +4.00000 q^{25} +1.79129 q^{26} -1.00000 q^{27} -1.20871 q^{28} +8.16515 q^{29} +5.37386 q^{30} +3.58258 q^{31} +6.04356 q^{32} +13.5826 q^{34} -3.00000 q^{35} +1.20871 q^{36} +1.00000 q^{37} -11.7913 q^{38} +1.00000 q^{39} +4.25227 q^{40} +11.1652 q^{41} -1.79129 q^{42} -1.58258 q^{43} +3.00000 q^{45} +10.0000 q^{46} +1.41742 q^{47} +4.95644 q^{48} +1.00000 q^{49} -7.16515 q^{50} +7.58258 q^{51} -1.20871 q^{52} -9.58258 q^{53} +1.79129 q^{54} -1.41742 q^{56} -6.58258 q^{57} -14.6261 q^{58} +4.58258 q^{59} -3.62614 q^{60} -10.0000 q^{61} -6.41742 q^{62} -1.00000 q^{63} -0.912878 q^{64} -3.00000 q^{65} +8.58258 q^{67} -9.16515 q^{68} +5.58258 q^{69} +5.37386 q^{70} +11.1652 q^{71} +1.41742 q^{72} -7.00000 q^{73} -1.79129 q^{74} -4.00000 q^{75} +7.95644 q^{76} -1.79129 q^{78} -7.16515 q^{79} -14.8693 q^{80} +1.00000 q^{81} -20.0000 q^{82} +11.5826 q^{83} +1.20871 q^{84} -22.7477 q^{85} +2.83485 q^{86} -8.16515 q^{87} +9.16515 q^{89} -5.37386 q^{90} +1.00000 q^{91} -6.74773 q^{92} -3.58258 q^{93} -2.53901 q^{94} +19.7477 q^{95} -6.04356 q^{96} -2.41742 q^{97} -1.79129 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + 3 q^{10} - 7 q^{12} - 2 q^{13} - q^{14} - 6 q^{15} + 13 q^{16} - 6 q^{17} + q^{18} + 4 q^{19} + 21 q^{20} + 2 q^{21} - 2 q^{23} - 12 q^{24} + 8 q^{25} - q^{26} - 2 q^{27} - 7 q^{28} - 2 q^{29} - 3 q^{30} - 2 q^{31} + 35 q^{32} + 18 q^{34} - 6 q^{35} + 7 q^{36} + 2 q^{37} - 19 q^{38} + 2 q^{39} + 36 q^{40} + 4 q^{41} + q^{42} + 6 q^{43} + 6 q^{45} + 20 q^{46} + 12 q^{47} - 13 q^{48} + 2 q^{49} + 4 q^{50} + 6 q^{51} - 7 q^{52} - 10 q^{53} - q^{54} - 12 q^{56} - 4 q^{57} - 43 q^{58} - 21 q^{60} - 20 q^{61} - 22 q^{62} - 2 q^{63} + 44 q^{64} - 6 q^{65} + 8 q^{67} + 2 q^{69} - 3 q^{70} + 4 q^{71} + 12 q^{72} - 14 q^{73} + q^{74} - 8 q^{75} - 7 q^{76} + q^{78} + 4 q^{79} + 39 q^{80} + 2 q^{81} - 40 q^{82} + 14 q^{83} + 7 q^{84} - 18 q^{85} + 24 q^{86} + 2 q^{87} + 3 q^{90} + 2 q^{91} + 14 q^{92} + 2 q^{93} + 27 q^{94} + 12 q^{95} - 35 q^{96} - 14 q^{97} + q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.79129 −1.26663 −0.633316 0.773893i \(-0.718307\pi\)
−0.633316 + 0.773893i \(0.718307\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.20871 0.604356
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 1.79129 0.731290
\(7\) −1.00000 −0.377964
\(8\) 1.41742 0.501135
\(9\) 1.00000 0.333333
\(10\) −5.37386 −1.69936
\(11\) 0 0
\(12\) −1.20871 −0.348925
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 1.79129 0.478742
\(15\) −3.00000 −0.774597
\(16\) −4.95644 −1.23911
\(17\) −7.58258 −1.83904 −0.919522 0.393038i \(-0.871424\pi\)
−0.919522 + 0.393038i \(0.871424\pi\)
\(18\) −1.79129 −0.422211
\(19\) 6.58258 1.51015 0.755073 0.655640i \(-0.227601\pi\)
0.755073 + 0.655640i \(0.227601\pi\)
\(20\) 3.62614 0.810829
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −5.58258 −1.16405 −0.582024 0.813172i \(-0.697739\pi\)
−0.582024 + 0.813172i \(0.697739\pi\)
\(24\) −1.41742 −0.289331
\(25\) 4.00000 0.800000
\(26\) 1.79129 0.351300
\(27\) −1.00000 −0.192450
\(28\) −1.20871 −0.228425
\(29\) 8.16515 1.51623 0.758115 0.652121i \(-0.226120\pi\)
0.758115 + 0.652121i \(0.226120\pi\)
\(30\) 5.37386 0.981129
\(31\) 3.58258 0.643450 0.321725 0.946833i \(-0.395737\pi\)
0.321725 + 0.946833i \(0.395737\pi\)
\(32\) 6.04356 1.06836
\(33\) 0 0
\(34\) 13.5826 2.32939
\(35\) −3.00000 −0.507093
\(36\) 1.20871 0.201452
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) −11.7913 −1.91280
\(39\) 1.00000 0.160128
\(40\) 4.25227 0.672343
\(41\) 11.1652 1.74370 0.871852 0.489770i \(-0.162919\pi\)
0.871852 + 0.489770i \(0.162919\pi\)
\(42\) −1.79129 −0.276402
\(43\) −1.58258 −0.241341 −0.120670 0.992693i \(-0.538504\pi\)
−0.120670 + 0.992693i \(0.538504\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 10.0000 1.47442
\(47\) 1.41742 0.206753 0.103376 0.994642i \(-0.467035\pi\)
0.103376 + 0.994642i \(0.467035\pi\)
\(48\) 4.95644 0.715400
\(49\) 1.00000 0.142857
\(50\) −7.16515 −1.01331
\(51\) 7.58258 1.06177
\(52\) −1.20871 −0.167618
\(53\) −9.58258 −1.31627 −0.658134 0.752901i \(-0.728654\pi\)
−0.658134 + 0.752901i \(0.728654\pi\)
\(54\) 1.79129 0.243763
\(55\) 0 0
\(56\) −1.41742 −0.189411
\(57\) −6.58258 −0.871883
\(58\) −14.6261 −1.92051
\(59\) 4.58258 0.596601 0.298300 0.954472i \(-0.403580\pi\)
0.298300 + 0.954472i \(0.403580\pi\)
\(60\) −3.62614 −0.468132
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −6.41742 −0.815014
\(63\) −1.00000 −0.125988
\(64\) −0.912878 −0.114110
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) 8.58258 1.04853 0.524264 0.851556i \(-0.324340\pi\)
0.524264 + 0.851556i \(0.324340\pi\)
\(68\) −9.16515 −1.11144
\(69\) 5.58258 0.672063
\(70\) 5.37386 0.642300
\(71\) 11.1652 1.32506 0.662530 0.749036i \(-0.269483\pi\)
0.662530 + 0.749036i \(0.269483\pi\)
\(72\) 1.41742 0.167045
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) −1.79129 −0.208233
\(75\) −4.00000 −0.461880
\(76\) 7.95644 0.912666
\(77\) 0 0
\(78\) −1.79129 −0.202823
\(79\) −7.16515 −0.806143 −0.403071 0.915169i \(-0.632057\pi\)
−0.403071 + 0.915169i \(0.632057\pi\)
\(80\) −14.8693 −1.66244
\(81\) 1.00000 0.111111
\(82\) −20.0000 −2.20863
\(83\) 11.5826 1.27135 0.635676 0.771956i \(-0.280721\pi\)
0.635676 + 0.771956i \(0.280721\pi\)
\(84\) 1.20871 0.131881
\(85\) −22.7477 −2.46734
\(86\) 2.83485 0.305690
\(87\) −8.16515 −0.875396
\(88\) 0 0
\(89\) 9.16515 0.971504 0.485752 0.874097i \(-0.338546\pi\)
0.485752 + 0.874097i \(0.338546\pi\)
\(90\) −5.37386 −0.566455
\(91\) 1.00000 0.104828
\(92\) −6.74773 −0.703499
\(93\) −3.58258 −0.371496
\(94\) −2.53901 −0.261879
\(95\) 19.7477 2.02607
\(96\) −6.04356 −0.616818
\(97\) −2.41742 −0.245452 −0.122726 0.992441i \(-0.539164\pi\)
−0.122726 + 0.992441i \(0.539164\pi\)
\(98\) −1.79129 −0.180947
\(99\) 0 0
\(100\) 4.83485 0.483485
\(101\) −11.5826 −1.15251 −0.576255 0.817270i \(-0.695486\pi\)
−0.576255 + 0.817270i \(0.695486\pi\)
\(102\) −13.5826 −1.34488
\(103\) −1.16515 −0.114806 −0.0574029 0.998351i \(-0.518282\pi\)
−0.0574029 + 0.998351i \(0.518282\pi\)
\(104\) −1.41742 −0.138990
\(105\) 3.00000 0.292770
\(106\) 17.1652 1.66723
\(107\) 12.5826 1.21640 0.608202 0.793782i \(-0.291891\pi\)
0.608202 + 0.793782i \(0.291891\pi\)
\(108\) −1.20871 −0.116308
\(109\) −3.58258 −0.343149 −0.171574 0.985171i \(-0.554885\pi\)
−0.171574 + 0.985171i \(0.554885\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 4.95644 0.468339
\(113\) 9.16515 0.862185 0.431092 0.902308i \(-0.358128\pi\)
0.431092 + 0.902308i \(0.358128\pi\)
\(114\) 11.7913 1.10436
\(115\) −16.7477 −1.56173
\(116\) 9.86932 0.916343
\(117\) −1.00000 −0.0924500
\(118\) −8.20871 −0.755673
\(119\) 7.58258 0.695094
\(120\) −4.25227 −0.388178
\(121\) 0 0
\(122\) 17.9129 1.62176
\(123\) −11.1652 −1.00673
\(124\) 4.33030 0.388873
\(125\) −3.00000 −0.268328
\(126\) 1.79129 0.159581
\(127\) 11.5826 1.02779 0.513894 0.857854i \(-0.328203\pi\)
0.513894 + 0.857854i \(0.328203\pi\)
\(128\) −10.4519 −0.923826
\(129\) 1.58258 0.139338
\(130\) 5.37386 0.471319
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) −6.58258 −0.570782
\(134\) −15.3739 −1.32810
\(135\) −3.00000 −0.258199
\(136\) −10.7477 −0.921610
\(137\) −11.5826 −0.989566 −0.494783 0.869016i \(-0.664752\pi\)
−0.494783 + 0.869016i \(0.664752\pi\)
\(138\) −10.0000 −0.851257
\(139\) −11.1652 −0.947016 −0.473508 0.880790i \(-0.657012\pi\)
−0.473508 + 0.880790i \(0.657012\pi\)
\(140\) −3.62614 −0.306464
\(141\) −1.41742 −0.119369
\(142\) −20.0000 −1.67836
\(143\) 0 0
\(144\) −4.95644 −0.413037
\(145\) 24.4955 2.03424
\(146\) 12.5390 1.03774
\(147\) −1.00000 −0.0824786
\(148\) 1.20871 0.0993555
\(149\) −6.16515 −0.505069 −0.252534 0.967588i \(-0.581264\pi\)
−0.252534 + 0.967588i \(0.581264\pi\)
\(150\) 7.16515 0.585032
\(151\) −3.58258 −0.291546 −0.145773 0.989318i \(-0.546567\pi\)
−0.145773 + 0.989318i \(0.546567\pi\)
\(152\) 9.33030 0.756787
\(153\) −7.58258 −0.613015
\(154\) 0 0
\(155\) 10.7477 0.863278
\(156\) 1.20871 0.0967744
\(157\) 19.1652 1.52955 0.764773 0.644300i \(-0.222851\pi\)
0.764773 + 0.644300i \(0.222851\pi\)
\(158\) 12.8348 1.02109
\(159\) 9.58258 0.759948
\(160\) 18.1307 1.43336
\(161\) 5.58258 0.439969
\(162\) −1.79129 −0.140737
\(163\) 8.58258 0.672239 0.336120 0.941819i \(-0.390885\pi\)
0.336120 + 0.941819i \(0.390885\pi\)
\(164\) 13.4955 1.05382
\(165\) 0 0
\(166\) −20.7477 −1.61034
\(167\) 4.74773 0.367390 0.183695 0.982983i \(-0.441194\pi\)
0.183695 + 0.982983i \(0.441194\pi\)
\(168\) 1.41742 0.109357
\(169\) −12.0000 −0.923077
\(170\) 40.7477 3.12521
\(171\) 6.58258 0.503382
\(172\) −1.91288 −0.145856
\(173\) 7.16515 0.544756 0.272378 0.962190i \(-0.412190\pi\)
0.272378 + 0.962190i \(0.412190\pi\)
\(174\) 14.6261 1.10880
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −4.58258 −0.344447
\(178\) −16.4174 −1.23054
\(179\) 14.3303 1.07110 0.535549 0.844504i \(-0.320105\pi\)
0.535549 + 0.844504i \(0.320105\pi\)
\(180\) 3.62614 0.270276
\(181\) −5.58258 −0.414950 −0.207475 0.978240i \(-0.566524\pi\)
−0.207475 + 0.978240i \(0.566524\pi\)
\(182\) −1.79129 −0.132779
\(183\) 10.0000 0.739221
\(184\) −7.91288 −0.583345
\(185\) 3.00000 0.220564
\(186\) 6.41742 0.470548
\(187\) 0 0
\(188\) 1.71326 0.124952
\(189\) 1.00000 0.0727393
\(190\) −35.3739 −2.56629
\(191\) 11.5826 0.838086 0.419043 0.907966i \(-0.362366\pi\)
0.419043 + 0.907966i \(0.362366\pi\)
\(192\) 0.912878 0.0658813
\(193\) 2.41742 0.174010 0.0870050 0.996208i \(-0.472270\pi\)
0.0870050 + 0.996208i \(0.472270\pi\)
\(194\) 4.33030 0.310898
\(195\) 3.00000 0.214834
\(196\) 1.20871 0.0863366
\(197\) −5.16515 −0.368002 −0.184001 0.982926i \(-0.558905\pi\)
−0.184001 + 0.982926i \(0.558905\pi\)
\(198\) 0 0
\(199\) −9.58258 −0.679291 −0.339645 0.940554i \(-0.610307\pi\)
−0.339645 + 0.940554i \(0.610307\pi\)
\(200\) 5.66970 0.400908
\(201\) −8.58258 −0.605368
\(202\) 20.7477 1.45980
\(203\) −8.16515 −0.573081
\(204\) 9.16515 0.641689
\(205\) 33.4955 2.33942
\(206\) 2.08712 0.145417
\(207\) −5.58258 −0.388016
\(208\) 4.95644 0.343667
\(209\) 0 0
\(210\) −5.37386 −0.370832
\(211\) 13.1652 0.906326 0.453163 0.891428i \(-0.350295\pi\)
0.453163 + 0.891428i \(0.350295\pi\)
\(212\) −11.5826 −0.795495
\(213\) −11.1652 −0.765024
\(214\) −22.5390 −1.54074
\(215\) −4.74773 −0.323792
\(216\) −1.41742 −0.0964435
\(217\) −3.58258 −0.243201
\(218\) 6.41742 0.434643
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 7.58258 0.510059
\(222\) 1.79129 0.120223
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) −6.04356 −0.403802
\(225\) 4.00000 0.266667
\(226\) −16.4174 −1.09207
\(227\) 22.0000 1.46019 0.730096 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) −7.95644 −0.526928
\(229\) 0.747727 0.0494112 0.0247056 0.999695i \(-0.492135\pi\)
0.0247056 + 0.999695i \(0.492135\pi\)
\(230\) 30.0000 1.97814
\(231\) 0 0
\(232\) 11.5735 0.759836
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 1.79129 0.117100
\(235\) 4.25227 0.277388
\(236\) 5.53901 0.360559
\(237\) 7.16515 0.465427
\(238\) −13.5826 −0.880428
\(239\) −16.5826 −1.07264 −0.536319 0.844015i \(-0.680186\pi\)
−0.536319 + 0.844015i \(0.680186\pi\)
\(240\) 14.8693 0.959810
\(241\) 10.1652 0.654795 0.327397 0.944887i \(-0.393828\pi\)
0.327397 + 0.944887i \(0.393828\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −12.0871 −0.773799
\(245\) 3.00000 0.191663
\(246\) 20.0000 1.27515
\(247\) −6.58258 −0.418839
\(248\) 5.07803 0.322455
\(249\) −11.5826 −0.734016
\(250\) 5.37386 0.339873
\(251\) 7.41742 0.468184 0.234092 0.972214i \(-0.424788\pi\)
0.234092 + 0.972214i \(0.424788\pi\)
\(252\) −1.20871 −0.0761417
\(253\) 0 0
\(254\) −20.7477 −1.30183
\(255\) 22.7477 1.42452
\(256\) 20.5481 1.28426
\(257\) 19.0000 1.18519 0.592594 0.805502i \(-0.298104\pi\)
0.592594 + 0.805502i \(0.298104\pi\)
\(258\) −2.83485 −0.176490
\(259\) −1.00000 −0.0621370
\(260\) −3.62614 −0.224883
\(261\) 8.16515 0.505410
\(262\) −28.6606 −1.77066
\(263\) 22.9129 1.41287 0.706434 0.707779i \(-0.250303\pi\)
0.706434 + 0.707779i \(0.250303\pi\)
\(264\) 0 0
\(265\) −28.7477 −1.76596
\(266\) 11.7913 0.722970
\(267\) −9.16515 −0.560898
\(268\) 10.3739 0.633685
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 5.37386 0.327043
\(271\) −5.41742 −0.329085 −0.164543 0.986370i \(-0.552615\pi\)
−0.164543 + 0.986370i \(0.552615\pi\)
\(272\) 37.5826 2.27878
\(273\) −1.00000 −0.0605228
\(274\) 20.7477 1.25342
\(275\) 0 0
\(276\) 6.74773 0.406165
\(277\) 19.1652 1.15152 0.575761 0.817618i \(-0.304706\pi\)
0.575761 + 0.817618i \(0.304706\pi\)
\(278\) 20.0000 1.19952
\(279\) 3.58258 0.214483
\(280\) −4.25227 −0.254122
\(281\) 27.3303 1.63039 0.815195 0.579187i \(-0.196630\pi\)
0.815195 + 0.579187i \(0.196630\pi\)
\(282\) 2.53901 0.151196
\(283\) −27.7477 −1.64943 −0.824716 0.565548i \(-0.808665\pi\)
−0.824716 + 0.565548i \(0.808665\pi\)
\(284\) 13.4955 0.800808
\(285\) −19.7477 −1.16975
\(286\) 0 0
\(287\) −11.1652 −0.659058
\(288\) 6.04356 0.356120
\(289\) 40.4955 2.38209
\(290\) −43.8784 −2.57663
\(291\) 2.41742 0.141712
\(292\) −8.46099 −0.495142
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 1.79129 0.104470
\(295\) 13.7477 0.800424
\(296\) 1.41742 0.0823861
\(297\) 0 0
\(298\) 11.0436 0.639736
\(299\) 5.58258 0.322849
\(300\) −4.83485 −0.279140
\(301\) 1.58258 0.0912181
\(302\) 6.41742 0.369281
\(303\) 11.5826 0.665402
\(304\) −32.6261 −1.87124
\(305\) −30.0000 −1.71780
\(306\) 13.5826 0.776464
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 1.16515 0.0662831
\(310\) −19.2523 −1.09346
\(311\) 14.3303 0.812597 0.406298 0.913740i \(-0.366819\pi\)
0.406298 + 0.913740i \(0.366819\pi\)
\(312\) 1.41742 0.0802458
\(313\) 19.5826 1.10687 0.553436 0.832891i \(-0.313316\pi\)
0.553436 + 0.832891i \(0.313316\pi\)
\(314\) −34.3303 −1.93737
\(315\) −3.00000 −0.169031
\(316\) −8.66061 −0.487197
\(317\) −22.4174 −1.25909 −0.629544 0.776965i \(-0.716758\pi\)
−0.629544 + 0.776965i \(0.716758\pi\)
\(318\) −17.1652 −0.962574
\(319\) 0 0
\(320\) −2.73864 −0.153094
\(321\) −12.5826 −0.702291
\(322\) −10.0000 −0.557278
\(323\) −49.9129 −2.77723
\(324\) 1.20871 0.0671507
\(325\) −4.00000 −0.221880
\(326\) −15.3739 −0.851480
\(327\) 3.58258 0.198117
\(328\) 15.8258 0.873831
\(329\) −1.41742 −0.0781451
\(330\) 0 0
\(331\) −3.16515 −0.173972 −0.0869862 0.996210i \(-0.527724\pi\)
−0.0869862 + 0.996210i \(0.527724\pi\)
\(332\) 14.0000 0.768350
\(333\) 1.00000 0.0547997
\(334\) −8.50455 −0.465348
\(335\) 25.7477 1.40675
\(336\) −4.95644 −0.270396
\(337\) −17.5826 −0.957784 −0.478892 0.877874i \(-0.658961\pi\)
−0.478892 + 0.877874i \(0.658961\pi\)
\(338\) 21.4955 1.16920
\(339\) −9.16515 −0.497783
\(340\) −27.4955 −1.49115
\(341\) 0 0
\(342\) −11.7913 −0.637600
\(343\) −1.00000 −0.0539949
\(344\) −2.24318 −0.120944
\(345\) 16.7477 0.901667
\(346\) −12.8348 −0.690006
\(347\) −26.3303 −1.41348 −0.706742 0.707471i \(-0.749836\pi\)
−0.706742 + 0.707471i \(0.749836\pi\)
\(348\) −9.86932 −0.529051
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) 7.16515 0.382993
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 24.1652 1.28618 0.643091 0.765790i \(-0.277652\pi\)
0.643091 + 0.765790i \(0.277652\pi\)
\(354\) 8.20871 0.436288
\(355\) 33.4955 1.77775
\(356\) 11.0780 0.587134
\(357\) −7.58258 −0.401312
\(358\) −25.6697 −1.35669
\(359\) 8.83485 0.466285 0.233143 0.972443i \(-0.425099\pi\)
0.233143 + 0.972443i \(0.425099\pi\)
\(360\) 4.25227 0.224114
\(361\) 24.3303 1.28054
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) 1.20871 0.0633537
\(365\) −21.0000 −1.09919
\(366\) −17.9129 −0.936321
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 27.6697 1.44238
\(369\) 11.1652 0.581235
\(370\) −5.37386 −0.279374
\(371\) 9.58258 0.497503
\(372\) −4.33030 −0.224516
\(373\) 34.7477 1.79917 0.899585 0.436747i \(-0.143869\pi\)
0.899585 + 0.436747i \(0.143869\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 2.00909 0.103611
\(377\) −8.16515 −0.420527
\(378\) −1.79129 −0.0921339
\(379\) −12.5826 −0.646323 −0.323162 0.946344i \(-0.604746\pi\)
−0.323162 + 0.946344i \(0.604746\pi\)
\(380\) 23.8693 1.22447
\(381\) −11.5826 −0.593393
\(382\) −20.7477 −1.06155
\(383\) 10.3303 0.527854 0.263927 0.964543i \(-0.414982\pi\)
0.263927 + 0.964543i \(0.414982\pi\)
\(384\) 10.4519 0.533371
\(385\) 0 0
\(386\) −4.33030 −0.220407
\(387\) −1.58258 −0.0804468
\(388\) −2.92197 −0.148341
\(389\) −26.3303 −1.33500 −0.667500 0.744610i \(-0.732635\pi\)
−0.667500 + 0.744610i \(0.732635\pi\)
\(390\) −5.37386 −0.272116
\(391\) 42.3303 2.14074
\(392\) 1.41742 0.0715907
\(393\) −16.0000 −0.807093
\(394\) 9.25227 0.466123
\(395\) −21.4955 −1.08155
\(396\) 0 0
\(397\) −31.5826 −1.58508 −0.792542 0.609817i \(-0.791243\pi\)
−0.792542 + 0.609817i \(0.791243\pi\)
\(398\) 17.1652 0.860411
\(399\) 6.58258 0.329541
\(400\) −19.8258 −0.991288
\(401\) 31.9129 1.59365 0.796827 0.604208i \(-0.206510\pi\)
0.796827 + 0.604208i \(0.206510\pi\)
\(402\) 15.3739 0.766779
\(403\) −3.58258 −0.178461
\(404\) −14.0000 −0.696526
\(405\) 3.00000 0.149071
\(406\) 14.6261 0.725883
\(407\) 0 0
\(408\) 10.7477 0.532092
\(409\) −8.33030 −0.411907 −0.205953 0.978562i \(-0.566030\pi\)
−0.205953 + 0.978562i \(0.566030\pi\)
\(410\) −60.0000 −2.96319
\(411\) 11.5826 0.571326
\(412\) −1.40833 −0.0693836
\(413\) −4.58258 −0.225494
\(414\) 10.0000 0.491473
\(415\) 34.7477 1.70570
\(416\) −6.04356 −0.296310
\(417\) 11.1652 0.546760
\(418\) 0 0
\(419\) 2.58258 0.126167 0.0630835 0.998008i \(-0.479907\pi\)
0.0630835 + 0.998008i \(0.479907\pi\)
\(420\) 3.62614 0.176937
\(421\) −33.6606 −1.64052 −0.820259 0.571993i \(-0.806171\pi\)
−0.820259 + 0.571993i \(0.806171\pi\)
\(422\) −23.5826 −1.14798
\(423\) 1.41742 0.0689175
\(424\) −13.5826 −0.659628
\(425\) −30.3303 −1.47124
\(426\) 20.0000 0.969003
\(427\) 10.0000 0.483934
\(428\) 15.2087 0.735141
\(429\) 0 0
\(430\) 8.50455 0.410126
\(431\) −17.7477 −0.854878 −0.427439 0.904044i \(-0.640584\pi\)
−0.427439 + 0.904044i \(0.640584\pi\)
\(432\) 4.95644 0.238467
\(433\) −11.1652 −0.536563 −0.268281 0.963341i \(-0.586456\pi\)
−0.268281 + 0.963341i \(0.586456\pi\)
\(434\) 6.41742 0.308046
\(435\) −24.4955 −1.17447
\(436\) −4.33030 −0.207384
\(437\) −36.7477 −1.75788
\(438\) −12.5390 −0.599137
\(439\) −17.4174 −0.831288 −0.415644 0.909527i \(-0.636444\pi\)
−0.415644 + 0.909527i \(0.636444\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −13.5826 −0.646057
\(443\) −23.1652 −1.10061 −0.550305 0.834964i \(-0.685488\pi\)
−0.550305 + 0.834964i \(0.685488\pi\)
\(444\) −1.20871 −0.0573629
\(445\) 27.4955 1.30341
\(446\) −10.7477 −0.508920
\(447\) 6.16515 0.291602
\(448\) 0.912878 0.0431295
\(449\) −18.3303 −0.865060 −0.432530 0.901619i \(-0.642379\pi\)
−0.432530 + 0.901619i \(0.642379\pi\)
\(450\) −7.16515 −0.337768
\(451\) 0 0
\(452\) 11.0780 0.521067
\(453\) 3.58258 0.168324
\(454\) −39.4083 −1.84952
\(455\) 3.00000 0.140642
\(456\) −9.33030 −0.436931
\(457\) 19.9129 0.931485 0.465743 0.884920i \(-0.345787\pi\)
0.465743 + 0.884920i \(0.345787\pi\)
\(458\) −1.33939 −0.0625858
\(459\) 7.58258 0.353924
\(460\) −20.2432 −0.943843
\(461\) −18.3303 −0.853727 −0.426864 0.904316i \(-0.640382\pi\)
−0.426864 + 0.904316i \(0.640382\pi\)
\(462\) 0 0
\(463\) 8.58258 0.398866 0.199433 0.979911i \(-0.436090\pi\)
0.199433 + 0.979911i \(0.436090\pi\)
\(464\) −40.4701 −1.87878
\(465\) −10.7477 −0.498414
\(466\) −25.0780 −1.16172
\(467\) −38.5826 −1.78539 −0.892694 0.450663i \(-0.851188\pi\)
−0.892694 + 0.450663i \(0.851188\pi\)
\(468\) −1.20871 −0.0558727
\(469\) −8.58258 −0.396307
\(470\) −7.61704 −0.351348
\(471\) −19.1652 −0.883084
\(472\) 6.49545 0.298978
\(473\) 0 0
\(474\) −12.8348 −0.589524
\(475\) 26.3303 1.20812
\(476\) 9.16515 0.420084
\(477\) −9.58258 −0.438756
\(478\) 29.7042 1.35864
\(479\) 15.5826 0.711986 0.355993 0.934489i \(-0.384143\pi\)
0.355993 + 0.934489i \(0.384143\pi\)
\(480\) −18.1307 −0.827549
\(481\) −1.00000 −0.0455961
\(482\) −18.2087 −0.829384
\(483\) −5.58258 −0.254016
\(484\) 0 0
\(485\) −7.25227 −0.329309
\(486\) 1.79129 0.0812545
\(487\) 10.3303 0.468111 0.234055 0.972223i \(-0.424800\pi\)
0.234055 + 0.972223i \(0.424800\pi\)
\(488\) −14.1742 −0.641638
\(489\) −8.58258 −0.388117
\(490\) −5.37386 −0.242766
\(491\) 22.9129 1.03404 0.517022 0.855972i \(-0.327041\pi\)
0.517022 + 0.855972i \(0.327041\pi\)
\(492\) −13.4955 −0.608422
\(493\) −61.9129 −2.78842
\(494\) 11.7913 0.530515
\(495\) 0 0
\(496\) −17.7568 −0.797305
\(497\) −11.1652 −0.500825
\(498\) 20.7477 0.929728
\(499\) 41.7477 1.86888 0.934442 0.356114i \(-0.115899\pi\)
0.934442 + 0.356114i \(0.115899\pi\)
\(500\) −3.62614 −0.162166
\(501\) −4.74773 −0.212113
\(502\) −13.2867 −0.593016
\(503\) 0.747727 0.0333395 0.0166698 0.999861i \(-0.494694\pi\)
0.0166698 + 0.999861i \(0.494694\pi\)
\(504\) −1.41742 −0.0631371
\(505\) −34.7477 −1.54625
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 14.0000 0.621150
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) −40.7477 −1.80434
\(511\) 7.00000 0.309662
\(512\) −15.9038 −0.702855
\(513\) −6.58258 −0.290628
\(514\) −34.0345 −1.50120
\(515\) −3.49545 −0.154028
\(516\) 1.91288 0.0842098
\(517\) 0 0
\(518\) 1.79129 0.0787047
\(519\) −7.16515 −0.314515
\(520\) −4.25227 −0.186475
\(521\) 15.8348 0.693737 0.346869 0.937914i \(-0.387245\pi\)
0.346869 + 0.937914i \(0.387245\pi\)
\(522\) −14.6261 −0.640169
\(523\) −15.4174 −0.674157 −0.337078 0.941477i \(-0.609439\pi\)
−0.337078 + 0.941477i \(0.609439\pi\)
\(524\) 19.3394 0.844845
\(525\) 4.00000 0.174574
\(526\) −41.0436 −1.78958
\(527\) −27.1652 −1.18333
\(528\) 0 0
\(529\) 8.16515 0.355007
\(530\) 51.4955 2.23682
\(531\) 4.58258 0.198867
\(532\) −7.95644 −0.344955
\(533\) −11.1652 −0.483616
\(534\) 16.4174 0.710451
\(535\) 37.7477 1.63198
\(536\) 12.1652 0.525455
\(537\) −14.3303 −0.618398
\(538\) −17.9129 −0.772279
\(539\) 0 0
\(540\) −3.62614 −0.156044
\(541\) −18.3303 −0.788081 −0.394041 0.919093i \(-0.628923\pi\)
−0.394041 + 0.919093i \(0.628923\pi\)
\(542\) 9.70417 0.416830
\(543\) 5.58258 0.239571
\(544\) −45.8258 −1.96476
\(545\) −10.7477 −0.460382
\(546\) 1.79129 0.0766600
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −14.0000 −0.598050
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 53.7477 2.28973
\(552\) 7.91288 0.336794
\(553\) 7.16515 0.304693
\(554\) −34.3303 −1.45855
\(555\) −3.00000 −0.127343
\(556\) −13.4955 −0.572335
\(557\) −9.33030 −0.395338 −0.197669 0.980269i \(-0.563337\pi\)
−0.197669 + 0.980269i \(0.563337\pi\)
\(558\) −6.41742 −0.271671
\(559\) 1.58258 0.0669358
\(560\) 14.8693 0.628343
\(561\) 0 0
\(562\) −48.9564 −2.06510
\(563\) 37.5826 1.58392 0.791958 0.610575i \(-0.209062\pi\)
0.791958 + 0.610575i \(0.209062\pi\)
\(564\) −1.71326 −0.0721412
\(565\) 27.4955 1.15674
\(566\) 49.7042 2.08922
\(567\) −1.00000 −0.0419961
\(568\) 15.8258 0.664034
\(569\) 26.6606 1.11767 0.558835 0.829279i \(-0.311248\pi\)
0.558835 + 0.829279i \(0.311248\pi\)
\(570\) 35.3739 1.48165
\(571\) −28.8348 −1.20670 −0.603350 0.797476i \(-0.706168\pi\)
−0.603350 + 0.797476i \(0.706168\pi\)
\(572\) 0 0
\(573\) −11.5826 −0.483869
\(574\) 20.0000 0.834784
\(575\) −22.3303 −0.931238
\(576\) −0.912878 −0.0380366
\(577\) −21.9129 −0.912245 −0.456123 0.889917i \(-0.650762\pi\)
−0.456123 + 0.889917i \(0.650762\pi\)
\(578\) −72.5390 −3.01723
\(579\) −2.41742 −0.100465
\(580\) 29.6080 1.22940
\(581\) −11.5826 −0.480526
\(582\) −4.33030 −0.179497
\(583\) 0 0
\(584\) −9.92197 −0.410574
\(585\) −3.00000 −0.124035
\(586\) 0 0
\(587\) 37.7477 1.55802 0.779008 0.627014i \(-0.215723\pi\)
0.779008 + 0.627014i \(0.215723\pi\)
\(588\) −1.20871 −0.0498464
\(589\) 23.5826 0.971703
\(590\) −24.6261 −1.01384
\(591\) 5.16515 0.212466
\(592\) −4.95644 −0.203708
\(593\) −16.0000 −0.657041 −0.328521 0.944497i \(-0.606550\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) 0 0
\(595\) 22.7477 0.932566
\(596\) −7.45189 −0.305241
\(597\) 9.58258 0.392189
\(598\) −10.0000 −0.408930
\(599\) 7.16515 0.292760 0.146380 0.989228i \(-0.453238\pi\)
0.146380 + 0.989228i \(0.453238\pi\)
\(600\) −5.66970 −0.231464
\(601\) −24.4955 −0.999190 −0.499595 0.866259i \(-0.666518\pi\)
−0.499595 + 0.866259i \(0.666518\pi\)
\(602\) −2.83485 −0.115540
\(603\) 8.58258 0.349510
\(604\) −4.33030 −0.176198
\(605\) 0 0
\(606\) −20.7477 −0.842819
\(607\) 21.7477 0.882713 0.441357 0.897332i \(-0.354497\pi\)
0.441357 + 0.897332i \(0.354497\pi\)
\(608\) 39.7822 1.61338
\(609\) 8.16515 0.330869
\(610\) 53.7386 2.17581
\(611\) −1.41742 −0.0573428
\(612\) −9.16515 −0.370479
\(613\) 26.7477 1.08033 0.540165 0.841559i \(-0.318362\pi\)
0.540165 + 0.841559i \(0.318362\pi\)
\(614\) 0 0
\(615\) −33.4955 −1.35067
\(616\) 0 0
\(617\) 2.83485 0.114127 0.0570634 0.998371i \(-0.481826\pi\)
0.0570634 + 0.998371i \(0.481826\pi\)
\(618\) −2.08712 −0.0839563
\(619\) −29.0780 −1.16874 −0.584372 0.811486i \(-0.698659\pi\)
−0.584372 + 0.811486i \(0.698659\pi\)
\(620\) 12.9909 0.521727
\(621\) 5.58258 0.224021
\(622\) −25.6697 −1.02926
\(623\) −9.16515 −0.367194
\(624\) −4.95644 −0.198416
\(625\) −29.0000 −1.16000
\(626\) −35.0780 −1.40200
\(627\) 0 0
\(628\) 23.1652 0.924390
\(629\) −7.58258 −0.302337
\(630\) 5.37386 0.214100
\(631\) 23.1652 0.922190 0.461095 0.887351i \(-0.347457\pi\)
0.461095 + 0.887351i \(0.347457\pi\)
\(632\) −10.1561 −0.403986
\(633\) −13.1652 −0.523268
\(634\) 40.1561 1.59480
\(635\) 34.7477 1.37892
\(636\) 11.5826 0.459279
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 11.1652 0.441687
\(640\) −31.3557 −1.23944
\(641\) 43.5826 1.72141 0.860704 0.509106i \(-0.170024\pi\)
0.860704 + 0.509106i \(0.170024\pi\)
\(642\) 22.5390 0.889544
\(643\) 38.2432 1.50816 0.754082 0.656780i \(-0.228082\pi\)
0.754082 + 0.656780i \(0.228082\pi\)
\(644\) 6.74773 0.265898
\(645\) 4.74773 0.186942
\(646\) 89.4083 3.51772
\(647\) −10.9129 −0.429030 −0.214515 0.976721i \(-0.568817\pi\)
−0.214515 + 0.976721i \(0.568817\pi\)
\(648\) 1.41742 0.0556817
\(649\) 0 0
\(650\) 7.16515 0.281040
\(651\) 3.58258 0.140412
\(652\) 10.3739 0.406272
\(653\) −30.3303 −1.18692 −0.593458 0.804865i \(-0.702238\pi\)
−0.593458 + 0.804865i \(0.702238\pi\)
\(654\) −6.41742 −0.250941
\(655\) 48.0000 1.87552
\(656\) −55.3394 −2.16064
\(657\) −7.00000 −0.273096
\(658\) 2.53901 0.0989811
\(659\) 28.5826 1.11342 0.556710 0.830707i \(-0.312064\pi\)
0.556710 + 0.830707i \(0.312064\pi\)
\(660\) 0 0
\(661\) −39.0780 −1.51996 −0.759980 0.649947i \(-0.774791\pi\)
−0.759980 + 0.649947i \(0.774791\pi\)
\(662\) 5.66970 0.220359
\(663\) −7.58258 −0.294483
\(664\) 16.4174 0.637120
\(665\) −19.7477 −0.765784
\(666\) −1.79129 −0.0694110
\(667\) −45.5826 −1.76496
\(668\) 5.73864 0.222034
\(669\) −6.00000 −0.231973
\(670\) −46.1216 −1.78183
\(671\) 0 0
\(672\) 6.04356 0.233135
\(673\) −11.2523 −0.433743 −0.216872 0.976200i \(-0.569585\pi\)
−0.216872 + 0.976200i \(0.569585\pi\)
\(674\) 31.4955 1.21316
\(675\) −4.00000 −0.153960
\(676\) −14.5045 −0.557867
\(677\) 45.1652 1.73584 0.867919 0.496706i \(-0.165457\pi\)
0.867919 + 0.496706i \(0.165457\pi\)
\(678\) 16.4174 0.630507
\(679\) 2.41742 0.0927722
\(680\) −32.2432 −1.23647
\(681\) −22.0000 −0.843042
\(682\) 0 0
\(683\) −33.0780 −1.26570 −0.632848 0.774276i \(-0.718114\pi\)
−0.632848 + 0.774276i \(0.718114\pi\)
\(684\) 7.95644 0.304222
\(685\) −34.7477 −1.32764
\(686\) 1.79129 0.0683917
\(687\) −0.747727 −0.0285276
\(688\) 7.84394 0.299047
\(689\) 9.58258 0.365067
\(690\) −30.0000 −1.14208
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 8.66061 0.329227
\(693\) 0 0
\(694\) 47.1652 1.79036
\(695\) −33.4955 −1.27055
\(696\) −11.5735 −0.438692
\(697\) −84.6606 −3.20675
\(698\) −26.8693 −1.01702
\(699\) −14.0000 −0.529529
\(700\) −4.83485 −0.182740
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) −1.79129 −0.0676078
\(703\) 6.58258 0.248267
\(704\) 0 0
\(705\) −4.25227 −0.160150
\(706\) −43.2867 −1.62912
\(707\) 11.5826 0.435608
\(708\) −5.53901 −0.208169
\(709\) 27.6606 1.03882 0.519408 0.854526i \(-0.326153\pi\)
0.519408 + 0.854526i \(0.326153\pi\)
\(710\) −60.0000 −2.25176
\(711\) −7.16515 −0.268714
\(712\) 12.9909 0.486855
\(713\) −20.0000 −0.749006
\(714\) 13.5826 0.508315
\(715\) 0 0
\(716\) 17.3212 0.647324
\(717\) 16.5826 0.619288
\(718\) −15.8258 −0.590612
\(719\) −14.0780 −0.525022 −0.262511 0.964929i \(-0.584551\pi\)
−0.262511 + 0.964929i \(0.584551\pi\)
\(720\) −14.8693 −0.554147
\(721\) 1.16515 0.0433925
\(722\) −43.5826 −1.62198
\(723\) −10.1652 −0.378046
\(724\) −6.74773 −0.250777
\(725\) 32.6606 1.21298
\(726\) 0 0
\(727\) −15.9129 −0.590176 −0.295088 0.955470i \(-0.595349\pi\)
−0.295088 + 0.955470i \(0.595349\pi\)
\(728\) 1.41742 0.0525332
\(729\) 1.00000 0.0370370
\(730\) 37.6170 1.39227
\(731\) 12.0000 0.443836
\(732\) 12.0871 0.446753
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −39.4083 −1.45459
\(735\) −3.00000 −0.110657
\(736\) −33.7386 −1.24362
\(737\) 0 0
\(738\) −20.0000 −0.736210
\(739\) 31.9129 1.17393 0.586967 0.809611i \(-0.300322\pi\)
0.586967 + 0.809611i \(0.300322\pi\)
\(740\) 3.62614 0.133299
\(741\) 6.58258 0.241817
\(742\) −17.1652 −0.630153
\(743\) 53.2432 1.95330 0.976651 0.214830i \(-0.0689198\pi\)
0.976651 + 0.214830i \(0.0689198\pi\)
\(744\) −5.07803 −0.186170
\(745\) −18.4955 −0.677621
\(746\) −62.2432 −2.27888
\(747\) 11.5826 0.423784
\(748\) 0 0
\(749\) −12.5826 −0.459757
\(750\) −5.37386 −0.196226
\(751\) −8.91288 −0.325236 −0.162618 0.986689i \(-0.551994\pi\)
−0.162618 + 0.986689i \(0.551994\pi\)
\(752\) −7.02538 −0.256189
\(753\) −7.41742 −0.270306
\(754\) 14.6261 0.532652
\(755\) −10.7477 −0.391150
\(756\) 1.20871 0.0439604
\(757\) 27.3303 0.993337 0.496668 0.867940i \(-0.334557\pi\)
0.496668 + 0.867940i \(0.334557\pi\)
\(758\) 22.5390 0.818654
\(759\) 0 0
\(760\) 27.9909 1.01534
\(761\) −42.3303 −1.53447 −0.767236 0.641365i \(-0.778369\pi\)
−0.767236 + 0.641365i \(0.778369\pi\)
\(762\) 20.7477 0.751611
\(763\) 3.58258 0.129698
\(764\) 14.0000 0.506502
\(765\) −22.7477 −0.822446
\(766\) −18.5045 −0.668596
\(767\) −4.58258 −0.165467
\(768\) −20.5481 −0.741466
\(769\) −6.49545 −0.234232 −0.117116 0.993118i \(-0.537365\pi\)
−0.117116 + 0.993118i \(0.537365\pi\)
\(770\) 0 0
\(771\) −19.0000 −0.684268
\(772\) 2.92197 0.105164
\(773\) 6.16515 0.221745 0.110873 0.993835i \(-0.464635\pi\)
0.110873 + 0.993835i \(0.464635\pi\)
\(774\) 2.83485 0.101897
\(775\) 14.3303 0.514760
\(776\) −3.42652 −0.123005
\(777\) 1.00000 0.0358748
\(778\) 47.1652 1.69095
\(779\) 73.4955 2.63325
\(780\) 3.62614 0.129837
\(781\) 0 0
\(782\) −75.8258 −2.71152
\(783\) −8.16515 −0.291799
\(784\) −4.95644 −0.177016
\(785\) 57.4955 2.05210
\(786\) 28.6606 1.02229
\(787\) −38.5826 −1.37532 −0.687660 0.726033i \(-0.741362\pi\)
−0.687660 + 0.726033i \(0.741362\pi\)
\(788\) −6.24318 −0.222404
\(789\) −22.9129 −0.815720
\(790\) 38.5045 1.36993
\(791\) −9.16515 −0.325875
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 56.5735 2.00772
\(795\) 28.7477 1.01958
\(796\) −11.5826 −0.410534
\(797\) −52.4955 −1.85948 −0.929742 0.368211i \(-0.879970\pi\)
−0.929742 + 0.368211i \(0.879970\pi\)
\(798\) −11.7913 −0.417407
\(799\) −10.7477 −0.380227
\(800\) 24.1742 0.854689
\(801\) 9.16515 0.323835
\(802\) −57.1652 −2.01857
\(803\) 0 0
\(804\) −10.3739 −0.365858
\(805\) 16.7477 0.590280
\(806\) 6.41742 0.226044
\(807\) −10.0000 −0.352017
\(808\) −16.4174 −0.577563
\(809\) −9.33030 −0.328036 −0.164018 0.986457i \(-0.552446\pi\)
−0.164018 + 0.986457i \(0.552446\pi\)
\(810\) −5.37386 −0.188818
\(811\) 2.25227 0.0790880 0.0395440 0.999218i \(-0.487409\pi\)
0.0395440 + 0.999218i \(0.487409\pi\)
\(812\) −9.86932 −0.346345
\(813\) 5.41742 0.189997
\(814\) 0 0
\(815\) 25.7477 0.901904
\(816\) −37.5826 −1.31565
\(817\) −10.4174 −0.364460
\(818\) 14.9220 0.521734
\(819\) 1.00000 0.0349428
\(820\) 40.4864 1.41385
\(821\) 47.0000 1.64031 0.820156 0.572140i \(-0.193887\pi\)
0.820156 + 0.572140i \(0.193887\pi\)
\(822\) −20.7477 −0.723660
\(823\) −30.5826 −1.06604 −0.533021 0.846102i \(-0.678943\pi\)
−0.533021 + 0.846102i \(0.678943\pi\)
\(824\) −1.65151 −0.0575332
\(825\) 0 0
\(826\) 8.20871 0.285618
\(827\) −8.91288 −0.309931 −0.154966 0.987920i \(-0.549527\pi\)
−0.154966 + 0.987920i \(0.549527\pi\)
\(828\) −6.74773 −0.234500
\(829\) −40.0000 −1.38926 −0.694629 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(830\) −62.2432 −2.16049
\(831\) −19.1652 −0.664832
\(832\) 0.912878 0.0316484
\(833\) −7.58258 −0.262721
\(834\) −20.0000 −0.692543
\(835\) 14.2432 0.492906
\(836\) 0 0
\(837\) −3.58258 −0.123832
\(838\) −4.62614 −0.159807
\(839\) 7.08712 0.244675 0.122337 0.992489i \(-0.460961\pi\)
0.122337 + 0.992489i \(0.460961\pi\)
\(840\) 4.25227 0.146717
\(841\) 37.6697 1.29896
\(842\) 60.2958 2.07793
\(843\) −27.3303 −0.941306
\(844\) 15.9129 0.547744
\(845\) −36.0000 −1.23844
\(846\) −2.53901 −0.0872931
\(847\) 0 0
\(848\) 47.4955 1.63100
\(849\) 27.7477 0.952300
\(850\) 54.3303 1.86351
\(851\) −5.58258 −0.191368
\(852\) −13.4955 −0.462347
\(853\) −17.1652 −0.587724 −0.293862 0.955848i \(-0.594941\pi\)
−0.293862 + 0.955848i \(0.594941\pi\)
\(854\) −17.9129 −0.612966
\(855\) 19.7477 0.675358
\(856\) 17.8348 0.609583
\(857\) 7.66970 0.261992 0.130996 0.991383i \(-0.458183\pi\)
0.130996 + 0.991383i \(0.458183\pi\)
\(858\) 0 0
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) −5.73864 −0.195686
\(861\) 11.1652 0.380507
\(862\) 31.7913 1.08282
\(863\) 14.4174 0.490775 0.245387 0.969425i \(-0.421085\pi\)
0.245387 + 0.969425i \(0.421085\pi\)
\(864\) −6.04356 −0.205606
\(865\) 21.4955 0.730867
\(866\) 20.0000 0.679628
\(867\) −40.4955 −1.37530
\(868\) −4.33030 −0.146980
\(869\) 0 0
\(870\) 43.8784 1.48762
\(871\) −8.58258 −0.290809
\(872\) −5.07803 −0.171964
\(873\) −2.41742 −0.0818174
\(874\) 65.8258 2.22659
\(875\) 3.00000 0.101419
\(876\) 8.46099 0.285870
\(877\) −9.49545 −0.320639 −0.160319 0.987065i \(-0.551252\pi\)
−0.160319 + 0.987065i \(0.551252\pi\)
\(878\) 31.1996 1.05294
\(879\) 0 0
\(880\) 0 0
\(881\) −37.6606 −1.26882 −0.634409 0.772998i \(-0.718756\pi\)
−0.634409 + 0.772998i \(0.718756\pi\)
\(882\) −1.79129 −0.0603158
\(883\) −55.7477 −1.87606 −0.938030 0.346554i \(-0.887352\pi\)
−0.938030 + 0.346554i \(0.887352\pi\)
\(884\) 9.16515 0.308257
\(885\) −13.7477 −0.462125
\(886\) 41.4955 1.39407
\(887\) −38.7477 −1.30102 −0.650511 0.759497i \(-0.725445\pi\)
−0.650511 + 0.759497i \(0.725445\pi\)
\(888\) −1.41742 −0.0475656
\(889\) −11.5826 −0.388467
\(890\) −49.2523 −1.65094
\(891\) 0 0
\(892\) 7.25227 0.242824
\(893\) 9.33030 0.312227
\(894\) −11.0436 −0.369352
\(895\) 42.9909 1.43703
\(896\) 10.4519 0.349173
\(897\) −5.58258 −0.186397
\(898\) 32.8348 1.09571
\(899\) 29.2523 0.975618
\(900\) 4.83485 0.161162
\(901\) 72.6606 2.42068
\(902\) 0 0
\(903\) −1.58258 −0.0526648
\(904\) 12.9909 0.432071
\(905\) −16.7477 −0.556713
\(906\) −6.41742 −0.213205
\(907\) 42.3303 1.40555 0.702777 0.711410i \(-0.251943\pi\)
0.702777 + 0.711410i \(0.251943\pi\)
\(908\) 26.5917 0.882475
\(909\) −11.5826 −0.384170
\(910\) −5.37386 −0.178142
\(911\) −3.49545 −0.115810 −0.0579048 0.998322i \(-0.518442\pi\)
−0.0579048 + 0.998322i \(0.518442\pi\)
\(912\) 32.6261 1.08036
\(913\) 0 0
\(914\) −35.6697 −1.17985
\(915\) 30.0000 0.991769
\(916\) 0.903787 0.0298620
\(917\) −16.0000 −0.528367
\(918\) −13.5826 −0.448292
\(919\) 8.08712 0.266770 0.133385 0.991064i \(-0.457415\pi\)
0.133385 + 0.991064i \(0.457415\pi\)
\(920\) −23.7386 −0.782640
\(921\) 0 0
\(922\) 32.8348 1.08136
\(923\) −11.1652 −0.367505
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) −15.3739 −0.505217
\(927\) −1.16515 −0.0382686
\(928\) 49.3466 1.61988
\(929\) −21.3303 −0.699825 −0.349912 0.936782i \(-0.613789\pi\)
−0.349912 + 0.936782i \(0.613789\pi\)
\(930\) 19.2523 0.631307
\(931\) 6.58258 0.215735
\(932\) 16.9220 0.554298
\(933\) −14.3303 −0.469153
\(934\) 69.1125 2.26143
\(935\) 0 0
\(936\) −1.41742 −0.0463300
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 15.3739 0.501974
\(939\) −19.5826 −0.639053
\(940\) 5.13977 0.167641
\(941\) −35.1652 −1.14635 −0.573176 0.819433i \(-0.694289\pi\)
−0.573176 + 0.819433i \(0.694289\pi\)
\(942\) 34.3303 1.11854
\(943\) −62.3303 −2.02975
\(944\) −22.7133 −0.739254
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) 45.1652 1.46767 0.733835 0.679328i \(-0.237728\pi\)
0.733835 + 0.679328i \(0.237728\pi\)
\(948\) 8.66061 0.281283
\(949\) 7.00000 0.227230
\(950\) −47.1652 −1.53024
\(951\) 22.4174 0.726935
\(952\) 10.7477 0.348336
\(953\) 28.1652 0.912359 0.456179 0.889888i \(-0.349218\pi\)
0.456179 + 0.889888i \(0.349218\pi\)
\(954\) 17.1652 0.555742
\(955\) 34.7477 1.12441
\(956\) −20.0436 −0.648255
\(957\) 0 0
\(958\) −27.9129 −0.901824
\(959\) 11.5826 0.374021
\(960\) 2.73864 0.0883891
\(961\) −18.1652 −0.585973
\(962\) 1.79129 0.0577534
\(963\) 12.5826 0.405468
\(964\) 12.2867 0.395729
\(965\) 7.25227 0.233459
\(966\) 10.0000 0.321745
\(967\) 13.1652 0.423363 0.211681 0.977339i \(-0.432106\pi\)
0.211681 + 0.977339i \(0.432106\pi\)
\(968\) 0 0
\(969\) 49.9129 1.60343
\(970\) 12.9909 0.417113
\(971\) −12.5826 −0.403794 −0.201897 0.979407i \(-0.564711\pi\)
−0.201897 + 0.979407i \(0.564711\pi\)
\(972\) −1.20871 −0.0387695
\(973\) 11.1652 0.357938
\(974\) −18.5045 −0.592924
\(975\) 4.00000 0.128103
\(976\) 49.5644 1.58652
\(977\) 23.2523 0.743906 0.371953 0.928252i \(-0.378688\pi\)
0.371953 + 0.928252i \(0.378688\pi\)
\(978\) 15.3739 0.491602
\(979\) 0 0
\(980\) 3.62614 0.115833
\(981\) −3.58258 −0.114383
\(982\) −41.0436 −1.30975
\(983\) −4.83485 −0.154208 −0.0771039 0.997023i \(-0.524567\pi\)
−0.0771039 + 0.997023i \(0.524567\pi\)
\(984\) −15.8258 −0.504507
\(985\) −15.4955 −0.493726
\(986\) 110.904 3.53190
\(987\) 1.41742 0.0451171
\(988\) −7.95644 −0.253128
\(989\) 8.83485 0.280932
\(990\) 0 0
\(991\) −20.2523 −0.643335 −0.321667 0.946853i \(-0.604243\pi\)
−0.321667 + 0.946853i \(0.604243\pi\)
\(992\) 21.6515 0.687436
\(993\) 3.16515 0.100443
\(994\) 20.0000 0.634361
\(995\) −28.7477 −0.911364
\(996\) −14.0000 −0.443607
\(997\) −10.6606 −0.337625 −0.168812 0.985648i \(-0.553993\pi\)
−0.168812 + 0.985648i \(0.553993\pi\)
\(998\) −74.7822 −2.36719
\(999\) −1.00000 −0.0316386
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 2541.2.a.z.1.1 2
3.2 odd 2 7623.2.a.bf.1.2 2
11.10 odd 2 231.2.a.b.1.2 2
33.32 even 2 693.2.a.j.1.1 2
44.43 even 2 3696.2.a.bl.1.2 2
55.54 odd 2 5775.2.a.bn.1.1 2
77.76 even 2 1617.2.a.o.1.2 2
231.230 odd 2 4851.2.a.ba.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.b.1.2 2 11.10 odd 2
693.2.a.j.1.1 2 33.32 even 2
1617.2.a.o.1.2 2 77.76 even 2
2541.2.a.z.1.1 2 1.1 even 1 trivial
3696.2.a.bl.1.2 2 44.43 even 2
4851.2.a.ba.1.1 2 231.230 odd 2
5775.2.a.bn.1.1 2 55.54 odd 2
7623.2.a.bf.1.2 2 3.2 odd 2