Properties

Label 5054.2.a.bb.1.5
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1528713.1
Defining polynomial: \( x^{6} - 3x^{5} - 3x^{4} + 7x^{3} + 3x^{2} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.725554\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.11161 q^{3} +1.00000 q^{4} -3.59028 q^{5} -2.11161 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.45891 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.11161 q^{3} +1.00000 q^{4} -3.59028 q^{5} -2.11161 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.45891 q^{9} +3.59028 q^{10} +3.24298 q^{11} +2.11161 q^{12} +0.613941 q^{13} +1.00000 q^{14} -7.58127 q^{15} +1.00000 q^{16} -4.69713 q^{17} -1.45891 q^{18} -3.59028 q^{20} -2.11161 q^{21} -3.24298 q^{22} +4.53089 q^{23} -2.11161 q^{24} +7.89009 q^{25} -0.613941 q^{26} -3.25419 q^{27} -1.00000 q^{28} +4.45024 q^{29} +7.58127 q^{30} -6.48937 q^{31} -1.00000 q^{32} +6.84792 q^{33} +4.69713 q^{34} +3.59028 q^{35} +1.45891 q^{36} +5.34592 q^{37} +1.29641 q^{39} +3.59028 q^{40} -1.08664 q^{41} +2.11161 q^{42} +5.44103 q^{43} +3.24298 q^{44} -5.23789 q^{45} -4.53089 q^{46} +11.6016 q^{47} +2.11161 q^{48} +1.00000 q^{49} -7.89009 q^{50} -9.91851 q^{51} +0.613941 q^{52} -6.12638 q^{53} +3.25419 q^{54} -11.6432 q^{55} +1.00000 q^{56} -4.45024 q^{58} -2.89177 q^{59} -7.58127 q^{60} -13.2269 q^{61} +6.48937 q^{62} -1.45891 q^{63} +1.00000 q^{64} -2.20422 q^{65} -6.84792 q^{66} -6.41158 q^{67} -4.69713 q^{68} +9.56748 q^{69} -3.59028 q^{70} +11.5793 q^{71} -1.45891 q^{72} -16.3204 q^{73} -5.34592 q^{74} +16.6608 q^{75} -3.24298 q^{77} -1.29641 q^{78} -6.43324 q^{79} -3.59028 q^{80} -11.2483 q^{81} +1.08664 q^{82} -8.52076 q^{83} -2.11161 q^{84} +16.8640 q^{85} -5.44103 q^{86} +9.39718 q^{87} -3.24298 q^{88} +1.49287 q^{89} +5.23789 q^{90} -0.613941 q^{91} +4.53089 q^{92} -13.7030 q^{93} -11.6016 q^{94} -2.11161 q^{96} +1.67786 q^{97} -1.00000 q^{98} +4.73121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 3 q^{3} + 6 q^{4} - 3 q^{5} - 3 q^{6} - 6 q^{7} - 6 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 3 q^{3} + 6 q^{4} - 3 q^{5} - 3 q^{6} - 6 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{10} + 3 q^{11} + 3 q^{12} + 6 q^{13} + 6 q^{14} - 6 q^{15} + 6 q^{16} - 9 q^{17} + 3 q^{18} - 3 q^{20} - 3 q^{21} - 3 q^{22} - 3 q^{24} + 3 q^{25} - 6 q^{26} + 3 q^{27} - 6 q^{28} + 6 q^{29} + 6 q^{30} + 3 q^{31} - 6 q^{32} + 6 q^{33} + 9 q^{34} + 3 q^{35} - 3 q^{36} - 3 q^{37} - 9 q^{39} + 3 q^{40} + 9 q^{41} + 3 q^{42} - 6 q^{43} + 3 q^{44} - 12 q^{45} + 9 q^{47} + 3 q^{48} + 6 q^{49} - 3 q^{50} - 27 q^{51} + 6 q^{52} + 6 q^{53} - 3 q^{54} - 24 q^{55} + 6 q^{56} - 6 q^{58} + 15 q^{59} - 6 q^{60} - 30 q^{61} - 3 q^{62} + 3 q^{63} + 6 q^{64} + 3 q^{65} - 6 q^{66} + 15 q^{67} - 9 q^{68} + 24 q^{69} - 3 q^{70} + 21 q^{71} + 3 q^{72} - 33 q^{73} + 3 q^{74} + 33 q^{75} - 3 q^{77} + 9 q^{78} - 30 q^{79} - 3 q^{80} - 18 q^{81} - 9 q^{82} - 33 q^{83} - 3 q^{84} - 18 q^{85} + 6 q^{86} + 15 q^{87} - 3 q^{88} - 12 q^{89} + 12 q^{90} - 6 q^{91} - 21 q^{93} - 9 q^{94} - 3 q^{96} - 18 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.11161 1.21914 0.609570 0.792732i \(-0.291342\pi\)
0.609570 + 0.792732i \(0.291342\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.59028 −1.60562 −0.802810 0.596235i \(-0.796663\pi\)
−0.802810 + 0.596235i \(0.796663\pi\)
\(6\) −2.11161 −0.862062
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.45891 0.486303
\(10\) 3.59028 1.13535
\(11\) 3.24298 0.977795 0.488898 0.872341i \(-0.337399\pi\)
0.488898 + 0.872341i \(0.337399\pi\)
\(12\) 2.11161 0.609570
\(13\) 0.613941 0.170277 0.0851383 0.996369i \(-0.472867\pi\)
0.0851383 + 0.996369i \(0.472867\pi\)
\(14\) 1.00000 0.267261
\(15\) −7.58127 −1.95748
\(16\) 1.00000 0.250000
\(17\) −4.69713 −1.13922 −0.569610 0.821915i \(-0.692906\pi\)
−0.569610 + 0.821915i \(0.692906\pi\)
\(18\) −1.45891 −0.343868
\(19\) 0 0
\(20\) −3.59028 −0.802810
\(21\) −2.11161 −0.460792
\(22\) −3.24298 −0.691406
\(23\) 4.53089 0.944756 0.472378 0.881396i \(-0.343396\pi\)
0.472378 + 0.881396i \(0.343396\pi\)
\(24\) −2.11161 −0.431031
\(25\) 7.89009 1.57802
\(26\) −0.613941 −0.120404
\(27\) −3.25419 −0.626269
\(28\) −1.00000 −0.188982
\(29\) 4.45024 0.826388 0.413194 0.910643i \(-0.364413\pi\)
0.413194 + 0.910643i \(0.364413\pi\)
\(30\) 7.58127 1.38415
\(31\) −6.48937 −1.16552 −0.582762 0.812643i \(-0.698028\pi\)
−0.582762 + 0.812643i \(0.698028\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.84792 1.19207
\(34\) 4.69713 0.805551
\(35\) 3.59028 0.606868
\(36\) 1.45891 0.243152
\(37\) 5.34592 0.878864 0.439432 0.898276i \(-0.355180\pi\)
0.439432 + 0.898276i \(0.355180\pi\)
\(38\) 0 0
\(39\) 1.29641 0.207591
\(40\) 3.59028 0.567673
\(41\) −1.08664 −0.169705 −0.0848525 0.996394i \(-0.527042\pi\)
−0.0848525 + 0.996394i \(0.527042\pi\)
\(42\) 2.11161 0.325829
\(43\) 5.44103 0.829750 0.414875 0.909878i \(-0.363825\pi\)
0.414875 + 0.909878i \(0.363825\pi\)
\(44\) 3.24298 0.488898
\(45\) −5.23789 −0.780818
\(46\) −4.53089 −0.668043
\(47\) 11.6016 1.69227 0.846133 0.532971i \(-0.178925\pi\)
0.846133 + 0.532971i \(0.178925\pi\)
\(48\) 2.11161 0.304785
\(49\) 1.00000 0.142857
\(50\) −7.89009 −1.11583
\(51\) −9.91851 −1.38887
\(52\) 0.613941 0.0851383
\(53\) −6.12638 −0.841523 −0.420762 0.907171i \(-0.638237\pi\)
−0.420762 + 0.907171i \(0.638237\pi\)
\(54\) 3.25419 0.442839
\(55\) −11.6432 −1.56997
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −4.45024 −0.584345
\(59\) −2.89177 −0.376477 −0.188238 0.982123i \(-0.560278\pi\)
−0.188238 + 0.982123i \(0.560278\pi\)
\(60\) −7.58127 −0.978738
\(61\) −13.2269 −1.69353 −0.846764 0.531969i \(-0.821452\pi\)
−0.846764 + 0.531969i \(0.821452\pi\)
\(62\) 6.48937 0.824150
\(63\) −1.45891 −0.183805
\(64\) 1.00000 0.125000
\(65\) −2.20422 −0.273400
\(66\) −6.84792 −0.842921
\(67\) −6.41158 −0.783299 −0.391650 0.920114i \(-0.628095\pi\)
−0.391650 + 0.920114i \(0.628095\pi\)
\(68\) −4.69713 −0.569610
\(69\) 9.56748 1.15179
\(70\) −3.59028 −0.429120
\(71\) 11.5793 1.37421 0.687103 0.726560i \(-0.258882\pi\)
0.687103 + 0.726560i \(0.258882\pi\)
\(72\) −1.45891 −0.171934
\(73\) −16.3204 −1.91015 −0.955077 0.296357i \(-0.904228\pi\)
−0.955077 + 0.296357i \(0.904228\pi\)
\(74\) −5.34592 −0.621451
\(75\) 16.6608 1.92382
\(76\) 0 0
\(77\) −3.24298 −0.369572
\(78\) −1.29641 −0.146789
\(79\) −6.43324 −0.723796 −0.361898 0.932218i \(-0.617871\pi\)
−0.361898 + 0.932218i \(0.617871\pi\)
\(80\) −3.59028 −0.401405
\(81\) −11.2483 −1.24981
\(82\) 1.08664 0.120000
\(83\) −8.52076 −0.935274 −0.467637 0.883921i \(-0.654895\pi\)
−0.467637 + 0.883921i \(0.654895\pi\)
\(84\) −2.11161 −0.230396
\(85\) 16.8640 1.82916
\(86\) −5.44103 −0.586722
\(87\) 9.39718 1.00748
\(88\) −3.24298 −0.345703
\(89\) 1.49287 0.158244 0.0791221 0.996865i \(-0.474788\pi\)
0.0791221 + 0.996865i \(0.474788\pi\)
\(90\) 5.23789 0.552122
\(91\) −0.613941 −0.0643585
\(92\) 4.53089 0.472378
\(93\) −13.7030 −1.42094
\(94\) −11.6016 −1.19661
\(95\) 0 0
\(96\) −2.11161 −0.215516
\(97\) 1.67786 0.170361 0.0851804 0.996366i \(-0.472853\pi\)
0.0851804 + 0.996366i \(0.472853\pi\)
\(98\) −1.00000 −0.101015
\(99\) 4.73121 0.475505
\(100\) 7.89009 0.789009
\(101\) 1.86854 0.185927 0.0929636 0.995670i \(-0.470366\pi\)
0.0929636 + 0.995670i \(0.470366\pi\)
\(102\) 9.91851 0.982079
\(103\) −20.1263 −1.98310 −0.991550 0.129727i \(-0.958590\pi\)
−0.991550 + 0.129727i \(0.958590\pi\)
\(104\) −0.613941 −0.0602019
\(105\) 7.58127 0.739857
\(106\) 6.12638 0.595047
\(107\) 9.37414 0.906233 0.453116 0.891451i \(-0.350312\pi\)
0.453116 + 0.891451i \(0.350312\pi\)
\(108\) −3.25419 −0.313134
\(109\) −9.28080 −0.888940 −0.444470 0.895794i \(-0.646608\pi\)
−0.444470 + 0.895794i \(0.646608\pi\)
\(110\) 11.6432 1.11014
\(111\) 11.2885 1.07146
\(112\) −1.00000 −0.0944911
\(113\) −8.67289 −0.815877 −0.407938 0.913009i \(-0.633752\pi\)
−0.407938 + 0.913009i \(0.633752\pi\)
\(114\) 0 0
\(115\) −16.2671 −1.51692
\(116\) 4.45024 0.413194
\(117\) 0.895684 0.0828060
\(118\) 2.89177 0.266209
\(119\) 4.69713 0.430585
\(120\) 7.58127 0.692073
\(121\) −0.483078 −0.0439162
\(122\) 13.2269 1.19750
\(123\) −2.29457 −0.206894
\(124\) −6.48937 −0.582762
\(125\) −10.3762 −0.928077
\(126\) 1.45891 0.129970
\(127\) 18.2693 1.62113 0.810567 0.585645i \(-0.199159\pi\)
0.810567 + 0.585645i \(0.199159\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.4894 1.01158
\(130\) 2.20422 0.193323
\(131\) −8.09149 −0.706957 −0.353478 0.935443i \(-0.615001\pi\)
−0.353478 + 0.935443i \(0.615001\pi\)
\(132\) 6.84792 0.596035
\(133\) 0 0
\(134\) 6.41158 0.553876
\(135\) 11.6834 1.00555
\(136\) 4.69713 0.402775
\(137\) 4.55482 0.389145 0.194572 0.980888i \(-0.437668\pi\)
0.194572 + 0.980888i \(0.437668\pi\)
\(138\) −9.56748 −0.814438
\(139\) −21.6945 −1.84010 −0.920051 0.391799i \(-0.871853\pi\)
−0.920051 + 0.391799i \(0.871853\pi\)
\(140\) 3.59028 0.303434
\(141\) 24.4981 2.06311
\(142\) −11.5793 −0.971710
\(143\) 1.99100 0.166496
\(144\) 1.45891 0.121576
\(145\) −15.9776 −1.32687
\(146\) 16.3204 1.35068
\(147\) 2.11161 0.174163
\(148\) 5.34592 0.439432
\(149\) 21.4072 1.75375 0.876874 0.480721i \(-0.159625\pi\)
0.876874 + 0.480721i \(0.159625\pi\)
\(150\) −16.6608 −1.36035
\(151\) 5.52022 0.449229 0.224615 0.974448i \(-0.427888\pi\)
0.224615 + 0.974448i \(0.427888\pi\)
\(152\) 0 0
\(153\) −6.85268 −0.554007
\(154\) 3.24298 0.261327
\(155\) 23.2986 1.87139
\(156\) 1.29641 0.103796
\(157\) −13.5838 −1.08410 −0.542052 0.840345i \(-0.682352\pi\)
−0.542052 + 0.840345i \(0.682352\pi\)
\(158\) 6.43324 0.511801
\(159\) −12.9365 −1.02594
\(160\) 3.59028 0.283836
\(161\) −4.53089 −0.357084
\(162\) 11.2483 0.883751
\(163\) −6.32223 −0.495195 −0.247598 0.968863i \(-0.579641\pi\)
−0.247598 + 0.968863i \(0.579641\pi\)
\(164\) −1.08664 −0.0848525
\(165\) −24.5859 −1.91401
\(166\) 8.52076 0.661339
\(167\) −14.1989 −1.09875 −0.549373 0.835577i \(-0.685133\pi\)
−0.549373 + 0.835577i \(0.685133\pi\)
\(168\) 2.11161 0.162914
\(169\) −12.6231 −0.971006
\(170\) −16.8640 −1.29341
\(171\) 0 0
\(172\) 5.44103 0.414875
\(173\) 7.38609 0.561554 0.280777 0.959773i \(-0.409408\pi\)
0.280777 + 0.959773i \(0.409408\pi\)
\(174\) −9.39718 −0.712398
\(175\) −7.89009 −0.596435
\(176\) 3.24298 0.244449
\(177\) −6.10631 −0.458978
\(178\) −1.49287 −0.111895
\(179\) −6.37886 −0.476778 −0.238389 0.971170i \(-0.576619\pi\)
−0.238389 + 0.971170i \(0.576619\pi\)
\(180\) −5.23789 −0.390409
\(181\) 22.2731 1.65555 0.827774 0.561062i \(-0.189607\pi\)
0.827774 + 0.561062i \(0.189607\pi\)
\(182\) 0.613941 0.0455083
\(183\) −27.9300 −2.06465
\(184\) −4.53089 −0.334022
\(185\) −19.1933 −1.41112
\(186\) 13.7030 1.00475
\(187\) −15.2327 −1.11392
\(188\) 11.6016 0.846133
\(189\) 3.25419 0.236707
\(190\) 0 0
\(191\) −23.4129 −1.69410 −0.847048 0.531516i \(-0.821623\pi\)
−0.847048 + 0.531516i \(0.821623\pi\)
\(192\) 2.11161 0.152393
\(193\) −15.1577 −1.09107 −0.545537 0.838087i \(-0.683674\pi\)
−0.545537 + 0.838087i \(0.683674\pi\)
\(194\) −1.67786 −0.120463
\(195\) −4.65445 −0.333312
\(196\) 1.00000 0.0714286
\(197\) −14.5729 −1.03828 −0.519138 0.854691i \(-0.673747\pi\)
−0.519138 + 0.854691i \(0.673747\pi\)
\(198\) −4.73121 −0.336233
\(199\) 7.05899 0.500398 0.250199 0.968194i \(-0.419504\pi\)
0.250199 + 0.968194i \(0.419504\pi\)
\(200\) −7.89009 −0.557913
\(201\) −13.5388 −0.954952
\(202\) −1.86854 −0.131470
\(203\) −4.45024 −0.312345
\(204\) −9.91851 −0.694435
\(205\) 3.90134 0.272482
\(206\) 20.1263 1.40226
\(207\) 6.61016 0.459438
\(208\) 0.613941 0.0425691
\(209\) 0 0
\(210\) −7.58127 −0.523158
\(211\) −5.69557 −0.392099 −0.196050 0.980594i \(-0.562811\pi\)
−0.196050 + 0.980594i \(0.562811\pi\)
\(212\) −6.12638 −0.420762
\(213\) 24.4509 1.67535
\(214\) −9.37414 −0.640803
\(215\) −19.5348 −1.33226
\(216\) 3.25419 0.221419
\(217\) 6.48937 0.440527
\(218\) 9.28080 0.628575
\(219\) −34.4623 −2.32875
\(220\) −11.6432 −0.784984
\(221\) −2.88376 −0.193983
\(222\) −11.2885 −0.757636
\(223\) 18.5836 1.24445 0.622226 0.782838i \(-0.286228\pi\)
0.622226 + 0.782838i \(0.286228\pi\)
\(224\) 1.00000 0.0668153
\(225\) 11.5109 0.767395
\(226\) 8.67289 0.576912
\(227\) 0.527433 0.0350070 0.0175035 0.999847i \(-0.494428\pi\)
0.0175035 + 0.999847i \(0.494428\pi\)
\(228\) 0 0
\(229\) −11.9310 −0.788426 −0.394213 0.919019i \(-0.628983\pi\)
−0.394213 + 0.919019i \(0.628983\pi\)
\(230\) 16.2671 1.07262
\(231\) −6.84792 −0.450560
\(232\) −4.45024 −0.292172
\(233\) −6.00187 −0.393195 −0.196598 0.980484i \(-0.562989\pi\)
−0.196598 + 0.980484i \(0.562989\pi\)
\(234\) −0.895684 −0.0585527
\(235\) −41.6529 −2.71714
\(236\) −2.89177 −0.188238
\(237\) −13.5845 −0.882409
\(238\) −4.69713 −0.304470
\(239\) −22.2435 −1.43882 −0.719408 0.694588i \(-0.755587\pi\)
−0.719408 + 0.694588i \(0.755587\pi\)
\(240\) −7.58127 −0.489369
\(241\) 19.8067 1.27586 0.637932 0.770093i \(-0.279790\pi\)
0.637932 + 0.770093i \(0.279790\pi\)
\(242\) 0.483078 0.0310534
\(243\) −13.9895 −0.897428
\(244\) −13.2269 −0.846764
\(245\) −3.59028 −0.229374
\(246\) 2.29457 0.146296
\(247\) 0 0
\(248\) 6.48937 0.412075
\(249\) −17.9925 −1.14023
\(250\) 10.3762 0.656249
\(251\) 6.27968 0.396370 0.198185 0.980165i \(-0.436495\pi\)
0.198185 + 0.980165i \(0.436495\pi\)
\(252\) −1.45891 −0.0919026
\(253\) 14.6936 0.923778
\(254\) −18.2693 −1.14632
\(255\) 35.6102 2.23000
\(256\) 1.00000 0.0625000
\(257\) −11.4959 −0.717096 −0.358548 0.933511i \(-0.616728\pi\)
−0.358548 + 0.933511i \(0.616728\pi\)
\(258\) −11.4894 −0.715296
\(259\) −5.34592 −0.332179
\(260\) −2.20422 −0.136700
\(261\) 6.49249 0.401875
\(262\) 8.09149 0.499894
\(263\) 4.04284 0.249292 0.124646 0.992201i \(-0.460220\pi\)
0.124646 + 0.992201i \(0.460220\pi\)
\(264\) −6.84792 −0.421460
\(265\) 21.9954 1.35117
\(266\) 0 0
\(267\) 3.15237 0.192922
\(268\) −6.41158 −0.391650
\(269\) −15.2503 −0.929826 −0.464913 0.885356i \(-0.653914\pi\)
−0.464913 + 0.885356i \(0.653914\pi\)
\(270\) −11.6834 −0.711031
\(271\) −17.9249 −1.08886 −0.544429 0.838807i \(-0.683254\pi\)
−0.544429 + 0.838807i \(0.683254\pi\)
\(272\) −4.69713 −0.284805
\(273\) −1.29641 −0.0784620
\(274\) −4.55482 −0.275167
\(275\) 25.5874 1.54298
\(276\) 9.56748 0.575895
\(277\) −3.58168 −0.215202 −0.107601 0.994194i \(-0.534317\pi\)
−0.107601 + 0.994194i \(0.534317\pi\)
\(278\) 21.6945 1.30115
\(279\) −9.46740 −0.566798
\(280\) −3.59028 −0.214560
\(281\) −7.09159 −0.423049 −0.211524 0.977373i \(-0.567843\pi\)
−0.211524 + 0.977373i \(0.567843\pi\)
\(282\) −24.4981 −1.45884
\(283\) −14.0809 −0.837021 −0.418511 0.908212i \(-0.637448\pi\)
−0.418511 + 0.908212i \(0.637448\pi\)
\(284\) 11.5793 0.687103
\(285\) 0 0
\(286\) −1.99100 −0.117730
\(287\) 1.08664 0.0641424
\(288\) −1.45891 −0.0859671
\(289\) 5.06300 0.297824
\(290\) 15.9776 0.938236
\(291\) 3.54299 0.207694
\(292\) −16.3204 −0.955077
\(293\) 22.8960 1.33760 0.668798 0.743444i \(-0.266809\pi\)
0.668798 + 0.743444i \(0.266809\pi\)
\(294\) −2.11161 −0.123152
\(295\) 10.3823 0.604479
\(296\) −5.34592 −0.310725
\(297\) −10.5533 −0.612363
\(298\) −21.4072 −1.24009
\(299\) 2.78170 0.160870
\(300\) 16.6608 0.961912
\(301\) −5.44103 −0.313616
\(302\) −5.52022 −0.317653
\(303\) 3.94564 0.226671
\(304\) 0 0
\(305\) 47.4881 2.71916
\(306\) 6.85268 0.391742
\(307\) −32.8057 −1.87232 −0.936161 0.351572i \(-0.885647\pi\)
−0.936161 + 0.351572i \(0.885647\pi\)
\(308\) −3.24298 −0.184786
\(309\) −42.4989 −2.41768
\(310\) −23.2986 −1.32327
\(311\) 13.6967 0.776671 0.388336 0.921518i \(-0.373050\pi\)
0.388336 + 0.921518i \(0.373050\pi\)
\(312\) −1.29641 −0.0733945
\(313\) 7.88847 0.445883 0.222941 0.974832i \(-0.428434\pi\)
0.222941 + 0.974832i \(0.428434\pi\)
\(314\) 13.5838 0.766577
\(315\) 5.23789 0.295122
\(316\) −6.43324 −0.361898
\(317\) 28.0164 1.57356 0.786780 0.617234i \(-0.211747\pi\)
0.786780 + 0.617234i \(0.211747\pi\)
\(318\) 12.9365 0.725446
\(319\) 14.4320 0.808039
\(320\) −3.59028 −0.200703
\(321\) 19.7946 1.10482
\(322\) 4.53089 0.252497
\(323\) 0 0
\(324\) −11.2483 −0.624906
\(325\) 4.84405 0.268699
\(326\) 6.32223 0.350156
\(327\) −19.5975 −1.08374
\(328\) 1.08664 0.0599998
\(329\) −11.6016 −0.639617
\(330\) 24.5859 1.35341
\(331\) −13.6501 −0.750278 −0.375139 0.926969i \(-0.622405\pi\)
−0.375139 + 0.926969i \(0.622405\pi\)
\(332\) −8.52076 −0.467637
\(333\) 7.79921 0.427394
\(334\) 14.1989 0.776931
\(335\) 23.0194 1.25768
\(336\) −2.11161 −0.115198
\(337\) −17.2621 −0.940324 −0.470162 0.882580i \(-0.655805\pi\)
−0.470162 + 0.882580i \(0.655805\pi\)
\(338\) 12.6231 0.686605
\(339\) −18.3138 −0.994668
\(340\) 16.8640 0.914578
\(341\) −21.0449 −1.13964
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −5.44103 −0.293361
\(345\) −34.3499 −1.84934
\(346\) −7.38609 −0.397078
\(347\) −2.69423 −0.144634 −0.0723170 0.997382i \(-0.523039\pi\)
−0.0723170 + 0.997382i \(0.523039\pi\)
\(348\) 9.39718 0.503742
\(349\) −5.06542 −0.271145 −0.135573 0.990767i \(-0.543287\pi\)
−0.135573 + 0.990767i \(0.543287\pi\)
\(350\) 7.89009 0.421743
\(351\) −1.99788 −0.106639
\(352\) −3.24298 −0.172851
\(353\) −21.7080 −1.15540 −0.577701 0.816249i \(-0.696050\pi\)
−0.577701 + 0.816249i \(0.696050\pi\)
\(354\) 6.10631 0.324547
\(355\) −41.5728 −2.20645
\(356\) 1.49287 0.0791221
\(357\) 9.91851 0.524943
\(358\) 6.37886 0.337133
\(359\) 5.02083 0.264989 0.132495 0.991184i \(-0.457701\pi\)
0.132495 + 0.991184i \(0.457701\pi\)
\(360\) 5.23789 0.276061
\(361\) 0 0
\(362\) −22.2731 −1.17065
\(363\) −1.02007 −0.0535400
\(364\) −0.613941 −0.0321792
\(365\) 58.5946 3.06698
\(366\) 27.9300 1.45993
\(367\) −7.51570 −0.392316 −0.196158 0.980572i \(-0.562847\pi\)
−0.196158 + 0.980572i \(0.562847\pi\)
\(368\) 4.53089 0.236189
\(369\) −1.58531 −0.0825280
\(370\) 19.1933 0.997814
\(371\) 6.12638 0.318066
\(372\) −13.7030 −0.710469
\(373\) 14.0820 0.729137 0.364569 0.931177i \(-0.381216\pi\)
0.364569 + 0.931177i \(0.381216\pi\)
\(374\) 15.2327 0.787664
\(375\) −21.9105 −1.13146
\(376\) −11.6016 −0.598307
\(377\) 2.73218 0.140715
\(378\) −3.25419 −0.167377
\(379\) 6.46372 0.332019 0.166010 0.986124i \(-0.446912\pi\)
0.166010 + 0.986124i \(0.446912\pi\)
\(380\) 0 0
\(381\) 38.5776 1.97639
\(382\) 23.4129 1.19791
\(383\) −18.5181 −0.946229 −0.473115 0.881001i \(-0.656870\pi\)
−0.473115 + 0.881001i \(0.656870\pi\)
\(384\) −2.11161 −0.107758
\(385\) 11.6432 0.593392
\(386\) 15.1577 0.771506
\(387\) 7.93797 0.403510
\(388\) 1.67786 0.0851804
\(389\) −37.2070 −1.88647 −0.943234 0.332128i \(-0.892233\pi\)
−0.943234 + 0.332128i \(0.892233\pi\)
\(390\) 4.65445 0.235687
\(391\) −21.2822 −1.07629
\(392\) −1.00000 −0.0505076
\(393\) −17.0861 −0.861880
\(394\) 14.5729 0.734172
\(395\) 23.0971 1.16214
\(396\) 4.73121 0.237752
\(397\) 34.3856 1.72576 0.862881 0.505407i \(-0.168658\pi\)
0.862881 + 0.505407i \(0.168658\pi\)
\(398\) −7.05899 −0.353835
\(399\) 0 0
\(400\) 7.89009 0.394504
\(401\) 24.7614 1.23653 0.618263 0.785971i \(-0.287837\pi\)
0.618263 + 0.785971i \(0.287837\pi\)
\(402\) 13.5388 0.675253
\(403\) −3.98409 −0.198462
\(404\) 1.86854 0.0929636
\(405\) 40.3846 2.00672
\(406\) 4.45024 0.220862
\(407\) 17.3367 0.859349
\(408\) 9.91851 0.491040
\(409\) 8.46451 0.418543 0.209271 0.977858i \(-0.432891\pi\)
0.209271 + 0.977858i \(0.432891\pi\)
\(410\) −3.90134 −0.192674
\(411\) 9.61802 0.474422
\(412\) −20.1263 −0.991550
\(413\) 2.89177 0.142295
\(414\) −6.61016 −0.324871
\(415\) 30.5919 1.50170
\(416\) −0.613941 −0.0301009
\(417\) −45.8103 −2.24334
\(418\) 0 0
\(419\) −10.9702 −0.535929 −0.267964 0.963429i \(-0.586351\pi\)
−0.267964 + 0.963429i \(0.586351\pi\)
\(420\) 7.58127 0.369928
\(421\) 32.1234 1.56560 0.782800 0.622273i \(-0.213791\pi\)
0.782800 + 0.622273i \(0.213791\pi\)
\(422\) 5.69557 0.277256
\(423\) 16.9257 0.822955
\(424\) 6.12638 0.297523
\(425\) −37.0607 −1.79771
\(426\) −24.4509 −1.18465
\(427\) 13.2269 0.640093
\(428\) 9.37414 0.453116
\(429\) 4.20422 0.202982
\(430\) 19.5348 0.942052
\(431\) −7.26547 −0.349966 −0.174983 0.984571i \(-0.555987\pi\)
−0.174983 + 0.984571i \(0.555987\pi\)
\(432\) −3.25419 −0.156567
\(433\) 18.2307 0.876111 0.438056 0.898948i \(-0.355667\pi\)
0.438056 + 0.898948i \(0.355667\pi\)
\(434\) −6.48937 −0.311500
\(435\) −33.7385 −1.61764
\(436\) −9.28080 −0.444470
\(437\) 0 0
\(438\) 34.4623 1.64667
\(439\) −32.5406 −1.55308 −0.776539 0.630069i \(-0.783027\pi\)
−0.776539 + 0.630069i \(0.783027\pi\)
\(440\) 11.6432 0.555068
\(441\) 1.45891 0.0694719
\(442\) 2.88376 0.137166
\(443\) −0.0189181 −0.000898825 0 −0.000449412 1.00000i \(-0.500143\pi\)
−0.000449412 1.00000i \(0.500143\pi\)
\(444\) 11.2885 0.535729
\(445\) −5.35982 −0.254080
\(446\) −18.5836 −0.879961
\(447\) 45.2037 2.13806
\(448\) −1.00000 −0.0472456
\(449\) 4.85684 0.229209 0.114604 0.993411i \(-0.463440\pi\)
0.114604 + 0.993411i \(0.463440\pi\)
\(450\) −11.5109 −0.542630
\(451\) −3.52396 −0.165937
\(452\) −8.67289 −0.407938
\(453\) 11.6566 0.547673
\(454\) −0.527433 −0.0247537
\(455\) 2.20422 0.103335
\(456\) 0 0
\(457\) 30.2842 1.41663 0.708317 0.705895i \(-0.249455\pi\)
0.708317 + 0.705895i \(0.249455\pi\)
\(458\) 11.9310 0.557501
\(459\) 15.2853 0.713458
\(460\) −16.2671 −0.758460
\(461\) 20.6428 0.961432 0.480716 0.876876i \(-0.340377\pi\)
0.480716 + 0.876876i \(0.340377\pi\)
\(462\) 6.84792 0.318594
\(463\) 21.7134 1.00911 0.504554 0.863380i \(-0.331657\pi\)
0.504554 + 0.863380i \(0.331657\pi\)
\(464\) 4.45024 0.206597
\(465\) 49.1977 2.28149
\(466\) 6.00187 0.278031
\(467\) 18.9511 0.876952 0.438476 0.898743i \(-0.355518\pi\)
0.438476 + 0.898743i \(0.355518\pi\)
\(468\) 0.895684 0.0414030
\(469\) 6.41158 0.296059
\(470\) 41.6529 1.92131
\(471\) −28.6837 −1.32167
\(472\) 2.89177 0.133105
\(473\) 17.6452 0.811325
\(474\) 13.5845 0.623958
\(475\) 0 0
\(476\) 4.69713 0.215292
\(477\) −8.93784 −0.409235
\(478\) 22.2435 1.01740
\(479\) 20.3473 0.929690 0.464845 0.885392i \(-0.346110\pi\)
0.464845 + 0.885392i \(0.346110\pi\)
\(480\) 7.58127 0.346036
\(481\) 3.28208 0.149650
\(482\) −19.8067 −0.902171
\(483\) −9.56748 −0.435336
\(484\) −0.483078 −0.0219581
\(485\) −6.02398 −0.273535
\(486\) 13.9895 0.634577
\(487\) 0.582139 0.0263793 0.0131896 0.999913i \(-0.495801\pi\)
0.0131896 + 0.999913i \(0.495801\pi\)
\(488\) 13.2269 0.598752
\(489\) −13.3501 −0.603713
\(490\) 3.59028 0.162192
\(491\) −33.5615 −1.51461 −0.757304 0.653063i \(-0.773484\pi\)
−0.757304 + 0.653063i \(0.773484\pi\)
\(492\) −2.29457 −0.103447
\(493\) −20.9033 −0.941439
\(494\) 0 0
\(495\) −16.9864 −0.763481
\(496\) −6.48937 −0.291381
\(497\) −11.5793 −0.519401
\(498\) 17.9925 0.806265
\(499\) −7.71505 −0.345373 −0.172687 0.984977i \(-0.555245\pi\)
−0.172687 + 0.984977i \(0.555245\pi\)
\(500\) −10.3762 −0.464038
\(501\) −29.9826 −1.33953
\(502\) −6.27968 −0.280276
\(503\) 17.1560 0.764949 0.382475 0.923966i \(-0.375072\pi\)
0.382475 + 0.923966i \(0.375072\pi\)
\(504\) 1.45891 0.0649850
\(505\) −6.70859 −0.298528
\(506\) −14.6936 −0.653209
\(507\) −26.6551 −1.18379
\(508\) 18.2693 0.810567
\(509\) −25.9509 −1.15025 −0.575126 0.818065i \(-0.695047\pi\)
−0.575126 + 0.818065i \(0.695047\pi\)
\(510\) −35.6102 −1.57685
\(511\) 16.3204 0.721971
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 11.4959 0.507064
\(515\) 72.2588 3.18411
\(516\) 11.4894 0.505791
\(517\) 37.6238 1.65469
\(518\) 5.34592 0.234886
\(519\) 15.5966 0.684613
\(520\) 2.20422 0.0966613
\(521\) −19.9715 −0.874969 −0.437485 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(522\) −6.49249 −0.284169
\(523\) 8.44182 0.369135 0.184568 0.982820i \(-0.440912\pi\)
0.184568 + 0.982820i \(0.440912\pi\)
\(524\) −8.09149 −0.353478
\(525\) −16.6608 −0.727137
\(526\) −4.04284 −0.176276
\(527\) 30.4814 1.32779
\(528\) 6.84792 0.298017
\(529\) −2.47105 −0.107437
\(530\) −21.9954 −0.955419
\(531\) −4.21884 −0.183082
\(532\) 0 0
\(533\) −0.667134 −0.0288968
\(534\) −3.15237 −0.136416
\(535\) −33.6558 −1.45507
\(536\) 6.41158 0.276938
\(537\) −13.4697 −0.581260
\(538\) 15.2503 0.657486
\(539\) 3.24298 0.139685
\(540\) 11.6834 0.502775
\(541\) −39.3246 −1.69070 −0.845349 0.534214i \(-0.820608\pi\)
−0.845349 + 0.534214i \(0.820608\pi\)
\(542\) 17.9249 0.769939
\(543\) 47.0322 2.01834
\(544\) 4.69713 0.201388
\(545\) 33.3206 1.42730
\(546\) 1.29641 0.0554810
\(547\) −20.7275 −0.886245 −0.443122 0.896461i \(-0.646129\pi\)
−0.443122 + 0.896461i \(0.646129\pi\)
\(548\) 4.55482 0.194572
\(549\) −19.2968 −0.823568
\(550\) −25.5874 −1.09105
\(551\) 0 0
\(552\) −9.56748 −0.407219
\(553\) 6.43324 0.273569
\(554\) 3.58168 0.152171
\(555\) −40.5289 −1.72036
\(556\) −21.6945 −0.920051
\(557\) −30.6400 −1.29826 −0.649130 0.760677i \(-0.724867\pi\)
−0.649130 + 0.760677i \(0.724867\pi\)
\(558\) 9.46740 0.400787
\(559\) 3.34047 0.141287
\(560\) 3.59028 0.151717
\(561\) −32.1655 −1.35803
\(562\) 7.09159 0.299141
\(563\) −19.3305 −0.814682 −0.407341 0.913276i \(-0.633544\pi\)
−0.407341 + 0.913276i \(0.633544\pi\)
\(564\) 24.4981 1.03156
\(565\) 31.1381 1.30999
\(566\) 14.0809 0.591863
\(567\) 11.2483 0.472385
\(568\) −11.5793 −0.485855
\(569\) −18.0853 −0.758174 −0.379087 0.925361i \(-0.623762\pi\)
−0.379087 + 0.925361i \(0.623762\pi\)
\(570\) 0 0
\(571\) 1.49157 0.0624203 0.0312102 0.999513i \(-0.490064\pi\)
0.0312102 + 0.999513i \(0.490064\pi\)
\(572\) 1.99100 0.0832478
\(573\) −49.4390 −2.06534
\(574\) −1.08664 −0.0453555
\(575\) 35.7491 1.49084
\(576\) 1.45891 0.0607879
\(577\) 26.2740 1.09380 0.546900 0.837198i \(-0.315808\pi\)
0.546900 + 0.837198i \(0.315808\pi\)
\(578\) −5.06300 −0.210593
\(579\) −32.0072 −1.33017
\(580\) −15.9776 −0.663433
\(581\) 8.52076 0.353500
\(582\) −3.54299 −0.146862
\(583\) −19.8677 −0.822838
\(584\) 16.3204 0.675342
\(585\) −3.21575 −0.132955
\(586\) −22.8960 −0.945823
\(587\) 14.2560 0.588408 0.294204 0.955743i \(-0.404946\pi\)
0.294204 + 0.955743i \(0.404946\pi\)
\(588\) 2.11161 0.0870815
\(589\) 0 0
\(590\) −10.3823 −0.427431
\(591\) −30.7723 −1.26580
\(592\) 5.34592 0.219716
\(593\) 26.0203 1.06852 0.534262 0.845319i \(-0.320589\pi\)
0.534262 + 0.845319i \(0.320589\pi\)
\(594\) 10.5533 0.433006
\(595\) −16.8640 −0.691356
\(596\) 21.4072 0.876874
\(597\) 14.9058 0.610056
\(598\) −2.78170 −0.113752
\(599\) −26.0530 −1.06450 −0.532248 0.846589i \(-0.678653\pi\)
−0.532248 + 0.846589i \(0.678653\pi\)
\(600\) −16.6608 −0.680175
\(601\) 26.5798 1.08421 0.542106 0.840310i \(-0.317627\pi\)
0.542106 + 0.840310i \(0.317627\pi\)
\(602\) 5.44103 0.221760
\(603\) −9.35392 −0.380921
\(604\) 5.52022 0.224615
\(605\) 1.73438 0.0705127
\(606\) −3.94564 −0.160281
\(607\) 10.3567 0.420367 0.210184 0.977662i \(-0.432594\pi\)
0.210184 + 0.977662i \(0.432594\pi\)
\(608\) 0 0
\(609\) −9.39718 −0.380793
\(610\) −47.4881 −1.92274
\(611\) 7.12270 0.288153
\(612\) −6.85268 −0.277003
\(613\) −13.1694 −0.531908 −0.265954 0.963986i \(-0.585687\pi\)
−0.265954 + 0.963986i \(0.585687\pi\)
\(614\) 32.8057 1.32393
\(615\) 8.23813 0.332193
\(616\) 3.24298 0.130663
\(617\) 6.28250 0.252924 0.126462 0.991971i \(-0.459638\pi\)
0.126462 + 0.991971i \(0.459638\pi\)
\(618\) 42.4989 1.70956
\(619\) −6.86393 −0.275885 −0.137942 0.990440i \(-0.544049\pi\)
−0.137942 + 0.990440i \(0.544049\pi\)
\(620\) 23.2986 0.935695
\(621\) −14.7444 −0.591671
\(622\) −13.6967 −0.549190
\(623\) −1.49287 −0.0598107
\(624\) 1.29641 0.0518978
\(625\) −2.19696 −0.0878784
\(626\) −7.88847 −0.315287
\(627\) 0 0
\(628\) −13.5838 −0.542052
\(629\) −25.1105 −1.00122
\(630\) −5.23789 −0.208682
\(631\) −34.6486 −1.37934 −0.689669 0.724125i \(-0.742244\pi\)
−0.689669 + 0.724125i \(0.742244\pi\)
\(632\) 6.43324 0.255901
\(633\) −12.0268 −0.478024
\(634\) −28.0164 −1.11267
\(635\) −65.5917 −2.60293
\(636\) −12.9365 −0.512968
\(637\) 0.613941 0.0243252
\(638\) −14.4320 −0.571370
\(639\) 16.8931 0.668280
\(640\) 3.59028 0.141918
\(641\) −2.33296 −0.0921463 −0.0460732 0.998938i \(-0.514671\pi\)
−0.0460732 + 0.998938i \(0.514671\pi\)
\(642\) −19.7946 −0.781229
\(643\) −42.1217 −1.66112 −0.830559 0.556931i \(-0.811979\pi\)
−0.830559 + 0.556931i \(0.811979\pi\)
\(644\) −4.53089 −0.178542
\(645\) −41.2500 −1.62422
\(646\) 0 0
\(647\) 18.7499 0.737133 0.368566 0.929601i \(-0.379849\pi\)
0.368566 + 0.929601i \(0.379849\pi\)
\(648\) 11.2483 0.441875
\(649\) −9.37797 −0.368117
\(650\) −4.84405 −0.189999
\(651\) 13.7030 0.537064
\(652\) −6.32223 −0.247598
\(653\) 12.8937 0.504569 0.252285 0.967653i \(-0.418818\pi\)
0.252285 + 0.967653i \(0.418818\pi\)
\(654\) 19.5975 0.766321
\(655\) 29.0507 1.13510
\(656\) −1.08664 −0.0424262
\(657\) −23.8099 −0.928914
\(658\) 11.6016 0.452277
\(659\) 18.6966 0.728318 0.364159 0.931337i \(-0.381357\pi\)
0.364159 + 0.931337i \(0.381357\pi\)
\(660\) −24.5859 −0.957006
\(661\) 34.7384 1.35117 0.675584 0.737283i \(-0.263892\pi\)
0.675584 + 0.737283i \(0.263892\pi\)
\(662\) 13.6501 0.530526
\(663\) −6.08938 −0.236492
\(664\) 8.52076 0.330669
\(665\) 0 0
\(666\) −7.79921 −0.302213
\(667\) 20.1635 0.780735
\(668\) −14.1989 −0.549373
\(669\) 39.2414 1.51716
\(670\) −23.0194 −0.889315
\(671\) −42.8945 −1.65592
\(672\) 2.11161 0.0814572
\(673\) −32.2228 −1.24210 −0.621049 0.783771i \(-0.713293\pi\)
−0.621049 + 0.783771i \(0.713293\pi\)
\(674\) 17.2621 0.664910
\(675\) −25.6758 −0.988263
\(676\) −12.6231 −0.485503
\(677\) −25.6438 −0.985570 −0.492785 0.870151i \(-0.664021\pi\)
−0.492785 + 0.870151i \(0.664021\pi\)
\(678\) 18.3138 0.703337
\(679\) −1.67786 −0.0643903
\(680\) −16.8640 −0.646704
\(681\) 1.11373 0.0426784
\(682\) 21.0449 0.805850
\(683\) −3.96087 −0.151558 −0.0757792 0.997125i \(-0.524144\pi\)
−0.0757792 + 0.997125i \(0.524144\pi\)
\(684\) 0 0
\(685\) −16.3531 −0.624819
\(686\) 1.00000 0.0381802
\(687\) −25.1938 −0.961202
\(688\) 5.44103 0.207437
\(689\) −3.76124 −0.143292
\(690\) 34.3499 1.30768
\(691\) 16.9238 0.643811 0.321905 0.946772i \(-0.395677\pi\)
0.321905 + 0.946772i \(0.395677\pi\)
\(692\) 7.38609 0.280777
\(693\) −4.73121 −0.179724
\(694\) 2.69423 0.102272
\(695\) 77.8892 2.95450
\(696\) −9.39718 −0.356199
\(697\) 5.10409 0.193331
\(698\) 5.06542 0.191729
\(699\) −12.6736 −0.479360
\(700\) −7.89009 −0.298217
\(701\) 16.8434 0.636166 0.318083 0.948063i \(-0.396961\pi\)
0.318083 + 0.948063i \(0.396961\pi\)
\(702\) 1.99788 0.0754051
\(703\) 0 0
\(704\) 3.24298 0.122224
\(705\) −87.9549 −3.31257
\(706\) 21.7080 0.816992
\(707\) −1.86854 −0.0702739
\(708\) −6.10631 −0.229489
\(709\) −28.7267 −1.07885 −0.539426 0.842033i \(-0.681359\pi\)
−0.539426 + 0.842033i \(0.681359\pi\)
\(710\) 41.5728 1.56020
\(711\) −9.38552 −0.351984
\(712\) −1.49287 −0.0559477
\(713\) −29.4026 −1.10114
\(714\) −9.91851 −0.371191
\(715\) −7.14823 −0.267329
\(716\) −6.37886 −0.238389
\(717\) −46.9698 −1.75412
\(718\) −5.02083 −0.187376
\(719\) −32.2712 −1.20351 −0.601757 0.798679i \(-0.705532\pi\)
−0.601757 + 0.798679i \(0.705532\pi\)
\(720\) −5.23789 −0.195205
\(721\) 20.1263 0.749541
\(722\) 0 0
\(723\) 41.8241 1.55546
\(724\) 22.2731 0.827774
\(725\) 35.1128 1.30406
\(726\) 1.02007 0.0378585
\(727\) 6.48738 0.240604 0.120302 0.992737i \(-0.461614\pi\)
0.120302 + 0.992737i \(0.461614\pi\)
\(728\) 0.613941 0.0227542
\(729\) 4.20448 0.155722
\(730\) −58.5946 −2.16868
\(731\) −25.5572 −0.945268
\(732\) −27.9300 −1.03232
\(733\) −15.0737 −0.556758 −0.278379 0.960471i \(-0.589797\pi\)
−0.278379 + 0.960471i \(0.589797\pi\)
\(734\) 7.51570 0.277410
\(735\) −7.58127 −0.279640
\(736\) −4.53089 −0.167011
\(737\) −20.7926 −0.765906
\(738\) 1.58531 0.0583561
\(739\) −21.8502 −0.803772 −0.401886 0.915690i \(-0.631645\pi\)
−0.401886 + 0.915690i \(0.631645\pi\)
\(740\) −19.1933 −0.705561
\(741\) 0 0
\(742\) −6.12638 −0.224907
\(743\) 41.7004 1.52984 0.764919 0.644127i \(-0.222779\pi\)
0.764919 + 0.644127i \(0.222779\pi\)
\(744\) 13.7030 0.502377
\(745\) −76.8578 −2.81585
\(746\) −14.0820 −0.515578
\(747\) −12.4310 −0.454827
\(748\) −15.2327 −0.556962
\(749\) −9.37414 −0.342524
\(750\) 21.9105 0.800060
\(751\) −36.4048 −1.32843 −0.664215 0.747542i \(-0.731234\pi\)
−0.664215 + 0.747542i \(0.731234\pi\)
\(752\) 11.6016 0.423067
\(753\) 13.2603 0.483230
\(754\) −2.73218 −0.0995002
\(755\) −19.8191 −0.721292
\(756\) 3.25419 0.118354
\(757\) 4.98234 0.181086 0.0905431 0.995893i \(-0.471140\pi\)
0.0905431 + 0.995893i \(0.471140\pi\)
\(758\) −6.46372 −0.234773
\(759\) 31.0272 1.12621
\(760\) 0 0
\(761\) 16.1769 0.586412 0.293206 0.956049i \(-0.405278\pi\)
0.293206 + 0.956049i \(0.405278\pi\)
\(762\) −38.5776 −1.39752
\(763\) 9.28080 0.335988
\(764\) −23.4129 −0.847048
\(765\) 24.6030 0.889524
\(766\) 18.5181 0.669085
\(767\) −1.77538 −0.0641052
\(768\) 2.11161 0.0761963
\(769\) 29.6375 1.06876 0.534378 0.845246i \(-0.320546\pi\)
0.534378 + 0.845246i \(0.320546\pi\)
\(770\) −11.6432 −0.419592
\(771\) −24.2749 −0.874241
\(772\) −15.1577 −0.545537
\(773\) 35.2879 1.26922 0.634608 0.772834i \(-0.281161\pi\)
0.634608 + 0.772834i \(0.281161\pi\)
\(774\) −7.93797 −0.285325
\(775\) −51.2017 −1.83922
\(776\) −1.67786 −0.0602316
\(777\) −11.2885 −0.404973
\(778\) 37.2070 1.33393
\(779\) 0 0
\(780\) −4.65445 −0.166656
\(781\) 37.5513 1.34369
\(782\) 21.2822 0.761048
\(783\) −14.4819 −0.517541
\(784\) 1.00000 0.0357143
\(785\) 48.7695 1.74066
\(786\) 17.0861 0.609441
\(787\) 48.1868 1.71767 0.858836 0.512250i \(-0.171188\pi\)
0.858836 + 0.512250i \(0.171188\pi\)
\(788\) −14.5729 −0.519138
\(789\) 8.53691 0.303922
\(790\) −23.0971 −0.821759
\(791\) 8.67289 0.308372
\(792\) −4.73121 −0.168116
\(793\) −8.12052 −0.288368
\(794\) −34.3856 −1.22030
\(795\) 46.4458 1.64726
\(796\) 7.05899 0.250199
\(797\) 10.0767 0.356934 0.178467 0.983946i \(-0.442886\pi\)
0.178467 + 0.983946i \(0.442886\pi\)
\(798\) 0 0
\(799\) −54.4942 −1.92787
\(800\) −7.89009 −0.278957
\(801\) 2.17797 0.0769546
\(802\) −24.7614 −0.874356
\(803\) −52.9266 −1.86774
\(804\) −13.5388 −0.477476
\(805\) 16.2671 0.573342
\(806\) 3.98409 0.140333
\(807\) −32.2027 −1.13359
\(808\) −1.86854 −0.0657352
\(809\) 37.6152 1.32248 0.661240 0.750174i \(-0.270030\pi\)
0.661240 + 0.750174i \(0.270030\pi\)
\(810\) −40.3846 −1.41897
\(811\) −1.02897 −0.0361321 −0.0180661 0.999837i \(-0.505751\pi\)
−0.0180661 + 0.999837i \(0.505751\pi\)
\(812\) −4.45024 −0.156173
\(813\) −37.8504 −1.32747
\(814\) −17.3367 −0.607652
\(815\) 22.6986 0.795096
\(816\) −9.91851 −0.347217
\(817\) 0 0
\(818\) −8.46451 −0.295954
\(819\) −0.895684 −0.0312977
\(820\) 3.90134 0.136241
\(821\) −29.6164 −1.03362 −0.516809 0.856101i \(-0.672880\pi\)
−0.516809 + 0.856101i \(0.672880\pi\)
\(822\) −9.61802 −0.335467
\(823\) −29.6541 −1.03368 −0.516839 0.856083i \(-0.672891\pi\)
−0.516839 + 0.856083i \(0.672891\pi\)
\(824\) 20.1263 0.701132
\(825\) 54.0307 1.88111
\(826\) −2.89177 −0.100618
\(827\) 5.98622 0.208161 0.104081 0.994569i \(-0.466810\pi\)
0.104081 + 0.994569i \(0.466810\pi\)
\(828\) 6.61016 0.229719
\(829\) 3.97321 0.137995 0.0689976 0.997617i \(-0.478020\pi\)
0.0689976 + 0.997617i \(0.478020\pi\)
\(830\) −30.5919 −1.06186
\(831\) −7.56312 −0.262362
\(832\) 0.613941 0.0212846
\(833\) −4.69713 −0.162746
\(834\) 45.8103 1.58628
\(835\) 50.9781 1.76417
\(836\) 0 0
\(837\) 21.1176 0.729931
\(838\) 10.9702 0.378959
\(839\) 44.1324 1.52362 0.761810 0.647801i \(-0.224311\pi\)
0.761810 + 0.647801i \(0.224311\pi\)
\(840\) −7.58127 −0.261579
\(841\) −9.19539 −0.317082
\(842\) −32.1234 −1.10705
\(843\) −14.9747 −0.515756
\(844\) −5.69557 −0.196050
\(845\) 45.3203 1.55907
\(846\) −16.9257 −0.581917
\(847\) 0.483078 0.0165988
\(848\) −6.12638 −0.210381
\(849\) −29.7334 −1.02045
\(850\) 37.0607 1.27117
\(851\) 24.2218 0.830312
\(852\) 24.4509 0.837675
\(853\) 1.76629 0.0604767 0.0302384 0.999543i \(-0.490373\pi\)
0.0302384 + 0.999543i \(0.490373\pi\)
\(854\) −13.2269 −0.452614
\(855\) 0 0
\(856\) −9.37414 −0.320402
\(857\) −42.8844 −1.46490 −0.732451 0.680819i \(-0.761624\pi\)
−0.732451 + 0.680819i \(0.761624\pi\)
\(858\) −4.20422 −0.143530
\(859\) 3.86858 0.131994 0.0659971 0.997820i \(-0.478977\pi\)
0.0659971 + 0.997820i \(0.478977\pi\)
\(860\) −19.5348 −0.666132
\(861\) 2.29457 0.0781986
\(862\) 7.26547 0.247463
\(863\) 32.7259 1.11400 0.557001 0.830512i \(-0.311952\pi\)
0.557001 + 0.830512i \(0.311952\pi\)
\(864\) 3.25419 0.110710
\(865\) −26.5181 −0.901642
\(866\) −18.2307 −0.619504
\(867\) 10.6911 0.363089
\(868\) 6.48937 0.220263
\(869\) −20.8629 −0.707725
\(870\) 33.7385 1.14384
\(871\) −3.93633 −0.133378
\(872\) 9.28080 0.314288
\(873\) 2.44784 0.0828470
\(874\) 0 0
\(875\) 10.3762 0.350780
\(876\) −34.4623 −1.16437
\(877\) −11.6549 −0.393559 −0.196779 0.980448i \(-0.563048\pi\)
−0.196779 + 0.980448i \(0.563048\pi\)
\(878\) 32.5406 1.09819
\(879\) 48.3474 1.63072
\(880\) −11.6432 −0.392492
\(881\) 52.9550 1.78410 0.892050 0.451937i \(-0.149267\pi\)
0.892050 + 0.451937i \(0.149267\pi\)
\(882\) −1.45891 −0.0491240
\(883\) −0.838696 −0.0282244 −0.0141122 0.999900i \(-0.504492\pi\)
−0.0141122 + 0.999900i \(0.504492\pi\)
\(884\) −2.88376 −0.0969913
\(885\) 21.9233 0.736945
\(886\) 0.0189181 0.000635565 0
\(887\) 48.3051 1.62193 0.810963 0.585097i \(-0.198944\pi\)
0.810963 + 0.585097i \(0.198944\pi\)
\(888\) −11.2885 −0.378818
\(889\) −18.2693 −0.612731
\(890\) 5.35982 0.179662
\(891\) −36.4781 −1.22206
\(892\) 18.5836 0.622226
\(893\) 0 0
\(894\) −45.2037 −1.51184
\(895\) 22.9019 0.765525
\(896\) 1.00000 0.0334077
\(897\) 5.87387 0.196123
\(898\) −4.85684 −0.162075
\(899\) −28.8792 −0.963176
\(900\) 11.5109 0.383697
\(901\) 28.7764 0.958681
\(902\) 3.52396 0.117335
\(903\) −11.4894 −0.382342
\(904\) 8.67289 0.288456
\(905\) −79.9667 −2.65818
\(906\) −11.6566 −0.387264
\(907\) −39.8456 −1.32305 −0.661526 0.749922i \(-0.730091\pi\)
−0.661526 + 0.749922i \(0.730091\pi\)
\(908\) 0.527433 0.0175035
\(909\) 2.72604 0.0904170
\(910\) −2.20422 −0.0730691
\(911\) 33.9742 1.12562 0.562808 0.826587i \(-0.309721\pi\)
0.562808 + 0.826587i \(0.309721\pi\)
\(912\) 0 0
\(913\) −27.6326 −0.914507
\(914\) −30.2842 −1.00171
\(915\) 100.277 3.31504
\(916\) −11.9310 −0.394213
\(917\) 8.09149 0.267205
\(918\) −15.2853 −0.504491
\(919\) 23.7930 0.784858 0.392429 0.919782i \(-0.371635\pi\)
0.392429 + 0.919782i \(0.371635\pi\)
\(920\) 16.2671 0.536312
\(921\) −69.2730 −2.28262
\(922\) −20.6428 −0.679835
\(923\) 7.10898 0.233995
\(924\) −6.84792 −0.225280
\(925\) 42.1798 1.38686
\(926\) −21.7134 −0.713547
\(927\) −29.3624 −0.964387
\(928\) −4.45024 −0.146086
\(929\) −6.90591 −0.226575 −0.113288 0.993562i \(-0.536138\pi\)
−0.113288 + 0.993562i \(0.536138\pi\)
\(930\) −49.1977 −1.61326
\(931\) 0 0
\(932\) −6.00187 −0.196598
\(933\) 28.9222 0.946871
\(934\) −18.9511 −0.620099
\(935\) 54.6896 1.78854
\(936\) −0.895684 −0.0292763
\(937\) 0.663778 0.0216847 0.0108423 0.999941i \(-0.496549\pi\)
0.0108423 + 0.999941i \(0.496549\pi\)
\(938\) −6.41158 −0.209346
\(939\) 16.6574 0.543593
\(940\) −41.6529 −1.35857
\(941\) 1.97654 0.0644334 0.0322167 0.999481i \(-0.489743\pi\)
0.0322167 + 0.999481i \(0.489743\pi\)
\(942\) 28.6837 0.934564
\(943\) −4.92345 −0.160330
\(944\) −2.89177 −0.0941192
\(945\) −11.6834 −0.380062
\(946\) −17.6452 −0.573694
\(947\) 42.1283 1.36899 0.684493 0.729020i \(-0.260024\pi\)
0.684493 + 0.729020i \(0.260024\pi\)
\(948\) −13.5845 −0.441205
\(949\) −10.0197 −0.325255
\(950\) 0 0
\(951\) 59.1599 1.91839
\(952\) −4.69713 −0.152235
\(953\) 50.5277 1.63675 0.818376 0.574683i \(-0.194875\pi\)
0.818376 + 0.574683i \(0.194875\pi\)
\(954\) 8.93784 0.289373
\(955\) 84.0587 2.72008
\(956\) −22.2435 −0.719408
\(957\) 30.4749 0.985112
\(958\) −20.3473 −0.657390
\(959\) −4.55482 −0.147083
\(960\) −7.58127 −0.244685
\(961\) 11.1119 0.358447
\(962\) −3.28208 −0.105818
\(963\) 13.6760 0.440704
\(964\) 19.8067 0.637932
\(965\) 54.4203 1.75185
\(966\) 9.56748 0.307829
\(967\) −38.8196 −1.24835 −0.624176 0.781284i \(-0.714565\pi\)
−0.624176 + 0.781284i \(0.714565\pi\)
\(968\) 0.483078 0.0155267
\(969\) 0 0
\(970\) 6.02398 0.193418
\(971\) 1.81105 0.0581194 0.0290597 0.999578i \(-0.490749\pi\)
0.0290597 + 0.999578i \(0.490749\pi\)
\(972\) −13.9895 −0.448714
\(973\) 21.6945 0.695493
\(974\) −0.582139 −0.0186530
\(975\) 10.2288 0.327582
\(976\) −13.2269 −0.423382
\(977\) −19.1940 −0.614069 −0.307035 0.951698i \(-0.599337\pi\)
−0.307035 + 0.951698i \(0.599337\pi\)
\(978\) 13.3501 0.426889
\(979\) 4.84136 0.154730
\(980\) −3.59028 −0.114687
\(981\) −13.5398 −0.432294
\(982\) 33.5615 1.07099
\(983\) 4.19927 0.133936 0.0669680 0.997755i \(-0.478667\pi\)
0.0669680 + 0.997755i \(0.478667\pi\)
\(984\) 2.29457 0.0731481
\(985\) 52.3207 1.66708
\(986\) 20.9033 0.665698
\(987\) −24.4981 −0.779783
\(988\) 0 0
\(989\) 24.6527 0.783911
\(990\) 16.9864 0.539862
\(991\) 9.13292 0.290117 0.145058 0.989423i \(-0.453663\pi\)
0.145058 + 0.989423i \(0.453663\pi\)
\(992\) 6.48937 0.206038
\(993\) −28.8237 −0.914694
\(994\) 11.5793 0.367272
\(995\) −25.3437 −0.803450
\(996\) −17.9925 −0.570115
\(997\) −23.2862 −0.737482 −0.368741 0.929532i \(-0.620211\pi\)
−0.368741 + 0.929532i \(0.620211\pi\)
\(998\) 7.71505 0.244216
\(999\) −17.3966 −0.550405
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 5054.2.a.bb.1.5 6
19.9 even 9 266.2.u.b.43.2 12
19.17 even 9 266.2.u.b.99.2 yes 12
19.18 odd 2 5054.2.a.bc.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.u.b.43.2 12 19.9 even 9
266.2.u.b.99.2 yes 12 19.17 even 9
5054.2.a.bb.1.5 6 1.1 even 1 trivial
5054.2.a.bc.1.2 6 19.18 odd 2