Properties

Label 1014.2.a.o.1.1
Level $1014$
Weight $2$
Character 1014.1
Self dual yes
Analytic conductor $8.097$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
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 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 1014.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.04892 q^{5} +1.00000 q^{6} -0.692021 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.04892 q^{5} +1.00000 q^{6} -0.692021 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.04892 q^{10} +4.85086 q^{11} +1.00000 q^{12} -0.692021 q^{14} -4.04892 q^{15} +1.00000 q^{16} +7.38404 q^{17} +1.00000 q^{18} +1.78017 q^{19} -4.04892 q^{20} -0.692021 q^{21} +4.85086 q^{22} +5.10992 q^{23} +1.00000 q^{24} +11.3937 q^{25} +1.00000 q^{27} -0.692021 q^{28} -3.34481 q^{29} -4.04892 q^{30} -0.972853 q^{31} +1.00000 q^{32} +4.85086 q^{33} +7.38404 q^{34} +2.80194 q^{35} +1.00000 q^{36} -1.28621 q^{37} +1.78017 q^{38} -4.04892 q^{40} -1.50604 q^{41} -0.692021 q^{42} -8.31767 q^{43} +4.85086 q^{44} -4.04892 q^{45} +5.10992 q^{46} +7.20775 q^{47} +1.00000 q^{48} -6.52111 q^{49} +11.3937 q^{50} +7.38404 q^{51} +13.4765 q^{53} +1.00000 q^{54} -19.6407 q^{55} -0.692021 q^{56} +1.78017 q^{57} -3.34481 q^{58} -1.30798 q^{59} -4.04892 q^{60} -0.396125 q^{61} -0.972853 q^{62} -0.692021 q^{63} +1.00000 q^{64} +4.85086 q^{66} -6.05429 q^{67} +7.38404 q^{68} +5.10992 q^{69} +2.80194 q^{70} +1.32975 q^{71} +1.00000 q^{72} +7.65279 q^{73} -1.28621 q^{74} +11.3937 q^{75} +1.78017 q^{76} -3.35690 q^{77} -8.33944 q^{79} -4.04892 q^{80} +1.00000 q^{81} -1.50604 q^{82} -15.3274 q^{83} -0.692021 q^{84} -29.8974 q^{85} -8.31767 q^{86} -3.34481 q^{87} +4.85086 q^{88} -3.10992 q^{89} -4.04892 q^{90} +5.10992 q^{92} -0.972853 q^{93} +7.20775 q^{94} -7.20775 q^{95} +1.00000 q^{96} +8.54288 q^{97} -6.52111 q^{98} +4.85086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + q^{11} + 3 q^{12} + 3 q^{14} - 3 q^{15} + 3 q^{16} + 12 q^{17} + 3 q^{18} + 4 q^{19} - 3 q^{20} + 3 q^{21} + q^{22} + 16 q^{23} + 3 q^{24} + 2 q^{25} + 3 q^{27} + 3 q^{28} + 13 q^{29} - 3 q^{30} - 9 q^{31} + 3 q^{32} + q^{33} + 12 q^{34} + 4 q^{35} + 3 q^{36} - 12 q^{37} + 4 q^{38} - 3 q^{40} - 14 q^{41} + 3 q^{42} - 8 q^{43} + q^{44} - 3 q^{45} + 16 q^{46} + 4 q^{47} + 3 q^{48} - 4 q^{49} + 2 q^{50} + 12 q^{51} + 15 q^{53} + 3 q^{54} - 22 q^{55} + 3 q^{56} + 4 q^{57} + 13 q^{58} - 9 q^{59} - 3 q^{60} - 10 q^{61} - 9 q^{62} + 3 q^{63} + 3 q^{64} + q^{66} - 6 q^{67} + 12 q^{68} + 16 q^{69} + 4 q^{70} + 6 q^{71} + 3 q^{72} + 5 q^{73} - 12 q^{74} + 2 q^{75} + 4 q^{76} - 6 q^{77} - 5 q^{79} - 3 q^{80} + 3 q^{81} - 14 q^{82} - 7 q^{83} + 3 q^{84} - 26 q^{85} - 8 q^{86} + 13 q^{87} + q^{88} - 10 q^{89} - 3 q^{90} + 16 q^{92} - 9 q^{93} + 4 q^{94} - 4 q^{95} + 3 q^{96} + 7 q^{97} - 4 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −4.04892 −1.81073 −0.905365 0.424633i \(-0.860403\pi\)
−0.905365 + 0.424633i \(0.860403\pi\)
\(6\) 1.00000 0.408248
\(7\) −0.692021 −0.261560 −0.130780 0.991411i \(-0.541748\pi\)
−0.130780 + 0.991411i \(0.541748\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −4.04892 −1.28038
\(11\) 4.85086 1.46259 0.731294 0.682062i \(-0.238917\pi\)
0.731294 + 0.682062i \(0.238917\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −0.692021 −0.184951
\(15\) −4.04892 −1.04543
\(16\) 1.00000 0.250000
\(17\) 7.38404 1.79089 0.895447 0.445169i \(-0.146856\pi\)
0.895447 + 0.445169i \(0.146856\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.78017 0.408398 0.204199 0.978929i \(-0.434541\pi\)
0.204199 + 0.978929i \(0.434541\pi\)
\(20\) −4.04892 −0.905365
\(21\) −0.692021 −0.151011
\(22\) 4.85086 1.03421
\(23\) 5.10992 1.06549 0.532746 0.846275i \(-0.321160\pi\)
0.532746 + 0.846275i \(0.321160\pi\)
\(24\) 1.00000 0.204124
\(25\) 11.3937 2.27875
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −0.692021 −0.130780
\(29\) −3.34481 −0.621116 −0.310558 0.950554i \(-0.600516\pi\)
−0.310558 + 0.950554i \(0.600516\pi\)
\(30\) −4.04892 −0.739228
\(31\) −0.972853 −0.174730 −0.0873648 0.996176i \(-0.527845\pi\)
−0.0873648 + 0.996176i \(0.527845\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.85086 0.844425
\(34\) 7.38404 1.26635
\(35\) 2.80194 0.473614
\(36\) 1.00000 0.166667
\(37\) −1.28621 −0.211451 −0.105726 0.994395i \(-0.533717\pi\)
−0.105726 + 0.994395i \(0.533717\pi\)
\(38\) 1.78017 0.288781
\(39\) 0 0
\(40\) −4.04892 −0.640190
\(41\) −1.50604 −0.235204 −0.117602 0.993061i \(-0.537521\pi\)
−0.117602 + 0.993061i \(0.537521\pi\)
\(42\) −0.692021 −0.106781
\(43\) −8.31767 −1.26843 −0.634216 0.773156i \(-0.718677\pi\)
−0.634216 + 0.773156i \(0.718677\pi\)
\(44\) 4.85086 0.731294
\(45\) −4.04892 −0.603577
\(46\) 5.10992 0.753416
\(47\) 7.20775 1.05136 0.525679 0.850683i \(-0.323811\pi\)
0.525679 + 0.850683i \(0.323811\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.52111 −0.931587
\(50\) 11.3937 1.61132
\(51\) 7.38404 1.03397
\(52\) 0 0
\(53\) 13.4765 1.85114 0.925570 0.378577i \(-0.123586\pi\)
0.925570 + 0.378577i \(0.123586\pi\)
\(54\) 1.00000 0.136083
\(55\) −19.6407 −2.64835
\(56\) −0.692021 −0.0924753
\(57\) 1.78017 0.235789
\(58\) −3.34481 −0.439196
\(59\) −1.30798 −0.170284 −0.0851422 0.996369i \(-0.527134\pi\)
−0.0851422 + 0.996369i \(0.527134\pi\)
\(60\) −4.04892 −0.522713
\(61\) −0.396125 −0.0507185 −0.0253593 0.999678i \(-0.508073\pi\)
−0.0253593 + 0.999678i \(0.508073\pi\)
\(62\) −0.972853 −0.123552
\(63\) −0.692021 −0.0871865
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.85086 0.597099
\(67\) −6.05429 −0.739650 −0.369825 0.929101i \(-0.620582\pi\)
−0.369825 + 0.929101i \(0.620582\pi\)
\(68\) 7.38404 0.895447
\(69\) 5.10992 0.615162
\(70\) 2.80194 0.334896
\(71\) 1.32975 0.157812 0.0789061 0.996882i \(-0.474857\pi\)
0.0789061 + 0.996882i \(0.474857\pi\)
\(72\) 1.00000 0.117851
\(73\) 7.65279 0.895692 0.447846 0.894111i \(-0.352191\pi\)
0.447846 + 0.894111i \(0.352191\pi\)
\(74\) −1.28621 −0.149519
\(75\) 11.3937 1.31563
\(76\) 1.78017 0.204199
\(77\) −3.35690 −0.382554
\(78\) 0 0
\(79\) −8.33944 −0.938260 −0.469130 0.883129i \(-0.655432\pi\)
−0.469130 + 0.883129i \(0.655432\pi\)
\(80\) −4.04892 −0.452683
\(81\) 1.00000 0.111111
\(82\) −1.50604 −0.166314
\(83\) −15.3274 −1.68240 −0.841198 0.540727i \(-0.818149\pi\)
−0.841198 + 0.540727i \(0.818149\pi\)
\(84\) −0.692021 −0.0755057
\(85\) −29.8974 −3.24283
\(86\) −8.31767 −0.896917
\(87\) −3.34481 −0.358602
\(88\) 4.85086 0.517103
\(89\) −3.10992 −0.329650 −0.164825 0.986323i \(-0.552706\pi\)
−0.164825 + 0.986323i \(0.552706\pi\)
\(90\) −4.04892 −0.426793
\(91\) 0 0
\(92\) 5.10992 0.532746
\(93\) −0.972853 −0.100880
\(94\) 7.20775 0.743423
\(95\) −7.20775 −0.739500
\(96\) 1.00000 0.102062
\(97\) 8.54288 0.867398 0.433699 0.901058i \(-0.357208\pi\)
0.433699 + 0.901058i \(0.357208\pi\)
\(98\) −6.52111 −0.658731
\(99\) 4.85086 0.487529
\(100\) 11.3937 1.13937
\(101\) −11.9976 −1.19381 −0.596903 0.802313i \(-0.703602\pi\)
−0.596903 + 0.802313i \(0.703602\pi\)
\(102\) 7.38404 0.731129
\(103\) −12.3230 −1.21423 −0.607113 0.794616i \(-0.707672\pi\)
−0.607113 + 0.794616i \(0.707672\pi\)
\(104\) 0 0
\(105\) 2.80194 0.273441
\(106\) 13.4765 1.30895
\(107\) 5.89977 0.570353 0.285176 0.958475i \(-0.407948\pi\)
0.285176 + 0.958475i \(0.407948\pi\)
\(108\) 1.00000 0.0962250
\(109\) −0.792249 −0.0758837 −0.0379418 0.999280i \(-0.512080\pi\)
−0.0379418 + 0.999280i \(0.512080\pi\)
\(110\) −19.6407 −1.87267
\(111\) −1.28621 −0.122081
\(112\) −0.692021 −0.0653899
\(113\) 6.21983 0.585113 0.292556 0.956248i \(-0.405494\pi\)
0.292556 + 0.956248i \(0.405494\pi\)
\(114\) 1.78017 0.166728
\(115\) −20.6896 −1.92932
\(116\) −3.34481 −0.310558
\(117\) 0 0
\(118\) −1.30798 −0.120409
\(119\) −5.10992 −0.468425
\(120\) −4.04892 −0.369614
\(121\) 12.5308 1.13916
\(122\) −0.396125 −0.0358634
\(123\) −1.50604 −0.135795
\(124\) −0.972853 −0.0873648
\(125\) −25.8877 −2.31547
\(126\) −0.692021 −0.0616502
\(127\) −6.00538 −0.532891 −0.266446 0.963850i \(-0.585849\pi\)
−0.266446 + 0.963850i \(0.585849\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.31767 −0.732330
\(130\) 0 0
\(131\) 8.81700 0.770345 0.385173 0.922845i \(-0.374142\pi\)
0.385173 + 0.922845i \(0.374142\pi\)
\(132\) 4.85086 0.422213
\(133\) −1.23191 −0.106821
\(134\) −6.05429 −0.523011
\(135\) −4.04892 −0.348475
\(136\) 7.38404 0.633176
\(137\) 15.7560 1.34613 0.673063 0.739585i \(-0.264978\pi\)
0.673063 + 0.739585i \(0.264978\pi\)
\(138\) 5.10992 0.434985
\(139\) −6.09783 −0.517212 −0.258606 0.965983i \(-0.583263\pi\)
−0.258606 + 0.965983i \(0.583263\pi\)
\(140\) 2.80194 0.236807
\(141\) 7.20775 0.607002
\(142\) 1.32975 0.111590
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 13.5429 1.12467
\(146\) 7.65279 0.633350
\(147\) −6.52111 −0.537852
\(148\) −1.28621 −0.105726
\(149\) 2.55257 0.209114 0.104557 0.994519i \(-0.466657\pi\)
0.104557 + 0.994519i \(0.466657\pi\)
\(150\) 11.3937 0.930294
\(151\) 17.7168 1.44177 0.720885 0.693054i \(-0.243735\pi\)
0.720885 + 0.693054i \(0.243735\pi\)
\(152\) 1.78017 0.144391
\(153\) 7.38404 0.596964
\(154\) −3.35690 −0.270506
\(155\) 3.93900 0.316388
\(156\) 0 0
\(157\) −6.31767 −0.504205 −0.252102 0.967701i \(-0.581122\pi\)
−0.252102 + 0.967701i \(0.581122\pi\)
\(158\) −8.33944 −0.663450
\(159\) 13.4765 1.06876
\(160\) −4.04892 −0.320095
\(161\) −3.53617 −0.278689
\(162\) 1.00000 0.0785674
\(163\) 14.5918 1.14292 0.571459 0.820631i \(-0.306378\pi\)
0.571459 + 0.820631i \(0.306378\pi\)
\(164\) −1.50604 −0.117602
\(165\) −19.6407 −1.52903
\(166\) −15.3274 −1.18963
\(167\) −19.5013 −1.50905 −0.754526 0.656270i \(-0.772133\pi\)
−0.754526 + 0.656270i \(0.772133\pi\)
\(168\) −0.692021 −0.0533906
\(169\) 0 0
\(170\) −29.8974 −2.29302
\(171\) 1.78017 0.136133
\(172\) −8.31767 −0.634216
\(173\) −9.29052 −0.706345 −0.353173 0.935558i \(-0.614897\pi\)
−0.353173 + 0.935558i \(0.614897\pi\)
\(174\) −3.34481 −0.253570
\(175\) −7.88471 −0.596028
\(176\) 4.85086 0.365647
\(177\) −1.30798 −0.0983137
\(178\) −3.10992 −0.233098
\(179\) 22.7928 1.70362 0.851808 0.523853i \(-0.175506\pi\)
0.851808 + 0.523853i \(0.175506\pi\)
\(180\) −4.04892 −0.301788
\(181\) 0.537500 0.0399520 0.0199760 0.999800i \(-0.493641\pi\)
0.0199760 + 0.999800i \(0.493641\pi\)
\(182\) 0 0
\(183\) −0.396125 −0.0292824
\(184\) 5.10992 0.376708
\(185\) 5.20775 0.382881
\(186\) −0.972853 −0.0713330
\(187\) 35.8189 2.61934
\(188\) 7.20775 0.525679
\(189\) −0.692021 −0.0503372
\(190\) −7.20775 −0.522905
\(191\) −9.79954 −0.709070 −0.354535 0.935043i \(-0.615361\pi\)
−0.354535 + 0.935043i \(0.615361\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.1957 −1.02183 −0.510913 0.859632i \(-0.670693\pi\)
−0.510913 + 0.859632i \(0.670693\pi\)
\(194\) 8.54288 0.613343
\(195\) 0 0
\(196\) −6.52111 −0.465793
\(197\) −3.00969 −0.214431 −0.107216 0.994236i \(-0.534194\pi\)
−0.107216 + 0.994236i \(0.534194\pi\)
\(198\) 4.85086 0.344735
\(199\) 12.8944 0.914059 0.457030 0.889451i \(-0.348913\pi\)
0.457030 + 0.889451i \(0.348913\pi\)
\(200\) 11.3937 0.805658
\(201\) −6.05429 −0.427037
\(202\) −11.9976 −0.844149
\(203\) 2.31468 0.162459
\(204\) 7.38404 0.516986
\(205\) 6.09783 0.425891
\(206\) −12.3230 −0.858587
\(207\) 5.10992 0.355164
\(208\) 0 0
\(209\) 8.63533 0.597319
\(210\) 2.80194 0.193352
\(211\) 7.79954 0.536943 0.268471 0.963288i \(-0.413482\pi\)
0.268471 + 0.963288i \(0.413482\pi\)
\(212\) 13.4765 0.925570
\(213\) 1.32975 0.0911129
\(214\) 5.89977 0.403300
\(215\) 33.6775 2.29679
\(216\) 1.00000 0.0680414
\(217\) 0.673235 0.0457022
\(218\) −0.792249 −0.0536579
\(219\) 7.65279 0.517128
\(220\) −19.6407 −1.32418
\(221\) 0 0
\(222\) −1.28621 −0.0863246
\(223\) −12.1957 −0.816682 −0.408341 0.912829i \(-0.633893\pi\)
−0.408341 + 0.912829i \(0.633893\pi\)
\(224\) −0.692021 −0.0462376
\(225\) 11.3937 0.759582
\(226\) 6.21983 0.413737
\(227\) 6.74333 0.447571 0.223785 0.974638i \(-0.428159\pi\)
0.223785 + 0.974638i \(0.428159\pi\)
\(228\) 1.78017 0.117894
\(229\) −19.8237 −1.30999 −0.654994 0.755634i \(-0.727329\pi\)
−0.654994 + 0.755634i \(0.727329\pi\)
\(230\) −20.6896 −1.36423
\(231\) −3.35690 −0.220868
\(232\) −3.34481 −0.219598
\(233\) −30.0301 −1.96734 −0.983670 0.179983i \(-0.942396\pi\)
−0.983670 + 0.179983i \(0.942396\pi\)
\(234\) 0 0
\(235\) −29.1836 −1.90373
\(236\) −1.30798 −0.0851422
\(237\) −8.33944 −0.541705
\(238\) −5.10992 −0.331227
\(239\) 22.0978 1.42939 0.714695 0.699436i \(-0.246565\pi\)
0.714695 + 0.699436i \(0.246565\pi\)
\(240\) −4.04892 −0.261356
\(241\) 10.1274 0.652362 0.326181 0.945307i \(-0.394238\pi\)
0.326181 + 0.945307i \(0.394238\pi\)
\(242\) 12.5308 0.805510
\(243\) 1.00000 0.0641500
\(244\) −0.396125 −0.0253593
\(245\) 26.4034 1.68685
\(246\) −1.50604 −0.0960217
\(247\) 0 0
\(248\) −0.972853 −0.0617762
\(249\) −15.3274 −0.971332
\(250\) −25.8877 −1.63728
\(251\) 5.54719 0.350135 0.175068 0.984556i \(-0.443986\pi\)
0.175068 + 0.984556i \(0.443986\pi\)
\(252\) −0.692021 −0.0435933
\(253\) 24.7875 1.55837
\(254\) −6.00538 −0.376811
\(255\) −29.8974 −1.87225
\(256\) 1.00000 0.0625000
\(257\) −13.7995 −0.860792 −0.430396 0.902640i \(-0.641626\pi\)
−0.430396 + 0.902640i \(0.641626\pi\)
\(258\) −8.31767 −0.517835
\(259\) 0.890084 0.0553071
\(260\) 0 0
\(261\) −3.34481 −0.207039
\(262\) 8.81700 0.544716
\(263\) −22.4698 −1.38555 −0.692773 0.721155i \(-0.743611\pi\)
−0.692773 + 0.721155i \(0.743611\pi\)
\(264\) 4.85086 0.298549
\(265\) −54.5652 −3.35192
\(266\) −1.23191 −0.0755335
\(267\) −3.10992 −0.190324
\(268\) −6.05429 −0.369825
\(269\) −26.0140 −1.58610 −0.793051 0.609156i \(-0.791509\pi\)
−0.793051 + 0.609156i \(0.791509\pi\)
\(270\) −4.04892 −0.246409
\(271\) 2.88471 0.175233 0.0876167 0.996154i \(-0.472075\pi\)
0.0876167 + 0.996154i \(0.472075\pi\)
\(272\) 7.38404 0.447723
\(273\) 0 0
\(274\) 15.7560 0.951855
\(275\) 55.2693 3.33287
\(276\) 5.10992 0.307581
\(277\) −1.46250 −0.0878731 −0.0439366 0.999034i \(-0.513990\pi\)
−0.0439366 + 0.999034i \(0.513990\pi\)
\(278\) −6.09783 −0.365724
\(279\) −0.972853 −0.0582432
\(280\) 2.80194 0.167448
\(281\) 5.68233 0.338980 0.169490 0.985532i \(-0.445788\pi\)
0.169490 + 0.985532i \(0.445788\pi\)
\(282\) 7.20775 0.429215
\(283\) −25.2078 −1.49845 −0.749223 0.662318i \(-0.769573\pi\)
−0.749223 + 0.662318i \(0.769573\pi\)
\(284\) 1.32975 0.0789061
\(285\) −7.20775 −0.426950
\(286\) 0 0
\(287\) 1.04221 0.0615199
\(288\) 1.00000 0.0589256
\(289\) 37.5241 2.20730
\(290\) 13.5429 0.795265
\(291\) 8.54288 0.500792
\(292\) 7.65279 0.447846
\(293\) −7.14914 −0.417658 −0.208829 0.977952i \(-0.566965\pi\)
−0.208829 + 0.977952i \(0.566965\pi\)
\(294\) −6.52111 −0.380319
\(295\) 5.29590 0.308339
\(296\) −1.28621 −0.0747593
\(297\) 4.85086 0.281475
\(298\) 2.55257 0.147866
\(299\) 0 0
\(300\) 11.3937 0.657817
\(301\) 5.75600 0.331771
\(302\) 17.7168 1.01949
\(303\) −11.9976 −0.689245
\(304\) 1.78017 0.102100
\(305\) 1.60388 0.0918376
\(306\) 7.38404 0.422118
\(307\) −17.9952 −1.02704 −0.513521 0.858077i \(-0.671659\pi\)
−0.513521 + 0.858077i \(0.671659\pi\)
\(308\) −3.35690 −0.191277
\(309\) −12.3230 −0.701033
\(310\) 3.93900 0.223720
\(311\) 3.32975 0.188813 0.0944064 0.995534i \(-0.469905\pi\)
0.0944064 + 0.995534i \(0.469905\pi\)
\(312\) 0 0
\(313\) −17.8834 −1.01083 −0.505414 0.862877i \(-0.668660\pi\)
−0.505414 + 0.862877i \(0.668660\pi\)
\(314\) −6.31767 −0.356527
\(315\) 2.80194 0.157871
\(316\) −8.33944 −0.469130
\(317\) 4.39373 0.246777 0.123388 0.992358i \(-0.460624\pi\)
0.123388 + 0.992358i \(0.460624\pi\)
\(318\) 13.4765 0.755725
\(319\) −16.2252 −0.908437
\(320\) −4.04892 −0.226341
\(321\) 5.89977 0.329293
\(322\) −3.53617 −0.197063
\(323\) 13.1448 0.731398
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 14.5918 0.808165
\(327\) −0.792249 −0.0438115
\(328\) −1.50604 −0.0831572
\(329\) −4.98792 −0.274993
\(330\) −19.6407 −1.08119
\(331\) −25.6775 −1.41137 −0.705683 0.708528i \(-0.749360\pi\)
−0.705683 + 0.708528i \(0.749360\pi\)
\(332\) −15.3274 −0.841198
\(333\) −1.28621 −0.0704838
\(334\) −19.5013 −1.06706
\(335\) 24.5133 1.33931
\(336\) −0.692021 −0.0377529
\(337\) −24.6504 −1.34279 −0.671396 0.741098i \(-0.734305\pi\)
−0.671396 + 0.741098i \(0.734305\pi\)
\(338\) 0 0
\(339\) 6.21983 0.337815
\(340\) −29.8974 −1.62141
\(341\) −4.71917 −0.255557
\(342\) 1.78017 0.0962604
\(343\) 9.35690 0.505225
\(344\) −8.31767 −0.448459
\(345\) −20.6896 −1.11389
\(346\) −9.29052 −0.499461
\(347\) 14.2959 0.767444 0.383722 0.923449i \(-0.374642\pi\)
0.383722 + 0.923449i \(0.374642\pi\)
\(348\) −3.34481 −0.179301
\(349\) 11.0616 0.592113 0.296057 0.955170i \(-0.404328\pi\)
0.296057 + 0.955170i \(0.404328\pi\)
\(350\) −7.88471 −0.421455
\(351\) 0 0
\(352\) 4.85086 0.258551
\(353\) −10.5047 −0.559109 −0.279555 0.960130i \(-0.590187\pi\)
−0.279555 + 0.960130i \(0.590187\pi\)
\(354\) −1.30798 −0.0695183
\(355\) −5.38404 −0.285755
\(356\) −3.10992 −0.164825
\(357\) −5.10992 −0.270445
\(358\) 22.7928 1.20464
\(359\) 5.10992 0.269691 0.134846 0.990867i \(-0.456946\pi\)
0.134846 + 0.990867i \(0.456946\pi\)
\(360\) −4.04892 −0.213397
\(361\) −15.8310 −0.833211
\(362\) 0.537500 0.0282504
\(363\) 12.5308 0.657696
\(364\) 0 0
\(365\) −30.9855 −1.62186
\(366\) −0.396125 −0.0207058
\(367\) −8.44803 −0.440983 −0.220492 0.975389i \(-0.570766\pi\)
−0.220492 + 0.975389i \(0.570766\pi\)
\(368\) 5.10992 0.266373
\(369\) −1.50604 −0.0784014
\(370\) 5.20775 0.270738
\(371\) −9.32603 −0.484183
\(372\) −0.972853 −0.0504401
\(373\) 7.69096 0.398223 0.199111 0.979977i \(-0.436194\pi\)
0.199111 + 0.979977i \(0.436194\pi\)
\(374\) 35.8189 1.85215
\(375\) −25.8877 −1.33683
\(376\) 7.20775 0.371711
\(377\) 0 0
\(378\) −0.692021 −0.0355937
\(379\) −11.4034 −0.585754 −0.292877 0.956150i \(-0.594613\pi\)
−0.292877 + 0.956150i \(0.594613\pi\)
\(380\) −7.20775 −0.369750
\(381\) −6.00538 −0.307665
\(382\) −9.79954 −0.501388
\(383\) 20.6703 1.05620 0.528100 0.849182i \(-0.322905\pi\)
0.528100 + 0.849182i \(0.322905\pi\)
\(384\) 1.00000 0.0510310
\(385\) 13.5918 0.692702
\(386\) −14.1957 −0.722541
\(387\) −8.31767 −0.422811
\(388\) 8.54288 0.433699
\(389\) 17.4776 0.886148 0.443074 0.896485i \(-0.353888\pi\)
0.443074 + 0.896485i \(0.353888\pi\)
\(390\) 0 0
\(391\) 37.7318 1.90818
\(392\) −6.52111 −0.329366
\(393\) 8.81700 0.444759
\(394\) −3.00969 −0.151626
\(395\) 33.7657 1.69894
\(396\) 4.85086 0.243765
\(397\) −19.3599 −0.971645 −0.485822 0.874058i \(-0.661480\pi\)
−0.485822 + 0.874058i \(0.661480\pi\)
\(398\) 12.8944 0.646338
\(399\) −1.23191 −0.0616728
\(400\) 11.3937 0.569687
\(401\) −14.4832 −0.723257 −0.361628 0.932322i \(-0.617779\pi\)
−0.361628 + 0.932322i \(0.617779\pi\)
\(402\) −6.05429 −0.301961
\(403\) 0 0
\(404\) −11.9976 −0.596903
\(405\) −4.04892 −0.201192
\(406\) 2.31468 0.114876
\(407\) −6.23921 −0.309266
\(408\) 7.38404 0.365565
\(409\) −18.8984 −0.934468 −0.467234 0.884134i \(-0.654749\pi\)
−0.467234 + 0.884134i \(0.654749\pi\)
\(410\) 6.09783 0.301151
\(411\) 15.7560 0.777186
\(412\) −12.3230 −0.607113
\(413\) 0.905149 0.0445395
\(414\) 5.10992 0.251139
\(415\) 62.0592 3.04637
\(416\) 0 0
\(417\) −6.09783 −0.298612
\(418\) 8.63533 0.422368
\(419\) −21.7603 −1.06306 −0.531531 0.847039i \(-0.678383\pi\)
−0.531531 + 0.847039i \(0.678383\pi\)
\(420\) 2.80194 0.136721
\(421\) 20.5918 1.00358 0.501791 0.864989i \(-0.332675\pi\)
0.501791 + 0.864989i \(0.332675\pi\)
\(422\) 7.79954 0.379676
\(423\) 7.20775 0.350453
\(424\) 13.4765 0.654477
\(425\) 84.1318 4.08099
\(426\) 1.32975 0.0644265
\(427\) 0.274127 0.0132659
\(428\) 5.89977 0.285176
\(429\) 0 0
\(430\) 33.6775 1.62408
\(431\) 34.5133 1.66245 0.831224 0.555937i \(-0.187640\pi\)
0.831224 + 0.555937i \(0.187640\pi\)
\(432\) 1.00000 0.0481125
\(433\) −2.12631 −0.102184 −0.0510920 0.998694i \(-0.516270\pi\)
−0.0510920 + 0.998694i \(0.516270\pi\)
\(434\) 0.673235 0.0323163
\(435\) 13.5429 0.649331
\(436\) −0.792249 −0.0379418
\(437\) 9.09651 0.435145
\(438\) 7.65279 0.365665
\(439\) 21.8321 1.04199 0.520994 0.853560i \(-0.325561\pi\)
0.520994 + 0.853560i \(0.325561\pi\)
\(440\) −19.6407 −0.936334
\(441\) −6.52111 −0.310529
\(442\) 0 0
\(443\) −7.54048 −0.358259 −0.179130 0.983825i \(-0.557328\pi\)
−0.179130 + 0.983825i \(0.557328\pi\)
\(444\) −1.28621 −0.0610407
\(445\) 12.5918 0.596908
\(446\) −12.1957 −0.577482
\(447\) 2.55257 0.120732
\(448\) −0.692021 −0.0326949
\(449\) 19.7560 0.932343 0.466172 0.884694i \(-0.345633\pi\)
0.466172 + 0.884694i \(0.345633\pi\)
\(450\) 11.3937 0.537106
\(451\) −7.30559 −0.344007
\(452\) 6.21983 0.292556
\(453\) 17.7168 0.832407
\(454\) 6.74333 0.316480
\(455\) 0 0
\(456\) 1.78017 0.0833640
\(457\) 23.8582 1.11604 0.558019 0.829828i \(-0.311562\pi\)
0.558019 + 0.829828i \(0.311562\pi\)
\(458\) −19.8237 −0.926301
\(459\) 7.38404 0.344658
\(460\) −20.6896 −0.964659
\(461\) −17.5773 −0.818657 −0.409329 0.912387i \(-0.634237\pi\)
−0.409329 + 0.912387i \(0.634237\pi\)
\(462\) −3.35690 −0.156177
\(463\) 23.8431 1.10808 0.554041 0.832489i \(-0.313085\pi\)
0.554041 + 0.832489i \(0.313085\pi\)
\(464\) −3.34481 −0.155279
\(465\) 3.93900 0.182667
\(466\) −30.0301 −1.39112
\(467\) 8.61058 0.398450 0.199225 0.979954i \(-0.436158\pi\)
0.199225 + 0.979954i \(0.436158\pi\)
\(468\) 0 0
\(469\) 4.18970 0.193462
\(470\) −29.1836 −1.34614
\(471\) −6.31767 −0.291103
\(472\) −1.30798 −0.0602046
\(473\) −40.3478 −1.85519
\(474\) −8.33944 −0.383043
\(475\) 20.2828 0.930636
\(476\) −5.10992 −0.234213
\(477\) 13.4765 0.617047
\(478\) 22.0978 1.01073
\(479\) 6.58104 0.300695 0.150348 0.988633i \(-0.451961\pi\)
0.150348 + 0.988633i \(0.451961\pi\)
\(480\) −4.04892 −0.184807
\(481\) 0 0
\(482\) 10.1274 0.461289
\(483\) −3.53617 −0.160901
\(484\) 12.5308 0.569582
\(485\) −34.5894 −1.57062
\(486\) 1.00000 0.0453609
\(487\) −23.2760 −1.05474 −0.527369 0.849636i \(-0.676822\pi\)
−0.527369 + 0.849636i \(0.676822\pi\)
\(488\) −0.396125 −0.0179317
\(489\) 14.5918 0.659864
\(490\) 26.4034 1.19278
\(491\) −15.5405 −0.701332 −0.350666 0.936501i \(-0.614045\pi\)
−0.350666 + 0.936501i \(0.614045\pi\)
\(492\) −1.50604 −0.0678976
\(493\) −24.6983 −1.11235
\(494\) 0 0
\(495\) −19.6407 −0.882784
\(496\) −0.972853 −0.0436824
\(497\) −0.920215 −0.0412773
\(498\) −15.3274 −0.686835
\(499\) 9.53617 0.426898 0.213449 0.976954i \(-0.431530\pi\)
0.213449 + 0.976954i \(0.431530\pi\)
\(500\) −25.8877 −1.15773
\(501\) −19.5013 −0.871252
\(502\) 5.54719 0.247583
\(503\) −13.8345 −0.616848 −0.308424 0.951249i \(-0.599802\pi\)
−0.308424 + 0.951249i \(0.599802\pi\)
\(504\) −0.692021 −0.0308251
\(505\) 48.5773 2.16166
\(506\) 24.7875 1.10194
\(507\) 0 0
\(508\) −6.00538 −0.266446
\(509\) −40.9638 −1.81569 −0.907843 0.419310i \(-0.862272\pi\)
−0.907843 + 0.419310i \(0.862272\pi\)
\(510\) −29.8974 −1.32388
\(511\) −5.29590 −0.234277
\(512\) 1.00000 0.0441942
\(513\) 1.78017 0.0785963
\(514\) −13.7995 −0.608672
\(515\) 49.8950 2.19864
\(516\) −8.31767 −0.366165
\(517\) 34.9638 1.53770
\(518\) 0.890084 0.0391080
\(519\) −9.29052 −0.407809
\(520\) 0 0
\(521\) 36.3672 1.59327 0.796637 0.604457i \(-0.206610\pi\)
0.796637 + 0.604457i \(0.206610\pi\)
\(522\) −3.34481 −0.146399
\(523\) −6.03013 −0.263679 −0.131840 0.991271i \(-0.542088\pi\)
−0.131840 + 0.991271i \(0.542088\pi\)
\(524\) 8.81700 0.385173
\(525\) −7.88471 −0.344117
\(526\) −22.4698 −0.979730
\(527\) −7.18359 −0.312922
\(528\) 4.85086 0.211106
\(529\) 3.11124 0.135271
\(530\) −54.5652 −2.37016
\(531\) −1.30798 −0.0567614
\(532\) −1.23191 −0.0534103
\(533\) 0 0
\(534\) −3.10992 −0.134579
\(535\) −23.8877 −1.03275
\(536\) −6.05429 −0.261506
\(537\) 22.7928 0.983584
\(538\) −26.0140 −1.12154
\(539\) −31.6329 −1.36253
\(540\) −4.04892 −0.174238
\(541\) −7.92154 −0.340574 −0.170287 0.985395i \(-0.554469\pi\)
−0.170287 + 0.985395i \(0.554469\pi\)
\(542\) 2.88471 0.123909
\(543\) 0.537500 0.0230663
\(544\) 7.38404 0.316588
\(545\) 3.20775 0.137405
\(546\) 0 0
\(547\) 18.4155 0.787390 0.393695 0.919241i \(-0.371197\pi\)
0.393695 + 0.919241i \(0.371197\pi\)
\(548\) 15.7560 0.673063
\(549\) −0.396125 −0.0169062
\(550\) 55.2693 2.35669
\(551\) −5.95433 −0.253663
\(552\) 5.10992 0.217492
\(553\) 5.77107 0.245411
\(554\) −1.46250 −0.0621357
\(555\) 5.20775 0.221057
\(556\) −6.09783 −0.258606
\(557\) −23.9758 −1.01589 −0.507944 0.861390i \(-0.669594\pi\)
−0.507944 + 0.861390i \(0.669594\pi\)
\(558\) −0.972853 −0.0411841
\(559\) 0 0
\(560\) 2.80194 0.118403
\(561\) 35.8189 1.51228
\(562\) 5.68233 0.239695
\(563\) −2.29291 −0.0966348 −0.0483174 0.998832i \(-0.515386\pi\)
−0.0483174 + 0.998832i \(0.515386\pi\)
\(564\) 7.20775 0.303501
\(565\) −25.1836 −1.05948
\(566\) −25.2078 −1.05956
\(567\) −0.692021 −0.0290622
\(568\) 1.32975 0.0557950
\(569\) −44.3430 −1.85896 −0.929478 0.368878i \(-0.879742\pi\)
−0.929478 + 0.368878i \(0.879742\pi\)
\(570\) −7.20775 −0.301899
\(571\) −15.2707 −0.639058 −0.319529 0.947577i \(-0.603525\pi\)
−0.319529 + 0.947577i \(0.603525\pi\)
\(572\) 0 0
\(573\) −9.79954 −0.409382
\(574\) 1.04221 0.0435011
\(575\) 58.2210 2.42798
\(576\) 1.00000 0.0416667
\(577\) 8.77048 0.365120 0.182560 0.983195i \(-0.441562\pi\)
0.182560 + 0.983195i \(0.441562\pi\)
\(578\) 37.5241 1.56080
\(579\) −14.1957 −0.589952
\(580\) 13.5429 0.562337
\(581\) 10.6069 0.440047
\(582\) 8.54288 0.354114
\(583\) 65.3726 2.70745
\(584\) 7.65279 0.316675
\(585\) 0 0
\(586\) −7.14914 −0.295328
\(587\) −38.1430 −1.57433 −0.787166 0.616742i \(-0.788452\pi\)
−0.787166 + 0.616742i \(0.788452\pi\)
\(588\) −6.52111 −0.268926
\(589\) −1.73184 −0.0713593
\(590\) 5.29590 0.218029
\(591\) −3.00969 −0.123802
\(592\) −1.28621 −0.0528628
\(593\) −37.9517 −1.55849 −0.779244 0.626720i \(-0.784397\pi\)
−0.779244 + 0.626720i \(0.784397\pi\)
\(594\) 4.85086 0.199033
\(595\) 20.6896 0.848192
\(596\) 2.55257 0.104557
\(597\) 12.8944 0.527732
\(598\) 0 0
\(599\) −3.57971 −0.146263 −0.0731315 0.997322i \(-0.523299\pi\)
−0.0731315 + 0.997322i \(0.523299\pi\)
\(600\) 11.3937 0.465147
\(601\) −5.71678 −0.233192 −0.116596 0.993179i \(-0.537198\pi\)
−0.116596 + 0.993179i \(0.537198\pi\)
\(602\) 5.75600 0.234597
\(603\) −6.05429 −0.246550
\(604\) 17.7168 0.720885
\(605\) −50.7362 −2.06272
\(606\) −11.9976 −0.487369
\(607\) 22.4286 0.910351 0.455175 0.890402i \(-0.349577\pi\)
0.455175 + 0.890402i \(0.349577\pi\)
\(608\) 1.78017 0.0721953
\(609\) 2.31468 0.0937957
\(610\) 1.60388 0.0649390
\(611\) 0 0
\(612\) 7.38404 0.298482
\(613\) −39.9603 −1.61398 −0.806991 0.590564i \(-0.798905\pi\)
−0.806991 + 0.590564i \(0.798905\pi\)
\(614\) −17.9952 −0.726228
\(615\) 6.09783 0.245888
\(616\) −3.35690 −0.135253
\(617\) 31.4470 1.26601 0.633003 0.774149i \(-0.281822\pi\)
0.633003 + 0.774149i \(0.281822\pi\)
\(618\) −12.3230 −0.495706
\(619\) −29.3685 −1.18042 −0.590210 0.807250i \(-0.700955\pi\)
−0.590210 + 0.807250i \(0.700955\pi\)
\(620\) 3.93900 0.158194
\(621\) 5.10992 0.205054
\(622\) 3.32975 0.133511
\(623\) 2.15213 0.0862232
\(624\) 0 0
\(625\) 47.8485 1.91394
\(626\) −17.8834 −0.714764
\(627\) 8.63533 0.344862
\(628\) −6.31767 −0.252102
\(629\) −9.49742 −0.378687
\(630\) 2.80194 0.111632
\(631\) 21.6799 0.863065 0.431532 0.902097i \(-0.357973\pi\)
0.431532 + 0.902097i \(0.357973\pi\)
\(632\) −8.33944 −0.331725
\(633\) 7.79954 0.310004
\(634\) 4.39373 0.174497
\(635\) 24.3153 0.964922
\(636\) 13.4765 0.534378
\(637\) 0 0
\(638\) −16.2252 −0.642362
\(639\) 1.32975 0.0526040
\(640\) −4.04892 −0.160048
\(641\) 14.0108 0.553391 0.276696 0.960958i \(-0.410761\pi\)
0.276696 + 0.960958i \(0.410761\pi\)
\(642\) 5.89977 0.232845
\(643\) 30.7875 1.21414 0.607070 0.794649i \(-0.292345\pi\)
0.607070 + 0.794649i \(0.292345\pi\)
\(644\) −3.53617 −0.139345
\(645\) 33.6775 1.32605
\(646\) 13.1448 0.517177
\(647\) −16.6025 −0.652713 −0.326357 0.945247i \(-0.605821\pi\)
−0.326357 + 0.945247i \(0.605821\pi\)
\(648\) 1.00000 0.0392837
\(649\) −6.34481 −0.249056
\(650\) 0 0
\(651\) 0.673235 0.0263862
\(652\) 14.5918 0.571459
\(653\) 28.5459 1.11709 0.558543 0.829476i \(-0.311361\pi\)
0.558543 + 0.829476i \(0.311361\pi\)
\(654\) −0.792249 −0.0309794
\(655\) −35.6993 −1.39489
\(656\) −1.50604 −0.0588010
\(657\) 7.65279 0.298564
\(658\) −4.98792 −0.194449
\(659\) 27.7187 1.07977 0.539884 0.841740i \(-0.318468\pi\)
0.539884 + 0.841740i \(0.318468\pi\)
\(660\) −19.6407 −0.764514
\(661\) 10.8009 0.420105 0.210053 0.977690i \(-0.432636\pi\)
0.210053 + 0.977690i \(0.432636\pi\)
\(662\) −25.6775 −0.997986
\(663\) 0 0
\(664\) −15.3274 −0.594817
\(665\) 4.98792 0.193423
\(666\) −1.28621 −0.0498396
\(667\) −17.0917 −0.661794
\(668\) −19.5013 −0.754526
\(669\) −12.1957 −0.471512
\(670\) 24.5133 0.947033
\(671\) −1.92154 −0.0741803
\(672\) −0.692021 −0.0266953
\(673\) 16.3260 0.629322 0.314661 0.949204i \(-0.398109\pi\)
0.314661 + 0.949204i \(0.398109\pi\)
\(674\) −24.6504 −0.949498
\(675\) 11.3937 0.438545
\(676\) 0 0
\(677\) 41.4252 1.59210 0.796050 0.605231i \(-0.206919\pi\)
0.796050 + 0.605231i \(0.206919\pi\)
\(678\) 6.21983 0.238871
\(679\) −5.91185 −0.226876
\(680\) −29.8974 −1.14651
\(681\) 6.74333 0.258405
\(682\) −4.71917 −0.180706
\(683\) 31.2325 1.19508 0.597539 0.801840i \(-0.296145\pi\)
0.597539 + 0.801840i \(0.296145\pi\)
\(684\) 1.78017 0.0680664
\(685\) −63.7948 −2.43747
\(686\) 9.35690 0.357248
\(687\) −19.8237 −0.756322
\(688\) −8.31767 −0.317108
\(689\) 0 0
\(690\) −20.6896 −0.787641
\(691\) 24.9638 0.949666 0.474833 0.880076i \(-0.342508\pi\)
0.474833 + 0.880076i \(0.342508\pi\)
\(692\) −9.29052 −0.353173
\(693\) −3.35690 −0.127518
\(694\) 14.2959 0.542665
\(695\) 24.6896 0.936531
\(696\) −3.34481 −0.126785
\(697\) −11.1207 −0.421225
\(698\) 11.0616 0.418687
\(699\) −30.0301 −1.13584
\(700\) −7.88471 −0.298014
\(701\) 8.17151 0.308634 0.154317 0.988021i \(-0.450682\pi\)
0.154317 + 0.988021i \(0.450682\pi\)
\(702\) 0 0
\(703\) −2.28967 −0.0863564
\(704\) 4.85086 0.182823
\(705\) −29.1836 −1.09912
\(706\) −10.5047 −0.395350
\(707\) 8.30260 0.312251
\(708\) −1.30798 −0.0491568
\(709\) −36.7982 −1.38199 −0.690993 0.722861i \(-0.742826\pi\)
−0.690993 + 0.722861i \(0.742826\pi\)
\(710\) −5.38404 −0.202060
\(711\) −8.33944 −0.312753
\(712\) −3.10992 −0.116549
\(713\) −4.97120 −0.186173
\(714\) −5.10992 −0.191234
\(715\) 0 0
\(716\) 22.7928 0.851808
\(717\) 22.0978 0.825259
\(718\) 5.10992 0.190700
\(719\) 35.2223 1.31357 0.656786 0.754077i \(-0.271916\pi\)
0.656786 + 0.754077i \(0.271916\pi\)
\(720\) −4.04892 −0.150894
\(721\) 8.52781 0.317592
\(722\) −15.8310 −0.589169
\(723\) 10.1274 0.376641
\(724\) 0.537500 0.0199760
\(725\) −38.1099 −1.41537
\(726\) 12.5308 0.465061
\(727\) 40.6872 1.50901 0.754503 0.656297i \(-0.227878\pi\)
0.754503 + 0.656297i \(0.227878\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −30.9855 −1.14683
\(731\) −61.4180 −2.27163
\(732\) −0.396125 −0.0146412
\(733\) 27.1400 1.00244 0.501220 0.865320i \(-0.332885\pi\)
0.501220 + 0.865320i \(0.332885\pi\)
\(734\) −8.44803 −0.311822
\(735\) 26.4034 0.973905
\(736\) 5.10992 0.188354
\(737\) −29.3685 −1.08180
\(738\) −1.50604 −0.0554381
\(739\) 3.72587 0.137058 0.0685292 0.997649i \(-0.478169\pi\)
0.0685292 + 0.997649i \(0.478169\pi\)
\(740\) 5.20775 0.191441
\(741\) 0 0
\(742\) −9.32603 −0.342369
\(743\) 21.8586 0.801915 0.400958 0.916097i \(-0.368677\pi\)
0.400958 + 0.916097i \(0.368677\pi\)
\(744\) −0.972853 −0.0356665
\(745\) −10.3351 −0.378650
\(746\) 7.69096 0.281586
\(747\) −15.3274 −0.560799
\(748\) 35.8189 1.30967
\(749\) −4.08277 −0.149181
\(750\) −25.8877 −0.945285
\(751\) 53.3642 1.94729 0.973644 0.228075i \(-0.0732432\pi\)
0.973644 + 0.228075i \(0.0732432\pi\)
\(752\) 7.20775 0.262840
\(753\) 5.54719 0.202151
\(754\) 0 0
\(755\) −71.7338 −2.61066
\(756\) −0.692021 −0.0251686
\(757\) −4.63533 −0.168474 −0.0842370 0.996446i \(-0.526845\pi\)
−0.0842370 + 0.996446i \(0.526845\pi\)
\(758\) −11.4034 −0.414191
\(759\) 24.7875 0.899728
\(760\) −7.20775 −0.261453
\(761\) 11.5603 0.419062 0.209531 0.977802i \(-0.432806\pi\)
0.209531 + 0.977802i \(0.432806\pi\)
\(762\) −6.00538 −0.217552
\(763\) 0.548253 0.0198481
\(764\) −9.79954 −0.354535
\(765\) −29.8974 −1.08094
\(766\) 20.6703 0.746847
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 14.7439 0.531679 0.265840 0.964017i \(-0.414351\pi\)
0.265840 + 0.964017i \(0.414351\pi\)
\(770\) 13.5918 0.489814
\(771\) −13.7995 −0.496978
\(772\) −14.1957 −0.510913
\(773\) −6.42268 −0.231008 −0.115504 0.993307i \(-0.536848\pi\)
−0.115504 + 0.993307i \(0.536848\pi\)
\(774\) −8.31767 −0.298972
\(775\) −11.0844 −0.398164
\(776\) 8.54288 0.306671
\(777\) 0.890084 0.0319316
\(778\) 17.4776 0.626601
\(779\) −2.68100 −0.0960570
\(780\) 0 0
\(781\) 6.45042 0.230814
\(782\) 37.7318 1.34929
\(783\) −3.34481 −0.119534
\(784\) −6.52111 −0.232897
\(785\) 25.5797 0.912979
\(786\) 8.81700 0.314492
\(787\) 43.4336 1.54824 0.774119 0.633040i \(-0.218193\pi\)
0.774119 + 0.633040i \(0.218193\pi\)
\(788\) −3.00969 −0.107216
\(789\) −22.4698 −0.799946
\(790\) 33.7657 1.20133
\(791\) −4.30426 −0.153042
\(792\) 4.85086 0.172368
\(793\) 0 0
\(794\) −19.3599 −0.687056
\(795\) −54.5652 −1.93523
\(796\) 12.8944 0.457030
\(797\) 21.0164 0.744439 0.372219 0.928145i \(-0.378597\pi\)
0.372219 + 0.928145i \(0.378597\pi\)
\(798\) −1.23191 −0.0436093
\(799\) 53.2223 1.88287
\(800\) 11.3937 0.402829
\(801\) −3.10992 −0.109883
\(802\) −14.4832 −0.511420
\(803\) 37.1226 1.31003
\(804\) −6.05429 −0.213518
\(805\) 14.3177 0.504631
\(806\) 0 0
\(807\) −26.0140 −0.915736
\(808\) −11.9976 −0.422074
\(809\) −8.32245 −0.292602 −0.146301 0.989240i \(-0.546737\pi\)
−0.146301 + 0.989240i \(0.546737\pi\)
\(810\) −4.04892 −0.142264
\(811\) 14.4638 0.507894 0.253947 0.967218i \(-0.418271\pi\)
0.253947 + 0.967218i \(0.418271\pi\)
\(812\) 2.31468 0.0812295
\(813\) 2.88471 0.101171
\(814\) −6.23921 −0.218684
\(815\) −59.0810 −2.06952
\(816\) 7.38404 0.258493
\(817\) −14.8068 −0.518026
\(818\) −18.8984 −0.660769
\(819\) 0 0
\(820\) 6.09783 0.212946
\(821\) 38.4161 1.34073 0.670365 0.742031i \(-0.266137\pi\)
0.670365 + 0.742031i \(0.266137\pi\)
\(822\) 15.7560 0.549554
\(823\) −40.9748 −1.42829 −0.714145 0.699997i \(-0.753184\pi\)
−0.714145 + 0.699997i \(0.753184\pi\)
\(824\) −12.3230 −0.429294
\(825\) 55.2693 1.92423
\(826\) 0.905149 0.0314942
\(827\) −3.51035 −0.122067 −0.0610335 0.998136i \(-0.519440\pi\)
−0.0610335 + 0.998136i \(0.519440\pi\)
\(828\) 5.10992 0.177582
\(829\) −13.2185 −0.459098 −0.229549 0.973297i \(-0.573725\pi\)
−0.229549 + 0.973297i \(0.573725\pi\)
\(830\) 62.0592 2.15411
\(831\) −1.46250 −0.0507336
\(832\) 0 0
\(833\) −48.1521 −1.66837
\(834\) −6.09783 −0.211151
\(835\) 78.9590 2.73249
\(836\) 8.63533 0.298659
\(837\) −0.972853 −0.0336267
\(838\) −21.7603 −0.751698
\(839\) −55.8491 −1.92812 −0.964062 0.265678i \(-0.914404\pi\)
−0.964062 + 0.265678i \(0.914404\pi\)
\(840\) 2.80194 0.0966760
\(841\) −17.8122 −0.614214
\(842\) 20.5918 0.709640
\(843\) 5.68233 0.195710
\(844\) 7.79954 0.268471
\(845\) 0 0
\(846\) 7.20775 0.247808
\(847\) −8.67158 −0.297959
\(848\) 13.4765 0.462785
\(849\) −25.2078 −0.865128
\(850\) 84.1318 2.88570
\(851\) −6.57242 −0.225300
\(852\) 1.32975 0.0455564
\(853\) −21.8103 −0.746770 −0.373385 0.927676i \(-0.621803\pi\)
−0.373385 + 0.927676i \(0.621803\pi\)
\(854\) 0.274127 0.00938042
\(855\) −7.20775 −0.246500
\(856\) 5.89977 0.201650
\(857\) 28.8961 0.987070 0.493535 0.869726i \(-0.335704\pi\)
0.493535 + 0.869726i \(0.335704\pi\)
\(858\) 0 0
\(859\) −17.2755 −0.589431 −0.294715 0.955585i \(-0.595225\pi\)
−0.294715 + 0.955585i \(0.595225\pi\)
\(860\) 33.6775 1.14839
\(861\) 1.04221 0.0355185
\(862\) 34.5133 1.17553
\(863\) −44.7741 −1.52413 −0.762063 0.647503i \(-0.775813\pi\)
−0.762063 + 0.647503i \(0.775813\pi\)
\(864\) 1.00000 0.0340207
\(865\) 37.6165 1.27900
\(866\) −2.12631 −0.0722549
\(867\) 37.5241 1.27438
\(868\) 0.673235 0.0228511
\(869\) −40.4534 −1.37229
\(870\) 13.5429 0.459147
\(871\) 0 0
\(872\) −0.792249 −0.0268289
\(873\) 8.54288 0.289133
\(874\) 9.09651 0.307694
\(875\) 17.9148 0.605632
\(876\) 7.65279 0.258564
\(877\) −40.2741 −1.35996 −0.679980 0.733230i \(-0.738012\pi\)
−0.679980 + 0.733230i \(0.738012\pi\)
\(878\) 21.8321 0.736797
\(879\) −7.14914 −0.241135
\(880\) −19.6407 −0.662088
\(881\) 9.40821 0.316971 0.158485 0.987361i \(-0.449339\pi\)
0.158485 + 0.987361i \(0.449339\pi\)
\(882\) −6.52111 −0.219577
\(883\) −51.2271 −1.72393 −0.861965 0.506968i \(-0.830766\pi\)
−0.861965 + 0.506968i \(0.830766\pi\)
\(884\) 0 0
\(885\) 5.29590 0.178020
\(886\) −7.54048 −0.253328
\(887\) −39.9215 −1.34043 −0.670217 0.742165i \(-0.733799\pi\)
−0.670217 + 0.742165i \(0.733799\pi\)
\(888\) −1.28621 −0.0431623
\(889\) 4.15585 0.139383
\(890\) 12.5918 0.422078
\(891\) 4.85086 0.162510
\(892\) −12.1957 −0.408341
\(893\) 12.8310 0.429373
\(894\) 2.55257 0.0853706
\(895\) −92.2863 −3.08479
\(896\) −0.692021 −0.0231188
\(897\) 0 0
\(898\) 19.7560 0.659266
\(899\) 3.25401 0.108527
\(900\) 11.3937 0.379791
\(901\) 99.5111 3.31519
\(902\) −7.30559 −0.243249
\(903\) 5.75600 0.191548
\(904\) 6.21983 0.206869
\(905\) −2.17629 −0.0723424
\(906\) 17.7168 0.588600
\(907\) −34.8310 −1.15654 −0.578272 0.815844i \(-0.696273\pi\)
−0.578272 + 0.815844i \(0.696273\pi\)
\(908\) 6.74333 0.223785
\(909\) −11.9976 −0.397936
\(910\) 0 0
\(911\) 13.2125 0.437751 0.218875 0.975753i \(-0.429761\pi\)
0.218875 + 0.975753i \(0.429761\pi\)
\(912\) 1.78017 0.0589472
\(913\) −74.3508 −2.46065
\(914\) 23.8582 0.789157
\(915\) 1.60388 0.0530225
\(916\) −19.8237 −0.654994
\(917\) −6.10156 −0.201491
\(918\) 7.38404 0.243710
\(919\) 21.0175 0.693302 0.346651 0.937994i \(-0.387319\pi\)
0.346651 + 0.937994i \(0.387319\pi\)
\(920\) −20.6896 −0.682117
\(921\) −17.9952 −0.592962
\(922\) −17.5773 −0.578878
\(923\) 0 0
\(924\) −3.35690 −0.110434
\(925\) −14.6547 −0.481844
\(926\) 23.8431 0.783532
\(927\) −12.3230 −0.404742
\(928\) −3.34481 −0.109799
\(929\) −26.9965 −0.885728 −0.442864 0.896589i \(-0.646038\pi\)
−0.442864 + 0.896589i \(0.646038\pi\)
\(930\) 3.93900 0.129165
\(931\) −11.6087 −0.380459
\(932\) −30.0301 −0.983670
\(933\) 3.32975 0.109011
\(934\) 8.61058 0.281747
\(935\) −145.028 −4.74292
\(936\) 0 0
\(937\) 23.1745 0.757078 0.378539 0.925585i \(-0.376427\pi\)
0.378539 + 0.925585i \(0.376427\pi\)
\(938\) 4.18970 0.136799
\(939\) −17.8834 −0.583602
\(940\) −29.1836 −0.951864
\(941\) 7.54048 0.245813 0.122906 0.992418i \(-0.460779\pi\)
0.122906 + 0.992418i \(0.460779\pi\)
\(942\) −6.31767 −0.205841
\(943\) −7.69574 −0.250608
\(944\) −1.30798 −0.0425711
\(945\) 2.80194 0.0911470
\(946\) −40.3478 −1.31182
\(947\) 20.6601 0.671363 0.335681 0.941976i \(-0.391033\pi\)
0.335681 + 0.941976i \(0.391033\pi\)
\(948\) −8.33944 −0.270852
\(949\) 0 0
\(950\) 20.2828 0.658059
\(951\) 4.39373 0.142477
\(952\) −5.10992 −0.165613
\(953\) −30.2935 −0.981303 −0.490651 0.871356i \(-0.663241\pi\)
−0.490651 + 0.871356i \(0.663241\pi\)
\(954\) 13.4765 0.436318
\(955\) 39.6775 1.28394
\(956\) 22.0978 0.714695
\(957\) −16.2252 −0.524487
\(958\) 6.58104 0.212624
\(959\) −10.9035 −0.352092
\(960\) −4.04892 −0.130678
\(961\) −30.0536 −0.969470
\(962\) 0 0
\(963\) 5.89977 0.190118
\(964\) 10.1274 0.326181
\(965\) 57.4771 1.85025
\(966\) −3.53617 −0.113774
\(967\) −20.7289 −0.666595 −0.333298 0.942822i \(-0.608161\pi\)
−0.333298 + 0.942822i \(0.608161\pi\)
\(968\) 12.5308 0.402755
\(969\) 13.1448 0.422273
\(970\) −34.5894 −1.11060
\(971\) −39.1094 −1.25508 −0.627541 0.778584i \(-0.715938\pi\)
−0.627541 + 0.778584i \(0.715938\pi\)
\(972\) 1.00000 0.0320750
\(973\) 4.21983 0.135282
\(974\) −23.2760 −0.745813
\(975\) 0 0
\(976\) −0.396125 −0.0126796
\(977\) 7.27545 0.232762 0.116381 0.993205i \(-0.462871\pi\)
0.116381 + 0.993205i \(0.462871\pi\)
\(978\) 14.5918 0.466594
\(979\) −15.0858 −0.482143
\(980\) 26.4034 0.843426
\(981\) −0.792249 −0.0252946
\(982\) −15.5405 −0.495917
\(983\) 53.4857 1.70593 0.852965 0.521969i \(-0.174802\pi\)
0.852965 + 0.521969i \(0.174802\pi\)
\(984\) −1.50604 −0.0480108
\(985\) 12.1860 0.388278
\(986\) −24.6983 −0.786553
\(987\) −4.98792 −0.158767
\(988\) 0 0
\(989\) −42.5026 −1.35150
\(990\) −19.6407 −0.624223
\(991\) 16.0575 0.510085 0.255042 0.966930i \(-0.417911\pi\)
0.255042 + 0.966930i \(0.417911\pi\)
\(992\) −0.972853 −0.0308881
\(993\) −25.6775 −0.814852
\(994\) −0.920215 −0.0291874
\(995\) −52.2083 −1.65512
\(996\) −15.3274 −0.485666
\(997\) 22.4263 0.710247 0.355123 0.934819i \(-0.384439\pi\)
0.355123 + 0.934819i \(0.384439\pi\)
\(998\) 9.53617 0.301862
\(999\) −1.28621 −0.0406938
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 1014.2.a.o.1.1 yes 3
3.2 odd 2 3042.2.a.bd.1.3 3
4.3 odd 2 8112.2.a.bz.1.1 3
13.2 odd 12 1014.2.i.g.823.6 12
13.3 even 3 1014.2.e.k.529.1 6
13.4 even 6 1014.2.e.m.991.3 6
13.5 odd 4 1014.2.b.g.337.3 6
13.6 odd 12 1014.2.i.g.361.3 12
13.7 odd 12 1014.2.i.g.361.4 12
13.8 odd 4 1014.2.b.g.337.4 6
13.9 even 3 1014.2.e.k.991.1 6
13.10 even 6 1014.2.e.m.529.3 6
13.11 odd 12 1014.2.i.g.823.1 12
13.12 even 2 1014.2.a.m.1.3 3
39.5 even 4 3042.2.b.r.1351.4 6
39.8 even 4 3042.2.b.r.1351.3 6
39.38 odd 2 3042.2.a.be.1.1 3
52.51 odd 2 8112.2.a.ce.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1014.2.a.m.1.3 3 13.12 even 2
1014.2.a.o.1.1 yes 3 1.1 even 1 trivial
1014.2.b.g.337.3 6 13.5 odd 4
1014.2.b.g.337.4 6 13.8 odd 4
1014.2.e.k.529.1 6 13.3 even 3
1014.2.e.k.991.1 6 13.9 even 3
1014.2.e.m.529.3 6 13.10 even 6
1014.2.e.m.991.3 6 13.4 even 6
1014.2.i.g.361.3 12 13.6 odd 12
1014.2.i.g.361.4 12 13.7 odd 12
1014.2.i.g.823.1 12 13.11 odd 12
1014.2.i.g.823.6 12 13.2 odd 12
3042.2.a.bd.1.3 3 3.2 odd 2
3042.2.a.be.1.1 3 39.38 odd 2
3042.2.b.r.1351.3 6 39.8 even 4
3042.2.b.r.1351.4 6 39.5 even 4
8112.2.a.bz.1.1 3 4.3 odd 2
8112.2.a.ce.1.3 3 52.51 odd 2