Properties

Label 5586.2.a.bg.1.2
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.6044345691\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5586.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.58579 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.58579 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.58579 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.82843 q^{13} +1.58579 q^{15} +1.00000 q^{16} -4.24264 q^{17} +1.00000 q^{18} +1.00000 q^{19} -1.58579 q^{20} -1.00000 q^{22} +3.41421 q^{23} -1.00000 q^{24} -2.48528 q^{25} +2.82843 q^{26} -1.00000 q^{27} +2.17157 q^{29} +1.58579 q^{30} -10.6569 q^{31} +1.00000 q^{32} +1.00000 q^{33} -4.24264 q^{34} +1.00000 q^{36} +0.242641 q^{37} +1.00000 q^{38} -2.82843 q^{39} -1.58579 q^{40} -0.585786 q^{41} +6.24264 q^{43} -1.00000 q^{44} -1.58579 q^{45} +3.41421 q^{46} +3.17157 q^{47} -1.00000 q^{48} -2.48528 q^{50} +4.24264 q^{51} +2.82843 q^{52} +1.82843 q^{53} -1.00000 q^{54} +1.58579 q^{55} -1.00000 q^{57} +2.17157 q^{58} +6.07107 q^{59} +1.58579 q^{60} -4.82843 q^{61} -10.6569 q^{62} +1.00000 q^{64} -4.48528 q^{65} +1.00000 q^{66} -2.00000 q^{67} -4.24264 q^{68} -3.41421 q^{69} -7.89949 q^{71} +1.00000 q^{72} -14.8284 q^{73} +0.242641 q^{74} +2.48528 q^{75} +1.00000 q^{76} -2.82843 q^{78} -0.171573 q^{79} -1.58579 q^{80} +1.00000 q^{81} -0.585786 q^{82} -17.1421 q^{83} +6.72792 q^{85} +6.24264 q^{86} -2.17157 q^{87} -1.00000 q^{88} +3.75736 q^{89} -1.58579 q^{90} +3.41421 q^{92} +10.6569 q^{93} +3.17157 q^{94} -1.58579 q^{95} -1.00000 q^{96} -5.58579 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 6q^{5} - 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 6q^{5} - 2q^{6} + 2q^{8} + 2q^{9} - 6q^{10} - 2q^{11} - 2q^{12} + 6q^{15} + 2q^{16} + 2q^{18} + 2q^{19} - 6q^{20} - 2q^{22} + 4q^{23} - 2q^{24} + 12q^{25} - 2q^{27} + 10q^{29} + 6q^{30} - 10q^{31} + 2q^{32} + 2q^{33} + 2q^{36} - 8q^{37} + 2q^{38} - 6q^{40} - 4q^{41} + 4q^{43} - 2q^{44} - 6q^{45} + 4q^{46} + 12q^{47} - 2q^{48} + 12q^{50} - 2q^{53} - 2q^{54} + 6q^{55} - 2q^{57} + 10q^{58} - 2q^{59} + 6q^{60} - 4q^{61} - 10q^{62} + 2q^{64} + 8q^{65} + 2q^{66} - 4q^{67} - 4q^{69} + 4q^{71} + 2q^{72} - 24q^{73} - 8q^{74} - 12q^{75} + 2q^{76} - 6q^{79} - 6q^{80} + 2q^{81} - 4q^{82} - 6q^{83} - 12q^{85} + 4q^{86} - 10q^{87} - 2q^{88} + 16q^{89} - 6q^{90} + 4q^{92} + 10q^{93} + 12q^{94} - 6q^{95} - 2q^{96} - 14q^{97} - 2q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.58579 −0.709185 −0.354593 0.935021i \(-0.615380\pi\)
−0.354593 + 0.935021i \(0.615380\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.58579 −0.501470
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 1.58579 0.409448
\(16\) 1.00000 0.250000
\(17\) −4.24264 −1.02899 −0.514496 0.857493i \(-0.672021\pi\)
−0.514496 + 0.857493i \(0.672021\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) −1.58579 −0.354593
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 3.41421 0.711913 0.355956 0.934503i \(-0.384155\pi\)
0.355956 + 0.934503i \(0.384155\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.48528 −0.497056
\(26\) 2.82843 0.554700
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.17157 0.403251 0.201625 0.979463i \(-0.435378\pi\)
0.201625 + 0.979463i \(0.435378\pi\)
\(30\) 1.58579 0.289524
\(31\) −10.6569 −1.91403 −0.957014 0.290043i \(-0.906331\pi\)
−0.957014 + 0.290043i \(0.906331\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −4.24264 −0.727607
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0.242641 0.0398899 0.0199449 0.999801i \(-0.493651\pi\)
0.0199449 + 0.999801i \(0.493651\pi\)
\(38\) 1.00000 0.162221
\(39\) −2.82843 −0.452911
\(40\) −1.58579 −0.250735
\(41\) −0.585786 −0.0914845 −0.0457422 0.998953i \(-0.514565\pi\)
−0.0457422 + 0.998953i \(0.514565\pi\)
\(42\) 0 0
\(43\) 6.24264 0.951994 0.475997 0.879447i \(-0.342087\pi\)
0.475997 + 0.879447i \(0.342087\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.58579 −0.236395
\(46\) 3.41421 0.503398
\(47\) 3.17157 0.462621 0.231311 0.972880i \(-0.425699\pi\)
0.231311 + 0.972880i \(0.425699\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −2.48528 −0.351472
\(51\) 4.24264 0.594089
\(52\) 2.82843 0.392232
\(53\) 1.82843 0.251154 0.125577 0.992084i \(-0.459922\pi\)
0.125577 + 0.992084i \(0.459922\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.58579 0.213827
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 2.17157 0.285141
\(59\) 6.07107 0.790386 0.395193 0.918598i \(-0.370678\pi\)
0.395193 + 0.918598i \(0.370678\pi\)
\(60\) 1.58579 0.204724
\(61\) −4.82843 −0.618217 −0.309108 0.951027i \(-0.600031\pi\)
−0.309108 + 0.951027i \(0.600031\pi\)
\(62\) −10.6569 −1.35342
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.48528 −0.556331
\(66\) 1.00000 0.123091
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −4.24264 −0.514496
\(69\) −3.41421 −0.411023
\(70\) 0 0
\(71\) −7.89949 −0.937498 −0.468749 0.883332i \(-0.655295\pi\)
−0.468749 + 0.883332i \(0.655295\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.8284 −1.73554 −0.867768 0.496969i \(-0.834446\pi\)
−0.867768 + 0.496969i \(0.834446\pi\)
\(74\) 0.242641 0.0282064
\(75\) 2.48528 0.286976
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −2.82843 −0.320256
\(79\) −0.171573 −0.0193035 −0.00965173 0.999953i \(-0.503072\pi\)
−0.00965173 + 0.999953i \(0.503072\pi\)
\(80\) −1.58579 −0.177296
\(81\) 1.00000 0.111111
\(82\) −0.585786 −0.0646893
\(83\) −17.1421 −1.88159 −0.940797 0.338971i \(-0.889921\pi\)
−0.940797 + 0.338971i \(0.889921\pi\)
\(84\) 0 0
\(85\) 6.72792 0.729746
\(86\) 6.24264 0.673161
\(87\) −2.17157 −0.232817
\(88\) −1.00000 −0.106600
\(89\) 3.75736 0.398279 0.199140 0.979971i \(-0.436185\pi\)
0.199140 + 0.979971i \(0.436185\pi\)
\(90\) −1.58579 −0.167157
\(91\) 0 0
\(92\) 3.41421 0.355956
\(93\) 10.6569 1.10506
\(94\) 3.17157 0.327123
\(95\) −1.58579 −0.162698
\(96\) −1.00000 −0.102062
\(97\) −5.58579 −0.567151 −0.283575 0.958950i \(-0.591521\pi\)
−0.283575 + 0.958950i \(0.591521\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −2.48528 −0.248528
\(101\) 14.9706 1.48963 0.744813 0.667273i \(-0.232539\pi\)
0.744813 + 0.667273i \(0.232539\pi\)
\(102\) 4.24264 0.420084
\(103\) −18.4853 −1.82141 −0.910704 0.413059i \(-0.864460\pi\)
−0.910704 + 0.413059i \(0.864460\pi\)
\(104\) 2.82843 0.277350
\(105\) 0 0
\(106\) 1.82843 0.177593
\(107\) 11.7279 1.13378 0.566891 0.823793i \(-0.308146\pi\)
0.566891 + 0.823793i \(0.308146\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 1.75736 0.168324 0.0841622 0.996452i \(-0.473179\pi\)
0.0841622 + 0.996452i \(0.473179\pi\)
\(110\) 1.58579 0.151199
\(111\) −0.242641 −0.0230304
\(112\) 0 0
\(113\) 10.2426 0.963547 0.481773 0.876296i \(-0.339993\pi\)
0.481773 + 0.876296i \(0.339993\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −5.41421 −0.504878
\(116\) 2.17157 0.201625
\(117\) 2.82843 0.261488
\(118\) 6.07107 0.558887
\(119\) 0 0
\(120\) 1.58579 0.144762
\(121\) −10.0000 −0.909091
\(122\) −4.82843 −0.437145
\(123\) 0.585786 0.0528186
\(124\) −10.6569 −0.957014
\(125\) 11.8701 1.06169
\(126\) 0 0
\(127\) −5.82843 −0.517189 −0.258595 0.965986i \(-0.583259\pi\)
−0.258595 + 0.965986i \(0.583259\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.24264 −0.549634
\(130\) −4.48528 −0.393385
\(131\) 4.31371 0.376890 0.188445 0.982084i \(-0.439655\pi\)
0.188445 + 0.982084i \(0.439655\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) 1.58579 0.136483
\(136\) −4.24264 −0.363803
\(137\) −0.343146 −0.0293169 −0.0146585 0.999893i \(-0.504666\pi\)
−0.0146585 + 0.999893i \(0.504666\pi\)
\(138\) −3.41421 −0.290637
\(139\) 7.89949 0.670026 0.335013 0.942213i \(-0.391259\pi\)
0.335013 + 0.942213i \(0.391259\pi\)
\(140\) 0 0
\(141\) −3.17157 −0.267095
\(142\) −7.89949 −0.662911
\(143\) −2.82843 −0.236525
\(144\) 1.00000 0.0833333
\(145\) −3.44365 −0.285980
\(146\) −14.8284 −1.22721
\(147\) 0 0
\(148\) 0.242641 0.0199449
\(149\) −18.4853 −1.51437 −0.757187 0.653199i \(-0.773427\pi\)
−0.757187 + 0.653199i \(0.773427\pi\)
\(150\) 2.48528 0.202922
\(151\) −11.3431 −0.923092 −0.461546 0.887116i \(-0.652705\pi\)
−0.461546 + 0.887116i \(0.652705\pi\)
\(152\) 1.00000 0.0811107
\(153\) −4.24264 −0.342997
\(154\) 0 0
\(155\) 16.8995 1.35740
\(156\) −2.82843 −0.226455
\(157\) −4.48528 −0.357964 −0.178982 0.983852i \(-0.557280\pi\)
−0.178982 + 0.983852i \(0.557280\pi\)
\(158\) −0.171573 −0.0136496
\(159\) −1.82843 −0.145004
\(160\) −1.58579 −0.125367
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −10.8284 −0.848148 −0.424074 0.905628i \(-0.639400\pi\)
−0.424074 + 0.905628i \(0.639400\pi\)
\(164\) −0.585786 −0.0457422
\(165\) −1.58579 −0.123453
\(166\) −17.1421 −1.33049
\(167\) −6.72792 −0.520622 −0.260311 0.965525i \(-0.583825\pi\)
−0.260311 + 0.965525i \(0.583825\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 6.72792 0.516008
\(171\) 1.00000 0.0764719
\(172\) 6.24264 0.475997
\(173\) −15.7990 −1.20117 −0.600587 0.799559i \(-0.705067\pi\)
−0.600587 + 0.799559i \(0.705067\pi\)
\(174\) −2.17157 −0.164627
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −6.07107 −0.456329
\(178\) 3.75736 0.281626
\(179\) −7.51472 −0.561676 −0.280838 0.959755i \(-0.590612\pi\)
−0.280838 + 0.959755i \(0.590612\pi\)
\(180\) −1.58579 −0.118198
\(181\) −20.4853 −1.52266 −0.761329 0.648365i \(-0.775453\pi\)
−0.761329 + 0.648365i \(0.775453\pi\)
\(182\) 0 0
\(183\) 4.82843 0.356928
\(184\) 3.41421 0.251699
\(185\) −0.384776 −0.0282893
\(186\) 10.6569 0.781398
\(187\) 4.24264 0.310253
\(188\) 3.17157 0.231311
\(189\) 0 0
\(190\) −1.58579 −0.115045
\(191\) −18.1421 −1.31272 −0.656359 0.754448i \(-0.727904\pi\)
−0.656359 + 0.754448i \(0.727904\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −23.7279 −1.70797 −0.853987 0.520295i \(-0.825822\pi\)
−0.853987 + 0.520295i \(0.825822\pi\)
\(194\) −5.58579 −0.401036
\(195\) 4.48528 0.321198
\(196\) 0 0
\(197\) 1.17157 0.0834711 0.0417356 0.999129i \(-0.486711\pi\)
0.0417356 + 0.999129i \(0.486711\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 8.48528 0.601506 0.300753 0.953702i \(-0.402762\pi\)
0.300753 + 0.953702i \(0.402762\pi\)
\(200\) −2.48528 −0.175736
\(201\) 2.00000 0.141069
\(202\) 14.9706 1.05333
\(203\) 0 0
\(204\) 4.24264 0.297044
\(205\) 0.928932 0.0648794
\(206\) −18.4853 −1.28793
\(207\) 3.41421 0.237304
\(208\) 2.82843 0.196116
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −15.0711 −1.03754 −0.518768 0.854915i \(-0.673609\pi\)
−0.518768 + 0.854915i \(0.673609\pi\)
\(212\) 1.82843 0.125577
\(213\) 7.89949 0.541264
\(214\) 11.7279 0.801704
\(215\) −9.89949 −0.675140
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 1.75736 0.119023
\(219\) 14.8284 1.00201
\(220\) 1.58579 0.106914
\(221\) −12.0000 −0.807207
\(222\) −0.242641 −0.0162850
\(223\) 23.4853 1.57269 0.786345 0.617787i \(-0.211971\pi\)
0.786345 + 0.617787i \(0.211971\pi\)
\(224\) 0 0
\(225\) −2.48528 −0.165685
\(226\) 10.2426 0.681330
\(227\) 1.10051 0.0730431 0.0365215 0.999333i \(-0.488372\pi\)
0.0365215 + 0.999333i \(0.488372\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −19.7990 −1.30835 −0.654177 0.756341i \(-0.726985\pi\)
−0.654177 + 0.756341i \(0.726985\pi\)
\(230\) −5.41421 −0.357003
\(231\) 0 0
\(232\) 2.17157 0.142571
\(233\) −5.17157 −0.338801 −0.169401 0.985547i \(-0.554183\pi\)
−0.169401 + 0.985547i \(0.554183\pi\)
\(234\) 2.82843 0.184900
\(235\) −5.02944 −0.328084
\(236\) 6.07107 0.395193
\(237\) 0.171573 0.0111449
\(238\) 0 0
\(239\) 28.2843 1.82956 0.914779 0.403955i \(-0.132365\pi\)
0.914779 + 0.403955i \(0.132365\pi\)
\(240\) 1.58579 0.102362
\(241\) 0.556349 0.0358376 0.0179188 0.999839i \(-0.494296\pi\)
0.0179188 + 0.999839i \(0.494296\pi\)
\(242\) −10.0000 −0.642824
\(243\) −1.00000 −0.0641500
\(244\) −4.82843 −0.309108
\(245\) 0 0
\(246\) 0.585786 0.0373484
\(247\) 2.82843 0.179969
\(248\) −10.6569 −0.676711
\(249\) 17.1421 1.08634
\(250\) 11.8701 0.750728
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) −3.41421 −0.214650
\(254\) −5.82843 −0.365708
\(255\) −6.72792 −0.421319
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −6.24264 −0.388650
\(259\) 0 0
\(260\) −4.48528 −0.278165
\(261\) 2.17157 0.134417
\(262\) 4.31371 0.266502
\(263\) −11.5563 −0.712595 −0.356298 0.934373i \(-0.615961\pi\)
−0.356298 + 0.934373i \(0.615961\pi\)
\(264\) 1.00000 0.0615457
\(265\) −2.89949 −0.178115
\(266\) 0 0
\(267\) −3.75736 −0.229947
\(268\) −2.00000 −0.122169
\(269\) 1.00000 0.0609711 0.0304855 0.999535i \(-0.490295\pi\)
0.0304855 + 0.999535i \(0.490295\pi\)
\(270\) 1.58579 0.0965079
\(271\) 8.89949 0.540606 0.270303 0.962775i \(-0.412876\pi\)
0.270303 + 0.962775i \(0.412876\pi\)
\(272\) −4.24264 −0.257248
\(273\) 0 0
\(274\) −0.343146 −0.0207302
\(275\) 2.48528 0.149868
\(276\) −3.41421 −0.205512
\(277\) −13.7574 −0.826600 −0.413300 0.910595i \(-0.635624\pi\)
−0.413300 + 0.910595i \(0.635624\pi\)
\(278\) 7.89949 0.473780
\(279\) −10.6569 −0.638009
\(280\) 0 0
\(281\) −1.31371 −0.0783693 −0.0391846 0.999232i \(-0.512476\pi\)
−0.0391846 + 0.999232i \(0.512476\pi\)
\(282\) −3.17157 −0.188864
\(283\) 21.5563 1.28139 0.640696 0.767795i \(-0.278646\pi\)
0.640696 + 0.767795i \(0.278646\pi\)
\(284\) −7.89949 −0.468749
\(285\) 1.58579 0.0939339
\(286\) −2.82843 −0.167248
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) −3.44365 −0.202218
\(291\) 5.58579 0.327445
\(292\) −14.8284 −0.867768
\(293\) −23.0000 −1.34367 −0.671837 0.740699i \(-0.734495\pi\)
−0.671837 + 0.740699i \(0.734495\pi\)
\(294\) 0 0
\(295\) −9.62742 −0.560530
\(296\) 0.242641 0.0141032
\(297\) 1.00000 0.0580259
\(298\) −18.4853 −1.07082
\(299\) 9.65685 0.558470
\(300\) 2.48528 0.143488
\(301\) 0 0
\(302\) −11.3431 −0.652725
\(303\) −14.9706 −0.860036
\(304\) 1.00000 0.0573539
\(305\) 7.65685 0.438430
\(306\) −4.24264 −0.242536
\(307\) 7.41421 0.423152 0.211576 0.977362i \(-0.432140\pi\)
0.211576 + 0.977362i \(0.432140\pi\)
\(308\) 0 0
\(309\) 18.4853 1.05159
\(310\) 16.8995 0.959827
\(311\) 22.2426 1.26126 0.630632 0.776082i \(-0.282796\pi\)
0.630632 + 0.776082i \(0.282796\pi\)
\(312\) −2.82843 −0.160128
\(313\) −3.82843 −0.216395 −0.108198 0.994129i \(-0.534508\pi\)
−0.108198 + 0.994129i \(0.534508\pi\)
\(314\) −4.48528 −0.253119
\(315\) 0 0
\(316\) −0.171573 −0.00965173
\(317\) −9.48528 −0.532746 −0.266373 0.963870i \(-0.585825\pi\)
−0.266373 + 0.963870i \(0.585825\pi\)
\(318\) −1.82843 −0.102533
\(319\) −2.17157 −0.121585
\(320\) −1.58579 −0.0886482
\(321\) −11.7279 −0.654589
\(322\) 0 0
\(323\) −4.24264 −0.236067
\(324\) 1.00000 0.0555556
\(325\) −7.02944 −0.389923
\(326\) −10.8284 −0.599731
\(327\) −1.75736 −0.0971822
\(328\) −0.585786 −0.0323446
\(329\) 0 0
\(330\) −1.58579 −0.0872947
\(331\) −4.38478 −0.241009 −0.120505 0.992713i \(-0.538451\pi\)
−0.120505 + 0.992713i \(0.538451\pi\)
\(332\) −17.1421 −0.940797
\(333\) 0.242641 0.0132966
\(334\) −6.72792 −0.368136
\(335\) 3.17157 0.173282
\(336\) 0 0
\(337\) 2.07107 0.112818 0.0564091 0.998408i \(-0.482035\pi\)
0.0564091 + 0.998408i \(0.482035\pi\)
\(338\) −5.00000 −0.271964
\(339\) −10.2426 −0.556304
\(340\) 6.72792 0.364873
\(341\) 10.6569 0.577101
\(342\) 1.00000 0.0540738
\(343\) 0 0
\(344\) 6.24264 0.336581
\(345\) 5.41421 0.291491
\(346\) −15.7990 −0.849359
\(347\) −34.0000 −1.82522 −0.912608 0.408836i \(-0.865935\pi\)
−0.912608 + 0.408836i \(0.865935\pi\)
\(348\) −2.17157 −0.116409
\(349\) −7.75736 −0.415242 −0.207621 0.978209i \(-0.566572\pi\)
−0.207621 + 0.978209i \(0.566572\pi\)
\(350\) 0 0
\(351\) −2.82843 −0.150970
\(352\) −1.00000 −0.0533002
\(353\) 14.1005 0.750494 0.375247 0.926925i \(-0.377558\pi\)
0.375247 + 0.926925i \(0.377558\pi\)
\(354\) −6.07107 −0.322674
\(355\) 12.5269 0.664859
\(356\) 3.75736 0.199140
\(357\) 0 0
\(358\) −7.51472 −0.397165
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) −1.58579 −0.0835783
\(361\) 1.00000 0.0526316
\(362\) −20.4853 −1.07668
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 23.5147 1.23082
\(366\) 4.82843 0.252386
\(367\) 5.24264 0.273664 0.136832 0.990594i \(-0.456308\pi\)
0.136832 + 0.990594i \(0.456308\pi\)
\(368\) 3.41421 0.177978
\(369\) −0.585786 −0.0304948
\(370\) −0.384776 −0.0200036
\(371\) 0 0
\(372\) 10.6569 0.552532
\(373\) 0.485281 0.0251269 0.0125635 0.999921i \(-0.496001\pi\)
0.0125635 + 0.999921i \(0.496001\pi\)
\(374\) 4.24264 0.219382
\(375\) −11.8701 −0.612967
\(376\) 3.17157 0.163561
\(377\) 6.14214 0.316336
\(378\) 0 0
\(379\) 3.31371 0.170214 0.0851069 0.996372i \(-0.472877\pi\)
0.0851069 + 0.996372i \(0.472877\pi\)
\(380\) −1.58579 −0.0813491
\(381\) 5.82843 0.298599
\(382\) −18.1421 −0.928232
\(383\) 15.6569 0.800028 0.400014 0.916509i \(-0.369005\pi\)
0.400014 + 0.916509i \(0.369005\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −23.7279 −1.20772
\(387\) 6.24264 0.317331
\(388\) −5.58579 −0.283575
\(389\) 23.1716 1.17485 0.587423 0.809280i \(-0.300143\pi\)
0.587423 + 0.809280i \(0.300143\pi\)
\(390\) 4.48528 0.227121
\(391\) −14.4853 −0.732552
\(392\) 0 0
\(393\) −4.31371 −0.217598
\(394\) 1.17157 0.0590230
\(395\) 0.272078 0.0136897
\(396\) −1.00000 −0.0502519
\(397\) −10.8284 −0.543463 −0.271732 0.962373i \(-0.587596\pi\)
−0.271732 + 0.962373i \(0.587596\pi\)
\(398\) 8.48528 0.425329
\(399\) 0 0
\(400\) −2.48528 −0.124264
\(401\) 14.1421 0.706225 0.353112 0.935581i \(-0.385123\pi\)
0.353112 + 0.935581i \(0.385123\pi\)
\(402\) 2.00000 0.0997509
\(403\) −30.1421 −1.50149
\(404\) 14.9706 0.744813
\(405\) −1.58579 −0.0787984
\(406\) 0 0
\(407\) −0.242641 −0.0120273
\(408\) 4.24264 0.210042
\(409\) 30.8995 1.52788 0.763941 0.645286i \(-0.223262\pi\)
0.763941 + 0.645286i \(0.223262\pi\)
\(410\) 0.928932 0.0458767
\(411\) 0.343146 0.0169261
\(412\) −18.4853 −0.910704
\(413\) 0 0
\(414\) 3.41421 0.167799
\(415\) 27.1838 1.33440
\(416\) 2.82843 0.138675
\(417\) −7.89949 −0.386840
\(418\) −1.00000 −0.0489116
\(419\) −30.9706 −1.51301 −0.756505 0.653987i \(-0.773095\pi\)
−0.756505 + 0.653987i \(0.773095\pi\)
\(420\) 0 0
\(421\) 15.1716 0.739417 0.369709 0.929148i \(-0.379458\pi\)
0.369709 + 0.929148i \(0.379458\pi\)
\(422\) −15.0711 −0.733648
\(423\) 3.17157 0.154207
\(424\) 1.82843 0.0887963
\(425\) 10.5442 0.511467
\(426\) 7.89949 0.382732
\(427\) 0 0
\(428\) 11.7279 0.566891
\(429\) 2.82843 0.136558
\(430\) −9.89949 −0.477396
\(431\) −14.1005 −0.679197 −0.339599 0.940570i \(-0.610291\pi\)
−0.339599 + 0.940570i \(0.610291\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −30.8284 −1.48152 −0.740760 0.671770i \(-0.765534\pi\)
−0.740760 + 0.671770i \(0.765534\pi\)
\(434\) 0 0
\(435\) 3.44365 0.165110
\(436\) 1.75736 0.0841622
\(437\) 3.41421 0.163324
\(438\) 14.8284 0.708530
\(439\) 22.4558 1.07176 0.535879 0.844294i \(-0.319980\pi\)
0.535879 + 0.844294i \(0.319980\pi\)
\(440\) 1.58579 0.0755994
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −14.6569 −0.696368 −0.348184 0.937426i \(-0.613202\pi\)
−0.348184 + 0.937426i \(0.613202\pi\)
\(444\) −0.242641 −0.0115152
\(445\) −5.95837 −0.282454
\(446\) 23.4853 1.11206
\(447\) 18.4853 0.874324
\(448\) 0 0
\(449\) 14.2426 0.672152 0.336076 0.941835i \(-0.390900\pi\)
0.336076 + 0.941835i \(0.390900\pi\)
\(450\) −2.48528 −0.117157
\(451\) 0.585786 0.0275836
\(452\) 10.2426 0.481773
\(453\) 11.3431 0.532947
\(454\) 1.10051 0.0516493
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 36.1127 1.68928 0.844640 0.535334i \(-0.179814\pi\)
0.844640 + 0.535334i \(0.179814\pi\)
\(458\) −19.7990 −0.925146
\(459\) 4.24264 0.198030
\(460\) −5.41421 −0.252439
\(461\) −17.3137 −0.806380 −0.403190 0.915116i \(-0.632099\pi\)
−0.403190 + 0.915116i \(0.632099\pi\)
\(462\) 0 0
\(463\) 1.65685 0.0770005 0.0385003 0.999259i \(-0.487742\pi\)
0.0385003 + 0.999259i \(0.487742\pi\)
\(464\) 2.17157 0.100813
\(465\) −16.8995 −0.783695
\(466\) −5.17157 −0.239568
\(467\) −15.1716 −0.702057 −0.351028 0.936365i \(-0.614168\pi\)
−0.351028 + 0.936365i \(0.614168\pi\)
\(468\) 2.82843 0.130744
\(469\) 0 0
\(470\) −5.02944 −0.231991
\(471\) 4.48528 0.206671
\(472\) 6.07107 0.279444
\(473\) −6.24264 −0.287037
\(474\) 0.171573 0.00788060
\(475\) −2.48528 −0.114033
\(476\) 0 0
\(477\) 1.82843 0.0837179
\(478\) 28.2843 1.29369
\(479\) 1.31371 0.0600249 0.0300124 0.999550i \(-0.490445\pi\)
0.0300124 + 0.999550i \(0.490445\pi\)
\(480\) 1.58579 0.0723809
\(481\) 0.686292 0.0312922
\(482\) 0.556349 0.0253410
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 8.85786 0.402215
\(486\) −1.00000 −0.0453609
\(487\) −38.7990 −1.75815 −0.879075 0.476683i \(-0.841839\pi\)
−0.879075 + 0.476683i \(0.841839\pi\)
\(488\) −4.82843 −0.218573
\(489\) 10.8284 0.489678
\(490\) 0 0
\(491\) −9.14214 −0.412579 −0.206289 0.978491i \(-0.566139\pi\)
−0.206289 + 0.978491i \(0.566139\pi\)
\(492\) 0.585786 0.0264093
\(493\) −9.21320 −0.414942
\(494\) 2.82843 0.127257
\(495\) 1.58579 0.0712758
\(496\) −10.6569 −0.478507
\(497\) 0 0
\(498\) 17.1421 0.768157
\(499\) 9.31371 0.416939 0.208469 0.978029i \(-0.433152\pi\)
0.208469 + 0.978029i \(0.433152\pi\)
\(500\) 11.8701 0.530845
\(501\) 6.72792 0.300581
\(502\) −21.0000 −0.937276
\(503\) 9.75736 0.435059 0.217530 0.976054i \(-0.430200\pi\)
0.217530 + 0.976054i \(0.430200\pi\)
\(504\) 0 0
\(505\) −23.7401 −1.05642
\(506\) −3.41421 −0.151780
\(507\) 5.00000 0.222058
\(508\) −5.82843 −0.258595
\(509\) −14.3137 −0.634444 −0.317222 0.948351i \(-0.602750\pi\)
−0.317222 + 0.948351i \(0.602750\pi\)
\(510\) −6.72792 −0.297917
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 6.00000 0.264649
\(515\) 29.3137 1.29172
\(516\) −6.24264 −0.274817
\(517\) −3.17157 −0.139486
\(518\) 0 0
\(519\) 15.7990 0.693499
\(520\) −4.48528 −0.196693
\(521\) 35.9411 1.57461 0.787305 0.616564i \(-0.211476\pi\)
0.787305 + 0.616564i \(0.211476\pi\)
\(522\) 2.17157 0.0950472
\(523\) 4.10051 0.179303 0.0896513 0.995973i \(-0.471425\pi\)
0.0896513 + 0.995973i \(0.471425\pi\)
\(524\) 4.31371 0.188445
\(525\) 0 0
\(526\) −11.5563 −0.503881
\(527\) 45.2132 1.96952
\(528\) 1.00000 0.0435194
\(529\) −11.3431 −0.493180
\(530\) −2.89949 −0.125946
\(531\) 6.07107 0.263462
\(532\) 0 0
\(533\) −1.65685 −0.0717663
\(534\) −3.75736 −0.162597
\(535\) −18.5980 −0.804061
\(536\) −2.00000 −0.0863868
\(537\) 7.51472 0.324284
\(538\) 1.00000 0.0431131
\(539\) 0 0
\(540\) 1.58579 0.0682414
\(541\) −30.3848 −1.30634 −0.653172 0.757210i \(-0.726562\pi\)
−0.653172 + 0.757210i \(0.726562\pi\)
\(542\) 8.89949 0.382266
\(543\) 20.4853 0.879108
\(544\) −4.24264 −0.181902
\(545\) −2.78680 −0.119373
\(546\) 0 0
\(547\) 10.7279 0.458693 0.229346 0.973345i \(-0.426341\pi\)
0.229346 + 0.973345i \(0.426341\pi\)
\(548\) −0.343146 −0.0146585
\(549\) −4.82843 −0.206072
\(550\) 2.48528 0.105973
\(551\) 2.17157 0.0925121
\(552\) −3.41421 −0.145319
\(553\) 0 0
\(554\) −13.7574 −0.584494
\(555\) 0.384776 0.0163328
\(556\) 7.89949 0.335013
\(557\) 24.4142 1.03446 0.517232 0.855845i \(-0.326963\pi\)
0.517232 + 0.855845i \(0.326963\pi\)
\(558\) −10.6569 −0.451141
\(559\) 17.6569 0.746805
\(560\) 0 0
\(561\) −4.24264 −0.179124
\(562\) −1.31371 −0.0554154
\(563\) −34.8995 −1.47084 −0.735419 0.677612i \(-0.763015\pi\)
−0.735419 + 0.677612i \(0.763015\pi\)
\(564\) −3.17157 −0.133547
\(565\) −16.2426 −0.683333
\(566\) 21.5563 0.906081
\(567\) 0 0
\(568\) −7.89949 −0.331455
\(569\) −22.6274 −0.948591 −0.474295 0.880366i \(-0.657297\pi\)
−0.474295 + 0.880366i \(0.657297\pi\)
\(570\) 1.58579 0.0664213
\(571\) 34.5269 1.44491 0.722453 0.691420i \(-0.243015\pi\)
0.722453 + 0.691420i \(0.243015\pi\)
\(572\) −2.82843 −0.118262
\(573\) 18.1421 0.757899
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 1.00000 0.0416667
\(577\) 15.4853 0.644661 0.322330 0.946627i \(-0.395534\pi\)
0.322330 + 0.946627i \(0.395534\pi\)
\(578\) 1.00000 0.0415945
\(579\) 23.7279 0.986099
\(580\) −3.44365 −0.142990
\(581\) 0 0
\(582\) 5.58579 0.231538
\(583\) −1.82843 −0.0757257
\(584\) −14.8284 −0.613605
\(585\) −4.48528 −0.185444
\(586\) −23.0000 −0.950121
\(587\) 23.6274 0.975208 0.487604 0.873065i \(-0.337871\pi\)
0.487604 + 0.873065i \(0.337871\pi\)
\(588\) 0 0
\(589\) −10.6569 −0.439108
\(590\) −9.62742 −0.396354
\(591\) −1.17157 −0.0481921
\(592\) 0.242641 0.00997247
\(593\) 28.7696 1.18142 0.590712 0.806883i \(-0.298847\pi\)
0.590712 + 0.806883i \(0.298847\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −18.4853 −0.757187
\(597\) −8.48528 −0.347279
\(598\) 9.65685 0.394898
\(599\) 38.2426 1.56255 0.781276 0.624186i \(-0.214569\pi\)
0.781276 + 0.624186i \(0.214569\pi\)
\(600\) 2.48528 0.101461
\(601\) −29.5858 −1.20683 −0.603415 0.797428i \(-0.706194\pi\)
−0.603415 + 0.797428i \(0.706194\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) −11.3431 −0.461546
\(605\) 15.8579 0.644714
\(606\) −14.9706 −0.608138
\(607\) −16.7990 −0.681850 −0.340925 0.940091i \(-0.610740\pi\)
−0.340925 + 0.940091i \(0.610740\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 7.65685 0.310017
\(611\) 8.97056 0.362910
\(612\) −4.24264 −0.171499
\(613\) −25.3553 −1.02409 −0.512046 0.858958i \(-0.671112\pi\)
−0.512046 + 0.858958i \(0.671112\pi\)
\(614\) 7.41421 0.299213
\(615\) −0.928932 −0.0374582
\(616\) 0 0
\(617\) 30.5269 1.22897 0.614484 0.788930i \(-0.289364\pi\)
0.614484 + 0.788930i \(0.289364\pi\)
\(618\) 18.4853 0.743587
\(619\) −5.79899 −0.233081 −0.116541 0.993186i \(-0.537180\pi\)
−0.116541 + 0.993186i \(0.537180\pi\)
\(620\) 16.8995 0.678700
\(621\) −3.41421 −0.137008
\(622\) 22.2426 0.891849
\(623\) 0 0
\(624\) −2.82843 −0.113228
\(625\) −6.39697 −0.255879
\(626\) −3.82843 −0.153015
\(627\) 1.00000 0.0399362
\(628\) −4.48528 −0.178982
\(629\) −1.02944 −0.0410464
\(630\) 0 0
\(631\) −6.55635 −0.261004 −0.130502 0.991448i \(-0.541659\pi\)
−0.130502 + 0.991448i \(0.541659\pi\)
\(632\) −0.171573 −0.00682480
\(633\) 15.0711 0.599021
\(634\) −9.48528 −0.376709
\(635\) 9.24264 0.366783
\(636\) −1.82843 −0.0725019
\(637\) 0 0
\(638\) −2.17157 −0.0859734
\(639\) −7.89949 −0.312499
\(640\) −1.58579 −0.0626837
\(641\) −18.5858 −0.734094 −0.367047 0.930202i \(-0.619631\pi\)
−0.367047 + 0.930202i \(0.619631\pi\)
\(642\) −11.7279 −0.462864
\(643\) 42.2426 1.66589 0.832944 0.553358i \(-0.186654\pi\)
0.832944 + 0.553358i \(0.186654\pi\)
\(644\) 0 0
\(645\) 9.89949 0.389792
\(646\) −4.24264 −0.166924
\(647\) 43.3553 1.70447 0.852237 0.523156i \(-0.175245\pi\)
0.852237 + 0.523156i \(0.175245\pi\)
\(648\) 1.00000 0.0392837
\(649\) −6.07107 −0.238310
\(650\) −7.02944 −0.275717
\(651\) 0 0
\(652\) −10.8284 −0.424074
\(653\) −26.2132 −1.02580 −0.512901 0.858448i \(-0.671429\pi\)
−0.512901 + 0.858448i \(0.671429\pi\)
\(654\) −1.75736 −0.0687182
\(655\) −6.84062 −0.267285
\(656\) −0.585786 −0.0228711
\(657\) −14.8284 −0.578512
\(658\) 0 0
\(659\) 2.97056 0.115717 0.0578583 0.998325i \(-0.481573\pi\)
0.0578583 + 0.998325i \(0.481573\pi\)
\(660\) −1.58579 −0.0617267
\(661\) 32.3848 1.25962 0.629811 0.776748i \(-0.283132\pi\)
0.629811 + 0.776748i \(0.283132\pi\)
\(662\) −4.38478 −0.170419
\(663\) 12.0000 0.466041
\(664\) −17.1421 −0.665244
\(665\) 0 0
\(666\) 0.242641 0.00940214
\(667\) 7.41421 0.287079
\(668\) −6.72792 −0.260311
\(669\) −23.4853 −0.907993
\(670\) 3.17157 0.122529
\(671\) 4.82843 0.186399
\(672\) 0 0
\(673\) −2.27208 −0.0875822 −0.0437911 0.999041i \(-0.513944\pi\)
−0.0437911 + 0.999041i \(0.513944\pi\)
\(674\) 2.07107 0.0797746
\(675\) 2.48528 0.0956585
\(676\) −5.00000 −0.192308
\(677\) −22.3137 −0.857585 −0.428793 0.903403i \(-0.641061\pi\)
−0.428793 + 0.903403i \(0.641061\pi\)
\(678\) −10.2426 −0.393366
\(679\) 0 0
\(680\) 6.72792 0.258004
\(681\) −1.10051 −0.0421714
\(682\) 10.6569 0.408072
\(683\) 31.2426 1.19547 0.597733 0.801695i \(-0.296068\pi\)
0.597733 + 0.801695i \(0.296068\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0.544156 0.0207911
\(686\) 0 0
\(687\) 19.7990 0.755379
\(688\) 6.24264 0.237998
\(689\) 5.17157 0.197021
\(690\) 5.41421 0.206116
\(691\) 20.4437 0.777713 0.388857 0.921298i \(-0.372870\pi\)
0.388857 + 0.921298i \(0.372870\pi\)
\(692\) −15.7990 −0.600587
\(693\) 0 0
\(694\) −34.0000 −1.29062
\(695\) −12.5269 −0.475173
\(696\) −2.17157 −0.0823133
\(697\) 2.48528 0.0941367
\(698\) −7.75736 −0.293620
\(699\) 5.17157 0.195607
\(700\) 0 0
\(701\) 41.0416 1.55012 0.775060 0.631887i \(-0.217719\pi\)
0.775060 + 0.631887i \(0.217719\pi\)
\(702\) −2.82843 −0.106752
\(703\) 0.242641 0.00915137
\(704\) −1.00000 −0.0376889
\(705\) 5.02944 0.189420
\(706\) 14.1005 0.530680
\(707\) 0 0
\(708\) −6.07107 −0.228165
\(709\) −50.3848 −1.89224 −0.946120 0.323816i \(-0.895034\pi\)
−0.946120 + 0.323816i \(0.895034\pi\)
\(710\) 12.5269 0.470127
\(711\) −0.171573 −0.00643449
\(712\) 3.75736 0.140813
\(713\) −36.3848 −1.36262
\(714\) 0 0
\(715\) 4.48528 0.167740
\(716\) −7.51472 −0.280838
\(717\) −28.2843 −1.05630
\(718\) 10.0000 0.373197
\(719\) 8.82843 0.329245 0.164622 0.986357i \(-0.447359\pi\)
0.164622 + 0.986357i \(0.447359\pi\)
\(720\) −1.58579 −0.0590988
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) −0.556349 −0.0206908
\(724\) −20.4853 −0.761329
\(725\) −5.39697 −0.200438
\(726\) 10.0000 0.371135
\(727\) 43.7279 1.62178 0.810889 0.585199i \(-0.198984\pi\)
0.810889 + 0.585199i \(0.198984\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 23.5147 0.870319
\(731\) −26.4853 −0.979594
\(732\) 4.82843 0.178464
\(733\) 34.1838 1.26261 0.631303 0.775536i \(-0.282520\pi\)
0.631303 + 0.775536i \(0.282520\pi\)
\(734\) 5.24264 0.193509
\(735\) 0 0
\(736\) 3.41421 0.125850
\(737\) 2.00000 0.0736709
\(738\) −0.585786 −0.0215631
\(739\) 33.5563 1.23439 0.617195 0.786810i \(-0.288269\pi\)
0.617195 + 0.786810i \(0.288269\pi\)
\(740\) −0.384776 −0.0141447
\(741\) −2.82843 −0.103905
\(742\) 0 0
\(743\) −48.9117 −1.79440 −0.897198 0.441629i \(-0.854401\pi\)
−0.897198 + 0.441629i \(0.854401\pi\)
\(744\) 10.6569 0.390699
\(745\) 29.3137 1.07397
\(746\) 0.485281 0.0177674
\(747\) −17.1421 −0.627198
\(748\) 4.24264 0.155126
\(749\) 0 0
\(750\) −11.8701 −0.433433
\(751\) −16.1127 −0.587961 −0.293980 0.955811i \(-0.594980\pi\)
−0.293980 + 0.955811i \(0.594980\pi\)
\(752\) 3.17157 0.115655
\(753\) 21.0000 0.765283
\(754\) 6.14214 0.223683
\(755\) 17.9878 0.654643
\(756\) 0 0
\(757\) 1.21320 0.0440946 0.0220473 0.999757i \(-0.492982\pi\)
0.0220473 + 0.999757i \(0.492982\pi\)
\(758\) 3.31371 0.120359
\(759\) 3.41421 0.123928
\(760\) −1.58579 −0.0575225
\(761\) 6.14214 0.222652 0.111326 0.993784i \(-0.464490\pi\)
0.111326 + 0.993784i \(0.464490\pi\)
\(762\) 5.82843 0.211142
\(763\) 0 0
\(764\) −18.1421 −0.656359
\(765\) 6.72792 0.243249
\(766\) 15.6569 0.565705
\(767\) 17.1716 0.620030
\(768\) −1.00000 −0.0360844
\(769\) 33.3431 1.20238 0.601192 0.799104i \(-0.294693\pi\)
0.601192 + 0.799104i \(0.294693\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) −23.7279 −0.853987
\(773\) −34.6274 −1.24546 −0.622731 0.782436i \(-0.713977\pi\)
−0.622731 + 0.782436i \(0.713977\pi\)
\(774\) 6.24264 0.224387
\(775\) 26.4853 0.951379
\(776\) −5.58579 −0.200518
\(777\) 0 0
\(778\) 23.1716 0.830741
\(779\) −0.585786 −0.0209880
\(780\) 4.48528 0.160599
\(781\) 7.89949 0.282666
\(782\) −14.4853 −0.517993
\(783\) −2.17157 −0.0776057
\(784\) 0 0
\(785\) 7.11270 0.253863
\(786\) −4.31371 −0.153865
\(787\) 18.6863 0.666094 0.333047 0.942910i \(-0.391923\pi\)
0.333047 + 0.942910i \(0.391923\pi\)
\(788\) 1.17157 0.0417356
\(789\) 11.5563 0.411417
\(790\) 0.272078 0.00968010
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) −13.6569 −0.484969
\(794\) −10.8284 −0.384286
\(795\) 2.89949 0.102834
\(796\) 8.48528 0.300753
\(797\) −9.00000 −0.318796 −0.159398 0.987214i \(-0.550955\pi\)
−0.159398 + 0.987214i \(0.550955\pi\)
\(798\) 0 0
\(799\) −13.4558 −0.476034
\(800\) −2.48528 −0.0878680
\(801\) 3.75736 0.132760
\(802\) 14.1421 0.499376
\(803\) 14.8284 0.523284
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) −30.1421 −1.06171
\(807\) −1.00000 −0.0352017
\(808\) 14.9706 0.526663
\(809\) −50.8701 −1.78850 −0.894248 0.447572i \(-0.852289\pi\)
−0.894248 + 0.447572i \(0.852289\pi\)
\(810\) −1.58579 −0.0557189
\(811\) 3.55635 0.124880 0.0624402 0.998049i \(-0.480112\pi\)
0.0624402 + 0.998049i \(0.480112\pi\)
\(812\) 0 0
\(813\) −8.89949 −0.312119
\(814\) −0.242641 −0.00850455
\(815\) 17.1716 0.601494
\(816\) 4.24264 0.148522
\(817\) 6.24264 0.218402
\(818\) 30.8995 1.08038
\(819\) 0 0
\(820\) 0.928932 0.0324397
\(821\) 31.1838 1.08832 0.544160 0.838981i \(-0.316848\pi\)
0.544160 + 0.838981i \(0.316848\pi\)
\(822\) 0.343146 0.0119686
\(823\) 24.2843 0.846496 0.423248 0.906014i \(-0.360890\pi\)
0.423248 + 0.906014i \(0.360890\pi\)
\(824\) −18.4853 −0.643965
\(825\) −2.48528 −0.0865264
\(826\) 0 0
\(827\) 27.7279 0.964194 0.482097 0.876118i \(-0.339875\pi\)
0.482097 + 0.876118i \(0.339875\pi\)
\(828\) 3.41421 0.118652
\(829\) −48.9706 −1.70082 −0.850409 0.526122i \(-0.823645\pi\)
−0.850409 + 0.526122i \(0.823645\pi\)
\(830\) 27.1838 0.943562
\(831\) 13.7574 0.477238
\(832\) 2.82843 0.0980581
\(833\) 0 0
\(834\) −7.89949 −0.273537
\(835\) 10.6690 0.369218
\(836\) −1.00000 −0.0345857
\(837\) 10.6569 0.368355
\(838\) −30.9706 −1.06986
\(839\) 53.2132 1.83712 0.918562 0.395277i \(-0.129351\pi\)
0.918562 + 0.395277i \(0.129351\pi\)
\(840\) 0 0
\(841\) −24.2843 −0.837389
\(842\) 15.1716 0.522847
\(843\) 1.31371 0.0452465
\(844\) −15.0711 −0.518768
\(845\) 7.92893 0.272764
\(846\) 3.17157 0.109041
\(847\) 0 0
\(848\) 1.82843 0.0627884
\(849\) −21.5563 −0.739812
\(850\) 10.5442 0.361662
\(851\) 0.828427 0.0283981
\(852\) 7.89949 0.270632
\(853\) 57.4975 1.96868 0.984338 0.176291i \(-0.0564101\pi\)
0.984338 + 0.176291i \(0.0564101\pi\)
\(854\) 0 0
\(855\) −1.58579 −0.0542328
\(856\) 11.7279 0.400852
\(857\) −18.6863 −0.638312 −0.319156 0.947702i \(-0.603399\pi\)
−0.319156 + 0.947702i \(0.603399\pi\)
\(858\) 2.82843 0.0965609
\(859\) −48.4264 −1.65229 −0.826144 0.563459i \(-0.809470\pi\)
−0.826144 + 0.563459i \(0.809470\pi\)
\(860\) −9.89949 −0.337570
\(861\) 0 0
\(862\) −14.1005 −0.480265
\(863\) 38.2426 1.30179 0.650897 0.759166i \(-0.274393\pi\)
0.650897 + 0.759166i \(0.274393\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 25.0538 0.851856
\(866\) −30.8284 −1.04759
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 0.171573 0.00582021
\(870\) 3.44365 0.116751
\(871\) −5.65685 −0.191675
\(872\) 1.75736 0.0595117
\(873\) −5.58579 −0.189050
\(874\) 3.41421 0.115487
\(875\) 0 0
\(876\) 14.8284 0.501006
\(877\) 40.7279 1.37528 0.687642 0.726050i \(-0.258646\pi\)
0.687642 + 0.726050i \(0.258646\pi\)
\(878\) 22.4558 0.757848
\(879\) 23.0000 0.775771
\(880\) 1.58579 0.0534568
\(881\) −23.9411 −0.806597 −0.403299 0.915068i \(-0.632136\pi\)
−0.403299 + 0.915068i \(0.632136\pi\)
\(882\) 0 0
\(883\) 46.8701 1.57730 0.788652 0.614840i \(-0.210780\pi\)
0.788652 + 0.614840i \(0.210780\pi\)
\(884\) −12.0000 −0.403604
\(885\) 9.62742 0.323622
\(886\) −14.6569 −0.492407
\(887\) −51.0711 −1.71480 −0.857399 0.514652i \(-0.827921\pi\)
−0.857399 + 0.514652i \(0.827921\pi\)
\(888\) −0.242641 −0.00814249
\(889\) 0 0
\(890\) −5.95837 −0.199725
\(891\) −1.00000 −0.0335013
\(892\) 23.4853 0.786345
\(893\) 3.17157 0.106133
\(894\) 18.4853 0.618240
\(895\) 11.9167 0.398333
\(896\) 0 0
\(897\) −9.65685 −0.322433
\(898\) 14.2426 0.475283
\(899\) −23.1421 −0.771833
\(900\) −2.48528 −0.0828427
\(901\) −7.75736 −0.258435
\(902\) 0.585786 0.0195046
\(903\) 0 0
\(904\) 10.2426 0.340665
\(905\) 32.4853 1.07985
\(906\) 11.3431 0.376851
\(907\) 16.1005 0.534608 0.267304 0.963612i \(-0.413867\pi\)
0.267304 + 0.963612i \(0.413867\pi\)
\(908\) 1.10051 0.0365215
\(909\) 14.9706 0.496542
\(910\) 0 0
\(911\) −28.8284 −0.955128 −0.477564 0.878597i \(-0.658480\pi\)
−0.477564 + 0.878597i \(0.658480\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 17.1421 0.567322
\(914\) 36.1127 1.19450
\(915\) −7.65685 −0.253128
\(916\) −19.7990 −0.654177
\(917\) 0 0
\(918\) 4.24264 0.140028
\(919\) −40.3431 −1.33080 −0.665399 0.746488i \(-0.731738\pi\)
−0.665399 + 0.746488i \(0.731738\pi\)
\(920\) −5.41421 −0.178501
\(921\) −7.41421 −0.244307
\(922\) −17.3137 −0.570197
\(923\) −22.3431 −0.735434
\(924\) 0 0
\(925\) −0.603030 −0.0198275
\(926\) 1.65685 0.0544476
\(927\) −18.4853 −0.607136
\(928\) 2.17157 0.0712854
\(929\) −2.92893 −0.0960951 −0.0480476 0.998845i \(-0.515300\pi\)
−0.0480476 + 0.998845i \(0.515300\pi\)
\(930\) −16.8995 −0.554156
\(931\) 0 0
\(932\) −5.17157 −0.169401
\(933\) −22.2426 −0.728191
\(934\) −15.1716 −0.496429
\(935\) −6.72792 −0.220027
\(936\) 2.82843 0.0924500
\(937\) 0.857864 0.0280252 0.0140126 0.999902i \(-0.495540\pi\)
0.0140126 + 0.999902i \(0.495540\pi\)
\(938\) 0 0
\(939\) 3.82843 0.124936
\(940\) −5.02944 −0.164042
\(941\) −26.3137 −0.857802 −0.428901 0.903351i \(-0.641099\pi\)
−0.428901 + 0.903351i \(0.641099\pi\)
\(942\) 4.48528 0.146138
\(943\) −2.00000 −0.0651290
\(944\) 6.07107 0.197596
\(945\) 0 0
\(946\) −6.24264 −0.202966
\(947\) −31.6569 −1.02871 −0.514355 0.857578i \(-0.671969\pi\)
−0.514355 + 0.857578i \(0.671969\pi\)
\(948\) 0.171573 0.00557243
\(949\) −41.9411 −1.36147
\(950\) −2.48528 −0.0806332
\(951\) 9.48528 0.307581
\(952\) 0 0
\(953\) 39.1716 1.26889 0.634446 0.772967i \(-0.281228\pi\)
0.634446 + 0.772967i \(0.281228\pi\)
\(954\) 1.82843 0.0591975
\(955\) 28.7696 0.930961
\(956\) 28.2843 0.914779
\(957\) 2.17157 0.0701970
\(958\) 1.31371 0.0424440
\(959\) 0 0
\(960\) 1.58579 0.0511810
\(961\) 82.5685 2.66350
\(962\) 0.686292 0.0221269
\(963\) 11.7279 0.377927
\(964\) 0.556349 0.0179188
\(965\) 37.6274 1.21127
\(966\) 0 0
\(967\) −30.5563 −0.982626 −0.491313 0.870983i \(-0.663483\pi\)
−0.491313 + 0.870983i \(0.663483\pi\)
\(968\) −10.0000 −0.321412
\(969\) 4.24264 0.136293
\(970\) 8.85786 0.284409
\(971\) 37.0416 1.18872 0.594361 0.804198i \(-0.297405\pi\)
0.594361 + 0.804198i \(0.297405\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −38.7990 −1.24320
\(975\) 7.02944 0.225122
\(976\) −4.82843 −0.154554
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 10.8284 0.346255
\(979\) −3.75736 −0.120086
\(980\) 0 0
\(981\) 1.75736 0.0561082
\(982\) −9.14214 −0.291737
\(983\) 48.9706 1.56192 0.780959 0.624582i \(-0.214731\pi\)
0.780959 + 0.624582i \(0.214731\pi\)
\(984\) 0.585786 0.0186742
\(985\) −1.85786 −0.0591965
\(986\) −9.21320 −0.293408
\(987\) 0 0
\(988\) 2.82843 0.0899843
\(989\) 21.3137 0.677737
\(990\) 1.58579 0.0503996
\(991\) −11.6274 −0.369357 −0.184679 0.982799i \(-0.559124\pi\)
−0.184679 + 0.982799i \(0.559124\pi\)
\(992\) −10.6569 −0.338355
\(993\) 4.38478 0.139147
\(994\) 0 0
\(995\) −13.4558 −0.426579
\(996\) 17.1421 0.543169
\(997\) 3.17157 0.100445 0.0502224 0.998738i \(-0.484007\pi\)
0.0502224 + 0.998738i \(0.484007\pi\)
\(998\) 9.31371 0.294820
\(999\) −0.242641 −0.00767681
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 5586.2.a.bg.1.2 2
7.2 even 3 798.2.j.h.571.1 yes 4
7.4 even 3 798.2.j.h.457.1 4
7.6 odd 2 5586.2.a.br.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.j.h.457.1 4 7.4 even 3
798.2.j.h.571.1 yes 4 7.2 even 3
5586.2.a.bg.1.2 2 1.1 even 1 trivial
5586.2.a.br.1.1 2 7.6 odd 2