Properties

Label 231.2.a.b.1.2
Level $231$
Weight $2$
Character 231.1
Self dual yes
Analytic conductor $1.845$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.84454428669\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 231.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.79129 q^{2} -1.00000 q^{3} +1.20871 q^{4} +3.00000 q^{5} -1.79129 q^{6} +1.00000 q^{7} -1.41742 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.79129 q^{2} -1.00000 q^{3} +1.20871 q^{4} +3.00000 q^{5} -1.79129 q^{6} +1.00000 q^{7} -1.41742 q^{8} +1.00000 q^{9} +5.37386 q^{10} -1.00000 q^{11} -1.20871 q^{12} +1.00000 q^{13} +1.79129 q^{14} -3.00000 q^{15} -4.95644 q^{16} +7.58258 q^{17} +1.79129 q^{18} -6.58258 q^{19} +3.62614 q^{20} -1.00000 q^{21} -1.79129 q^{22} -5.58258 q^{23} +1.41742 q^{24} +4.00000 q^{25} +1.79129 q^{26} -1.00000 q^{27} +1.20871 q^{28} -8.16515 q^{29} -5.37386 q^{30} +3.58258 q^{31} -6.04356 q^{32} +1.00000 q^{33} +13.5826 q^{34} +3.00000 q^{35} +1.20871 q^{36} +1.00000 q^{37} -11.7913 q^{38} -1.00000 q^{39} -4.25227 q^{40} -11.1652 q^{41} -1.79129 q^{42} +1.58258 q^{43} -1.20871 q^{44} +3.00000 q^{45} -10.0000 q^{46} +1.41742 q^{47} +4.95644 q^{48} +1.00000 q^{49} +7.16515 q^{50} -7.58258 q^{51} +1.20871 q^{52} -9.58258 q^{53} -1.79129 q^{54} -3.00000 q^{55} -1.41742 q^{56} +6.58258 q^{57} -14.6261 q^{58} +4.58258 q^{59} -3.62614 q^{60} +10.0000 q^{61} +6.41742 q^{62} +1.00000 q^{63} -0.912878 q^{64} +3.00000 q^{65} +1.79129 q^{66} +8.58258 q^{67} +9.16515 q^{68} +5.58258 q^{69} +5.37386 q^{70} +11.1652 q^{71} -1.41742 q^{72} +7.00000 q^{73} +1.79129 q^{74} -4.00000 q^{75} -7.95644 q^{76} -1.00000 q^{77} -1.79129 q^{78} +7.16515 q^{79} -14.8693 q^{80} +1.00000 q^{81} -20.0000 q^{82} -11.5826 q^{83} -1.20871 q^{84} +22.7477 q^{85} +2.83485 q^{86} +8.16515 q^{87} +1.41742 q^{88} +9.16515 q^{89} +5.37386 q^{90} +1.00000 q^{91} -6.74773 q^{92} -3.58258 q^{93} +2.53901 q^{94} -19.7477 q^{95} +6.04356 q^{96} -2.41742 q^{97} +1.79129 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} + q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} + q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9} - 3 q^{10} - 2 q^{11} - 7 q^{12} + 2 q^{13} - q^{14} - 6 q^{15} + 13 q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + 21 q^{20} - 2 q^{21} + q^{22} - 2 q^{23} + 12 q^{24} + 8 q^{25} - q^{26} - 2 q^{27} + 7 q^{28} + 2 q^{29} + 3 q^{30} - 2 q^{31} - 35 q^{32} + 2 q^{33} + 18 q^{34} + 6 q^{35} + 7 q^{36} + 2 q^{37} - 19 q^{38} - 2 q^{39} - 36 q^{40} - 4 q^{41} + q^{42} - 6 q^{43} - 7 q^{44} + 6 q^{45} - 20 q^{46} + 12 q^{47} - 13 q^{48} + 2 q^{49} - 4 q^{50} - 6 q^{51} + 7 q^{52} - 10 q^{53} + q^{54} - 6 q^{55} - 12 q^{56} + 4 q^{57} - 43 q^{58} - 21 q^{60} + 20 q^{61} + 22 q^{62} + 2 q^{63} + 44 q^{64} + 6 q^{65} - q^{66} + 8 q^{67} + 2 q^{69} - 3 q^{70} + 4 q^{71} - 12 q^{72} + 14 q^{73} - q^{74} - 8 q^{75} + 7 q^{76} - 2 q^{77} + q^{78} - 4 q^{79} + 39 q^{80} + 2 q^{81} - 40 q^{82} - 14 q^{83} - 7 q^{84} + 18 q^{85} + 24 q^{86} - 2 q^{87} + 12 q^{88} - 3 q^{90} + 2 q^{91} + 14 q^{92} + 2 q^{93} - 27 q^{94} - 12 q^{95} + 35 q^{96} - 14 q^{97} - q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.79129 1.26663 0.633316 0.773893i \(-0.281693\pi\)
0.633316 + 0.773893i \(0.281693\pi\)
\(3\) −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
\(11\) −1.00000 −0.301511
\(12\) −1.20871 −0.348925
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 1.79129 0.478742
\(15\) −3.00000 −0.774597
\(16\) −4.95644 −1.23911
\(17\) 7.58258 1.83904 0.919522 0.393038i \(-0.128576\pi\)
0.919522 + 0.393038i \(0.128576\pi\)
\(18\) 1.79129 0.422211
\(19\) −6.58258 −1.51015 −0.755073 0.655640i \(-0.772399\pi\)
−0.755073 + 0.655640i \(0.772399\pi\)
\(20\) 3.62614 0.810829
\(21\) −1.00000 −0.218218
\(22\) −1.79129 −0.381904
\(23\) −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 \(-0.773880\pi\)
−0.758115 + 0.652121i \(0.773880\pi\)
\(30\) −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
\(33\) 1.00000 0.174078
\(34\) 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 \(-0.837081\pi\)
−0.871852 + 0.489770i \(0.837081\pi\)
\(42\) −1.79129 −0.276402
\(43\) 1.58258 0.241341 0.120670 0.992693i \(-0.461496\pi\)
0.120670 + 0.992693i \(0.461496\pi\)
\(44\) −1.20871 −0.182220
\(45\) 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
\(55\) −3.00000 −0.404520
\(56\) −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 \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 6.41742 0.815014
\(63\) 1.00000 0.125988
\(64\) −0.912878 −0.114110
\(65\) 3.00000 0.372104
\(66\) 1.79129 0.220492
\(67\) 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 \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 1.79129 0.208233
\(75\) −4.00000 −0.461880
\(76\) −7.95644 −0.912666
\(77\) −1.00000 −0.113961
\(78\) −1.79129 −0.202823
\(79\) 7.16515 0.806143 0.403071 0.915169i \(-0.367943\pi\)
0.403071 + 0.915169i \(0.367943\pi\)
\(80\) −14.8693 −1.66244
\(81\) 1.00000 0.111111
\(82\) −20.0000 −2.20863
\(83\) −11.5826 −1.27135 −0.635676 0.771956i \(-0.719279\pi\)
−0.635676 + 0.771956i \(0.719279\pi\)
\(84\) −1.20871 −0.131881
\(85\) 22.7477 2.46734
\(86\) 2.83485 0.305690
\(87\) 8.16515 0.875396
\(88\) 1.41742 0.151098
\(89\) 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
\(99\) −1.00000 −0.100504
\(100\) 4.83485 0.483485
\(101\) 11.5826 1.15251 0.576255 0.817270i \(-0.304514\pi\)
0.576255 + 0.817270i \(0.304514\pi\)
\(102\) −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 \(-0.708109\pi\)
−0.608202 + 0.793782i \(0.708109\pi\)
\(108\) −1.20871 −0.116308
\(109\) 3.58258 0.343149 0.171574 0.985171i \(-0.445115\pi\)
0.171574 + 0.985171i \(0.445115\pi\)
\(110\) −5.37386 −0.512378
\(111\) −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
\(121\) 1.00000 0.0909091
\(122\) 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 \(-0.671797\pi\)
−0.513894 + 0.857854i \(0.671797\pi\)
\(128\) 10.4519 0.923826
\(129\) −1.58258 −0.139338
\(130\) 5.37386 0.471319
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 1.20871 0.105205
\(133\) −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 \(-0.342988\pi\)
0.473508 + 0.880790i \(0.342988\pi\)
\(140\) 3.62614 0.306464
\(141\) −1.41742 −0.119369
\(142\) 20.0000 1.67836
\(143\) −1.00000 −0.0836242
\(144\) −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 \(-0.418736\pi\)
0.252534 + 0.967588i \(0.418736\pi\)
\(150\) −7.16515 −0.585032
\(151\) 3.58258 0.291546 0.145773 0.989318i \(-0.453433\pi\)
0.145773 + 0.989318i \(0.453433\pi\)
\(152\) 9.33030 0.756787
\(153\) 7.58258 0.613015
\(154\) −1.79129 −0.144346
\(155\) 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
\(165\) 3.00000 0.233550
\(166\) −20.7477 −1.61034
\(167\) −4.74773 −0.367390 −0.183695 0.982983i \(-0.558806\pi\)
−0.183695 + 0.982983i \(0.558806\pi\)
\(168\) 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 \(-0.587810\pi\)
−0.272378 + 0.962190i \(0.587810\pi\)
\(174\) 14.6261 1.10880
\(175\) 4.00000 0.302372
\(176\) 4.95644 0.373606
\(177\) −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
\(187\) −7.58258 −0.554493
\(188\) 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 \(-0.527730\pi\)
−0.0870050 + 0.996208i \(0.527730\pi\)
\(194\) −4.33030 −0.310898
\(195\) −3.00000 −0.214834
\(196\) 1.20871 0.0863366
\(197\) 5.16515 0.368002 0.184001 0.982926i \(-0.441095\pi\)
0.184001 + 0.982926i \(0.441095\pi\)
\(198\) −1.79129 −0.127301
\(199\) −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
\(209\) 6.58258 0.455326
\(210\) −5.37386 −0.370832
\(211\) −13.1652 −0.906326 −0.453163 0.891428i \(-0.649705\pi\)
−0.453163 + 0.891428i \(0.649705\pi\)
\(212\) −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
\(220\) −3.62614 −0.244474
\(221\) 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 \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) 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
\(231\) 1.00000 0.0657952
\(232\) 11.5735 0.759836
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 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 \(-0.319814\pi\)
0.536319 + 0.844015i \(0.319814\pi\)
\(240\) 14.8693 0.959810
\(241\) −10.1652 −0.654795 −0.327397 0.944887i \(-0.606172\pi\)
−0.327397 + 0.944887i \(0.606172\pi\)
\(242\) 1.79129 0.115148
\(243\) −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
\(253\) 5.58258 0.350974
\(254\) −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 \(-0.749697\pi\)
−0.706434 + 0.707779i \(0.749697\pi\)
\(264\) −1.41742 −0.0872364
\(265\) −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 \(-0.447385\pi\)
0.164543 + 0.986370i \(0.447385\pi\)
\(272\) −37.5826 −2.27878
\(273\) −1.00000 −0.0605228
\(274\) −20.7477 −1.25342
\(275\) −4.00000 −0.241209
\(276\) 6.74773 0.406165
\(277\) −19.1652 −1.15152 −0.575761 0.817618i \(-0.695294\pi\)
−0.575761 + 0.817618i \(0.695294\pi\)
\(278\) 20.0000 1.19952
\(279\) 3.58258 0.214483
\(280\) −4.25227 −0.254122
\(281\) −27.3303 −1.63039 −0.815195 0.579187i \(-0.803370\pi\)
−0.815195 + 0.579187i \(0.803370\pi\)
\(282\) −2.53901 −0.151196
\(283\) 27.7477 1.64943 0.824716 0.565548i \(-0.191335\pi\)
0.824716 + 0.565548i \(0.191335\pi\)
\(284\) 13.4955 0.800808
\(285\) 19.7477 1.16975
\(286\) −1.79129 −0.105921
\(287\) −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
\(297\) 1.00000 0.0580259
\(298\) 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 \(0.5\pi\)
\(308\) −1.20871 −0.0688728
\(309\) 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
\(319\) 8.16515 0.457161
\(320\) −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
\(330\) 5.37386 0.295821
\(331\) −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 \(-0.341039\pi\)
0.478892 + 0.877874i \(0.341039\pi\)
\(338\) −21.4955 −1.16920
\(339\) −9.16515 −0.497783
\(340\) 27.4955 1.49115
\(341\) −3.58258 −0.194007
\(342\) −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 \(-0.250164\pi\)
0.706742 + 0.707471i \(0.250164\pi\)
\(348\) 9.86932 0.529051
\(349\) −15.0000 −0.802932 −0.401466 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) 7.16515 0.382993
\(351\) −1.00000 −0.0533761
\(352\) 6.04356 0.322123
\(353\) 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 \(-0.574901\pi\)
−0.233143 + 0.972443i \(0.574901\pi\)
\(360\) −4.25227 −0.224114
\(361\) 24.3303 1.28054
\(362\) −10.0000 −0.525588
\(363\) −1.00000 −0.0524864
\(364\) 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 \(-0.856131\pi\)
−0.899585 + 0.436747i \(0.856131\pi\)
\(374\) −13.5826 −0.702338
\(375\) 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
\(385\) −3.00000 −0.152894
\(386\) −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
\(396\) −1.20871 −0.0607401
\(397\) −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
\(407\) −1.00000 −0.0495682
\(408\) 10.7477 0.532092
\(409\) 8.33030 0.411907 0.205953 0.978562i \(-0.433970\pi\)
0.205953 + 0.978562i \(0.433970\pi\)
\(410\) −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
\(418\) 11.7913 0.576731
\(419\) 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
\(429\) 1.00000 0.0482805
\(430\) 8.50455 0.410126
\(431\) 17.7477 0.854878 0.427439 0.904044i \(-0.359416\pi\)
0.427439 + 0.904044i \(0.359416\pi\)
\(432\) 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 \(-0.363556\pi\)
0.415644 + 0.909527i \(0.363556\pi\)
\(440\) 4.25227 0.202719
\(441\) 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
\(451\) 11.1652 0.525746
\(452\) 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 \(-0.654213\pi\)
−0.465743 + 0.884920i \(0.654213\pi\)
\(458\) 1.33939 0.0625858
\(459\) −7.58258 −0.353924
\(460\) −20.2432 −0.943843
\(461\) 18.3303 0.853727 0.426864 0.904316i \(-0.359618\pi\)
0.426864 + 0.904316i \(0.359618\pi\)
\(462\) 1.79129 0.0833383
\(463\) 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
\(473\) −1.58258 −0.0727669
\(474\) −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 \(-0.615857\pi\)
−0.355993 + 0.934489i \(0.615857\pi\)
\(480\) 18.1307 0.827549
\(481\) 1.00000 0.0455961
\(482\) −18.2087 −0.829384
\(483\) 5.58258 0.254016
\(484\) 1.20871 0.0549415
\(485\) −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 \(-0.672959\pi\)
−0.517022 + 0.855972i \(0.672959\pi\)
\(492\) 13.4955 0.608422
\(493\) −61.9129 −2.78842
\(494\) −11.7913 −0.530515
\(495\) −3.00000 −0.134840
\(496\) −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 \(-0.505306\pi\)
−0.0166698 + 0.999861i \(0.505306\pi\)
\(504\) −1.41742 −0.0631371
\(505\) 34.7477 1.54625
\(506\) 10.0000 0.444554
\(507\) 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
\(517\) −1.41742 −0.0623382
\(518\) 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 \(-0.390561\pi\)
0.337078 + 0.941477i \(0.390561\pi\)
\(524\) −19.3394 −0.844845
\(525\) −4.00000 −0.174574
\(526\) −41.0436 −1.78958
\(527\) 27.1652 1.18333
\(528\) −4.95644 −0.215701
\(529\) 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
\(539\) −1.00000 −0.0430730
\(540\) −3.62614 −0.156044
\(541\) 18.3303 0.788081 0.394041 0.919093i \(-0.371077\pi\)
0.394041 + 0.919093i \(0.371077\pi\)
\(542\) 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 \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −14.0000 −0.598050
\(549\) 10.0000 0.426790
\(550\) −7.16515 −0.305523
\(551\) 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 \(-0.436663\pi\)
0.197669 + 0.980269i \(0.436663\pi\)
\(558\) 6.41742 0.271671
\(559\) 1.58258 0.0669358
\(560\) −14.8693 −0.628343
\(561\) 7.58258 0.320137
\(562\) −48.9564 −2.06510
\(563\) −37.5826 −1.58392 −0.791958 0.610575i \(-0.790938\pi\)
−0.791958 + 0.610575i \(0.790938\pi\)
\(564\) −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 \(-0.688752\pi\)
−0.558835 + 0.829279i \(0.688752\pi\)
\(570\) 35.3739 1.48165
\(571\) 28.8348 1.20670 0.603350 0.797476i \(-0.293832\pi\)
0.603350 + 0.797476i \(0.293832\pi\)
\(572\) −1.20871 −0.0505388
\(573\) −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
\(583\) 9.58258 0.396870
\(584\) −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 \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 1.79129 0.0734974
\(595\) 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 \(-0.333482\pi\)
0.499595 + 0.866259i \(0.333482\pi\)
\(602\) 2.83485 0.115540
\(603\) 8.58258 0.349510
\(604\) 4.33030 0.176198
\(605\) 3.00000 0.121967
\(606\) −20.7477 −0.842819
\(607\) −21.7477 −0.882713 −0.441357 0.897332i \(-0.645503\pi\)
−0.441357 + 0.897332i \(0.645503\pi\)
\(608\) 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 \(-0.681638\pi\)
−0.540165 + 0.841559i \(0.681638\pi\)
\(614\) 0 0
\(615\) 33.4955 1.35067
\(616\) 1.41742 0.0571097
\(617\) 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
\(627\) −6.58258 −0.262883
\(628\) 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
\(638\) 14.6261 0.579054
\(639\) 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
\(649\) −4.58258 −0.179882
\(650\) 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 \(-0.687936\pi\)
−0.556710 + 0.830707i \(0.687936\pi\)
\(660\) 3.62614 0.141147
\(661\) −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
\(671\) −10.0000 −0.386046
\(672\) 6.04356 0.233135
\(673\) 11.2523 0.433743 0.216872 0.976200i \(-0.430415\pi\)
0.216872 + 0.976200i \(0.430415\pi\)
\(674\) 31.4955 1.21316
\(675\) −4.00000 −0.153960
\(676\) −14.5045 −0.557867
\(677\) −45.1652 −1.73584 −0.867919 0.496706i \(-0.834543\pi\)
−0.867919 + 0.496706i \(0.834543\pi\)
\(678\) −16.4174 −0.630507
\(679\) −2.41742 −0.0927722
\(680\) −32.2432 −1.23647
\(681\) 22.0000 0.843042
\(682\) −6.41742 −0.245736
\(683\) −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
\(693\) −1.00000 −0.0379869
\(694\) 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 \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) −1.79129 −0.0676078
\(703\) −6.58258 −0.248267
\(704\) 0.912878 0.0344054
\(705\) −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
\(715\) −3.00000 −0.112194
\(716\) 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
\(726\) −1.79129 −0.0664809
\(727\) −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 \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 39.4083 1.45459
\(735\) −3.00000 −0.110657
\(736\) 33.7386 1.24362
\(737\) −8.58258 −0.316143
\(738\) −20.0000 −0.736210
\(739\) −31.9129 −1.17393 −0.586967 0.809611i \(-0.699678\pi\)
−0.586967 + 0.809611i \(0.699678\pi\)
\(740\) 3.62614 0.133299
\(741\) 6.58258 0.241817
\(742\) −17.1652 −0.630153
\(743\) −53.2432 −1.95330 −0.976651 0.214830i \(-0.931080\pi\)
−0.976651 + 0.214830i \(0.931080\pi\)
\(744\) 5.07803 0.186170
\(745\) 18.4955 0.677621
\(746\) −62.2432 −2.27888
\(747\) −11.5826 −0.423784
\(748\) −9.16515 −0.335111
\(749\) −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
\(759\) −5.58258 −0.202635
\(760\) 27.9909 1.01534
\(761\) 42.3303 1.53447 0.767236 0.641365i \(-0.221631\pi\)
0.767236 + 0.641365i \(0.221631\pi\)
\(762\) 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 \(-0.462635\pi\)
0.117116 + 0.993118i \(0.462635\pi\)
\(770\) −5.37386 −0.193661
\(771\) −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
\(781\) −11.1652 −0.399521
\(782\) −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 \(-0.258638\pi\)
0.687660 + 0.726033i \(0.258638\pi\)
\(788\) 6.24318 0.222404
\(789\) 22.9129 0.815720
\(790\) 38.5045 1.36993
\(791\) 9.16515 0.325875
\(792\) 1.41742 0.0503660
\(793\) 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
\(803\) −7.00000 −0.247025
\(804\) −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 \(-0.447554\pi\)
0.164018 + 0.986457i \(0.447554\pi\)
\(810\) 5.37386 0.188818
\(811\) −2.25227 −0.0790880 −0.0395440 0.999218i \(-0.512591\pi\)
−0.0395440 + 0.999218i \(0.512591\pi\)
\(812\) −9.86932 −0.346345
\(813\) −5.41742 −0.189997
\(814\) −1.79129 −0.0627846
\(815\) 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 \(-0.806113\pi\)
−0.820156 + 0.572140i \(0.806113\pi\)
\(822\) 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
\(825\) 4.00000 0.139262
\(826\) 8.20871 0.285618
\(827\) 8.91288 0.309931 0.154966 0.987920i \(-0.450473\pi\)
0.154966 + 0.987920i \(0.450473\pi\)
\(828\) −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
\(836\) 7.95644 0.275179
\(837\) −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
\(847\) 1.00000 0.0343604
\(848\) 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 \(-0.405059\pi\)
0.293862 + 0.955848i \(0.405059\pi\)
\(854\) 17.9129 0.612966
\(855\) −19.7477 −0.675358
\(856\) 17.8348 0.609583
\(857\) −7.66970 −0.261992 −0.130996 0.991383i \(-0.541817\pi\)
−0.130996 + 0.991383i \(0.541817\pi\)
\(858\) 1.79129 0.0611536
\(859\) 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
\(869\) −7.16515 −0.243061
\(870\) 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 \(-0.448748\pi\)
0.160319 + 0.987065i \(0.448748\pi\)
\(878\) 31.1996 1.05294
\(879\) 0 0
\(880\) 14.8693 0.501245
\(881\) −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 \(-0.274555\pi\)
0.650511 + 0.759497i \(0.274555\pi\)
\(888\) 1.41742 0.0475656
\(889\) −11.5826 −0.388467
\(890\) 49.2523 1.65094
\(891\) −1.00000 −0.0335013
\(892\) 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
\(902\) 20.0000 0.665927
\(903\) −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
\(913\) 11.5826 0.383327
\(914\) −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 \(-0.542585\pi\)
−0.133385 + 0.991064i \(0.542585\pi\)
\(920\) 23.7386 0.782640
\(921\) 0 0
\(922\) 32.8348 1.08136
\(923\) 11.1652 0.367505
\(924\) 1.20871 0.0397637
\(925\) 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
\(935\) −22.7477 −0.743930
\(936\) −1.41742 −0.0463300
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 15.3739 0.501974
\(939\) −19.5826 −0.639053
\(940\) 5.13977 0.167641
\(941\) 35.1652 1.14635 0.573176 0.819433i \(-0.305711\pi\)
0.573176 + 0.819433i \(0.305711\pi\)
\(942\) −34.3303 −1.11854
\(943\) 62.3303 2.02975
\(944\) −22.7133 −0.739254
\(945\) −3.00000 −0.0975900
\(946\) −2.83485 −0.0921689
\(947\) 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 \(-0.650782\pi\)
−0.456179 + 0.889888i \(0.650782\pi\)
\(954\) −17.1652 −0.555742
\(955\) 34.7477 1.12441
\(956\) 20.0436 0.648255
\(957\) −8.16515 −0.263942
\(958\) −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 \(-0.567894\pi\)
−0.211681 + 0.977339i \(0.567894\pi\)
\(968\) −1.41742 −0.0455577
\(969\) 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
\(979\) −9.16515 −0.292920
\(980\) 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
\(990\) −5.37386 −0.170793
\(991\) −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 \(-0.446007\pi\)
0.168812 + 0.985648i \(0.446007\pi\)
\(998\) 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 231.2.a.b.1.2 2
3.2 odd 2 693.2.a.j.1.1 2
4.3 odd 2 3696.2.a.bl.1.2 2
5.4 even 2 5775.2.a.bn.1.1 2
7.6 odd 2 1617.2.a.o.1.2 2
11.10 odd 2 2541.2.a.z.1.1 2
21.20 even 2 4851.2.a.ba.1.1 2
33.32 even 2 7623.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.b.1.2 2 1.1 even 1 trivial
693.2.a.j.1.1 2 3.2 odd 2
1617.2.a.o.1.2 2 7.6 odd 2
2541.2.a.z.1.1 2 11.10 odd 2
3696.2.a.bl.1.2 2 4.3 odd 2
4851.2.a.ba.1.1 2 21.20 even 2
5775.2.a.bn.1.1 2 5.4 even 2
7623.2.a.bf.1.2 2 33.32 even 2