Properties

Label 722.2.a.m.1.4
Level $722$
Weight $2$
Character 722.1
Self dual yes
Analytic conductor $5.765$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.76519902594\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \(x^{4} - 5 x^{2} + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.17557\) of defining polynomial
Character \(\chi\) \(=\) 722.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.52015 q^{3} +1.00000 q^{4} -2.45965 q^{5} -2.52015 q^{6} -2.79360 q^{7} -1.00000 q^{8} +3.35114 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.52015 q^{3} +1.00000 q^{4} -2.45965 q^{5} -2.52015 q^{6} -2.79360 q^{7} -1.00000 q^{8} +3.35114 q^{9} +2.45965 q^{10} -1.67853 q^{11} +2.52015 q^{12} -6.34458 q^{13} +2.79360 q^{14} -6.19868 q^{15} +1.00000 q^{16} +4.96917 q^{17} -3.35114 q^{18} -2.45965 q^{20} -7.04029 q^{21} +1.67853 q^{22} -2.49890 q^{23} -2.52015 q^{24} +1.04988 q^{25} +6.34458 q^{26} +0.884927 q^{27} -2.79360 q^{28} -5.93179 q^{29} +6.19868 q^{30} -7.28408 q^{31} -1.00000 q^{32} -4.23015 q^{33} -4.96917 q^{34} +6.87129 q^{35} +3.35114 q^{36} -0.550972 q^{37} -15.9893 q^{39} +2.45965 q^{40} +2.60741 q^{41} +7.04029 q^{42} +2.87129 q^{43} -1.67853 q^{44} -8.24263 q^{45} +2.49890 q^{46} +0.745593 q^{47} +2.52015 q^{48} +0.804226 q^{49} -1.04988 q^{50} +12.5231 q^{51} -6.34458 q^{52} +1.47027 q^{53} -0.884927 q^{54} +4.12860 q^{55} +2.79360 q^{56} +5.93179 q^{58} +4.96261 q^{59} -6.19868 q^{60} +9.33686 q^{61} +7.28408 q^{62} -9.36176 q^{63} +1.00000 q^{64} +15.6054 q^{65} +4.23015 q^{66} -11.5604 q^{67} +4.96917 q^{68} -6.29761 q^{69} -6.87129 q^{70} +6.99698 q^{71} -3.35114 q^{72} -6.18619 q^{73} +0.550972 q^{74} +2.64584 q^{75} +4.68915 q^{77} +15.9893 q^{78} +5.91158 q^{79} -2.45965 q^{80} -7.82328 q^{81} -2.60741 q^{82} +15.1773 q^{83} -7.04029 q^{84} -12.2224 q^{85} -2.87129 q^{86} -14.9490 q^{87} +1.67853 q^{88} -6.90502 q^{89} +8.24263 q^{90} +17.7242 q^{91} -2.49890 q^{92} -18.3570 q^{93} -0.745593 q^{94} -2.52015 q^{96} -14.3874 q^{97} -0.804226 q^{98} -5.62500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 2q^{3} + 4q^{4} - 2q^{5} + 2q^{6} - 2q^{7} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 2q^{3} + 4q^{4} - 2q^{5} + 2q^{6} - 2q^{7} - 4q^{8} + 4q^{9} + 2q^{10} + 2q^{11} - 2q^{12} - 18q^{13} + 2q^{14} - 4q^{15} + 4q^{16} + 6q^{17} - 4q^{18} - 2q^{20} - 4q^{21} - 2q^{22} - 10q^{23} + 2q^{24} + 6q^{25} + 18q^{26} + 4q^{27} - 2q^{28} + 2q^{29} + 4q^{30} - 26q^{31} - 4q^{32} - 16q^{33} - 6q^{34} + 6q^{35} + 4q^{36} - 4q^{37} - 6q^{39} + 2q^{40} + 12q^{41} + 4q^{42} - 10q^{43} + 2q^{44} - 22q^{45} + 10q^{46} - 12q^{47} - 2q^{48} - 12q^{49} - 6q^{50} + 2q^{51} - 18q^{52} - 8q^{53} - 4q^{54} - 26q^{55} + 2q^{56} - 2q^{58} + 8q^{59} - 4q^{60} + 26q^{62} - 22q^{63} + 4q^{64} + 4q^{65} + 16q^{66} - 10q^{67} + 6q^{68} + 20q^{69} - 6q^{70} - 4q^{72} - 14q^{73} + 4q^{74} - 8q^{75} + 4q^{77} + 6q^{78} - 22q^{79} - 2q^{80} - 4q^{81} - 12q^{82} - 12q^{83} - 4q^{84} - 18q^{85} + 10q^{86} - 26q^{87} - 2q^{88} + 16q^{89} + 22q^{90} + 4q^{91} - 10q^{92} + 8q^{93} + 12q^{94} + 2q^{96} - 28q^{97} + 12q^{98} + 22q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.52015 1.45501 0.727504 0.686104i \(-0.240680\pi\)
0.727504 + 0.686104i \(0.240680\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.45965 −1.09999 −0.549994 0.835168i \(-0.685370\pi\)
−0.549994 + 0.835168i \(0.685370\pi\)
\(6\) −2.52015 −1.02885
\(7\) −2.79360 −1.05588 −0.527942 0.849281i \(-0.677036\pi\)
−0.527942 + 0.849281i \(0.677036\pi\)
\(8\) −1.00000 −0.353553
\(9\) 3.35114 1.11705
\(10\) 2.45965 0.777809
\(11\) −1.67853 −0.506096 −0.253048 0.967454i \(-0.581433\pi\)
−0.253048 + 0.967454i \(0.581433\pi\)
\(12\) 2.52015 0.727504
\(13\) −6.34458 −1.75967 −0.879834 0.475280i \(-0.842347\pi\)
−0.879834 + 0.475280i \(0.842347\pi\)
\(14\) 2.79360 0.746622
\(15\) −6.19868 −1.60049
\(16\) 1.00000 0.250000
\(17\) 4.96917 1.20520 0.602601 0.798043i \(-0.294131\pi\)
0.602601 + 0.798043i \(0.294131\pi\)
\(18\) −3.35114 −0.789872
\(19\) 0 0
\(20\) −2.45965 −0.549994
\(21\) −7.04029 −1.53632
\(22\) 1.67853 0.357864
\(23\) −2.49890 −0.521057 −0.260529 0.965466i \(-0.583897\pi\)
−0.260529 + 0.965466i \(0.583897\pi\)
\(24\) −2.52015 −0.514423
\(25\) 1.04988 0.209975
\(26\) 6.34458 1.24427
\(27\) 0.884927 0.170304
\(28\) −2.79360 −0.527942
\(29\) −5.93179 −1.10150 −0.550752 0.834669i \(-0.685659\pi\)
−0.550752 + 0.834669i \(0.685659\pi\)
\(30\) 6.19868 1.13172
\(31\) −7.28408 −1.30826 −0.654130 0.756382i \(-0.726965\pi\)
−0.654130 + 0.756382i \(0.726965\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.23015 −0.736374
\(34\) −4.96917 −0.852206
\(35\) 6.87129 1.16146
\(36\) 3.35114 0.558524
\(37\) −0.550972 −0.0905792 −0.0452896 0.998974i \(-0.514421\pi\)
−0.0452896 + 0.998974i \(0.514421\pi\)
\(38\) 0 0
\(39\) −15.9893 −2.56033
\(40\) 2.45965 0.388905
\(41\) 2.60741 0.407209 0.203605 0.979053i \(-0.434734\pi\)
0.203605 + 0.979053i \(0.434734\pi\)
\(42\) 7.04029 1.08634
\(43\) 2.87129 0.437867 0.218934 0.975740i \(-0.429742\pi\)
0.218934 + 0.975740i \(0.429742\pi\)
\(44\) −1.67853 −0.253048
\(45\) −8.24263 −1.22874
\(46\) 2.49890 0.368443
\(47\) 0.745593 0.108756 0.0543780 0.998520i \(-0.482682\pi\)
0.0543780 + 0.998520i \(0.482682\pi\)
\(48\) 2.52015 0.363752
\(49\) 0.804226 0.114889
\(50\) −1.04988 −0.148475
\(51\) 12.5231 1.75358
\(52\) −6.34458 −0.879834
\(53\) 1.47027 0.201957 0.100979 0.994889i \(-0.467803\pi\)
0.100979 + 0.994889i \(0.467803\pi\)
\(54\) −0.884927 −0.120423
\(55\) 4.12860 0.556700
\(56\) 2.79360 0.373311
\(57\) 0 0
\(58\) 5.93179 0.778882
\(59\) 4.96261 0.646077 0.323038 0.946386i \(-0.395296\pi\)
0.323038 + 0.946386i \(0.395296\pi\)
\(60\) −6.19868 −0.800246
\(61\) 9.33686 1.19546 0.597731 0.801697i \(-0.296069\pi\)
0.597731 + 0.801697i \(0.296069\pi\)
\(62\) 7.28408 0.925079
\(63\) −9.36176 −1.17947
\(64\) 1.00000 0.125000
\(65\) 15.6054 1.93562
\(66\) 4.23015 0.520695
\(67\) −11.5604 −1.41233 −0.706166 0.708046i \(-0.749577\pi\)
−0.706166 + 0.708046i \(0.749577\pi\)
\(68\) 4.96917 0.602601
\(69\) −6.29761 −0.758143
\(70\) −6.87129 −0.821276
\(71\) 6.99698 0.830389 0.415195 0.909733i \(-0.363714\pi\)
0.415195 + 0.909733i \(0.363714\pi\)
\(72\) −3.35114 −0.394936
\(73\) −6.18619 −0.724039 −0.362020 0.932171i \(-0.617913\pi\)
−0.362020 + 0.932171i \(0.617913\pi\)
\(74\) 0.550972 0.0640492
\(75\) 2.64584 0.305515
\(76\) 0 0
\(77\) 4.68915 0.534379
\(78\) 15.9893 1.81043
\(79\) 5.91158 0.665105 0.332552 0.943085i \(-0.392090\pi\)
0.332552 + 0.943085i \(0.392090\pi\)
\(80\) −2.45965 −0.274997
\(81\) −7.82328 −0.869253
\(82\) −2.60741 −0.287941
\(83\) 15.1773 1.66593 0.832964 0.553328i \(-0.186642\pi\)
0.832964 + 0.553328i \(0.186642\pi\)
\(84\) −7.04029 −0.768159
\(85\) −12.2224 −1.32571
\(86\) −2.87129 −0.309619
\(87\) −14.9490 −1.60270
\(88\) 1.67853 0.178932
\(89\) −6.90502 −0.731930 −0.365965 0.930629i \(-0.619261\pi\)
−0.365965 + 0.930629i \(0.619261\pi\)
\(90\) 8.24263 0.868850
\(91\) 17.7242 1.85800
\(92\) −2.49890 −0.260529
\(93\) −18.3570 −1.90353
\(94\) −0.745593 −0.0769021
\(95\) 0 0
\(96\) −2.52015 −0.257211
\(97\) −14.3874 −1.46082 −0.730408 0.683011i \(-0.760670\pi\)
−0.730408 + 0.683011i \(0.760670\pi\)
\(98\) −0.804226 −0.0812391
\(99\) −5.62500 −0.565333
\(100\) 1.04988 0.104988
\(101\) 11.2829 1.12269 0.561347 0.827581i \(-0.310283\pi\)
0.561347 + 0.827581i \(0.310283\pi\)
\(102\) −12.5231 −1.23997
\(103\) −14.8280 −1.46104 −0.730522 0.682889i \(-0.760723\pi\)
−0.730522 + 0.682889i \(0.760723\pi\)
\(104\) 6.34458 0.622137
\(105\) 17.3167 1.68993
\(106\) −1.47027 −0.142805
\(107\) 2.82849 0.273440 0.136720 0.990610i \(-0.456344\pi\)
0.136720 + 0.990610i \(0.456344\pi\)
\(108\) 0.884927 0.0851521
\(109\) −1.72539 −0.165262 −0.0826312 0.996580i \(-0.526332\pi\)
−0.0826312 + 0.996580i \(0.526332\pi\)
\(110\) −4.12860 −0.393646
\(111\) −1.38853 −0.131793
\(112\) −2.79360 −0.263971
\(113\) 0.427785 0.0402426 0.0201213 0.999798i \(-0.493595\pi\)
0.0201213 + 0.999798i \(0.493595\pi\)
\(114\) 0 0
\(115\) 6.14643 0.573157
\(116\) −5.93179 −0.550752
\(117\) −21.2616 −1.96563
\(118\) −4.96261 −0.456845
\(119\) −13.8819 −1.27255
\(120\) 6.19868 0.565859
\(121\) −8.18253 −0.743867
\(122\) −9.33686 −0.845320
\(123\) 6.57106 0.592493
\(124\) −7.28408 −0.654130
\(125\) 9.71592 0.869018
\(126\) 9.36176 0.834012
\(127\) 1.13632 0.100832 0.0504159 0.998728i \(-0.483945\pi\)
0.0504159 + 0.998728i \(0.483945\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.23607 0.637100
\(130\) −15.6054 −1.36869
\(131\) −17.8415 −1.55882 −0.779410 0.626515i \(-0.784481\pi\)
−0.779410 + 0.626515i \(0.784481\pi\)
\(132\) −4.23015 −0.368187
\(133\) 0 0
\(134\) 11.5604 0.998670
\(135\) −2.17661 −0.187333
\(136\) −4.96917 −0.426103
\(137\) −15.9156 −1.35976 −0.679882 0.733321i \(-0.737969\pi\)
−0.679882 + 0.733321i \(0.737969\pi\)
\(138\) 6.29761 0.536088
\(139\) 10.1202 0.858382 0.429191 0.903214i \(-0.358799\pi\)
0.429191 + 0.903214i \(0.358799\pi\)
\(140\) 6.87129 0.580730
\(141\) 1.87901 0.158241
\(142\) −6.99698 −0.587174
\(143\) 10.6496 0.890562
\(144\) 3.35114 0.279262
\(145\) 14.5901 1.21164
\(146\) 6.18619 0.511973
\(147\) 2.02677 0.167165
\(148\) −0.550972 −0.0452896
\(149\) 9.39851 0.769956 0.384978 0.922926i \(-0.374209\pi\)
0.384978 + 0.922926i \(0.374209\pi\)
\(150\) −2.64584 −0.216032
\(151\) 10.2632 0.835210 0.417605 0.908629i \(-0.362870\pi\)
0.417605 + 0.908629i \(0.362870\pi\)
\(152\) 0 0
\(153\) 16.6524 1.34627
\(154\) −4.68915 −0.377863
\(155\) 17.9163 1.43907
\(156\) −15.9893 −1.28017
\(157\) 2.97387 0.237341 0.118671 0.992934i \(-0.462137\pi\)
0.118671 + 0.992934i \(0.462137\pi\)
\(158\) −5.91158 −0.470300
\(159\) 3.70530 0.293849
\(160\) 2.45965 0.194452
\(161\) 6.98095 0.550176
\(162\) 7.82328 0.614655
\(163\) 5.59191 0.437992 0.218996 0.975726i \(-0.429722\pi\)
0.218996 + 0.975726i \(0.429722\pi\)
\(164\) 2.60741 0.203605
\(165\) 10.4047 0.810003
\(166\) −15.1773 −1.17799
\(167\) 7.08361 0.548146 0.274073 0.961709i \(-0.411629\pi\)
0.274073 + 0.961709i \(0.411629\pi\)
\(168\) 7.04029 0.543170
\(169\) 27.2537 2.09643
\(170\) 12.2224 0.937418
\(171\) 0 0
\(172\) 2.87129 0.218934
\(173\) 23.0208 1.75024 0.875120 0.483907i \(-0.160783\pi\)
0.875120 + 0.483907i \(0.160783\pi\)
\(174\) 14.9490 1.13328
\(175\) −2.93294 −0.221709
\(176\) −1.67853 −0.126524
\(177\) 12.5065 0.940047
\(178\) 6.90502 0.517553
\(179\) −8.21471 −0.613996 −0.306998 0.951710i \(-0.599325\pi\)
−0.306998 + 0.951710i \(0.599325\pi\)
\(180\) −8.24263 −0.614370
\(181\) 12.9999 0.966274 0.483137 0.875545i \(-0.339497\pi\)
0.483137 + 0.875545i \(0.339497\pi\)
\(182\) −17.7242 −1.31381
\(183\) 23.5303 1.73941
\(184\) 2.49890 0.184222
\(185\) 1.35520 0.0996361
\(186\) 18.3570 1.34600
\(187\) −8.34092 −0.609948
\(188\) 0.745593 0.0543780
\(189\) −2.47214 −0.179821
\(190\) 0 0
\(191\) 6.57479 0.475735 0.237868 0.971298i \(-0.423552\pi\)
0.237868 + 0.971298i \(0.423552\pi\)
\(192\) 2.52015 0.181876
\(193\) −26.8826 −1.93505 −0.967527 0.252769i \(-0.918659\pi\)
−0.967527 + 0.252769i \(0.918659\pi\)
\(194\) 14.3874 1.03295
\(195\) 39.3280 2.81634
\(196\) 0.804226 0.0574447
\(197\) −9.84940 −0.701741 −0.350870 0.936424i \(-0.614114\pi\)
−0.350870 + 0.936424i \(0.614114\pi\)
\(198\) 5.62500 0.399751
\(199\) −20.4661 −1.45080 −0.725402 0.688326i \(-0.758346\pi\)
−0.725402 + 0.688326i \(0.758346\pi\)
\(200\) −1.04988 −0.0742374
\(201\) −29.1340 −2.05495
\(202\) −11.2829 −0.793864
\(203\) 16.5711 1.16306
\(204\) 12.5231 0.876789
\(205\) −6.41332 −0.447926
\(206\) 14.8280 1.03311
\(207\) −8.37418 −0.582046
\(208\) −6.34458 −0.439917
\(209\) 0 0
\(210\) −17.3167 −1.19496
\(211\) −8.11630 −0.558749 −0.279374 0.960182i \(-0.590127\pi\)
−0.279374 + 0.960182i \(0.590127\pi\)
\(212\) 1.47027 0.100979
\(213\) 17.6334 1.20822
\(214\) −2.82849 −0.193351
\(215\) −7.06236 −0.481649
\(216\) −0.884927 −0.0602117
\(217\) 20.3488 1.38137
\(218\) 1.72539 0.116858
\(219\) −15.5901 −1.05348
\(220\) 4.12860 0.278350
\(221\) −31.5273 −2.12076
\(222\) 1.38853 0.0931921
\(223\) −10.7540 −0.720143 −0.360071 0.932925i \(-0.617248\pi\)
−0.360071 + 0.932925i \(0.617248\pi\)
\(224\) 2.79360 0.186656
\(225\) 3.51828 0.234552
\(226\) −0.427785 −0.0284558
\(227\) −27.0936 −1.79827 −0.899133 0.437676i \(-0.855802\pi\)
−0.899133 + 0.437676i \(0.855802\pi\)
\(228\) 0 0
\(229\) 9.46557 0.625503 0.312751 0.949835i \(-0.398749\pi\)
0.312751 + 0.949835i \(0.398749\pi\)
\(230\) −6.14643 −0.405283
\(231\) 11.8174 0.777525
\(232\) 5.93179 0.389441
\(233\) 22.8621 1.49775 0.748873 0.662713i \(-0.230595\pi\)
0.748873 + 0.662713i \(0.230595\pi\)
\(234\) 21.2616 1.38991
\(235\) −1.83390 −0.119630
\(236\) 4.96261 0.323038
\(237\) 14.8981 0.967733
\(238\) 13.8819 0.899831
\(239\) 8.66611 0.560564 0.280282 0.959918i \(-0.409572\pi\)
0.280282 + 0.959918i \(0.409572\pi\)
\(240\) −6.19868 −0.400123
\(241\) −5.65826 −0.364480 −0.182240 0.983254i \(-0.558335\pi\)
−0.182240 + 0.983254i \(0.558335\pi\)
\(242\) 8.18253 0.525993
\(243\) −22.3706 −1.43507
\(244\) 9.33686 0.597731
\(245\) −1.97811 −0.126377
\(246\) −6.57106 −0.418956
\(247\) 0 0
\(248\) 7.28408 0.462539
\(249\) 38.2491 2.42394
\(250\) −9.71592 −0.614489
\(251\) −13.5552 −0.855599 −0.427799 0.903874i \(-0.640711\pi\)
−0.427799 + 0.903874i \(0.640711\pi\)
\(252\) −9.36176 −0.589736
\(253\) 4.19449 0.263705
\(254\) −1.13632 −0.0712988
\(255\) −30.8023 −1.92892
\(256\) 1.00000 0.0625000
\(257\) −10.0393 −0.626231 −0.313116 0.949715i \(-0.601373\pi\)
−0.313116 + 0.949715i \(0.601373\pi\)
\(258\) −7.23607 −0.450498
\(259\) 1.53920 0.0956411
\(260\) 15.6054 0.967808
\(261\) −19.8782 −1.23043
\(262\) 17.8415 1.10225
\(263\) −18.8262 −1.16087 −0.580436 0.814306i \(-0.697118\pi\)
−0.580436 + 0.814306i \(0.697118\pi\)
\(264\) 4.23015 0.260347
\(265\) −3.61635 −0.222151
\(266\) 0 0
\(267\) −17.4017 −1.06496
\(268\) −11.5604 −0.706166
\(269\) 16.3827 0.998870 0.499435 0.866351i \(-0.333541\pi\)
0.499435 + 0.866351i \(0.333541\pi\)
\(270\) 2.17661 0.132464
\(271\) −2.76684 −0.168073 −0.0840367 0.996463i \(-0.526781\pi\)
−0.0840367 + 0.996463i \(0.526781\pi\)
\(272\) 4.96917 0.301300
\(273\) 44.6677 2.70341
\(274\) 15.9156 0.961499
\(275\) −1.76225 −0.106268
\(276\) −6.29761 −0.379071
\(277\) 24.5321 1.47399 0.736996 0.675897i \(-0.236244\pi\)
0.736996 + 0.675897i \(0.236244\pi\)
\(278\) −10.1202 −0.606967
\(279\) −24.4100 −1.46139
\(280\) −6.87129 −0.410638
\(281\) −10.1704 −0.606713 −0.303356 0.952877i \(-0.598107\pi\)
−0.303356 + 0.952877i \(0.598107\pi\)
\(282\) −1.87901 −0.111893
\(283\) 27.6678 1.64468 0.822340 0.568997i \(-0.192668\pi\)
0.822340 + 0.568997i \(0.192668\pi\)
\(284\) 6.99698 0.415195
\(285\) 0 0
\(286\) −10.6496 −0.629722
\(287\) −7.28408 −0.429966
\(288\) −3.35114 −0.197468
\(289\) 7.69270 0.452512
\(290\) −14.5901 −0.856761
\(291\) −36.2583 −2.12550
\(292\) −6.18619 −0.362020
\(293\) −23.8386 −1.39267 −0.696333 0.717719i \(-0.745186\pi\)
−0.696333 + 0.717719i \(0.745186\pi\)
\(294\) −2.02677 −0.118204
\(295\) −12.2063 −0.710677
\(296\) 0.550972 0.0320246
\(297\) −1.48538 −0.0861904
\(298\) −9.39851 −0.544441
\(299\) 15.8545 0.916889
\(300\) 2.64584 0.152758
\(301\) −8.02124 −0.462337
\(302\) −10.2632 −0.590583
\(303\) 28.4346 1.63353
\(304\) 0 0
\(305\) −22.9654 −1.31500
\(306\) −16.6524 −0.951955
\(307\) −25.9884 −1.48324 −0.741619 0.670821i \(-0.765942\pi\)
−0.741619 + 0.670821i \(0.765942\pi\)
\(308\) 4.68915 0.267189
\(309\) −37.3687 −2.12583
\(310\) −17.9163 −1.01758
\(311\) −23.1043 −1.31013 −0.655063 0.755574i \(-0.727358\pi\)
−0.655063 + 0.755574i \(0.727358\pi\)
\(312\) 15.9893 0.905214
\(313\) 12.2647 0.693242 0.346621 0.938005i \(-0.387329\pi\)
0.346621 + 0.938005i \(0.387329\pi\)
\(314\) −2.97387 −0.167825
\(315\) 23.0267 1.29741
\(316\) 5.91158 0.332552
\(317\) 0.272305 0.0152942 0.00764708 0.999971i \(-0.497566\pi\)
0.00764708 + 0.999971i \(0.497566\pi\)
\(318\) −3.70530 −0.207783
\(319\) 9.95669 0.557468
\(320\) −2.45965 −0.137499
\(321\) 7.12820 0.397857
\(322\) −6.98095 −0.389033
\(323\) 0 0
\(324\) −7.82328 −0.434626
\(325\) −6.66102 −0.369487
\(326\) −5.59191 −0.309707
\(327\) −4.34824 −0.240458
\(328\) −2.60741 −0.143970
\(329\) −2.08289 −0.114834
\(330\) −10.4047 −0.572759
\(331\) 19.9930 1.09891 0.549457 0.835522i \(-0.314835\pi\)
0.549457 + 0.835522i \(0.314835\pi\)
\(332\) 15.1773 0.832964
\(333\) −1.84638 −0.101181
\(334\) −7.08361 −0.387598
\(335\) 28.4346 1.55355
\(336\) −7.04029 −0.384080
\(337\) 17.2824 0.941433 0.470717 0.882284i \(-0.343995\pi\)
0.470717 + 0.882284i \(0.343995\pi\)
\(338\) −27.2537 −1.48240
\(339\) 1.07808 0.0585533
\(340\) −12.2224 −0.662854
\(341\) 12.2266 0.662105
\(342\) 0 0
\(343\) 17.3085 0.934573
\(344\) −2.87129 −0.154809
\(345\) 15.4899 0.833948
\(346\) −23.0208 −1.23761
\(347\) −18.5456 −0.995579 −0.497789 0.867298i \(-0.665855\pi\)
−0.497789 + 0.867298i \(0.665855\pi\)
\(348\) −14.9490 −0.801349
\(349\) −20.1210 −1.07705 −0.538527 0.842609i \(-0.681019\pi\)
−0.538527 + 0.842609i \(0.681019\pi\)
\(350\) 2.93294 0.156772
\(351\) −5.61449 −0.299679
\(352\) 1.67853 0.0894660
\(353\) 24.0557 1.28035 0.640177 0.768227i \(-0.278861\pi\)
0.640177 + 0.768227i \(0.278861\pi\)
\(354\) −12.5065 −0.664713
\(355\) −17.2101 −0.913419
\(356\) −6.90502 −0.365965
\(357\) −34.9845 −1.85157
\(358\) 8.21471 0.434161
\(359\) −10.8535 −0.572824 −0.286412 0.958107i \(-0.592463\pi\)
−0.286412 + 0.958107i \(0.592463\pi\)
\(360\) 8.24263 0.434425
\(361\) 0 0
\(362\) −12.9999 −0.683259
\(363\) −20.6212 −1.08233
\(364\) 17.7242 0.929002
\(365\) 15.2159 0.796435
\(366\) −23.5303 −1.22995
\(367\) −32.6121 −1.70234 −0.851168 0.524893i \(-0.824105\pi\)
−0.851168 + 0.524893i \(0.824105\pi\)
\(368\) −2.49890 −0.130264
\(369\) 8.73781 0.454872
\(370\) −1.35520 −0.0704534
\(371\) −4.10736 −0.213243
\(372\) −18.3570 −0.951764
\(373\) 5.30198 0.274526 0.137263 0.990535i \(-0.456169\pi\)
0.137263 + 0.990535i \(0.456169\pi\)
\(374\) 8.34092 0.431299
\(375\) 24.4855 1.26443
\(376\) −0.745593 −0.0384510
\(377\) 37.6347 1.93828
\(378\) 2.47214 0.127153
\(379\) −24.6656 −1.26699 −0.633494 0.773748i \(-0.718380\pi\)
−0.633494 + 0.773748i \(0.718380\pi\)
\(380\) 0 0
\(381\) 2.86368 0.146711
\(382\) −6.57479 −0.336396
\(383\) −3.64416 −0.186208 −0.0931039 0.995656i \(-0.529679\pi\)
−0.0931039 + 0.995656i \(0.529679\pi\)
\(384\) −2.52015 −0.128606
\(385\) −11.5337 −0.587810
\(386\) 26.8826 1.36829
\(387\) 9.62209 0.489118
\(388\) −14.3874 −0.730408
\(389\) 2.79353 0.141638 0.0708189 0.997489i \(-0.477439\pi\)
0.0708189 + 0.997489i \(0.477439\pi\)
\(390\) −39.3280 −1.99145
\(391\) −12.4175 −0.627979
\(392\) −0.804226 −0.0406196
\(393\) −44.9632 −2.26809
\(394\) 9.84940 0.496206
\(395\) −14.5404 −0.731608
\(396\) −5.62500 −0.282667
\(397\) −39.5101 −1.98296 −0.991478 0.130273i \(-0.958415\pi\)
−0.991478 + 0.130273i \(0.958415\pi\)
\(398\) 20.4661 1.02587
\(399\) 0 0
\(400\) 1.04988 0.0524938
\(401\) 18.6215 0.929912 0.464956 0.885334i \(-0.346070\pi\)
0.464956 + 0.885334i \(0.346070\pi\)
\(402\) 29.1340 1.45307
\(403\) 46.2144 2.30210
\(404\) 11.2829 0.561347
\(405\) 19.2425 0.956168
\(406\) −16.5711 −0.822408
\(407\) 0.924824 0.0458418
\(408\) −12.5231 −0.619983
\(409\) 17.1433 0.847681 0.423840 0.905737i \(-0.360682\pi\)
0.423840 + 0.905737i \(0.360682\pi\)
\(410\) 6.41332 0.316731
\(411\) −40.1098 −1.97847
\(412\) −14.8280 −0.730522
\(413\) −13.8636 −0.682182
\(414\) 8.37418 0.411568
\(415\) −37.3309 −1.83250
\(416\) 6.34458 0.311068
\(417\) 25.5043 1.24895
\(418\) 0 0
\(419\) −18.5042 −0.903989 −0.451995 0.892021i \(-0.649287\pi\)
−0.451995 + 0.892021i \(0.649287\pi\)
\(420\) 17.3167 0.844966
\(421\) 16.1996 0.789520 0.394760 0.918784i \(-0.370828\pi\)
0.394760 + 0.918784i \(0.370828\pi\)
\(422\) 8.11630 0.395095
\(423\) 2.49859 0.121486
\(424\) −1.47027 −0.0714027
\(425\) 5.21702 0.253062
\(426\) −17.6334 −0.854342
\(427\) −26.0835 −1.26227
\(428\) 2.82849 0.136720
\(429\) 26.8385 1.29577
\(430\) 7.06236 0.340577
\(431\) −12.6997 −0.611721 −0.305861 0.952076i \(-0.598944\pi\)
−0.305861 + 0.952076i \(0.598944\pi\)
\(432\) 0.884927 0.0425761
\(433\) −14.2350 −0.684092 −0.342046 0.939683i \(-0.611120\pi\)
−0.342046 + 0.939683i \(0.611120\pi\)
\(434\) −20.3488 −0.976775
\(435\) 36.7692 1.76295
\(436\) −1.72539 −0.0826312
\(437\) 0 0
\(438\) 15.5901 0.744924
\(439\) −28.1900 −1.34544 −0.672718 0.739899i \(-0.734873\pi\)
−0.672718 + 0.739899i \(0.734873\pi\)
\(440\) −4.12860 −0.196823
\(441\) 2.69507 0.128337
\(442\) 31.5273 1.49960
\(443\) −32.9944 −1.56761 −0.783805 0.621007i \(-0.786724\pi\)
−0.783805 + 0.621007i \(0.786724\pi\)
\(444\) −1.38853 −0.0658967
\(445\) 16.9839 0.805115
\(446\) 10.7540 0.509218
\(447\) 23.6856 1.12029
\(448\) −2.79360 −0.131985
\(449\) 17.1975 0.811601 0.405801 0.913962i \(-0.366993\pi\)
0.405801 + 0.913962i \(0.366993\pi\)
\(450\) −3.51828 −0.165853
\(451\) −4.37662 −0.206087
\(452\) 0.427785 0.0201213
\(453\) 25.8649 1.21524
\(454\) 27.0936 1.27157
\(455\) −43.5954 −2.04378
\(456\) 0 0
\(457\) −31.1517 −1.45722 −0.728608 0.684931i \(-0.759832\pi\)
−0.728608 + 0.684931i \(0.759832\pi\)
\(458\) −9.46557 −0.442297
\(459\) 4.39736 0.205251
\(460\) 6.14643 0.286579
\(461\) 15.5410 0.723816 0.361908 0.932214i \(-0.382125\pi\)
0.361908 + 0.932214i \(0.382125\pi\)
\(462\) −11.8174 −0.549793
\(463\) 20.6648 0.960374 0.480187 0.877166i \(-0.340569\pi\)
0.480187 + 0.877166i \(0.340569\pi\)
\(464\) −5.93179 −0.275376
\(465\) 45.1517 2.09386
\(466\) −22.8621 −1.05907
\(467\) −28.4830 −1.31803 −0.659017 0.752128i \(-0.729028\pi\)
−0.659017 + 0.752128i \(0.729028\pi\)
\(468\) −21.2616 −0.982816
\(469\) 32.2953 1.49126
\(470\) 1.83390 0.0845914
\(471\) 7.49460 0.345333
\(472\) −4.96261 −0.228423
\(473\) −4.81955 −0.221603
\(474\) −14.8981 −0.684290
\(475\) 0 0
\(476\) −13.8819 −0.636276
\(477\) 4.92709 0.225596
\(478\) −8.66611 −0.396379
\(479\) −10.4726 −0.478507 −0.239254 0.970957i \(-0.576903\pi\)
−0.239254 + 0.970957i \(0.576903\pi\)
\(480\) 6.19868 0.282930
\(481\) 3.49568 0.159389
\(482\) 5.65826 0.257727
\(483\) 17.5930 0.800510
\(484\) −8.18253 −0.371933
\(485\) 35.3879 1.60688
\(486\) 22.3706 1.01475
\(487\) −2.29894 −0.104175 −0.0520875 0.998643i \(-0.516587\pi\)
−0.0520875 + 0.998643i \(0.516587\pi\)
\(488\) −9.33686 −0.422660
\(489\) 14.0924 0.637282
\(490\) 1.97811 0.0893621
\(491\) 12.1612 0.548826 0.274413 0.961612i \(-0.411516\pi\)
0.274413 + 0.961612i \(0.411516\pi\)
\(492\) 6.57106 0.296246
\(493\) −29.4761 −1.32754
\(494\) 0 0
\(495\) 13.8355 0.621860
\(496\) −7.28408 −0.327065
\(497\) −19.5468 −0.876794
\(498\) −38.2491 −1.71398
\(499\) 15.1348 0.677528 0.338764 0.940871i \(-0.389991\pi\)
0.338764 + 0.940871i \(0.389991\pi\)
\(500\) 9.71592 0.434509
\(501\) 17.8517 0.797556
\(502\) 13.5552 0.605000
\(503\) −0.573683 −0.0255793 −0.0127896 0.999918i \(-0.504071\pi\)
−0.0127896 + 0.999918i \(0.504071\pi\)
\(504\) 9.36176 0.417006
\(505\) −27.7520 −1.23495
\(506\) −4.19449 −0.186468
\(507\) 68.6832 3.05033
\(508\) 1.13632 0.0504159
\(509\) 26.2163 1.16202 0.581008 0.813898i \(-0.302659\pi\)
0.581008 + 0.813898i \(0.302659\pi\)
\(510\) 30.8023 1.36395
\(511\) 17.2818 0.764501
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 10.0393 0.442813
\(515\) 36.4716 1.60713
\(516\) 7.23607 0.318550
\(517\) −1.25150 −0.0550410
\(518\) −1.53920 −0.0676285
\(519\) 58.0158 2.54661
\(520\) −15.6054 −0.684344
\(521\) 12.9637 0.567948 0.283974 0.958832i \(-0.408347\pi\)
0.283974 + 0.958832i \(0.408347\pi\)
\(522\) 19.8782 0.870047
\(523\) −17.7793 −0.777432 −0.388716 0.921358i \(-0.627081\pi\)
−0.388716 + 0.921358i \(0.627081\pi\)
\(524\) −17.8415 −0.779410
\(525\) −7.39144 −0.322589
\(526\) 18.8262 0.820861
\(527\) −36.1959 −1.57672
\(528\) −4.23015 −0.184093
\(529\) −16.7555 −0.728499
\(530\) 3.61635 0.157084
\(531\) 16.6304 0.721698
\(532\) 0 0
\(533\) −16.5429 −0.716554
\(534\) 17.4017 0.753043
\(535\) −6.95709 −0.300781
\(536\) 11.5604 0.499335
\(537\) −20.7023 −0.893369
\(538\) −16.3827 −0.706307
\(539\) −1.34992 −0.0581451
\(540\) −2.17661 −0.0936664
\(541\) −5.34418 −0.229764 −0.114882 0.993379i \(-0.536649\pi\)
−0.114882 + 0.993379i \(0.536649\pi\)
\(542\) 2.76684 0.118846
\(543\) 32.7616 1.40594
\(544\) −4.96917 −0.213052
\(545\) 4.24385 0.181787
\(546\) −44.6677 −1.91160
\(547\) −5.10112 −0.218108 −0.109054 0.994036i \(-0.534782\pi\)
−0.109054 + 0.994036i \(0.534782\pi\)
\(548\) −15.9156 −0.679882
\(549\) 31.2891 1.33539
\(550\) 1.76225 0.0751426
\(551\) 0 0
\(552\) 6.29761 0.268044
\(553\) −16.5146 −0.702273
\(554\) −24.5321 −1.04227
\(555\) 3.41530 0.144971
\(556\) 10.1202 0.429191
\(557\) −10.7235 −0.454370 −0.227185 0.973852i \(-0.572952\pi\)
−0.227185 + 0.973852i \(0.572952\pi\)
\(558\) 24.4100 1.03336
\(559\) −18.2171 −0.770502
\(560\) 6.87129 0.290365
\(561\) −21.0203 −0.887479
\(562\) 10.1704 0.429011
\(563\) −9.76285 −0.411455 −0.205728 0.978609i \(-0.565956\pi\)
−0.205728 + 0.978609i \(0.565956\pi\)
\(564\) 1.87901 0.0791204
\(565\) −1.05220 −0.0442664
\(566\) −27.6678 −1.16296
\(567\) 21.8551 0.917830
\(568\) −6.99698 −0.293587
\(569\) 8.84194 0.370674 0.185337 0.982675i \(-0.440662\pi\)
0.185337 + 0.982675i \(0.440662\pi\)
\(570\) 0 0
\(571\) 22.8411 0.955871 0.477935 0.878395i \(-0.341385\pi\)
0.477935 + 0.878395i \(0.341385\pi\)
\(572\) 10.6496 0.445281
\(573\) 16.5694 0.692198
\(574\) 7.28408 0.304032
\(575\) −2.62354 −0.109409
\(576\) 3.35114 0.139631
\(577\) −14.0176 −0.583560 −0.291780 0.956486i \(-0.594247\pi\)
−0.291780 + 0.956486i \(0.594247\pi\)
\(578\) −7.69270 −0.319974
\(579\) −67.7482 −2.81552
\(580\) 14.5901 0.605821
\(581\) −42.3994 −1.75903
\(582\) 36.2583 1.50296
\(583\) −2.46790 −0.102210
\(584\) 6.18619 0.255986
\(585\) 52.2960 2.16217
\(586\) 23.8386 0.984763
\(587\) 21.0356 0.868232 0.434116 0.900857i \(-0.357061\pi\)
0.434116 + 0.900857i \(0.357061\pi\)
\(588\) 2.02677 0.0835825
\(589\) 0 0
\(590\) 12.2063 0.502525
\(591\) −24.8219 −1.02104
\(592\) −0.550972 −0.0226448
\(593\) −34.9193 −1.43396 −0.716981 0.697093i \(-0.754477\pi\)
−0.716981 + 0.697093i \(0.754477\pi\)
\(594\) 1.48538 0.0609458
\(595\) 34.1446 1.39979
\(596\) 9.39851 0.384978
\(597\) −51.5776 −2.11093
\(598\) −15.8545 −0.648338
\(599\) −21.5106 −0.878901 −0.439450 0.898267i \(-0.644827\pi\)
−0.439450 + 0.898267i \(0.644827\pi\)
\(600\) −2.64584 −0.108016
\(601\) 14.3417 0.585012 0.292506 0.956264i \(-0.405511\pi\)
0.292506 + 0.956264i \(0.405511\pi\)
\(602\) 8.02124 0.326921
\(603\) −38.7407 −1.57764
\(604\) 10.2632 0.417605
\(605\) 20.1262 0.818245
\(606\) −28.4346 −1.15508
\(607\) 10.3488 0.420046 0.210023 0.977696i \(-0.432646\pi\)
0.210023 + 0.977696i \(0.432646\pi\)
\(608\) 0 0
\(609\) 41.7615 1.69226
\(610\) 22.9654 0.929842
\(611\) −4.73047 −0.191375
\(612\) 16.6524 0.673134
\(613\) 19.5674 0.790320 0.395160 0.918612i \(-0.370689\pi\)
0.395160 + 0.918612i \(0.370689\pi\)
\(614\) 25.9884 1.04881
\(615\) −16.1625 −0.651735
\(616\) −4.68915 −0.188931
\(617\) −5.92522 −0.238540 −0.119270 0.992862i \(-0.538055\pi\)
−0.119270 + 0.992862i \(0.538055\pi\)
\(618\) 37.3687 1.50319
\(619\) 8.03958 0.323138 0.161569 0.986861i \(-0.448345\pi\)
0.161569 + 0.986861i \(0.448345\pi\)
\(620\) 17.9163 0.719535
\(621\) −2.21135 −0.0887383
\(622\) 23.1043 0.926400
\(623\) 19.2899 0.772833
\(624\) −15.9893 −0.640083
\(625\) −29.1471 −1.16589
\(626\) −12.2647 −0.490196
\(627\) 0 0
\(628\) 2.97387 0.118671
\(629\) −2.73788 −0.109166
\(630\) −23.0267 −0.917404
\(631\) −12.9006 −0.513567 −0.256783 0.966469i \(-0.582663\pi\)
−0.256783 + 0.966469i \(0.582663\pi\)
\(632\) −5.91158 −0.235150
\(633\) −20.4543 −0.812984
\(634\) −0.272305 −0.0108146
\(635\) −2.79494 −0.110914
\(636\) 3.70530 0.146925
\(637\) −5.10247 −0.202167
\(638\) −9.95669 −0.394189
\(639\) 23.4479 0.927584
\(640\) 2.45965 0.0972262
\(641\) 28.8639 1.14006 0.570028 0.821625i \(-0.306932\pi\)
0.570028 + 0.821625i \(0.306932\pi\)
\(642\) −7.12820 −0.281328
\(643\) 26.3394 1.03873 0.519363 0.854554i \(-0.326169\pi\)
0.519363 + 0.854554i \(0.326169\pi\)
\(644\) 6.98095 0.275088
\(645\) −17.7982 −0.700803
\(646\) 0 0
\(647\) −3.04601 −0.119751 −0.0598756 0.998206i \(-0.519070\pi\)
−0.0598756 + 0.998206i \(0.519070\pi\)
\(648\) 7.82328 0.307327
\(649\) −8.32990 −0.326977
\(650\) 6.66102 0.261267
\(651\) 51.2821 2.00990
\(652\) 5.59191 0.218996
\(653\) −15.1973 −0.594717 −0.297359 0.954766i \(-0.596106\pi\)
−0.297359 + 0.954766i \(0.596106\pi\)
\(654\) 4.34824 0.170030
\(655\) 43.8838 1.71468
\(656\) 2.60741 0.101802
\(657\) −20.7308 −0.808786
\(658\) 2.08289 0.0811996
\(659\) −17.8741 −0.696275 −0.348138 0.937443i \(-0.613186\pi\)
−0.348138 + 0.937443i \(0.613186\pi\)
\(660\) 10.4047 0.405001
\(661\) −37.8525 −1.47229 −0.736146 0.676823i \(-0.763356\pi\)
−0.736146 + 0.676823i \(0.763356\pi\)
\(662\) −19.9930 −0.777050
\(663\) −79.4535 −3.08572
\(664\) −15.1773 −0.588994
\(665\) 0 0
\(666\) 1.84638 0.0715460
\(667\) 14.8230 0.573947
\(668\) 7.08361 0.274073
\(669\) −27.1017 −1.04781
\(670\) −28.4346 −1.09853
\(671\) −15.6722 −0.605019
\(672\) 7.04029 0.271585
\(673\) 8.28461 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(674\) −17.2824 −0.665694
\(675\) 0.929063 0.0357597
\(676\) 27.2537 1.04822
\(677\) 18.3233 0.704223 0.352111 0.935958i \(-0.385464\pi\)
0.352111 + 0.935958i \(0.385464\pi\)
\(678\) −1.07808 −0.0414034
\(679\) 40.1926 1.54245
\(680\) 12.2224 0.468709
\(681\) −68.2799 −2.61649
\(682\) −12.2266 −0.468179
\(683\) 49.2217 1.88342 0.941709 0.336429i \(-0.109219\pi\)
0.941709 + 0.336429i \(0.109219\pi\)
\(684\) 0 0
\(685\) 39.1469 1.49573
\(686\) −17.3085 −0.660843
\(687\) 23.8546 0.910111
\(688\) 2.87129 0.109467
\(689\) −9.32825 −0.355378
\(690\) −15.4899 −0.589690
\(691\) 16.6779 0.634458 0.317229 0.948349i \(-0.397248\pi\)
0.317229 + 0.948349i \(0.397248\pi\)
\(692\) 23.0208 0.875120
\(693\) 15.7140 0.596926
\(694\) 18.5456 0.703981
\(695\) −24.8921 −0.944210
\(696\) 14.9490 0.566639
\(697\) 12.9567 0.490770
\(698\) 20.1210 0.761592
\(699\) 57.6159 2.17923
\(700\) −2.93294 −0.110855
\(701\) −30.9066 −1.16733 −0.583663 0.811996i \(-0.698381\pi\)
−0.583663 + 0.811996i \(0.698381\pi\)
\(702\) 5.61449 0.211905
\(703\) 0 0
\(704\) −1.67853 −0.0632620
\(705\) −4.62169 −0.174063
\(706\) −24.0557 −0.905348
\(707\) −31.5200 −1.18543
\(708\) 12.5065 0.470023
\(709\) 40.1056 1.50620 0.753098 0.657908i \(-0.228558\pi\)
0.753098 + 0.657908i \(0.228558\pi\)
\(710\) 17.2101 0.645884
\(711\) 19.8105 0.742953
\(712\) 6.90502 0.258776
\(713\) 18.2022 0.681678
\(714\) 34.9845 1.30926
\(715\) −26.1942 −0.979608
\(716\) −8.21471 −0.306998
\(717\) 21.8399 0.815625
\(718\) 10.8535 0.405048
\(719\) −45.6529 −1.70256 −0.851282 0.524708i \(-0.824175\pi\)
−0.851282 + 0.524708i \(0.824175\pi\)
\(720\) −8.24263 −0.307185
\(721\) 41.4235 1.54269
\(722\) 0 0
\(723\) −14.2596 −0.530322
\(724\) 12.9999 0.483137
\(725\) −6.22764 −0.231289
\(726\) 20.6212 0.765324
\(727\) −15.1811 −0.563034 −0.281517 0.959556i \(-0.590838\pi\)
−0.281517 + 0.959556i \(0.590838\pi\)
\(728\) −17.7242 −0.656904
\(729\) −32.9073 −1.21879
\(730\) −15.2159 −0.563164
\(731\) 14.2679 0.527719
\(732\) 23.5303 0.869703
\(733\) −26.9986 −0.997217 −0.498608 0.866827i \(-0.666155\pi\)
−0.498608 + 0.866827i \(0.666155\pi\)
\(734\) 32.6121 1.20373
\(735\) −4.98514 −0.183880
\(736\) 2.49890 0.0921108
\(737\) 19.4046 0.714776
\(738\) −8.73781 −0.321643
\(739\) −5.20782 −0.191573 −0.0957864 0.995402i \(-0.530537\pi\)
−0.0957864 + 0.995402i \(0.530537\pi\)
\(740\) 1.35520 0.0498181
\(741\) 0 0
\(742\) 4.10736 0.150786
\(743\) −22.1079 −0.811059 −0.405529 0.914082i \(-0.632913\pi\)
−0.405529 + 0.914082i \(0.632913\pi\)
\(744\) 18.3570 0.672998
\(745\) −23.1170 −0.846943
\(746\) −5.30198 −0.194119
\(747\) 50.8613 1.86092
\(748\) −8.34092 −0.304974
\(749\) −7.90167 −0.288721
\(750\) −24.4855 −0.894086
\(751\) 12.4960 0.455984 0.227992 0.973663i \(-0.426784\pi\)
0.227992 + 0.973663i \(0.426784\pi\)
\(752\) 0.745593 0.0271890
\(753\) −34.1612 −1.24490
\(754\) −37.6347 −1.37057
\(755\) −25.2440 −0.918722
\(756\) −2.47214 −0.0899107
\(757\) −30.9988 −1.12667 −0.563336 0.826228i \(-0.690482\pi\)
−0.563336 + 0.826228i \(0.690482\pi\)
\(758\) 24.6656 0.895895
\(759\) 10.5707 0.383693
\(760\) 0 0
\(761\) 1.96398 0.0711941 0.0355971 0.999366i \(-0.488667\pi\)
0.0355971 + 0.999366i \(0.488667\pi\)
\(762\) −2.86368 −0.103740
\(763\) 4.82006 0.174498
\(764\) 6.57479 0.237868
\(765\) −40.9591 −1.48088
\(766\) 3.64416 0.131669
\(767\) −31.4857 −1.13688
\(768\) 2.52015 0.0909380
\(769\) −22.3322 −0.805318 −0.402659 0.915350i \(-0.631914\pi\)
−0.402659 + 0.915350i \(0.631914\pi\)
\(770\) 11.5337 0.415645
\(771\) −25.3004 −0.911172
\(772\) −26.8826 −0.967527
\(773\) −27.3553 −0.983901 −0.491951 0.870623i \(-0.663716\pi\)
−0.491951 + 0.870623i \(0.663716\pi\)
\(774\) −9.62209 −0.345859
\(775\) −7.64738 −0.274702
\(776\) 14.3874 0.516477
\(777\) 3.87901 0.139159
\(778\) −2.79353 −0.100153
\(779\) 0 0
\(780\) 39.3280 1.40817
\(781\) −11.7447 −0.420257
\(782\) 12.4175 0.444049
\(783\) −5.24920 −0.187591
\(784\) 0.804226 0.0287224
\(785\) −7.31469 −0.261072
\(786\) 44.9632 1.60378
\(787\) −7.78962 −0.277670 −0.138835 0.990316i \(-0.544336\pi\)
−0.138835 + 0.990316i \(0.544336\pi\)
\(788\) −9.84940 −0.350870
\(789\) −47.4447 −1.68908
\(790\) 14.5404 0.517325
\(791\) −1.19506 −0.0424915
\(792\) 5.62500 0.199876
\(793\) −59.2384 −2.10362
\(794\) 39.5101 1.40216
\(795\) −9.11374 −0.323231
\(796\) −20.4661 −0.725402
\(797\) 6.09655 0.215951 0.107975 0.994154i \(-0.465563\pi\)
0.107975 + 0.994154i \(0.465563\pi\)
\(798\) 0 0
\(799\) 3.70498 0.131073
\(800\) −1.04988 −0.0371187
\(801\) −23.1397 −0.817601
\(802\) −18.6215 −0.657547
\(803\) 10.3837 0.366433
\(804\) −29.1340 −1.02748
\(805\) −17.1707 −0.605187
\(806\) −46.2144 −1.62783
\(807\) 41.2868 1.45336
\(808\) −11.2829 −0.396932
\(809\) 56.8455 1.99858 0.999291 0.0376413i \(-0.0119844\pi\)
0.999291 + 0.0376413i \(0.0119844\pi\)
\(810\) −19.2425 −0.676113
\(811\) 20.9209 0.734631 0.367316 0.930096i \(-0.380277\pi\)
0.367316 + 0.930096i \(0.380277\pi\)
\(812\) 16.5711 0.581530
\(813\) −6.97283 −0.244548
\(814\) −0.924824 −0.0324151
\(815\) −13.7541 −0.481786
\(816\) 12.5231 0.438394
\(817\) 0 0
\(818\) −17.1433 −0.599401
\(819\) 59.3964 2.07548
\(820\) −6.41332 −0.223963
\(821\) 16.2582 0.567416 0.283708 0.958911i \(-0.408435\pi\)
0.283708 + 0.958911i \(0.408435\pi\)
\(822\) 40.1098 1.39899
\(823\) −17.5328 −0.611157 −0.305578 0.952167i \(-0.598850\pi\)
−0.305578 + 0.952167i \(0.598850\pi\)
\(824\) 14.8280 0.516557
\(825\) −4.44113 −0.154620
\(826\) 13.8636 0.482375
\(827\) 17.0254 0.592030 0.296015 0.955183i \(-0.404342\pi\)
0.296015 + 0.955183i \(0.404342\pi\)
\(828\) −8.37418 −0.291023
\(829\) 5.76207 0.200125 0.100062 0.994981i \(-0.468096\pi\)
0.100062 + 0.994981i \(0.468096\pi\)
\(830\) 37.3309 1.29577
\(831\) 61.8246 2.14467
\(832\) −6.34458 −0.219959
\(833\) 3.99634 0.138465
\(834\) −25.5043 −0.883142
\(835\) −17.4232 −0.602954
\(836\) 0 0
\(837\) −6.44588 −0.222802
\(838\) 18.5042 0.639217
\(839\) −8.89954 −0.307246 −0.153623 0.988130i \(-0.549094\pi\)
−0.153623 + 0.988130i \(0.549094\pi\)
\(840\) −17.3167 −0.597481
\(841\) 6.18608 0.213313
\(842\) −16.1996 −0.558275
\(843\) −25.6308 −0.882772
\(844\) −8.11630 −0.279374
\(845\) −67.0344 −2.30605
\(846\) −2.49859 −0.0859032
\(847\) 22.8588 0.785436
\(848\) 1.47027 0.0504893
\(849\) 69.7269 2.39302
\(850\) −5.21702 −0.178942
\(851\) 1.37683 0.0471970
\(852\) 17.6334 0.604111
\(853\) 2.78915 0.0954987 0.0477493 0.998859i \(-0.484795\pi\)
0.0477493 + 0.998859i \(0.484795\pi\)
\(854\) 26.0835 0.892559
\(855\) 0 0
\(856\) −2.82849 −0.0966757
\(857\) −16.3945 −0.560025 −0.280012 0.959996i \(-0.590339\pi\)
−0.280012 + 0.959996i \(0.590339\pi\)
\(858\) −26.8385 −0.916251
\(859\) 9.52505 0.324990 0.162495 0.986709i \(-0.448046\pi\)
0.162495 + 0.986709i \(0.448046\pi\)
\(860\) −7.06236 −0.240825
\(861\) −18.3570 −0.625603
\(862\) 12.6997 0.432552
\(863\) 50.6764 1.72505 0.862523 0.506018i \(-0.168883\pi\)
0.862523 + 0.506018i \(0.168883\pi\)
\(864\) −0.884927 −0.0301058
\(865\) −56.6231 −1.92524
\(866\) 14.2350 0.483726
\(867\) 19.3867 0.658408
\(868\) 20.3488 0.690684
\(869\) −9.92278 −0.336607
\(870\) −36.7692 −1.24659
\(871\) 73.3461 2.48524
\(872\) 1.72539 0.0584291
\(873\) −48.2141 −1.63180
\(874\) 0 0
\(875\) −27.1424 −0.917582
\(876\) −15.5901 −0.526741
\(877\) −19.2578 −0.650291 −0.325146 0.945664i \(-0.605413\pi\)
−0.325146 + 0.945664i \(0.605413\pi\)
\(878\) 28.1900 0.951367
\(879\) −60.0768 −2.02634
\(880\) 4.12860 0.139175
\(881\) 19.1076 0.643750 0.321875 0.946782i \(-0.395687\pi\)
0.321875 + 0.946782i \(0.395687\pi\)
\(882\) −2.69507 −0.0907479
\(883\) 49.9980 1.68257 0.841283 0.540595i \(-0.181801\pi\)
0.841283 + 0.540595i \(0.181801\pi\)
\(884\) −31.5273 −1.06038
\(885\) −30.7616 −1.03404
\(886\) 32.9944 1.10847
\(887\) −39.6195 −1.33029 −0.665146 0.746713i \(-0.731631\pi\)
−0.665146 + 0.746713i \(0.731631\pi\)
\(888\) 1.38853 0.0465960
\(889\) −3.17442 −0.106467
\(890\) −16.9839 −0.569302
\(891\) 13.1316 0.439926
\(892\) −10.7540 −0.360071
\(893\) 0 0
\(894\) −23.6856 −0.792166
\(895\) 20.2053 0.675389
\(896\) 2.79360 0.0933278
\(897\) 39.9556 1.33408
\(898\) −17.1975 −0.573889
\(899\) 43.2076 1.44105
\(900\) 3.51828 0.117276
\(901\) 7.30603 0.243399
\(902\) 4.37662 0.145726
\(903\) −20.2147 −0.672703
\(904\) −0.427785 −0.0142279
\(905\) −31.9752 −1.06289
\(906\) −25.8649 −0.859302
\(907\) −55.2545 −1.83470 −0.917348 0.398087i \(-0.869674\pi\)
−0.917348 + 0.398087i \(0.869674\pi\)
\(908\) −27.0936 −0.899133
\(909\) 37.8107 1.25410
\(910\) 43.5954 1.44517
\(911\) −27.7087 −0.918031 −0.459015 0.888428i \(-0.651798\pi\)
−0.459015 + 0.888428i \(0.651798\pi\)
\(912\) 0 0
\(913\) −25.4756 −0.843120
\(914\) 31.1517 1.03041
\(915\) −57.8762 −1.91333
\(916\) 9.46557 0.312751
\(917\) 49.8421 1.64593
\(918\) −4.39736 −0.145134
\(919\) 27.2500 0.898894 0.449447 0.893307i \(-0.351621\pi\)
0.449447 + 0.893307i \(0.351621\pi\)
\(920\) −6.14643 −0.202642
\(921\) −65.4947 −2.15812
\(922\) −15.5410 −0.511815
\(923\) −44.3929 −1.46121
\(924\) 11.8174 0.388762
\(925\) −0.578452 −0.0190194
\(926\) −20.6648 −0.679087
\(927\) −49.6906 −1.63205
\(928\) 5.93179 0.194720
\(929\) 44.4410 1.45806 0.729031 0.684481i \(-0.239971\pi\)
0.729031 + 0.684481i \(0.239971\pi\)
\(930\) −45.1517 −1.48058
\(931\) 0 0
\(932\) 22.8621 0.748873
\(933\) −58.2263 −1.90624
\(934\) 28.4830 0.931991
\(935\) 20.5157 0.670936
\(936\) 21.2616 0.694956
\(937\) −18.2097 −0.594885 −0.297442 0.954740i \(-0.596134\pi\)
−0.297442 + 0.954740i \(0.596134\pi\)
\(938\) −32.2953 −1.05448
\(939\) 30.9088 1.00867
\(940\) −1.83390 −0.0598152
\(941\) 40.2680 1.31270 0.656350 0.754457i \(-0.272099\pi\)
0.656350 + 0.754457i \(0.272099\pi\)
\(942\) −7.49460 −0.244187
\(943\) −6.51567 −0.212180
\(944\) 4.96261 0.161519
\(945\) 6.08059 0.197802
\(946\) 4.81955 0.156697
\(947\) 6.36561 0.206854 0.103427 0.994637i \(-0.467019\pi\)
0.103427 + 0.994637i \(0.467019\pi\)
\(948\) 14.8981 0.483866
\(949\) 39.2488 1.27407
\(950\) 0 0
\(951\) 0.686248 0.0222531
\(952\) 13.8819 0.449915
\(953\) 32.6944 1.05907 0.529537 0.848287i \(-0.322366\pi\)
0.529537 + 0.848287i \(0.322366\pi\)
\(954\) −4.92709 −0.159520
\(955\) −16.1717 −0.523303
\(956\) 8.66611 0.280282
\(957\) 25.0923 0.811119
\(958\) 10.4726 0.338356
\(959\) 44.4620 1.43575
\(960\) −6.19868 −0.200061
\(961\) 22.0578 0.711542
\(962\) −3.49568 −0.112705
\(963\) 9.47866 0.305445
\(964\) −5.65826 −0.182240
\(965\) 66.1218 2.12854
\(966\) −17.5930 −0.566046
\(967\) 8.55936 0.275250 0.137625 0.990484i \(-0.456053\pi\)
0.137625 + 0.990484i \(0.456053\pi\)
\(968\) 8.18253 0.262997
\(969\) 0 0
\(970\) −35.3879 −1.13624
\(971\) 41.8789 1.34396 0.671978 0.740571i \(-0.265445\pi\)
0.671978 + 0.740571i \(0.265445\pi\)
\(972\) −22.3706 −0.717537
\(973\) −28.2718 −0.906351
\(974\) 2.29894 0.0736628
\(975\) −16.7867 −0.537606
\(976\) 9.33686 0.298866
\(977\) 6.18021 0.197722 0.0988612 0.995101i \(-0.468480\pi\)
0.0988612 + 0.995101i \(0.468480\pi\)
\(978\) −14.0924 −0.450626
\(979\) 11.5903 0.370427
\(980\) −1.97811 −0.0631885
\(981\) −5.78203 −0.184606
\(982\) −12.1612 −0.388079
\(983\) −33.5742 −1.07085 −0.535425 0.844583i \(-0.679848\pi\)
−0.535425 + 0.844583i \(0.679848\pi\)
\(984\) −6.57106 −0.209478
\(985\) 24.2261 0.771907
\(986\) 29.4761 0.938710
\(987\) −5.24920 −0.167084
\(988\) 0 0
\(989\) −7.17507 −0.228154
\(990\) −13.8355 −0.439722
\(991\) 46.1746 1.46679 0.733393 0.679805i \(-0.237936\pi\)
0.733393 + 0.679805i \(0.237936\pi\)
\(992\) 7.28408 0.231270
\(993\) 50.3853 1.59893
\(994\) 19.5468 0.619987
\(995\) 50.3394 1.59587
\(996\) 38.2491 1.21197
\(997\) −0.819130 −0.0259421 −0.0129711 0.999916i \(-0.504129\pi\)
−0.0129711 + 0.999916i \(0.504129\pi\)
\(998\) −15.1348 −0.479085
\(999\) −0.487570 −0.0154260
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 722.2.a.m.1.4 4
3.2 odd 2 6498.2.a.ca.1.4 4
4.3 odd 2 5776.2.a.bv.1.1 4
19.2 odd 18 722.2.e.r.99.1 24
19.3 odd 18 722.2.e.r.389.1 24
19.4 even 9 722.2.e.s.415.1 24
19.5 even 9 722.2.e.s.595.1 24
19.6 even 9 722.2.e.s.245.4 24
19.7 even 3 722.2.c.n.429.1 8
19.8 odd 6 722.2.c.m.653.4 8
19.9 even 9 722.2.e.s.423.4 24
19.10 odd 18 722.2.e.r.423.1 24
19.11 even 3 722.2.c.n.653.1 8
19.12 odd 6 722.2.c.m.429.4 8
19.13 odd 18 722.2.e.r.245.1 24
19.14 odd 18 722.2.e.r.595.4 24
19.15 odd 18 722.2.e.r.415.4 24
19.16 even 9 722.2.e.s.389.4 24
19.17 even 9 722.2.e.s.99.4 24
19.18 odd 2 722.2.a.n.1.1 yes 4
57.56 even 2 6498.2.a.bx.1.4 4
76.75 even 2 5776.2.a.bt.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
722.2.a.m.1.4 4 1.1 even 1 trivial
722.2.a.n.1.1 yes 4 19.18 odd 2
722.2.c.m.429.4 8 19.12 odd 6
722.2.c.m.653.4 8 19.8 odd 6
722.2.c.n.429.1 8 19.7 even 3
722.2.c.n.653.1 8 19.11 even 3
722.2.e.r.99.1 24 19.2 odd 18
722.2.e.r.245.1 24 19.13 odd 18
722.2.e.r.389.1 24 19.3 odd 18
722.2.e.r.415.4 24 19.15 odd 18
722.2.e.r.423.1 24 19.10 odd 18
722.2.e.r.595.4 24 19.14 odd 18
722.2.e.s.99.4 24 19.17 even 9
722.2.e.s.245.4 24 19.6 even 9
722.2.e.s.389.4 24 19.16 even 9
722.2.e.s.415.1 24 19.4 even 9
722.2.e.s.423.4 24 19.9 even 9
722.2.e.s.595.1 24 19.5 even 9
5776.2.a.bt.1.4 4 76.75 even 2
5776.2.a.bv.1.1 4 4.3 odd 2
6498.2.a.bx.1.4 4 57.56 even 2
6498.2.a.ca.1.4 4 3.2 odd 2