Properties

Label 7098.2.a.cr.1.3
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.48406561.1
Defining polynomial: \(x^{6} - 3 x^{5} - 17 x^{4} + 39 x^{3} + 111 x^{2} - 131 x - 281\)
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 1.3
Root \(2.83411\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.83411 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.83411 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.83411 q^{10} -0.198062 q^{11} +1.00000 q^{12} +1.00000 q^{14} -2.83411 q^{15} +1.00000 q^{16} +0.445042 q^{17} -1.00000 q^{18} +1.01785 q^{19} -2.83411 q^{20} -1.00000 q^{21} +0.198062 q^{22} -5.03709 q^{23} -1.00000 q^{24} +3.03217 q^{25} +1.00000 q^{27} -1.00000 q^{28} +4.34507 q^{29} +2.83411 q^{30} -2.21481 q^{31} -1.00000 q^{32} -0.198062 q^{33} -0.445042 q^{34} +2.83411 q^{35} +1.00000 q^{36} +7.43142 q^{37} -1.01785 q^{38} +2.83411 q^{40} +0.443500 q^{41} +1.00000 q^{42} +7.64733 q^{43} -0.198062 q^{44} -2.83411 q^{45} +5.03709 q^{46} -3.63873 q^{47} +1.00000 q^{48} +1.00000 q^{49} -3.03217 q^{50} +0.445042 q^{51} +6.50916 q^{53} -1.00000 q^{54} +0.561330 q^{55} +1.00000 q^{56} +1.01785 q^{57} -4.34507 q^{58} -5.45031 q^{59} -2.83411 q^{60} +11.3069 q^{61} +2.21481 q^{62} -1.00000 q^{63} +1.00000 q^{64} +0.198062 q^{66} -14.5407 q^{67} +0.445042 q^{68} -5.03709 q^{69} -2.83411 q^{70} -5.80083 q^{71} -1.00000 q^{72} +0.641997 q^{73} -7.43142 q^{74} +3.03217 q^{75} +1.01785 q^{76} +0.198062 q^{77} +1.67721 q^{79} -2.83411 q^{80} +1.00000 q^{81} -0.443500 q^{82} -6.42808 q^{83} -1.00000 q^{84} -1.26130 q^{85} -7.64733 q^{86} +4.34507 q^{87} +0.198062 q^{88} +10.2278 q^{89} +2.83411 q^{90} -5.03709 q^{92} -2.21481 q^{93} +3.63873 q^{94} -2.88471 q^{95} -1.00000 q^{96} +9.69272 q^{97} -1.00000 q^{98} -0.198062 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + 6q^{3} + 6q^{4} - 3q^{5} - 6q^{6} - 6q^{7} - 6q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + 6q^{3} + 6q^{4} - 3q^{5} - 6q^{6} - 6q^{7} - 6q^{8} + 6q^{9} + 3q^{10} - 10q^{11} + 6q^{12} + 6q^{14} - 3q^{15} + 6q^{16} + 2q^{17} - 6q^{18} - 2q^{19} - 3q^{20} - 6q^{21} + 10q^{22} + 4q^{23} - 6q^{24} + 13q^{25} + 6q^{27} - 6q^{28} + 2q^{29} + 3q^{30} - 9q^{31} - 6q^{32} - 10q^{33} - 2q^{34} + 3q^{35} + 6q^{36} - 7q^{37} + 2q^{38} + 3q^{40} - 11q^{41} + 6q^{42} - 5q^{43} - 10q^{44} - 3q^{45} - 4q^{46} - 5q^{47} + 6q^{48} + 6q^{49} - 13q^{50} + 2q^{51} - 6q^{53} - 6q^{54} + 5q^{55} + 6q^{56} - 2q^{57} - 2q^{58} - 28q^{59} - 3q^{60} + 23q^{61} + 9q^{62} - 6q^{63} + 6q^{64} + 10q^{66} + 10q^{67} + 2q^{68} + 4q^{69} - 3q^{70} - 21q^{71} - 6q^{72} + 7q^{73} + 7q^{74} + 13q^{75} - 2q^{76} + 10q^{77} - 14q^{79} - 3q^{80} + 6q^{81} + 11q^{82} - 17q^{83} - 6q^{84} - q^{85} + 5q^{86} + 2q^{87} + 10q^{88} - 17q^{89} + 3q^{90} + 4q^{92} - 9q^{93} + 5q^{94} - 22q^{95} - 6q^{96} - 6q^{98} - 10q^{99} + O(q^{100}) \)

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
\(5\) −2.83411 −1.26745 −0.633726 0.773558i \(-0.718475\pi\)
−0.633726 + 0.773558i \(0.718475\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.83411 0.896224
\(11\) −0.198062 −0.0597180 −0.0298590 0.999554i \(-0.509506\pi\)
−0.0298590 + 0.999554i \(0.509506\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) −2.83411 −0.731764
\(16\) 1.00000 0.250000
\(17\) 0.445042 0.107939 0.0539693 0.998543i \(-0.482813\pi\)
0.0539693 + 0.998543i \(0.482813\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.01785 0.233512 0.116756 0.993161i \(-0.462751\pi\)
0.116756 + 0.993161i \(0.462751\pi\)
\(20\) −2.83411 −0.633726
\(21\) −1.00000 −0.218218
\(22\) 0.198062 0.0422270
\(23\) −5.03709 −1.05031 −0.525153 0.851008i \(-0.675992\pi\)
−0.525153 + 0.851008i \(0.675992\pi\)
\(24\) −1.00000 −0.204124
\(25\) 3.03217 0.606434
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 4.34507 0.806858 0.403429 0.915011i \(-0.367818\pi\)
0.403429 + 0.915011i \(0.367818\pi\)
\(30\) 2.83411 0.517435
\(31\) −2.21481 −0.397791 −0.198896 0.980021i \(-0.563736\pi\)
−0.198896 + 0.980021i \(0.563736\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.198062 −0.0344782
\(34\) −0.445042 −0.0763241
\(35\) 2.83411 0.479052
\(36\) 1.00000 0.166667
\(37\) 7.43142 1.22172 0.610859 0.791740i \(-0.290824\pi\)
0.610859 + 0.791740i \(0.290824\pi\)
\(38\) −1.01785 −0.165118
\(39\) 0 0
\(40\) 2.83411 0.448112
\(41\) 0.443500 0.0692631 0.0346315 0.999400i \(-0.488974\pi\)
0.0346315 + 0.999400i \(0.488974\pi\)
\(42\) 1.00000 0.154303
\(43\) 7.64733 1.16621 0.583104 0.812398i \(-0.301838\pi\)
0.583104 + 0.812398i \(0.301838\pi\)
\(44\) −0.198062 −0.0298590
\(45\) −2.83411 −0.422484
\(46\) 5.03709 0.742678
\(47\) −3.63873 −0.530763 −0.265381 0.964144i \(-0.585498\pi\)
−0.265381 + 0.964144i \(0.585498\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −3.03217 −0.428814
\(51\) 0.445042 0.0623183
\(52\) 0 0
\(53\) 6.50916 0.894102 0.447051 0.894508i \(-0.352474\pi\)
0.447051 + 0.894508i \(0.352474\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0.561330 0.0756897
\(56\) 1.00000 0.133631
\(57\) 1.01785 0.134818
\(58\) −4.34507 −0.570535
\(59\) −5.45031 −0.709570 −0.354785 0.934948i \(-0.615446\pi\)
−0.354785 + 0.934948i \(0.615446\pi\)
\(60\) −2.83411 −0.365882
\(61\) 11.3069 1.44771 0.723853 0.689954i \(-0.242369\pi\)
0.723853 + 0.689954i \(0.242369\pi\)
\(62\) 2.21481 0.281281
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.198062 0.0243798
\(67\) −14.5407 −1.77643 −0.888217 0.459425i \(-0.848056\pi\)
−0.888217 + 0.459425i \(0.848056\pi\)
\(68\) 0.445042 0.0539693
\(69\) −5.03709 −0.606394
\(70\) −2.83411 −0.338741
\(71\) −5.80083 −0.688432 −0.344216 0.938890i \(-0.611855\pi\)
−0.344216 + 0.938890i \(0.611855\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0.641997 0.0751401 0.0375701 0.999294i \(-0.488038\pi\)
0.0375701 + 0.999294i \(0.488038\pi\)
\(74\) −7.43142 −0.863885
\(75\) 3.03217 0.350125
\(76\) 1.01785 0.116756
\(77\) 0.198062 0.0225713
\(78\) 0 0
\(79\) 1.67721 0.188701 0.0943505 0.995539i \(-0.469923\pi\)
0.0943505 + 0.995539i \(0.469923\pi\)
\(80\) −2.83411 −0.316863
\(81\) 1.00000 0.111111
\(82\) −0.443500 −0.0489764
\(83\) −6.42808 −0.705573 −0.352787 0.935704i \(-0.614766\pi\)
−0.352787 + 0.935704i \(0.614766\pi\)
\(84\) −1.00000 −0.109109
\(85\) −1.26130 −0.136807
\(86\) −7.64733 −0.824633
\(87\) 4.34507 0.465840
\(88\) 0.198062 0.0211135
\(89\) 10.2278 1.08415 0.542074 0.840330i \(-0.317639\pi\)
0.542074 + 0.840330i \(0.317639\pi\)
\(90\) 2.83411 0.298741
\(91\) 0 0
\(92\) −5.03709 −0.525153
\(93\) −2.21481 −0.229665
\(94\) 3.63873 0.375306
\(95\) −2.88471 −0.295965
\(96\) −1.00000 −0.102062
\(97\) 9.69272 0.984146 0.492073 0.870554i \(-0.336239\pi\)
0.492073 + 0.870554i \(0.336239\pi\)
\(98\) −1.00000 −0.101015
\(99\) −0.198062 −0.0199060
\(100\) 3.03217 0.303217
\(101\) −0.496795 −0.0494329 −0.0247165 0.999695i \(-0.507868\pi\)
−0.0247165 + 0.999695i \(0.507868\pi\)
\(102\) −0.445042 −0.0440657
\(103\) −9.61959 −0.947846 −0.473923 0.880566i \(-0.657163\pi\)
−0.473923 + 0.880566i \(0.657163\pi\)
\(104\) 0 0
\(105\) 2.83411 0.276581
\(106\) −6.50916 −0.632226
\(107\) −10.8776 −1.05157 −0.525787 0.850616i \(-0.676229\pi\)
−0.525787 + 0.850616i \(0.676229\pi\)
\(108\) 1.00000 0.0962250
\(109\) −8.51454 −0.815545 −0.407773 0.913084i \(-0.633694\pi\)
−0.407773 + 0.913084i \(0.633694\pi\)
\(110\) −0.561330 −0.0535207
\(111\) 7.43142 0.705359
\(112\) −1.00000 −0.0944911
\(113\) −6.97708 −0.656349 −0.328174 0.944617i \(-0.606433\pi\)
−0.328174 + 0.944617i \(0.606433\pi\)
\(114\) −1.01785 −0.0953307
\(115\) 14.2757 1.33121
\(116\) 4.34507 0.403429
\(117\) 0 0
\(118\) 5.45031 0.501741
\(119\) −0.445042 −0.0407969
\(120\) 2.83411 0.258718
\(121\) −10.9608 −0.996434
\(122\) −11.3069 −1.02368
\(123\) 0.443500 0.0399891
\(124\) −2.21481 −0.198896
\(125\) 5.57704 0.498826
\(126\) 1.00000 0.0890871
\(127\) −3.64802 −0.323709 −0.161855 0.986815i \(-0.551748\pi\)
−0.161855 + 0.986815i \(0.551748\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.64733 0.673310
\(130\) 0 0
\(131\) −2.42438 −0.211819 −0.105909 0.994376i \(-0.533775\pi\)
−0.105909 + 0.994376i \(0.533775\pi\)
\(132\) −0.198062 −0.0172391
\(133\) −1.01785 −0.0882591
\(134\) 14.5407 1.25613
\(135\) −2.83411 −0.243921
\(136\) −0.445042 −0.0381620
\(137\) −21.5583 −1.84185 −0.920927 0.389735i \(-0.872567\pi\)
−0.920927 + 0.389735i \(0.872567\pi\)
\(138\) 5.03709 0.428785
\(139\) −2.63351 −0.223371 −0.111686 0.993744i \(-0.535625\pi\)
−0.111686 + 0.993744i \(0.535625\pi\)
\(140\) 2.83411 0.239526
\(141\) −3.63873 −0.306436
\(142\) 5.80083 0.486795
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −12.3144 −1.02265
\(146\) −0.641997 −0.0531321
\(147\) 1.00000 0.0824786
\(148\) 7.43142 0.610859
\(149\) −9.09027 −0.744704 −0.372352 0.928092i \(-0.621449\pi\)
−0.372352 + 0.928092i \(0.621449\pi\)
\(150\) −3.03217 −0.247576
\(151\) 21.2010 1.72531 0.862657 0.505790i \(-0.168799\pi\)
0.862657 + 0.505790i \(0.168799\pi\)
\(152\) −1.01785 −0.0825588
\(153\) 0.445042 0.0359795
\(154\) −0.198062 −0.0159603
\(155\) 6.27701 0.504181
\(156\) 0 0
\(157\) 9.99173 0.797427 0.398713 0.917076i \(-0.369457\pi\)
0.398713 + 0.917076i \(0.369457\pi\)
\(158\) −1.67721 −0.133432
\(159\) 6.50916 0.516210
\(160\) 2.83411 0.224056
\(161\) 5.03709 0.396978
\(162\) −1.00000 −0.0785674
\(163\) −3.18547 −0.249506 −0.124753 0.992188i \(-0.539814\pi\)
−0.124753 + 0.992188i \(0.539814\pi\)
\(164\) 0.443500 0.0346315
\(165\) 0.561330 0.0436995
\(166\) 6.42808 0.498915
\(167\) 18.5456 1.43510 0.717550 0.696507i \(-0.245263\pi\)
0.717550 + 0.696507i \(0.245263\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 1.26130 0.0967371
\(171\) 1.01785 0.0778372
\(172\) 7.64733 0.583104
\(173\) −3.53929 −0.269087 −0.134544 0.990908i \(-0.542957\pi\)
−0.134544 + 0.990908i \(0.542957\pi\)
\(174\) −4.34507 −0.329399
\(175\) −3.03217 −0.229211
\(176\) −0.198062 −0.0149295
\(177\) −5.45031 −0.409670
\(178\) −10.2278 −0.766609
\(179\) 3.24260 0.242363 0.121182 0.992630i \(-0.461332\pi\)
0.121182 + 0.992630i \(0.461332\pi\)
\(180\) −2.83411 −0.211242
\(181\) 23.4047 1.73966 0.869829 0.493354i \(-0.164229\pi\)
0.869829 + 0.493354i \(0.164229\pi\)
\(182\) 0 0
\(183\) 11.3069 0.835833
\(184\) 5.03709 0.371339
\(185\) −21.0614 −1.54847
\(186\) 2.21481 0.162398
\(187\) −0.0881460 −0.00644587
\(188\) −3.63873 −0.265381
\(189\) −1.00000 −0.0727393
\(190\) 2.88471 0.209279
\(191\) −15.4891 −1.12075 −0.560376 0.828239i \(-0.689343\pi\)
−0.560376 + 0.828239i \(0.689343\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.7419 0.773217 0.386609 0.922244i \(-0.373646\pi\)
0.386609 + 0.922244i \(0.373646\pi\)
\(194\) −9.69272 −0.695896
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 1.63996 0.116843 0.0584213 0.998292i \(-0.481393\pi\)
0.0584213 + 0.998292i \(0.481393\pi\)
\(198\) 0.198062 0.0140757
\(199\) −12.4549 −0.882907 −0.441454 0.897284i \(-0.645537\pi\)
−0.441454 + 0.897284i \(0.645537\pi\)
\(200\) −3.03217 −0.214407
\(201\) −14.5407 −1.02562
\(202\) 0.496795 0.0349544
\(203\) −4.34507 −0.304964
\(204\) 0.445042 0.0311592
\(205\) −1.25693 −0.0877876
\(206\) 9.61959 0.670228
\(207\) −5.03709 −0.350102
\(208\) 0 0
\(209\) −0.201598 −0.0139448
\(210\) −2.83411 −0.195572
\(211\) −26.3734 −1.81562 −0.907808 0.419385i \(-0.862246\pi\)
−0.907808 + 0.419385i \(0.862246\pi\)
\(212\) 6.50916 0.447051
\(213\) −5.80083 −0.397466
\(214\) 10.8776 0.743575
\(215\) −21.6734 −1.47811
\(216\) −1.00000 −0.0680414
\(217\) 2.21481 0.150351
\(218\) 8.51454 0.576677
\(219\) 0.641997 0.0433822
\(220\) 0.561330 0.0378449
\(221\) 0 0
\(222\) −7.43142 −0.498764
\(223\) −9.07058 −0.607411 −0.303705 0.952766i \(-0.598224\pi\)
−0.303705 + 0.952766i \(0.598224\pi\)
\(224\) 1.00000 0.0668153
\(225\) 3.03217 0.202145
\(226\) 6.97708 0.464109
\(227\) −20.0785 −1.33266 −0.666328 0.745659i \(-0.732135\pi\)
−0.666328 + 0.745659i \(0.732135\pi\)
\(228\) 1.01785 0.0674090
\(229\) −2.94223 −0.194428 −0.0972141 0.995263i \(-0.530993\pi\)
−0.0972141 + 0.995263i \(0.530993\pi\)
\(230\) −14.2757 −0.941309
\(231\) 0.198062 0.0130315
\(232\) −4.34507 −0.285268
\(233\) −11.8698 −0.777614 −0.388807 0.921319i \(-0.627113\pi\)
−0.388807 + 0.921319i \(0.627113\pi\)
\(234\) 0 0
\(235\) 10.3125 0.672716
\(236\) −5.45031 −0.354785
\(237\) 1.67721 0.108947
\(238\) 0.445042 0.0288478
\(239\) 21.0447 1.36127 0.680635 0.732622i \(-0.261704\pi\)
0.680635 + 0.732622i \(0.261704\pi\)
\(240\) −2.83411 −0.182941
\(241\) −17.4505 −1.12408 −0.562042 0.827109i \(-0.689984\pi\)
−0.562042 + 0.827109i \(0.689984\pi\)
\(242\) 10.9608 0.704585
\(243\) 1.00000 0.0641500
\(244\) 11.3069 0.723853
\(245\) −2.83411 −0.181065
\(246\) −0.443500 −0.0282765
\(247\) 0 0
\(248\) 2.21481 0.140641
\(249\) −6.42808 −0.407363
\(250\) −5.57704 −0.352723
\(251\) −6.73221 −0.424934 −0.212467 0.977168i \(-0.568150\pi\)
−0.212467 + 0.977168i \(0.568150\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0.997657 0.0627221
\(254\) 3.64802 0.228897
\(255\) −1.26130 −0.0789855
\(256\) 1.00000 0.0625000
\(257\) −13.6497 −0.851446 −0.425723 0.904854i \(-0.639980\pi\)
−0.425723 + 0.904854i \(0.639980\pi\)
\(258\) −7.64733 −0.476102
\(259\) −7.43142 −0.461766
\(260\) 0 0
\(261\) 4.34507 0.268953
\(262\) 2.42438 0.149779
\(263\) −5.22545 −0.322215 −0.161108 0.986937i \(-0.551507\pi\)
−0.161108 + 0.986937i \(0.551507\pi\)
\(264\) 0.198062 0.0121899
\(265\) −18.4477 −1.13323
\(266\) 1.01785 0.0624086
\(267\) 10.2278 0.625934
\(268\) −14.5407 −0.888217
\(269\) −17.4972 −1.06683 −0.533413 0.845855i \(-0.679091\pi\)
−0.533413 + 0.845855i \(0.679091\pi\)
\(270\) 2.83411 0.172478
\(271\) 25.9779 1.57804 0.789021 0.614366i \(-0.210588\pi\)
0.789021 + 0.614366i \(0.210588\pi\)
\(272\) 0.445042 0.0269846
\(273\) 0 0
\(274\) 21.5583 1.30239
\(275\) −0.600559 −0.0362150
\(276\) −5.03709 −0.303197
\(277\) −7.25383 −0.435840 −0.217920 0.975967i \(-0.569927\pi\)
−0.217920 + 0.975967i \(0.569927\pi\)
\(278\) 2.63351 0.157947
\(279\) −2.21481 −0.132597
\(280\) −2.83411 −0.169370
\(281\) −8.97769 −0.535564 −0.267782 0.963480i \(-0.586291\pi\)
−0.267782 + 0.963480i \(0.586291\pi\)
\(282\) 3.63873 0.216683
\(283\) −11.4182 −0.678740 −0.339370 0.940653i \(-0.610214\pi\)
−0.339370 + 0.940653i \(0.610214\pi\)
\(284\) −5.80083 −0.344216
\(285\) −2.88471 −0.170875
\(286\) 0 0
\(287\) −0.443500 −0.0261790
\(288\) −1.00000 −0.0589256
\(289\) −16.8019 −0.988349
\(290\) 12.3144 0.723126
\(291\) 9.69272 0.568197
\(292\) 0.641997 0.0375701
\(293\) 0.481985 0.0281579 0.0140789 0.999901i \(-0.495518\pi\)
0.0140789 + 0.999901i \(0.495518\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 15.4468 0.899345
\(296\) −7.43142 −0.431942
\(297\) −0.198062 −0.0114927
\(298\) 9.09027 0.526585
\(299\) 0 0
\(300\) 3.03217 0.175062
\(301\) −7.64733 −0.440785
\(302\) −21.2010 −1.21998
\(303\) −0.496795 −0.0285401
\(304\) 1.01785 0.0583779
\(305\) −32.0451 −1.83490
\(306\) −0.445042 −0.0254414
\(307\) 12.1290 0.692239 0.346119 0.938190i \(-0.387499\pi\)
0.346119 + 0.938190i \(0.387499\pi\)
\(308\) 0.198062 0.0112856
\(309\) −9.61959 −0.547239
\(310\) −6.27701 −0.356510
\(311\) −30.0623 −1.70468 −0.852338 0.522992i \(-0.824816\pi\)
−0.852338 + 0.522992i \(0.824816\pi\)
\(312\) 0 0
\(313\) 16.6595 0.941648 0.470824 0.882227i \(-0.343957\pi\)
0.470824 + 0.882227i \(0.343957\pi\)
\(314\) −9.99173 −0.563866
\(315\) 2.83411 0.159684
\(316\) 1.67721 0.0943505
\(317\) 11.1386 0.625605 0.312803 0.949818i \(-0.398732\pi\)
0.312803 + 0.949818i \(0.398732\pi\)
\(318\) −6.50916 −0.365016
\(319\) −0.860594 −0.0481840
\(320\) −2.83411 −0.158431
\(321\) −10.8776 −0.607127
\(322\) −5.03709 −0.280706
\(323\) 0.452987 0.0252049
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 3.18547 0.176427
\(327\) −8.51454 −0.470855
\(328\) −0.443500 −0.0244882
\(329\) 3.63873 0.200609
\(330\) −0.561330 −0.0309002
\(331\) −20.8141 −1.14404 −0.572022 0.820238i \(-0.693841\pi\)
−0.572022 + 0.820238i \(0.693841\pi\)
\(332\) −6.42808 −0.352787
\(333\) 7.43142 0.407239
\(334\) −18.5456 −1.01477
\(335\) 41.2100 2.25154
\(336\) −1.00000 −0.0545545
\(337\) 5.98060 0.325784 0.162892 0.986644i \(-0.447918\pi\)
0.162892 + 0.986644i \(0.447918\pi\)
\(338\) 0 0
\(339\) −6.97708 −0.378943
\(340\) −1.26130 −0.0684034
\(341\) 0.438670 0.0237553
\(342\) −1.01785 −0.0550392
\(343\) −1.00000 −0.0539949
\(344\) −7.64733 −0.412317
\(345\) 14.2757 0.768575
\(346\) 3.53929 0.190274
\(347\) −23.4377 −1.25820 −0.629100 0.777324i \(-0.716576\pi\)
−0.629100 + 0.777324i \(0.716576\pi\)
\(348\) 4.34507 0.232920
\(349\) 0.694217 0.0371606 0.0185803 0.999827i \(-0.494085\pi\)
0.0185803 + 0.999827i \(0.494085\pi\)
\(350\) 3.03217 0.162076
\(351\) 0 0
\(352\) 0.198062 0.0105568
\(353\) 19.7833 1.05296 0.526480 0.850188i \(-0.323512\pi\)
0.526480 + 0.850188i \(0.323512\pi\)
\(354\) 5.45031 0.289681
\(355\) 16.4402 0.872554
\(356\) 10.2278 0.542074
\(357\) −0.445042 −0.0235541
\(358\) −3.24260 −0.171377
\(359\) −30.7362 −1.62219 −0.811097 0.584911i \(-0.801129\pi\)
−0.811097 + 0.584911i \(0.801129\pi\)
\(360\) 2.83411 0.149371
\(361\) −17.9640 −0.945472
\(362\) −23.4047 −1.23012
\(363\) −10.9608 −0.575291
\(364\) 0 0
\(365\) −1.81949 −0.0952365
\(366\) −11.3069 −0.591023
\(367\) −22.0009 −1.14844 −0.574218 0.818702i \(-0.694694\pi\)
−0.574218 + 0.818702i \(0.694694\pi\)
\(368\) −5.03709 −0.262576
\(369\) 0.443500 0.0230877
\(370\) 21.0614 1.09493
\(371\) −6.50916 −0.337939
\(372\) −2.21481 −0.114832
\(373\) 11.8812 0.615186 0.307593 0.951518i \(-0.400476\pi\)
0.307593 + 0.951518i \(0.400476\pi\)
\(374\) 0.0881460 0.00455792
\(375\) 5.57704 0.287997
\(376\) 3.63873 0.187653
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) 2.13559 0.109698 0.0548490 0.998495i \(-0.482532\pi\)
0.0548490 + 0.998495i \(0.482532\pi\)
\(380\) −2.88471 −0.147982
\(381\) −3.64802 −0.186894
\(382\) 15.4891 0.792491
\(383\) −31.0649 −1.58734 −0.793670 0.608348i \(-0.791832\pi\)
−0.793670 + 0.608348i \(0.791832\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −0.561330 −0.0286080
\(386\) −10.7419 −0.546747
\(387\) 7.64733 0.388736
\(388\) 9.69272 0.492073
\(389\) 32.2971 1.63753 0.818764 0.574130i \(-0.194660\pi\)
0.818764 + 0.574130i \(0.194660\pi\)
\(390\) 0 0
\(391\) −2.24171 −0.113368
\(392\) −1.00000 −0.0505076
\(393\) −2.42438 −0.122294
\(394\) −1.63996 −0.0826201
\(395\) −4.75340 −0.239169
\(396\) −0.198062 −0.00995300
\(397\) 0.324869 0.0163047 0.00815235 0.999967i \(-0.497405\pi\)
0.00815235 + 0.999967i \(0.497405\pi\)
\(398\) 12.4549 0.624310
\(399\) −1.01785 −0.0509564
\(400\) 3.03217 0.151609
\(401\) −32.5831 −1.62712 −0.813560 0.581481i \(-0.802474\pi\)
−0.813560 + 0.581481i \(0.802474\pi\)
\(402\) 14.5407 0.725226
\(403\) 0 0
\(404\) −0.496795 −0.0247165
\(405\) −2.83411 −0.140828
\(406\) 4.34507 0.215642
\(407\) −1.47188 −0.0729586
\(408\) −0.445042 −0.0220329
\(409\) 0.898604 0.0444331 0.0222165 0.999753i \(-0.492928\pi\)
0.0222165 + 0.999753i \(0.492928\pi\)
\(410\) 1.25693 0.0620752
\(411\) −21.5583 −1.06339
\(412\) −9.61959 −0.473923
\(413\) 5.45031 0.268192
\(414\) 5.03709 0.247559
\(415\) 18.2179 0.894280
\(416\) 0 0
\(417\) −2.63351 −0.128964
\(418\) 0.201598 0.00986050
\(419\) −20.1679 −0.985266 −0.492633 0.870237i \(-0.663965\pi\)
−0.492633 + 0.870237i \(0.663965\pi\)
\(420\) 2.83411 0.138290
\(421\) 3.31432 0.161530 0.0807650 0.996733i \(-0.474264\pi\)
0.0807650 + 0.996733i \(0.474264\pi\)
\(422\) 26.3734 1.28383
\(423\) −3.63873 −0.176921
\(424\) −6.50916 −0.316113
\(425\) 1.34944 0.0654576
\(426\) 5.80083 0.281051
\(427\) −11.3069 −0.547181
\(428\) −10.8776 −0.525787
\(429\) 0 0
\(430\) 21.6734 1.04518
\(431\) 9.71137 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(432\) 1.00000 0.0481125
\(433\) −38.4791 −1.84919 −0.924594 0.380955i \(-0.875595\pi\)
−0.924594 + 0.380955i \(0.875595\pi\)
\(434\) −2.21481 −0.106314
\(435\) −12.3144 −0.590430
\(436\) −8.51454 −0.407773
\(437\) −5.12702 −0.245258
\(438\) −0.641997 −0.0306758
\(439\) −1.86358 −0.0889437 −0.0444718 0.999011i \(-0.514160\pi\)
−0.0444718 + 0.999011i \(0.514160\pi\)
\(440\) −0.561330 −0.0267604
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −14.9638 −0.710949 −0.355475 0.934686i \(-0.615681\pi\)
−0.355475 + 0.934686i \(0.615681\pi\)
\(444\) 7.43142 0.352680
\(445\) −28.9868 −1.37411
\(446\) 9.07058 0.429504
\(447\) −9.09027 −0.429955
\(448\) −1.00000 −0.0472456
\(449\) 38.0913 1.79764 0.898821 0.438317i \(-0.144425\pi\)
0.898821 + 0.438317i \(0.144425\pi\)
\(450\) −3.03217 −0.142938
\(451\) −0.0878406 −0.00413625
\(452\) −6.97708 −0.328174
\(453\) 21.2010 0.996110
\(454\) 20.0785 0.942330
\(455\) 0 0
\(456\) −1.01785 −0.0476654
\(457\) −4.68245 −0.219036 −0.109518 0.993985i \(-0.534931\pi\)
−0.109518 + 0.993985i \(0.534931\pi\)
\(458\) 2.94223 0.137481
\(459\) 0.445042 0.0207728
\(460\) 14.2757 0.665606
\(461\) 23.2754 1.08405 0.542023 0.840364i \(-0.317659\pi\)
0.542023 + 0.840364i \(0.317659\pi\)
\(462\) −0.198062 −0.00921469
\(463\) −12.3208 −0.572597 −0.286299 0.958140i \(-0.592425\pi\)
−0.286299 + 0.958140i \(0.592425\pi\)
\(464\) 4.34507 0.201715
\(465\) 6.27701 0.291089
\(466\) 11.8698 0.549856
\(467\) −3.26780 −0.151216 −0.0756080 0.997138i \(-0.524090\pi\)
−0.0756080 + 0.997138i \(0.524090\pi\)
\(468\) 0 0
\(469\) 14.5407 0.671429
\(470\) −10.3125 −0.475682
\(471\) 9.99173 0.460395
\(472\) 5.45031 0.250871
\(473\) −1.51465 −0.0696436
\(474\) −1.67721 −0.0770369
\(475\) 3.08631 0.141609
\(476\) −0.445042 −0.0203985
\(477\) 6.50916 0.298034
\(478\) −21.0447 −0.962564
\(479\) 11.2400 0.513568 0.256784 0.966469i \(-0.417337\pi\)
0.256784 + 0.966469i \(0.417337\pi\)
\(480\) 2.83411 0.129359
\(481\) 0 0
\(482\) 17.4505 0.794847
\(483\) 5.03709 0.229195
\(484\) −10.9608 −0.498217
\(485\) −27.4702 −1.24736
\(486\) −1.00000 −0.0453609
\(487\) −10.0813 −0.456829 −0.228414 0.973564i \(-0.573354\pi\)
−0.228414 + 0.973564i \(0.573354\pi\)
\(488\) −11.3069 −0.511841
\(489\) −3.18547 −0.144052
\(490\) 2.83411 0.128032
\(491\) 17.6228 0.795307 0.397654 0.917536i \(-0.369825\pi\)
0.397654 + 0.917536i \(0.369825\pi\)
\(492\) 0.443500 0.0199945
\(493\) 1.93374 0.0870911
\(494\) 0 0
\(495\) 0.561330 0.0252299
\(496\) −2.21481 −0.0994479
\(497\) 5.80083 0.260203
\(498\) 6.42808 0.288049
\(499\) 38.6205 1.72889 0.864446 0.502726i \(-0.167670\pi\)
0.864446 + 0.502726i \(0.167670\pi\)
\(500\) 5.57704 0.249413
\(501\) 18.5456 0.828555
\(502\) 6.73221 0.300473
\(503\) −29.6739 −1.32309 −0.661547 0.749904i \(-0.730100\pi\)
−0.661547 + 0.749904i \(0.730100\pi\)
\(504\) 1.00000 0.0445435
\(505\) 1.40797 0.0626539
\(506\) −0.997657 −0.0443513
\(507\) 0 0
\(508\) −3.64802 −0.161855
\(509\) −16.8438 −0.746590 −0.373295 0.927713i \(-0.621772\pi\)
−0.373295 + 0.927713i \(0.621772\pi\)
\(510\) 1.26130 0.0558512
\(511\) −0.641997 −0.0284003
\(512\) −1.00000 −0.0441942
\(513\) 1.01785 0.0449393
\(514\) 13.6497 0.602063
\(515\) 27.2630 1.20135
\(516\) 7.64733 0.336655
\(517\) 0.720694 0.0316961
\(518\) 7.43142 0.326518
\(519\) −3.53929 −0.155358
\(520\) 0 0
\(521\) −2.68289 −0.117540 −0.0587699 0.998272i \(-0.518718\pi\)
−0.0587699 + 0.998272i \(0.518718\pi\)
\(522\) −4.34507 −0.190178
\(523\) 12.3515 0.540095 0.270048 0.962847i \(-0.412961\pi\)
0.270048 + 0.962847i \(0.412961\pi\)
\(524\) −2.42438 −0.105909
\(525\) −3.03217 −0.132335
\(526\) 5.22545 0.227840
\(527\) −0.985683 −0.0429370
\(528\) −0.198062 −0.00861955
\(529\) 2.37224 0.103141
\(530\) 18.4477 0.801316
\(531\) −5.45031 −0.236523
\(532\) −1.01785 −0.0441295
\(533\) 0 0
\(534\) −10.2278 −0.442602
\(535\) 30.8282 1.33282
\(536\) 14.5407 0.628064
\(537\) 3.24260 0.139929
\(538\) 17.4972 0.754359
\(539\) −0.198062 −0.00853115
\(540\) −2.83411 −0.121961
\(541\) −40.8196 −1.75497 −0.877487 0.479601i \(-0.840782\pi\)
−0.877487 + 0.479601i \(0.840782\pi\)
\(542\) −25.9779 −1.11584
\(543\) 23.4047 1.00439
\(544\) −0.445042 −0.0190810
\(545\) 24.1311 1.03366
\(546\) 0 0
\(547\) 7.74509 0.331156 0.165578 0.986197i \(-0.447051\pi\)
0.165578 + 0.986197i \(0.447051\pi\)
\(548\) −21.5583 −0.920927
\(549\) 11.3069 0.482569
\(550\) 0.600559 0.0256079
\(551\) 4.42264 0.188411
\(552\) 5.03709 0.214393
\(553\) −1.67721 −0.0713223
\(554\) 7.25383 0.308186
\(555\) −21.0614 −0.894009
\(556\) −2.63351 −0.111686
\(557\) −26.9144 −1.14040 −0.570199 0.821507i \(-0.693134\pi\)
−0.570199 + 0.821507i \(0.693134\pi\)
\(558\) 2.21481 0.0937603
\(559\) 0 0
\(560\) 2.83411 0.119763
\(561\) −0.0881460 −0.00372153
\(562\) 8.97769 0.378701
\(563\) 31.1150 1.31134 0.655670 0.755048i \(-0.272386\pi\)
0.655670 + 0.755048i \(0.272386\pi\)
\(564\) −3.63873 −0.153218
\(565\) 19.7738 0.831890
\(566\) 11.4182 0.479942
\(567\) −1.00000 −0.0419961
\(568\) 5.80083 0.243397
\(569\) −27.4196 −1.14949 −0.574745 0.818333i \(-0.694899\pi\)
−0.574745 + 0.818333i \(0.694899\pi\)
\(570\) 2.88471 0.120827
\(571\) −8.99728 −0.376524 −0.188262 0.982119i \(-0.560285\pi\)
−0.188262 + 0.982119i \(0.560285\pi\)
\(572\) 0 0
\(573\) −15.4891 −0.647066
\(574\) 0.443500 0.0185113
\(575\) −15.2733 −0.636941
\(576\) 1.00000 0.0416667
\(577\) −39.9868 −1.66467 −0.832337 0.554270i \(-0.812998\pi\)
−0.832337 + 0.554270i \(0.812998\pi\)
\(578\) 16.8019 0.698868
\(579\) 10.7419 0.446417
\(580\) −12.3144 −0.511327
\(581\) 6.42808 0.266682
\(582\) −9.69272 −0.401776
\(583\) −1.28922 −0.0533940
\(584\) −0.641997 −0.0265660
\(585\) 0 0
\(586\) −0.481985 −0.0199106
\(587\) −18.8078 −0.776280 −0.388140 0.921600i \(-0.626882\pi\)
−0.388140 + 0.921600i \(0.626882\pi\)
\(588\) 1.00000 0.0412393
\(589\) −2.25435 −0.0928889
\(590\) −15.4468 −0.635933
\(591\) 1.63996 0.0674591
\(592\) 7.43142 0.305429
\(593\) −24.5201 −1.00692 −0.503459 0.864019i \(-0.667940\pi\)
−0.503459 + 0.864019i \(0.667940\pi\)
\(594\) 0.198062 0.00812659
\(595\) 1.26130 0.0517081
\(596\) −9.09027 −0.372352
\(597\) −12.4549 −0.509747
\(598\) 0 0
\(599\) 21.7347 0.888056 0.444028 0.896013i \(-0.353549\pi\)
0.444028 + 0.896013i \(0.353549\pi\)
\(600\) −3.03217 −0.123788
\(601\) 28.6388 1.16820 0.584100 0.811682i \(-0.301448\pi\)
0.584100 + 0.811682i \(0.301448\pi\)
\(602\) 7.64733 0.311682
\(603\) −14.5407 −0.592144
\(604\) 21.2010 0.862657
\(605\) 31.0640 1.26293
\(606\) 0.496795 0.0201809
\(607\) −7.42544 −0.301389 −0.150695 0.988580i \(-0.548151\pi\)
−0.150695 + 0.988580i \(0.548151\pi\)
\(608\) −1.01785 −0.0412794
\(609\) −4.34507 −0.176071
\(610\) 32.0451 1.29747
\(611\) 0 0
\(612\) 0.445042 0.0179898
\(613\) −10.7404 −0.433802 −0.216901 0.976194i \(-0.569595\pi\)
−0.216901 + 0.976194i \(0.569595\pi\)
\(614\) −12.1290 −0.489487
\(615\) −1.25693 −0.0506842
\(616\) −0.198062 −0.00798016
\(617\) −42.1071 −1.69517 −0.847583 0.530662i \(-0.821943\pi\)
−0.847583 + 0.530662i \(0.821943\pi\)
\(618\) 9.61959 0.386957
\(619\) 26.8812 1.08045 0.540224 0.841521i \(-0.318339\pi\)
0.540224 + 0.841521i \(0.318339\pi\)
\(620\) 6.27701 0.252091
\(621\) −5.03709 −0.202131
\(622\) 30.0623 1.20539
\(623\) −10.2278 −0.409770
\(624\) 0 0
\(625\) −30.9668 −1.23867
\(626\) −16.6595 −0.665846
\(627\) −0.201598 −0.00805106
\(628\) 9.99173 0.398713
\(629\) 3.30729 0.131870
\(630\) −2.83411 −0.112914
\(631\) 31.2192 1.24282 0.621409 0.783487i \(-0.286561\pi\)
0.621409 + 0.783487i \(0.286561\pi\)
\(632\) −1.67721 −0.0667159
\(633\) −26.3734 −1.04825
\(634\) −11.1386 −0.442370
\(635\) 10.3389 0.410286
\(636\) 6.50916 0.258105
\(637\) 0 0
\(638\) 0.860594 0.0340712
\(639\) −5.80083 −0.229477
\(640\) 2.83411 0.112028
\(641\) −35.5823 −1.40542 −0.702709 0.711478i \(-0.748026\pi\)
−0.702709 + 0.711478i \(0.748026\pi\)
\(642\) 10.8776 0.429303
\(643\) 29.8634 1.17770 0.588849 0.808243i \(-0.299581\pi\)
0.588849 + 0.808243i \(0.299581\pi\)
\(644\) 5.03709 0.198489
\(645\) −21.6734 −0.853388
\(646\) −0.452987 −0.0178226
\(647\) 33.3922 1.31278 0.656391 0.754421i \(-0.272082\pi\)
0.656391 + 0.754421i \(0.272082\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 1.07950 0.0423741
\(650\) 0 0
\(651\) 2.21481 0.0868052
\(652\) −3.18547 −0.124753
\(653\) 4.97027 0.194502 0.0972509 0.995260i \(-0.468995\pi\)
0.0972509 + 0.995260i \(0.468995\pi\)
\(654\) 8.51454 0.332945
\(655\) 6.87095 0.268470
\(656\) 0.443500 0.0173158
\(657\) 0.641997 0.0250467
\(658\) −3.63873 −0.141852
\(659\) 30.1774 1.17554 0.587772 0.809027i \(-0.300005\pi\)
0.587772 + 0.809027i \(0.300005\pi\)
\(660\) 0.561330 0.0218497
\(661\) 26.0514 1.01328 0.506641 0.862157i \(-0.330887\pi\)
0.506641 + 0.862157i \(0.330887\pi\)
\(662\) 20.8141 0.808962
\(663\) 0 0
\(664\) 6.42808 0.249458
\(665\) 2.88471 0.111864
\(666\) −7.43142 −0.287962
\(667\) −21.8865 −0.847448
\(668\) 18.5456 0.717550
\(669\) −9.07058 −0.350689
\(670\) −41.2100 −1.59208
\(671\) −2.23948 −0.0864541
\(672\) 1.00000 0.0385758
\(673\) −32.4744 −1.25180 −0.625898 0.779905i \(-0.715267\pi\)
−0.625898 + 0.779905i \(0.715267\pi\)
\(674\) −5.98060 −0.230364
\(675\) 3.03217 0.116708
\(676\) 0 0
\(677\) 24.4172 0.938428 0.469214 0.883084i \(-0.344537\pi\)
0.469214 + 0.883084i \(0.344537\pi\)
\(678\) 6.97708 0.267953
\(679\) −9.69272 −0.371972
\(680\) 1.26130 0.0483685
\(681\) −20.0785 −0.769410
\(682\) −0.438670 −0.0167975
\(683\) −0.310216 −0.0118701 −0.00593505 0.999982i \(-0.501889\pi\)
−0.00593505 + 0.999982i \(0.501889\pi\)
\(684\) 1.01785 0.0389186
\(685\) 61.0987 2.33446
\(686\) 1.00000 0.0381802
\(687\) −2.94223 −0.112253
\(688\) 7.64733 0.291552
\(689\) 0 0
\(690\) −14.2757 −0.543465
\(691\) −30.0144 −1.14180 −0.570901 0.821019i \(-0.693406\pi\)
−0.570901 + 0.821019i \(0.693406\pi\)
\(692\) −3.53929 −0.134544
\(693\) 0.198062 0.00752376
\(694\) 23.4377 0.889682
\(695\) 7.46365 0.283112
\(696\) −4.34507 −0.164699
\(697\) 0.197376 0.00747615
\(698\) −0.694217 −0.0262765
\(699\) −11.8698 −0.448956
\(700\) −3.03217 −0.114605
\(701\) −1.92672 −0.0727712 −0.0363856 0.999338i \(-0.511584\pi\)
−0.0363856 + 0.999338i \(0.511584\pi\)
\(702\) 0 0
\(703\) 7.56409 0.285285
\(704\) −0.198062 −0.00746475
\(705\) 10.3125 0.388393
\(706\) −19.7833 −0.744555
\(707\) 0.496795 0.0186839
\(708\) −5.45031 −0.204835
\(709\) −29.3694 −1.10299 −0.551495 0.834178i \(-0.685942\pi\)
−0.551495 + 0.834178i \(0.685942\pi\)
\(710\) −16.4402 −0.616989
\(711\) 1.67721 0.0629003
\(712\) −10.2278 −0.383305
\(713\) 11.1562 0.417802
\(714\) 0.445042 0.0166553
\(715\) 0 0
\(716\) 3.24260 0.121182
\(717\) 21.0447 0.785930
\(718\) 30.7362 1.14706
\(719\) −24.8982 −0.928548 −0.464274 0.885692i \(-0.653685\pi\)
−0.464274 + 0.885692i \(0.653685\pi\)
\(720\) −2.83411 −0.105621
\(721\) 9.61959 0.358252
\(722\) 17.9640 0.668550
\(723\) −17.4505 −0.648990
\(724\) 23.4047 0.869829
\(725\) 13.1750 0.489306
\(726\) 10.9608 0.406792
\(727\) 26.5085 0.983147 0.491574 0.870836i \(-0.336422\pi\)
0.491574 + 0.870836i \(0.336422\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 1.81949 0.0673424
\(731\) 3.40338 0.125879
\(732\) 11.3069 0.417917
\(733\) −14.7617 −0.545235 −0.272618 0.962122i \(-0.587889\pi\)
−0.272618 + 0.962122i \(0.587889\pi\)
\(734\) 22.0009 0.812067
\(735\) −2.83411 −0.104538
\(736\) 5.03709 0.185669
\(737\) 2.87997 0.106085
\(738\) −0.443500 −0.0163255
\(739\) 51.4949 1.89427 0.947135 0.320835i \(-0.103964\pi\)
0.947135 + 0.320835i \(0.103964\pi\)
\(740\) −21.0614 −0.774234
\(741\) 0 0
\(742\) 6.50916 0.238959
\(743\) −0.0267377 −0.000980911 0 −0.000490455 1.00000i \(-0.500156\pi\)
−0.000490455 1.00000i \(0.500156\pi\)
\(744\) 2.21481 0.0811988
\(745\) 25.7628 0.943876
\(746\) −11.8812 −0.435002
\(747\) −6.42808 −0.235191
\(748\) −0.0881460 −0.00322294
\(749\) 10.8776 0.397458
\(750\) −5.57704 −0.203645
\(751\) 18.8526 0.687941 0.343970 0.938980i \(-0.388228\pi\)
0.343970 + 0.938980i \(0.388228\pi\)
\(752\) −3.63873 −0.132691
\(753\) −6.73221 −0.245335
\(754\) 0 0
\(755\) −60.0860 −2.18675
\(756\) −1.00000 −0.0363696
\(757\) 8.24623 0.299714 0.149857 0.988708i \(-0.452119\pi\)
0.149857 + 0.988708i \(0.452119\pi\)
\(758\) −2.13559 −0.0775682
\(759\) 0.997657 0.0362126
\(760\) 2.88471 0.104639
\(761\) 0.884412 0.0320599 0.0160300 0.999872i \(-0.494897\pi\)
0.0160300 + 0.999872i \(0.494897\pi\)
\(762\) 3.64802 0.132154
\(763\) 8.51454 0.308247
\(764\) −15.4891 −0.560376
\(765\) −1.26130 −0.0456023
\(766\) 31.0649 1.12242
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 26.0436 0.939155 0.469578 0.882891i \(-0.344406\pi\)
0.469578 + 0.882891i \(0.344406\pi\)
\(770\) 0.561330 0.0202289
\(771\) −13.6497 −0.491582
\(772\) 10.7419 0.386609
\(773\) −22.3416 −0.803570 −0.401785 0.915734i \(-0.631610\pi\)
−0.401785 + 0.915734i \(0.631610\pi\)
\(774\) −7.64733 −0.274878
\(775\) −6.71568 −0.241234
\(776\) −9.69272 −0.347948
\(777\) −7.43142 −0.266601
\(778\) −32.2971 −1.15791
\(779\) 0.451418 0.0161737
\(780\) 0 0
\(781\) 1.14893 0.0411118
\(782\) 2.24171 0.0801635
\(783\) 4.34507 0.155280
\(784\) 1.00000 0.0357143
\(785\) −28.3176 −1.01070
\(786\) 2.42438 0.0864747
\(787\) −16.1249 −0.574792 −0.287396 0.957812i \(-0.592790\pi\)
−0.287396 + 0.957812i \(0.592790\pi\)
\(788\) 1.63996 0.0584213
\(789\) −5.22545 −0.186031
\(790\) 4.75340 0.169118
\(791\) 6.97708 0.248077
\(792\) 0.198062 0.00703784
\(793\) 0 0
\(794\) −0.324869 −0.0115292
\(795\) −18.4477 −0.654272
\(796\) −12.4549 −0.441454
\(797\) −31.8969 −1.12985 −0.564923 0.825143i \(-0.691094\pi\)
−0.564923 + 0.825143i \(0.691094\pi\)
\(798\) 1.01785 0.0360316
\(799\) −1.61939 −0.0572897
\(800\) −3.03217 −0.107203
\(801\) 10.2278 0.361383
\(802\) 32.5831 1.15055
\(803\) −0.127155 −0.00448722
\(804\) −14.5407 −0.512812
\(805\) −14.2757 −0.503151
\(806\) 0 0
\(807\) −17.4972 −0.615932
\(808\) 0.496795 0.0174772
\(809\) 18.0337 0.634032 0.317016 0.948420i \(-0.397319\pi\)
0.317016 + 0.948420i \(0.397319\pi\)
\(810\) 2.83411 0.0995804
\(811\) −45.5552 −1.59966 −0.799830 0.600227i \(-0.795077\pi\)
−0.799830 + 0.600227i \(0.795077\pi\)
\(812\) −4.34507 −0.152482
\(813\) 25.9779 0.911083
\(814\) 1.47188 0.0515895
\(815\) 9.02798 0.316236
\(816\) 0.445042 0.0155796
\(817\) 7.78387 0.272323
\(818\) −0.898604 −0.0314189
\(819\) 0 0
\(820\) −1.25693 −0.0438938
\(821\) −0.104162 −0.00363529 −0.00181764 0.999998i \(-0.500579\pi\)
−0.00181764 + 0.999998i \(0.500579\pi\)
\(822\) 21.5583 0.751934
\(823\) −20.2077 −0.704397 −0.352198 0.935925i \(-0.614566\pi\)
−0.352198 + 0.935925i \(0.614566\pi\)
\(824\) 9.61959 0.335114
\(825\) −0.600559 −0.0209088
\(826\) −5.45031 −0.189640
\(827\) 27.7803 0.966015 0.483008 0.875616i \(-0.339544\pi\)
0.483008 + 0.875616i \(0.339544\pi\)
\(828\) −5.03709 −0.175051
\(829\) −50.2962 −1.74686 −0.873430 0.486950i \(-0.838109\pi\)
−0.873430 + 0.486950i \(0.838109\pi\)
\(830\) −18.2179 −0.632351
\(831\) −7.25383 −0.251633
\(832\) 0 0
\(833\) 0.445042 0.0154198
\(834\) 2.63351 0.0911910
\(835\) −52.5602 −1.81892
\(836\) −0.201598 −0.00697242
\(837\) −2.21481 −0.0765550
\(838\) 20.1679 0.696688
\(839\) 7.00881 0.241971 0.120986 0.992654i \(-0.461395\pi\)
0.120986 + 0.992654i \(0.461395\pi\)
\(840\) −2.83411 −0.0977860
\(841\) −10.1204 −0.348980
\(842\) −3.31432 −0.114219
\(843\) −8.97769 −0.309208
\(844\) −26.3734 −0.907808
\(845\) 0 0
\(846\) 3.63873 0.125102
\(847\) 10.9608 0.376617
\(848\) 6.50916 0.223526
\(849\) −11.4182 −0.391871
\(850\) −1.34944 −0.0462855
\(851\) −37.4327 −1.28318
\(852\) −5.80083 −0.198733
\(853\) −50.7948 −1.73918 −0.869591 0.493773i \(-0.835617\pi\)
−0.869591 + 0.493773i \(0.835617\pi\)
\(854\) 11.3069 0.386916
\(855\) −2.88471 −0.0986549
\(856\) 10.8776 0.371788
\(857\) 33.5289 1.14533 0.572663 0.819791i \(-0.305910\pi\)
0.572663 + 0.819791i \(0.305910\pi\)
\(858\) 0 0
\(859\) 16.3516 0.557909 0.278954 0.960304i \(-0.410012\pi\)
0.278954 + 0.960304i \(0.410012\pi\)
\(860\) −21.6734 −0.739056
\(861\) −0.443500 −0.0151144
\(862\) −9.71137 −0.330771
\(863\) 3.70831 0.126232 0.0631162 0.998006i \(-0.479896\pi\)
0.0631162 + 0.998006i \(0.479896\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 10.0307 0.341055
\(866\) 38.4791 1.30757
\(867\) −16.8019 −0.570624
\(868\) 2.21481 0.0751755
\(869\) −0.332192 −0.0112689
\(870\) 12.3144 0.417497
\(871\) 0 0
\(872\) 8.51454 0.288339
\(873\) 9.69272 0.328049
\(874\) 5.12702 0.173424
\(875\) −5.57704 −0.188538
\(876\) 0.641997 0.0216911
\(877\) 54.7168 1.84766 0.923828 0.382809i \(-0.125043\pi\)
0.923828 + 0.382809i \(0.125043\pi\)
\(878\) 1.86358 0.0628927
\(879\) 0.481985 0.0162570
\(880\) 0.561330 0.0189224
\(881\) 57.8325 1.94843 0.974213 0.225632i \(-0.0724446\pi\)
0.974213 + 0.225632i \(0.0724446\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 2.25108 0.0757549 0.0378774 0.999282i \(-0.487940\pi\)
0.0378774 + 0.999282i \(0.487940\pi\)
\(884\) 0 0
\(885\) 15.4468 0.519237
\(886\) 14.9638 0.502717
\(887\) 36.8882 1.23859 0.619293 0.785160i \(-0.287419\pi\)
0.619293 + 0.785160i \(0.287419\pi\)
\(888\) −7.43142 −0.249382
\(889\) 3.64802 0.122351
\(890\) 28.9868 0.971640
\(891\) −0.198062 −0.00663534
\(892\) −9.07058 −0.303705
\(893\) −3.70369 −0.123939
\(894\) 9.09027 0.304024
\(895\) −9.18989 −0.307184
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −38.0913 −1.27112
\(899\) −9.62349 −0.320961
\(900\) 3.03217 0.101072
\(901\) 2.89685 0.0965081
\(902\) 0.0878406 0.00292477
\(903\) −7.64733 −0.254487
\(904\) 6.97708 0.232054
\(905\) −66.3314 −2.20493
\(906\) −21.2010 −0.704356
\(907\) −17.3337 −0.575556 −0.287778 0.957697i \(-0.592917\pi\)
−0.287778 + 0.957697i \(0.592917\pi\)
\(908\) −20.0785 −0.666328
\(909\) −0.496795 −0.0164776
\(910\) 0 0
\(911\) −21.0683 −0.698023 −0.349011 0.937118i \(-0.613483\pi\)
−0.349011 + 0.937118i \(0.613483\pi\)
\(912\) 1.01785 0.0337045
\(913\) 1.27316 0.0421354
\(914\) 4.68245 0.154882
\(915\) −32.0451 −1.05938
\(916\) −2.94223 −0.0972141
\(917\) 2.42438 0.0800600
\(918\) −0.445042 −0.0146886
\(919\) 3.15373 0.104032 0.0520159 0.998646i \(-0.483435\pi\)
0.0520159 + 0.998646i \(0.483435\pi\)
\(920\) −14.2757 −0.470654
\(921\) 12.1290 0.399664
\(922\) −23.2754 −0.766536
\(923\) 0 0
\(924\) 0.198062 0.00651577
\(925\) 22.5333 0.740891
\(926\) 12.3208 0.404887
\(927\) −9.61959 −0.315949
\(928\) −4.34507 −0.142634
\(929\) 0.341011 0.0111882 0.00559411 0.999984i \(-0.498219\pi\)
0.00559411 + 0.999984i \(0.498219\pi\)
\(930\) −6.27701 −0.205831
\(931\) 1.01785 0.0333588
\(932\) −11.8698 −0.388807
\(933\) −30.0623 −0.984195
\(934\) 3.26780 0.106926
\(935\) 0.249815 0.00816983
\(936\) 0 0
\(937\) −25.3681 −0.828740 −0.414370 0.910109i \(-0.635998\pi\)
−0.414370 + 0.910109i \(0.635998\pi\)
\(938\) −14.5407 −0.474772
\(939\) 16.6595 0.543661
\(940\) 10.3125 0.336358
\(941\) 48.2127 1.57169 0.785845 0.618424i \(-0.212229\pi\)
0.785845 + 0.618424i \(0.212229\pi\)
\(942\) −9.99173 −0.325548
\(943\) −2.23395 −0.0727474
\(944\) −5.45031 −0.177392
\(945\) 2.83411 0.0921936
\(946\) 1.51465 0.0492455
\(947\) −46.3365 −1.50573 −0.752867 0.658173i \(-0.771329\pi\)
−0.752867 + 0.658173i \(0.771329\pi\)
\(948\) 1.67721 0.0544733
\(949\) 0 0
\(950\) −3.08631 −0.100133
\(951\) 11.1386 0.361193
\(952\) 0.445042 0.0144239
\(953\) −2.30904 −0.0747971 −0.0373986 0.999300i \(-0.511907\pi\)
−0.0373986 + 0.999300i \(0.511907\pi\)
\(954\) −6.50916 −0.210742
\(955\) 43.8978 1.42050
\(956\) 21.0447 0.680635
\(957\) −0.860594 −0.0278190
\(958\) −11.2400 −0.363148
\(959\) 21.5583 0.696155
\(960\) −2.83411 −0.0914705
\(961\) −26.0946 −0.841762
\(962\) 0 0
\(963\) −10.8776 −0.350525
\(964\) −17.4505 −0.562042
\(965\) −30.4437 −0.980016
\(966\) −5.03709 −0.162066
\(967\) 50.8169 1.63416 0.817081 0.576523i \(-0.195591\pi\)
0.817081 + 0.576523i \(0.195591\pi\)
\(968\) 10.9608 0.352293
\(969\) 0.452987 0.0145521
\(970\) 27.4702 0.882015
\(971\) 32.9502 1.05742 0.528711 0.848802i \(-0.322676\pi\)
0.528711 + 0.848802i \(0.322676\pi\)
\(972\) 1.00000 0.0320750
\(973\) 2.63351 0.0844264
\(974\) 10.0813 0.323027
\(975\) 0 0
\(976\) 11.3069 0.361926
\(977\) −4.75978 −0.152279 −0.0761394 0.997097i \(-0.524259\pi\)
−0.0761394 + 0.997097i \(0.524259\pi\)
\(978\) 3.18547 0.101860
\(979\) −2.02575 −0.0647432
\(980\) −2.83411 −0.0905323
\(981\) −8.51454 −0.271848
\(982\) −17.6228 −0.562367
\(983\) 27.2726 0.869861 0.434931 0.900464i \(-0.356773\pi\)
0.434931 + 0.900464i \(0.356773\pi\)
\(984\) −0.443500 −0.0141383
\(985\) −4.64783 −0.148092
\(986\) −1.93374 −0.0615827
\(987\) 3.63873 0.115822
\(988\) 0 0
\(989\) −38.5203 −1.22487
\(990\) −0.561330 −0.0178402
\(991\) −32.9413 −1.04642 −0.523208 0.852205i \(-0.675265\pi\)
−0.523208 + 0.852205i \(0.675265\pi\)
\(992\) 2.21481 0.0703203
\(993\) −20.8141 −0.660515
\(994\) −5.80083 −0.183991
\(995\) 35.2987 1.11904
\(996\) −6.42808 −0.203681
\(997\) −25.7062 −0.814122 −0.407061 0.913401i \(-0.633446\pi\)
−0.407061 + 0.913401i \(0.633446\pi\)
\(998\) −38.6205 −1.22251
\(999\) 7.43142 0.235120
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 7098.2.a.cr.1.3 6
13.12 even 2 7098.2.a.ct.1.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cr.1.3 6 1.1 even 1 trivial
7098.2.a.ct.1.4 yes 6 13.12 even 2