Properties

Label 5390.2.a.ca.1.3
Level $5390$
Weight $2$
Character 5390.1
Self dual yes
Analytic conductor $43.039$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.0393666895\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 5390.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.24914 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.24914 q^{6} +1.00000 q^{8} +2.05863 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.24914 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.24914 q^{6} +1.00000 q^{8} +2.05863 q^{9} -1.00000 q^{10} +1.00000 q^{11} +2.24914 q^{12} -0.941367 q^{13} -2.24914 q^{15} +1.00000 q^{16} -6.49828 q^{17} +2.05863 q^{18} +4.36641 q^{19} -1.00000 q^{20} +1.00000 q^{22} +6.24914 q^{23} +2.24914 q^{24} +1.00000 q^{25} -0.941367 q^{26} -2.11727 q^{27} +8.74742 q^{29} -2.24914 q^{30} +9.55691 q^{31} +1.00000 q^{32} +2.24914 q^{33} -6.49828 q^{34} +2.05863 q^{36} +4.24914 q^{37} +4.36641 q^{38} -2.11727 q^{39} -1.00000 q^{40} +2.13187 q^{41} +7.67418 q^{43} +1.00000 q^{44} -2.05863 q^{45} +6.24914 q^{46} -11.1138 q^{47} +2.24914 q^{48} +1.00000 q^{50} -14.6155 q^{51} -0.941367 q^{52} -4.74742 q^{53} -2.11727 q^{54} -1.00000 q^{55} +9.82066 q^{57} +8.74742 q^{58} +1.88273 q^{59} -2.24914 q^{60} -9.11383 q^{61} +9.55691 q^{62} +1.00000 q^{64} +0.941367 q^{65} +2.24914 q^{66} +12.9966 q^{67} -6.49828 q^{68} +14.0552 q^{69} -14.6155 q^{71} +2.05863 q^{72} +10.4983 q^{73} +4.24914 q^{74} +2.24914 q^{75} +4.36641 q^{76} -2.11727 q^{78} +8.36641 q^{79} -1.00000 q^{80} -10.9379 q^{81} +2.13187 q^{82} +8.49828 q^{83} +6.49828 q^{85} +7.67418 q^{86} +19.6742 q^{87} +1.00000 q^{88} -12.3810 q^{89} -2.05863 q^{90} +6.24914 q^{92} +21.4948 q^{93} -11.1138 q^{94} -4.36641 q^{95} +2.24914 q^{96} +15.3630 q^{97} +2.05863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 2q^{3} + 3q^{4} - 3q^{5} - 2q^{6} + 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 2q^{3} + 3q^{4} - 3q^{5} - 2q^{6} + 3q^{8} + 7q^{9} - 3q^{10} + 3q^{11} - 2q^{12} - 2q^{13} + 2q^{15} + 3q^{16} - 2q^{17} + 7q^{18} + 6q^{19} - 3q^{20} + 3q^{22} + 10q^{23} - 2q^{24} + 3q^{25} - 2q^{26} - 8q^{27} + 2q^{30} + 12q^{31} + 3q^{32} - 2q^{33} - 2q^{34} + 7q^{36} + 4q^{37} + 6q^{38} - 8q^{39} - 3q^{40} - 4q^{41} + 8q^{43} + 3q^{44} - 7q^{45} + 10q^{46} - 2q^{48} + 3q^{50} - 28q^{51} - 2q^{52} + 12q^{53} - 8q^{54} - 3q^{55} - 8q^{57} + 4q^{59} + 2q^{60} + 6q^{61} + 12q^{62} + 3q^{64} + 2q^{65} - 2q^{66} + 4q^{67} - 2q^{68} + 8q^{69} - 28q^{71} + 7q^{72} + 14q^{73} + 4q^{74} - 2q^{75} + 6q^{76} - 8q^{78} + 18q^{79} - 3q^{80} + 3q^{81} - 4q^{82} + 8q^{83} + 2q^{85} + 8q^{86} + 44q^{87} + 3q^{88} - 18q^{89} - 7q^{90} + 10q^{92} + 12q^{93} - 6q^{95} - 2q^{96} + 4q^{97} + 7q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.24914 1.29854 0.649271 0.760557i \(-0.275074\pi\)
0.649271 + 0.760557i \(0.275074\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.24914 0.918208
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 2.05863 0.686211
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 2.24914 0.649271
\(13\) −0.941367 −0.261088 −0.130544 0.991443i \(-0.541672\pi\)
−0.130544 + 0.991443i \(0.541672\pi\)
\(14\) 0 0
\(15\) −2.24914 −0.580726
\(16\) 1.00000 0.250000
\(17\) −6.49828 −1.57606 −0.788032 0.615634i \(-0.788900\pi\)
−0.788032 + 0.615634i \(0.788900\pi\)
\(18\) 2.05863 0.485224
\(19\) 4.36641 1.00172 0.500861 0.865528i \(-0.333017\pi\)
0.500861 + 0.865528i \(0.333017\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 6.24914 1.30304 0.651518 0.758633i \(-0.274133\pi\)
0.651518 + 0.758633i \(0.274133\pi\)
\(24\) 2.24914 0.459104
\(25\) 1.00000 0.200000
\(26\) −0.941367 −0.184617
\(27\) −2.11727 −0.407468
\(28\) 0 0
\(29\) 8.74742 1.62436 0.812178 0.583410i \(-0.198282\pi\)
0.812178 + 0.583410i \(0.198282\pi\)
\(30\) −2.24914 −0.410635
\(31\) 9.55691 1.71647 0.858236 0.513255i \(-0.171560\pi\)
0.858236 + 0.513255i \(0.171560\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.24914 0.391525
\(34\) −6.49828 −1.11445
\(35\) 0 0
\(36\) 2.05863 0.343106
\(37\) 4.24914 0.698554 0.349277 0.937019i \(-0.386427\pi\)
0.349277 + 0.937019i \(0.386427\pi\)
\(38\) 4.36641 0.708325
\(39\) −2.11727 −0.339034
\(40\) −1.00000 −0.158114
\(41\) 2.13187 0.332943 0.166471 0.986046i \(-0.446763\pi\)
0.166471 + 0.986046i \(0.446763\pi\)
\(42\) 0 0
\(43\) 7.67418 1.17030 0.585151 0.810925i \(-0.301035\pi\)
0.585151 + 0.810925i \(0.301035\pi\)
\(44\) 1.00000 0.150756
\(45\) −2.05863 −0.306883
\(46\) 6.24914 0.921386
\(47\) −11.1138 −1.62112 −0.810559 0.585657i \(-0.800837\pi\)
−0.810559 + 0.585657i \(0.800837\pi\)
\(48\) 2.24914 0.324635
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −14.6155 −2.04659
\(52\) −0.941367 −0.130544
\(53\) −4.74742 −0.652109 −0.326054 0.945351i \(-0.605719\pi\)
−0.326054 + 0.945351i \(0.605719\pi\)
\(54\) −2.11727 −0.288123
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 9.82066 1.30078
\(58\) 8.74742 1.14859
\(59\) 1.88273 0.245111 0.122556 0.992462i \(-0.460891\pi\)
0.122556 + 0.992462i \(0.460891\pi\)
\(60\) −2.24914 −0.290363
\(61\) −9.11383 −1.16691 −0.583453 0.812147i \(-0.698299\pi\)
−0.583453 + 0.812147i \(0.698299\pi\)
\(62\) 9.55691 1.21373
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.941367 0.116762
\(66\) 2.24914 0.276850
\(67\) 12.9966 1.58778 0.793891 0.608060i \(-0.208052\pi\)
0.793891 + 0.608060i \(0.208052\pi\)
\(68\) −6.49828 −0.788032
\(69\) 14.0552 1.69205
\(70\) 0 0
\(71\) −14.6155 −1.73455 −0.867273 0.497833i \(-0.834129\pi\)
−0.867273 + 0.497833i \(0.834129\pi\)
\(72\) 2.05863 0.242612
\(73\) 10.4983 1.22873 0.614365 0.789022i \(-0.289412\pi\)
0.614365 + 0.789022i \(0.289412\pi\)
\(74\) 4.24914 0.493953
\(75\) 2.24914 0.259708
\(76\) 4.36641 0.500861
\(77\) 0 0
\(78\) −2.11727 −0.239733
\(79\) 8.36641 0.941294 0.470647 0.882322i \(-0.344020\pi\)
0.470647 + 0.882322i \(0.344020\pi\)
\(80\) −1.00000 −0.111803
\(81\) −10.9379 −1.21533
\(82\) 2.13187 0.235426
\(83\) 8.49828 0.932808 0.466404 0.884572i \(-0.345549\pi\)
0.466404 + 0.884572i \(0.345549\pi\)
\(84\) 0 0
\(85\) 6.49828 0.704838
\(86\) 7.67418 0.827528
\(87\) 19.6742 2.10929
\(88\) 1.00000 0.106600
\(89\) −12.3810 −1.31238 −0.656192 0.754594i \(-0.727834\pi\)
−0.656192 + 0.754594i \(0.727834\pi\)
\(90\) −2.05863 −0.216999
\(91\) 0 0
\(92\) 6.24914 0.651518
\(93\) 21.4948 2.22891
\(94\) −11.1138 −1.14630
\(95\) −4.36641 −0.447984
\(96\) 2.24914 0.229552
\(97\) 15.3630 1.55987 0.779937 0.625859i \(-0.215251\pi\)
0.779937 + 0.625859i \(0.215251\pi\)
\(98\) 0 0
\(99\) 2.05863 0.206900
\(100\) 1.00000 0.100000
\(101\) 16.8793 1.67955 0.839776 0.542932i \(-0.182686\pi\)
0.839776 + 0.542932i \(0.182686\pi\)
\(102\) −14.6155 −1.44715
\(103\) 6.61555 0.651849 0.325925 0.945396i \(-0.394324\pi\)
0.325925 + 0.945396i \(0.394324\pi\)
\(104\) −0.941367 −0.0923086
\(105\) 0 0
\(106\) −4.74742 −0.461110
\(107\) −14.5535 −1.40694 −0.703469 0.710726i \(-0.748367\pi\)
−0.703469 + 0.710726i \(0.748367\pi\)
\(108\) −2.11727 −0.203734
\(109\) −0.249141 −0.0238633 −0.0119317 0.999929i \(-0.503798\pi\)
−0.0119317 + 0.999929i \(0.503798\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 9.55691 0.907102
\(112\) 0 0
\(113\) 10.9966 1.03447 0.517235 0.855844i \(-0.326961\pi\)
0.517235 + 0.855844i \(0.326961\pi\)
\(114\) 9.82066 0.919789
\(115\) −6.24914 −0.582735
\(116\) 8.74742 0.812178
\(117\) −1.93793 −0.179162
\(118\) 1.88273 0.173320
\(119\) 0 0
\(120\) −2.24914 −0.205318
\(121\) 1.00000 0.0909091
\(122\) −9.11383 −0.825127
\(123\) 4.79488 0.432340
\(124\) 9.55691 0.858236
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −5.88273 −0.522008 −0.261004 0.965338i \(-0.584054\pi\)
−0.261004 + 0.965338i \(0.584054\pi\)
\(128\) 1.00000 0.0883883
\(129\) 17.2603 1.51969
\(130\) 0.941367 0.0825633
\(131\) 1.25258 0.109438 0.0547191 0.998502i \(-0.482574\pi\)
0.0547191 + 0.998502i \(0.482574\pi\)
\(132\) 2.24914 0.195763
\(133\) 0 0
\(134\) 12.9966 1.12273
\(135\) 2.11727 0.182225
\(136\) −6.49828 −0.557223
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 14.0552 1.19646
\(139\) −10.9820 −0.931477 −0.465739 0.884922i \(-0.654211\pi\)
−0.465739 + 0.884922i \(0.654211\pi\)
\(140\) 0 0
\(141\) −24.9966 −2.10509
\(142\) −14.6155 −1.22651
\(143\) −0.941367 −0.0787210
\(144\) 2.05863 0.171553
\(145\) −8.74742 −0.726434
\(146\) 10.4983 0.868844
\(147\) 0 0
\(148\) 4.24914 0.349277
\(149\) 0.0146079 0.00119673 0.000598363 1.00000i \(-0.499810\pi\)
0.000598363 1.00000i \(0.499810\pi\)
\(150\) 2.24914 0.183642
\(151\) −15.2457 −1.24068 −0.620339 0.784334i \(-0.713005\pi\)
−0.620339 + 0.784334i \(0.713005\pi\)
\(152\) 4.36641 0.354162
\(153\) −13.3776 −1.08151
\(154\) 0 0
\(155\) −9.55691 −0.767630
\(156\) −2.11727 −0.169517
\(157\) 5.50172 0.439085 0.219542 0.975603i \(-0.429544\pi\)
0.219542 + 0.975603i \(0.429544\pi\)
\(158\) 8.36641 0.665596
\(159\) −10.6776 −0.846790
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −10.9379 −0.859365
\(163\) −18.2277 −1.42770 −0.713850 0.700298i \(-0.753050\pi\)
−0.713850 + 0.700298i \(0.753050\pi\)
\(164\) 2.13187 0.166471
\(165\) −2.24914 −0.175095
\(166\) 8.49828 0.659595
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −12.1138 −0.931833
\(170\) 6.49828 0.498395
\(171\) 8.98883 0.687393
\(172\) 7.67418 0.585151
\(173\) −0.117266 −0.00891559 −0.00445780 0.999990i \(-0.501419\pi\)
−0.00445780 + 0.999990i \(0.501419\pi\)
\(174\) 19.6742 1.49150
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 4.23453 0.318287
\(178\) −12.3810 −0.927996
\(179\) −22.5535 −1.68573 −0.842863 0.538128i \(-0.819132\pi\)
−0.842863 + 0.538128i \(0.819132\pi\)
\(180\) −2.05863 −0.153441
\(181\) −20.8793 −1.55195 −0.775973 0.630766i \(-0.782741\pi\)
−0.775973 + 0.630766i \(0.782741\pi\)
\(182\) 0 0
\(183\) −20.4983 −1.51528
\(184\) 6.24914 0.460693
\(185\) −4.24914 −0.312403
\(186\) 21.4948 1.57608
\(187\) −6.49828 −0.475201
\(188\) −11.1138 −0.810559
\(189\) 0 0
\(190\) −4.36641 −0.316772
\(191\) 3.11383 0.225309 0.112654 0.993634i \(-0.464065\pi\)
0.112654 + 0.993634i \(0.464065\pi\)
\(192\) 2.24914 0.162318
\(193\) −6.17246 −0.444304 −0.222152 0.975012i \(-0.571308\pi\)
−0.222152 + 0.975012i \(0.571308\pi\)
\(194\) 15.3630 1.10300
\(195\) 2.11727 0.151621
\(196\) 0 0
\(197\) 6.73281 0.479693 0.239847 0.970811i \(-0.422903\pi\)
0.239847 + 0.970811i \(0.422903\pi\)
\(198\) 2.05863 0.146301
\(199\) 13.2311 0.937927 0.468964 0.883217i \(-0.344627\pi\)
0.468964 + 0.883217i \(0.344627\pi\)
\(200\) 1.00000 0.0707107
\(201\) 29.2311 2.06180
\(202\) 16.8793 1.18762
\(203\) 0 0
\(204\) −14.6155 −1.02329
\(205\) −2.13187 −0.148897
\(206\) 6.61555 0.460927
\(207\) 12.8647 0.894158
\(208\) −0.941367 −0.0652720
\(209\) 4.36641 0.302031
\(210\) 0 0
\(211\) 23.1138 1.59122 0.795611 0.605808i \(-0.207150\pi\)
0.795611 + 0.605808i \(0.207150\pi\)
\(212\) −4.74742 −0.326054
\(213\) −32.8724 −2.25238
\(214\) −14.5535 −0.994855
\(215\) −7.67418 −0.523375
\(216\) −2.11727 −0.144062
\(217\) 0 0
\(218\) −0.249141 −0.0168739
\(219\) 23.6121 1.59556
\(220\) −1.00000 −0.0674200
\(221\) 6.11727 0.411492
\(222\) 9.55691 0.641418
\(223\) −10.3810 −0.695164 −0.347582 0.937650i \(-0.612997\pi\)
−0.347582 + 0.937650i \(0.612997\pi\)
\(224\) 0 0
\(225\) 2.05863 0.137242
\(226\) 10.9966 0.731480
\(227\) −16.1104 −1.06928 −0.534642 0.845079i \(-0.679554\pi\)
−0.534642 + 0.845079i \(0.679554\pi\)
\(228\) 9.82066 0.650389
\(229\) −24.2897 −1.60511 −0.802555 0.596578i \(-0.796527\pi\)
−0.802555 + 0.596578i \(0.796527\pi\)
\(230\) −6.24914 −0.412056
\(231\) 0 0
\(232\) 8.74742 0.574296
\(233\) 6.88617 0.451128 0.225564 0.974228i \(-0.427578\pi\)
0.225564 + 0.974228i \(0.427578\pi\)
\(234\) −1.93793 −0.126686
\(235\) 11.1138 0.724986
\(236\) 1.88273 0.122556
\(237\) 18.8172 1.22231
\(238\) 0 0
\(239\) −11.1353 −0.720283 −0.360142 0.932898i \(-0.617272\pi\)
−0.360142 + 0.932898i \(0.617272\pi\)
\(240\) −2.24914 −0.145181
\(241\) −2.36641 −0.152434 −0.0762168 0.997091i \(-0.524284\pi\)
−0.0762168 + 0.997091i \(0.524284\pi\)
\(242\) 1.00000 0.0642824
\(243\) −18.2491 −1.17068
\(244\) −9.11383 −0.583453
\(245\) 0 0
\(246\) 4.79488 0.305711
\(247\) −4.11039 −0.261538
\(248\) 9.55691 0.606865
\(249\) 19.1138 1.21129
\(250\) −1.00000 −0.0632456
\(251\) −25.1070 −1.58474 −0.792368 0.610043i \(-0.791152\pi\)
−0.792368 + 0.610043i \(0.791152\pi\)
\(252\) 0 0
\(253\) 6.24914 0.392880
\(254\) −5.88273 −0.369116
\(255\) 14.6155 0.915261
\(256\) 1.00000 0.0625000
\(257\) 2.86469 0.178694 0.0893472 0.996001i \(-0.471522\pi\)
0.0893472 + 0.996001i \(0.471522\pi\)
\(258\) 17.2603 1.07458
\(259\) 0 0
\(260\) 0.941367 0.0583811
\(261\) 18.0077 1.11465
\(262\) 1.25258 0.0773846
\(263\) 3.76547 0.232189 0.116094 0.993238i \(-0.462963\pi\)
0.116094 + 0.993238i \(0.462963\pi\)
\(264\) 2.24914 0.138425
\(265\) 4.74742 0.291632
\(266\) 0 0
\(267\) −27.8466 −1.70419
\(268\) 12.9966 0.793891
\(269\) −8.94137 −0.545165 −0.272582 0.962132i \(-0.587878\pi\)
−0.272582 + 0.962132i \(0.587878\pi\)
\(270\) 2.11727 0.128853
\(271\) −21.4948 −1.30572 −0.652859 0.757479i \(-0.726431\pi\)
−0.652859 + 0.757479i \(0.726431\pi\)
\(272\) −6.49828 −0.394016
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 1.00000 0.0603023
\(276\) 14.0552 0.846023
\(277\) 7.64820 0.459536 0.229768 0.973245i \(-0.426203\pi\)
0.229768 + 0.973245i \(0.426203\pi\)
\(278\) −10.9820 −0.658654
\(279\) 19.6742 1.17786
\(280\) 0 0
\(281\) −28.6155 −1.70706 −0.853530 0.521043i \(-0.825543\pi\)
−0.853530 + 0.521043i \(0.825543\pi\)
\(282\) −24.9966 −1.48852
\(283\) 2.87930 0.171156 0.0855782 0.996331i \(-0.472726\pi\)
0.0855782 + 0.996331i \(0.472726\pi\)
\(284\) −14.6155 −0.867273
\(285\) −9.82066 −0.581726
\(286\) −0.941367 −0.0556642
\(287\) 0 0
\(288\) 2.05863 0.121306
\(289\) 25.2277 1.48398
\(290\) −8.74742 −0.513666
\(291\) 34.5535 2.02556
\(292\) 10.4983 0.614365
\(293\) 12.9414 0.756043 0.378021 0.925797i \(-0.376605\pi\)
0.378021 + 0.925797i \(0.376605\pi\)
\(294\) 0 0
\(295\) −1.88273 −0.109617
\(296\) 4.24914 0.246976
\(297\) −2.11727 −0.122856
\(298\) 0.0146079 0.000846213 0
\(299\) −5.88273 −0.340207
\(300\) 2.24914 0.129854
\(301\) 0 0
\(302\) −15.2457 −0.877292
\(303\) 37.9639 2.18097
\(304\) 4.36641 0.250431
\(305\) 9.11383 0.521856
\(306\) −13.3776 −0.764745
\(307\) −0.498281 −0.0284384 −0.0142192 0.999899i \(-0.504526\pi\)
−0.0142192 + 0.999899i \(0.504526\pi\)
\(308\) 0 0
\(309\) 14.8793 0.846454
\(310\) −9.55691 −0.542796
\(311\) 23.4396 1.32914 0.664570 0.747226i \(-0.268615\pi\)
0.664570 + 0.747226i \(0.268615\pi\)
\(312\) −2.11727 −0.119867
\(313\) −9.36984 −0.529615 −0.264807 0.964301i \(-0.585308\pi\)
−0.264807 + 0.964301i \(0.585308\pi\)
\(314\) 5.50172 0.310480
\(315\) 0 0
\(316\) 8.36641 0.470647
\(317\) −9.97852 −0.560449 −0.280225 0.959934i \(-0.590409\pi\)
−0.280225 + 0.959934i \(0.590409\pi\)
\(318\) −10.6776 −0.598771
\(319\) 8.74742 0.489762
\(320\) −1.00000 −0.0559017
\(321\) −32.7328 −1.82697
\(322\) 0 0
\(323\) −28.3741 −1.57878
\(324\) −10.9379 −0.607663
\(325\) −0.941367 −0.0522176
\(326\) −18.2277 −1.00954
\(327\) −0.560352 −0.0309875
\(328\) 2.13187 0.117713
\(329\) 0 0
\(330\) −2.24914 −0.123811
\(331\) −20.4362 −1.12328 −0.561638 0.827383i \(-0.689829\pi\)
−0.561638 + 0.827383i \(0.689829\pi\)
\(332\) 8.49828 0.466404
\(333\) 8.74742 0.479356
\(334\) 8.00000 0.437741
\(335\) −12.9966 −0.710078
\(336\) 0 0
\(337\) 7.88273 0.429400 0.214700 0.976680i \(-0.431123\pi\)
0.214700 + 0.976680i \(0.431123\pi\)
\(338\) −12.1138 −0.658905
\(339\) 24.7328 1.34330
\(340\) 6.49828 0.352419
\(341\) 9.55691 0.517536
\(342\) 8.98883 0.486060
\(343\) 0 0
\(344\) 7.67418 0.413764
\(345\) −14.0552 −0.756706
\(346\) −0.117266 −0.00630428
\(347\) 13.5569 0.727773 0.363887 0.931443i \(-0.381450\pi\)
0.363887 + 0.931443i \(0.381450\pi\)
\(348\) 19.6742 1.05465
\(349\) 10.7328 0.574514 0.287257 0.957853i \(-0.407257\pi\)
0.287257 + 0.957853i \(0.407257\pi\)
\(350\) 0 0
\(351\) 1.99312 0.106385
\(352\) 1.00000 0.0533002
\(353\) 20.8578 1.11015 0.555075 0.831801i \(-0.312690\pi\)
0.555075 + 0.831801i \(0.312690\pi\)
\(354\) 4.23453 0.225063
\(355\) 14.6155 0.775713
\(356\) −12.3810 −0.656192
\(357\) 0 0
\(358\) −22.5535 −1.19199
\(359\) 0.366407 0.0193382 0.00966911 0.999953i \(-0.496922\pi\)
0.00966911 + 0.999953i \(0.496922\pi\)
\(360\) −2.05863 −0.108499
\(361\) 0.0655089 0.00344783
\(362\) −20.8793 −1.09739
\(363\) 2.24914 0.118049
\(364\) 0 0
\(365\) −10.4983 −0.549505
\(366\) −20.4983 −1.07146
\(367\) −20.8432 −1.08801 −0.544003 0.839083i \(-0.683092\pi\)
−0.544003 + 0.839083i \(0.683092\pi\)
\(368\) 6.24914 0.325759
\(369\) 4.38875 0.228469
\(370\) −4.24914 −0.220902
\(371\) 0 0
\(372\) 21.4948 1.11446
\(373\) −5.37758 −0.278440 −0.139220 0.990261i \(-0.544460\pi\)
−0.139220 + 0.990261i \(0.544460\pi\)
\(374\) −6.49828 −0.336018
\(375\) −2.24914 −0.116145
\(376\) −11.1138 −0.573152
\(377\) −8.23453 −0.424100
\(378\) 0 0
\(379\) 3.53093 0.181372 0.0906860 0.995880i \(-0.471094\pi\)
0.0906860 + 0.995880i \(0.471094\pi\)
\(380\) −4.36641 −0.223992
\(381\) −13.2311 −0.677850
\(382\) 3.11383 0.159317
\(383\) −1.38445 −0.0707422 −0.0353711 0.999374i \(-0.511261\pi\)
−0.0353711 + 0.999374i \(0.511261\pi\)
\(384\) 2.24914 0.114776
\(385\) 0 0
\(386\) −6.17246 −0.314170
\(387\) 15.7983 0.803074
\(388\) 15.3630 0.779937
\(389\) 15.7294 0.797511 0.398756 0.917057i \(-0.369442\pi\)
0.398756 + 0.917057i \(0.369442\pi\)
\(390\) 2.11727 0.107212
\(391\) −40.6087 −2.05367
\(392\) 0 0
\(393\) 2.81722 0.142110
\(394\) 6.73281 0.339194
\(395\) −8.36641 −0.420960
\(396\) 2.05863 0.103450
\(397\) 17.6121 0.883926 0.441963 0.897033i \(-0.354282\pi\)
0.441963 + 0.897033i \(0.354282\pi\)
\(398\) 13.2311 0.663215
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −32.8172 −1.63881 −0.819407 0.573212i \(-0.805697\pi\)
−0.819407 + 0.573212i \(0.805697\pi\)
\(402\) 29.2311 1.45791
\(403\) −8.99656 −0.448151
\(404\) 16.8793 0.839776
\(405\) 10.9379 0.543510
\(406\) 0 0
\(407\) 4.24914 0.210622
\(408\) −14.6155 −0.723577
\(409\) 33.3561 1.64935 0.824676 0.565605i \(-0.191357\pi\)
0.824676 + 0.565605i \(0.191357\pi\)
\(410\) −2.13187 −0.105286
\(411\) 22.4914 1.10942
\(412\) 6.61555 0.325925
\(413\) 0 0
\(414\) 12.8647 0.632265
\(415\) −8.49828 −0.417164
\(416\) −0.941367 −0.0461543
\(417\) −24.7000 −1.20956
\(418\) 4.36641 0.213568
\(419\) −12.3449 −0.603089 −0.301544 0.953452i \(-0.597502\pi\)
−0.301544 + 0.953452i \(0.597502\pi\)
\(420\) 0 0
\(421\) −8.87930 −0.432750 −0.216375 0.976310i \(-0.569423\pi\)
−0.216375 + 0.976310i \(0.569423\pi\)
\(422\) 23.1138 1.12516
\(423\) −22.8793 −1.11243
\(424\) −4.74742 −0.230555
\(425\) −6.49828 −0.315213
\(426\) −32.8724 −1.59267
\(427\) 0 0
\(428\) −14.5535 −0.703469
\(429\) −2.11727 −0.102223
\(430\) −7.67418 −0.370082
\(431\) 18.2784 0.880437 0.440219 0.897891i \(-0.354901\pi\)
0.440219 + 0.897891i \(0.354901\pi\)
\(432\) −2.11727 −0.101867
\(433\) −6.86469 −0.329896 −0.164948 0.986302i \(-0.552746\pi\)
−0.164948 + 0.986302i \(0.552746\pi\)
\(434\) 0 0
\(435\) −19.6742 −0.943305
\(436\) −0.249141 −0.0119317
\(437\) 27.2863 1.30528
\(438\) 23.6121 1.12823
\(439\) 11.8466 0.565409 0.282705 0.959207i \(-0.408768\pi\)
0.282705 + 0.959207i \(0.408768\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 6.11727 0.290969
\(443\) 17.8827 0.849634 0.424817 0.905279i \(-0.360338\pi\)
0.424817 + 0.905279i \(0.360338\pi\)
\(444\) 9.55691 0.453551
\(445\) 12.3810 0.586916
\(446\) −10.3810 −0.491555
\(447\) 0.0328552 0.00155400
\(448\) 0 0
\(449\) −16.8172 −0.793654 −0.396827 0.917893i \(-0.629889\pi\)
−0.396827 + 0.917893i \(0.629889\pi\)
\(450\) 2.05863 0.0970449
\(451\) 2.13187 0.100386
\(452\) 10.9966 0.517235
\(453\) −34.2897 −1.61107
\(454\) −16.1104 −0.756098
\(455\) 0 0
\(456\) 9.82066 0.459895
\(457\) −12.2277 −0.571986 −0.285993 0.958232i \(-0.592323\pi\)
−0.285993 + 0.958232i \(0.592323\pi\)
\(458\) −24.2897 −1.13498
\(459\) 13.7586 0.642196
\(460\) −6.24914 −0.291368
\(461\) 16.6155 0.773863 0.386932 0.922108i \(-0.373535\pi\)
0.386932 + 0.922108i \(0.373535\pi\)
\(462\) 0 0
\(463\) 18.4837 0.859009 0.429505 0.903065i \(-0.358688\pi\)
0.429505 + 0.903065i \(0.358688\pi\)
\(464\) 8.74742 0.406089
\(465\) −21.4948 −0.996799
\(466\) 6.88617 0.318996
\(467\) −24.3956 −1.12889 −0.564447 0.825469i \(-0.690911\pi\)
−0.564447 + 0.825469i \(0.690911\pi\)
\(468\) −1.93793 −0.0895808
\(469\) 0 0
\(470\) 11.1138 0.512643
\(471\) 12.3741 0.570170
\(472\) 1.88273 0.0866598
\(473\) 7.67418 0.352859
\(474\) 18.8172 0.864304
\(475\) 4.36641 0.200344
\(476\) 0 0
\(477\) −9.77320 −0.447484
\(478\) −11.1353 −0.509317
\(479\) 27.7655 1.26864 0.634318 0.773072i \(-0.281281\pi\)
0.634318 + 0.773072i \(0.281281\pi\)
\(480\) −2.24914 −0.102659
\(481\) −4.00000 −0.182384
\(482\) −2.36641 −0.107787
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −15.3630 −0.697596
\(486\) −18.2491 −0.827798
\(487\) 10.0958 0.457484 0.228742 0.973487i \(-0.426539\pi\)
0.228742 + 0.973487i \(0.426539\pi\)
\(488\) −9.11383 −0.412564
\(489\) −40.9966 −1.85393
\(490\) 0 0
\(491\) −32.1104 −1.44912 −0.724561 0.689211i \(-0.757957\pi\)
−0.724561 + 0.689211i \(0.757957\pi\)
\(492\) 4.79488 0.216170
\(493\) −56.8432 −2.56009
\(494\) −4.11039 −0.184935
\(495\) −2.05863 −0.0925287
\(496\) 9.55691 0.429118
\(497\) 0 0
\(498\) 19.1138 0.856511
\(499\) −8.79488 −0.393713 −0.196857 0.980432i \(-0.563073\pi\)
−0.196857 + 0.980432i \(0.563073\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 17.9931 0.803874
\(502\) −25.1070 −1.12058
\(503\) 15.0034 0.668970 0.334485 0.942401i \(-0.391438\pi\)
0.334485 + 0.942401i \(0.391438\pi\)
\(504\) 0 0
\(505\) −16.8793 −0.751119
\(506\) 6.24914 0.277808
\(507\) −27.2457 −1.21002
\(508\) −5.88273 −0.261004
\(509\) −21.8759 −0.969630 −0.484815 0.874617i \(-0.661113\pi\)
−0.484815 + 0.874617i \(0.661113\pi\)
\(510\) 14.6155 0.647187
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −9.24485 −0.408170
\(514\) 2.86469 0.126356
\(515\) −6.61555 −0.291516
\(516\) 17.2603 0.759843
\(517\) −11.1138 −0.488786
\(518\) 0 0
\(519\) −0.263748 −0.0115773
\(520\) 0.941367 0.0412817
\(521\) −14.0292 −0.614631 −0.307316 0.951608i \(-0.599431\pi\)
−0.307316 + 0.951608i \(0.599431\pi\)
\(522\) 18.0077 0.788177
\(523\) −1.14992 −0.0502825 −0.0251412 0.999684i \(-0.508004\pi\)
−0.0251412 + 0.999684i \(0.508004\pi\)
\(524\) 1.25258 0.0547191
\(525\) 0 0
\(526\) 3.76547 0.164182
\(527\) −62.1035 −2.70527
\(528\) 2.24914 0.0978813
\(529\) 16.0518 0.697902
\(530\) 4.74742 0.206215
\(531\) 3.87586 0.168198
\(532\) 0 0
\(533\) −2.00688 −0.0869274
\(534\) −27.8466 −1.20504
\(535\) 14.5535 0.629202
\(536\) 12.9966 0.561366
\(537\) −50.7259 −2.18899
\(538\) −8.94137 −0.385490
\(539\) 0 0
\(540\) 2.11727 0.0911126
\(541\) −6.47680 −0.278459 −0.139230 0.990260i \(-0.544463\pi\)
−0.139230 + 0.990260i \(0.544463\pi\)
\(542\) −21.4948 −0.923283
\(543\) −46.9605 −2.01527
\(544\) −6.49828 −0.278612
\(545\) 0.249141 0.0106720
\(546\) 0 0
\(547\) 12.9966 0.555693 0.277846 0.960626i \(-0.410379\pi\)
0.277846 + 0.960626i \(0.410379\pi\)
\(548\) 10.0000 0.427179
\(549\) −18.7620 −0.800744
\(550\) 1.00000 0.0426401
\(551\) 38.1948 1.62715
\(552\) 14.0552 0.598229
\(553\) 0 0
\(554\) 7.64820 0.324941
\(555\) −9.55691 −0.405668
\(556\) −10.9820 −0.465739
\(557\) −16.4914 −0.698763 −0.349382 0.936981i \(-0.613608\pi\)
−0.349382 + 0.936981i \(0.613608\pi\)
\(558\) 19.6742 0.832874
\(559\) −7.22422 −0.305552
\(560\) 0 0
\(561\) −14.6155 −0.617069
\(562\) −28.6155 −1.20707
\(563\) 16.7620 0.706435 0.353218 0.935541i \(-0.385088\pi\)
0.353218 + 0.935541i \(0.385088\pi\)
\(564\) −24.9966 −1.05255
\(565\) −10.9966 −0.462629
\(566\) 2.87930 0.121026
\(567\) 0 0
\(568\) −14.6155 −0.613255
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −9.82066 −0.411342
\(571\) 12.7328 0.532852 0.266426 0.963855i \(-0.414157\pi\)
0.266426 + 0.963855i \(0.414157\pi\)
\(572\) −0.941367 −0.0393605
\(573\) 7.00344 0.292573
\(574\) 0 0
\(575\) 6.24914 0.260607
\(576\) 2.05863 0.0857764
\(577\) 11.8613 0.493790 0.246895 0.969042i \(-0.420590\pi\)
0.246895 + 0.969042i \(0.420590\pi\)
\(578\) 25.2277 1.04933
\(579\) −13.8827 −0.576947
\(580\) −8.74742 −0.363217
\(581\) 0 0
\(582\) 34.5535 1.43229
\(583\) −4.74742 −0.196618
\(584\) 10.4983 0.434422
\(585\) 1.93793 0.0801235
\(586\) 12.9414 0.534603
\(587\) 8.60094 0.354999 0.177499 0.984121i \(-0.443199\pi\)
0.177499 + 0.984121i \(0.443199\pi\)
\(588\) 0 0
\(589\) 41.7294 1.71943
\(590\) −1.88273 −0.0775109
\(591\) 15.1430 0.622902
\(592\) 4.24914 0.174639
\(593\) −10.7328 −0.440744 −0.220372 0.975416i \(-0.570727\pi\)
−0.220372 + 0.975416i \(0.570727\pi\)
\(594\) −2.11727 −0.0868725
\(595\) 0 0
\(596\) 0.0146079 0.000598363 0
\(597\) 29.7586 1.21794
\(598\) −5.88273 −0.240563
\(599\) 16.0812 0.657059 0.328529 0.944494i \(-0.393447\pi\)
0.328529 + 0.944494i \(0.393447\pi\)
\(600\) 2.24914 0.0918208
\(601\) 34.0889 1.39052 0.695258 0.718760i \(-0.255290\pi\)
0.695258 + 0.718760i \(0.255290\pi\)
\(602\) 0 0
\(603\) 26.7552 1.08955
\(604\) −15.2457 −0.620339
\(605\) −1.00000 −0.0406558
\(606\) 37.9639 1.54218
\(607\) −11.6742 −0.473840 −0.236920 0.971529i \(-0.576138\pi\)
−0.236920 + 0.971529i \(0.576138\pi\)
\(608\) 4.36641 0.177081
\(609\) 0 0
\(610\) 9.11383 0.369008
\(611\) 10.4622 0.423255
\(612\) −13.3776 −0.540756
\(613\) −34.2637 −1.38390 −0.691950 0.721946i \(-0.743248\pi\)
−0.691950 + 0.721946i \(0.743248\pi\)
\(614\) −0.498281 −0.0201090
\(615\) −4.79488 −0.193348
\(616\) 0 0
\(617\) 39.3707 1.58500 0.792502 0.609869i \(-0.208778\pi\)
0.792502 + 0.609869i \(0.208778\pi\)
\(618\) 14.8793 0.598533
\(619\) −16.1104 −0.647531 −0.323766 0.946137i \(-0.604949\pi\)
−0.323766 + 0.946137i \(0.604949\pi\)
\(620\) −9.55691 −0.383815
\(621\) −13.2311 −0.530946
\(622\) 23.4396 0.939844
\(623\) 0 0
\(624\) −2.11727 −0.0847585
\(625\) 1.00000 0.0400000
\(626\) −9.36984 −0.374494
\(627\) 9.82066 0.392199
\(628\) 5.50172 0.219542
\(629\) −27.6121 −1.10097
\(630\) 0 0
\(631\) 13.8827 0.552663 0.276331 0.961062i \(-0.410881\pi\)
0.276331 + 0.961062i \(0.410881\pi\)
\(632\) 8.36641 0.332798
\(633\) 51.9862 2.06627
\(634\) −9.97852 −0.396298
\(635\) 5.88273 0.233449
\(636\) −10.6776 −0.423395
\(637\) 0 0
\(638\) 8.74742 0.346314
\(639\) −30.0881 −1.19026
\(640\) −1.00000 −0.0395285
\(641\) −39.3415 −1.55390 −0.776948 0.629565i \(-0.783233\pi\)
−0.776948 + 0.629565i \(0.783233\pi\)
\(642\) −32.7328 −1.29186
\(643\) −38.3595 −1.51275 −0.756376 0.654137i \(-0.773032\pi\)
−0.756376 + 0.654137i \(0.773032\pi\)
\(644\) 0 0
\(645\) −17.2603 −0.679624
\(646\) −28.3741 −1.11637
\(647\) −25.7294 −1.01153 −0.505763 0.862672i \(-0.668789\pi\)
−0.505763 + 0.862672i \(0.668789\pi\)
\(648\) −10.9379 −0.429682
\(649\) 1.88273 0.0739038
\(650\) −0.941367 −0.0369234
\(651\) 0 0
\(652\) −18.2277 −0.713850
\(653\) 18.4768 0.723053 0.361526 0.932362i \(-0.382256\pi\)
0.361526 + 0.932362i \(0.382256\pi\)
\(654\) −0.560352 −0.0219115
\(655\) −1.25258 −0.0489423
\(656\) 2.13187 0.0832357
\(657\) 21.6121 0.843169
\(658\) 0 0
\(659\) −39.3776 −1.53393 −0.766966 0.641687i \(-0.778235\pi\)
−0.766966 + 0.641687i \(0.778235\pi\)
\(660\) −2.24914 −0.0875477
\(661\) 34.2208 1.33103 0.665517 0.746383i \(-0.268211\pi\)
0.665517 + 0.746383i \(0.268211\pi\)
\(662\) −20.4362 −0.794276
\(663\) 13.7586 0.534339
\(664\) 8.49828 0.329797
\(665\) 0 0
\(666\) 8.74742 0.338956
\(667\) 54.6639 2.11659
\(668\) 8.00000 0.309529
\(669\) −23.3484 −0.902700
\(670\) −12.9966 −0.502101
\(671\) −9.11383 −0.351835
\(672\) 0 0
\(673\) −24.4622 −0.942948 −0.471474 0.881880i \(-0.656278\pi\)
−0.471474 + 0.881880i \(0.656278\pi\)
\(674\) 7.88273 0.303632
\(675\) −2.11727 −0.0814936
\(676\) −12.1138 −0.465916
\(677\) 39.1070 1.50300 0.751501 0.659732i \(-0.229330\pi\)
0.751501 + 0.659732i \(0.229330\pi\)
\(678\) 24.7328 0.949858
\(679\) 0 0
\(680\) 6.49828 0.249198
\(681\) −36.2345 −1.38851
\(682\) 9.55691 0.365953
\(683\) 23.3776 0.894518 0.447259 0.894404i \(-0.352400\pi\)
0.447259 + 0.894404i \(0.352400\pi\)
\(684\) 8.98883 0.343697
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) −54.6310 −2.08430
\(688\) 7.67418 0.292575
\(689\) 4.46907 0.170258
\(690\) −14.0552 −0.535072
\(691\) −8.49828 −0.323290 −0.161645 0.986849i \(-0.551680\pi\)
−0.161645 + 0.986849i \(0.551680\pi\)
\(692\) −0.117266 −0.00445780
\(693\) 0 0
\(694\) 13.5569 0.514613
\(695\) 10.9820 0.416569
\(696\) 19.6742 0.745748
\(697\) −13.8535 −0.524739
\(698\) 10.7328 0.406243
\(699\) 15.4880 0.585809
\(700\) 0 0
\(701\) 14.9751 0.565601 0.282800 0.959179i \(-0.408737\pi\)
0.282800 + 0.959179i \(0.408737\pi\)
\(702\) 1.99312 0.0752256
\(703\) 18.5535 0.699758
\(704\) 1.00000 0.0376889
\(705\) 24.9966 0.941425
\(706\) 20.8578 0.784994
\(707\) 0 0
\(708\) 4.23453 0.159143
\(709\) 40.7259 1.52949 0.764747 0.644330i \(-0.222864\pi\)
0.764747 + 0.644330i \(0.222864\pi\)
\(710\) 14.6155 0.548512
\(711\) 17.2234 0.645927
\(712\) −12.3810 −0.463998
\(713\) 59.7225 2.23663
\(714\) 0 0
\(715\) 0.941367 0.0352051
\(716\) −22.5535 −0.842863
\(717\) −25.0449 −0.935318
\(718\) 0.366407 0.0136742
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −2.05863 −0.0767207
\(721\) 0 0
\(722\) 0.0655089 0.00243799
\(723\) −5.32238 −0.197942
\(724\) −20.8793 −0.775973
\(725\) 8.74742 0.324871
\(726\) 2.24914 0.0834734
\(727\) 51.6413 1.91527 0.957635 0.287984i \(-0.0929848\pi\)
0.957635 + 0.287984i \(0.0929848\pi\)
\(728\) 0 0
\(729\) −8.23109 −0.304855
\(730\) −10.4983 −0.388559
\(731\) −49.8690 −1.84447
\(732\) −20.4983 −0.757638
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) −20.8432 −0.769337
\(735\) 0 0
\(736\) 6.24914 0.230346
\(737\) 12.9966 0.478735
\(738\) 4.38875 0.161552
\(739\) −49.1070 −1.80643 −0.903214 0.429190i \(-0.858799\pi\)
−0.903214 + 0.429190i \(0.858799\pi\)
\(740\) −4.24914 −0.156202
\(741\) −9.24485 −0.339618
\(742\) 0 0
\(743\) −53.2311 −1.95286 −0.976430 0.215836i \(-0.930752\pi\)
−0.976430 + 0.215836i \(0.930752\pi\)
\(744\) 21.4948 0.788039
\(745\) −0.0146079 −0.000535192 0
\(746\) −5.37758 −0.196887
\(747\) 17.4948 0.640103
\(748\) −6.49828 −0.237601
\(749\) 0 0
\(750\) −2.24914 −0.0821270
\(751\) −31.6121 −1.15354 −0.576771 0.816906i \(-0.695688\pi\)
−0.576771 + 0.816906i \(0.695688\pi\)
\(752\) −11.1138 −0.405280
\(753\) −56.4691 −2.05785
\(754\) −8.23453 −0.299884
\(755\) 15.2457 0.554848
\(756\) 0 0
\(757\) −1.24570 −0.0452758 −0.0226379 0.999744i \(-0.507206\pi\)
−0.0226379 + 0.999744i \(0.507206\pi\)
\(758\) 3.53093 0.128249
\(759\) 14.0552 0.510171
\(760\) −4.36641 −0.158386
\(761\) 10.6009 0.384284 0.192142 0.981367i \(-0.438457\pi\)
0.192142 + 0.981367i \(0.438457\pi\)
\(762\) −13.2311 −0.479312
\(763\) 0 0
\(764\) 3.11383 0.112654
\(765\) 13.3776 0.483667
\(766\) −1.38445 −0.0500223
\(767\) −1.77234 −0.0639956
\(768\) 2.24914 0.0811589
\(769\) 19.1284 0.689789 0.344895 0.938641i \(-0.387915\pi\)
0.344895 + 0.938641i \(0.387915\pi\)
\(770\) 0 0
\(771\) 6.44309 0.232042
\(772\) −6.17246 −0.222152
\(773\) −39.9931 −1.43845 −0.719226 0.694776i \(-0.755504\pi\)
−0.719226 + 0.694776i \(0.755504\pi\)
\(774\) 15.7983 0.567859
\(775\) 9.55691 0.343294
\(776\) 15.3630 0.551498
\(777\) 0 0
\(778\) 15.7294 0.563925
\(779\) 9.30863 0.333516
\(780\) 2.11727 0.0758103
\(781\) −14.6155 −0.522985
\(782\) −40.6087 −1.45216
\(783\) −18.5206 −0.661873
\(784\) 0 0
\(785\) −5.50172 −0.196365
\(786\) 2.81722 0.100487
\(787\) −37.9931 −1.35431 −0.677154 0.735841i \(-0.736787\pi\)
−0.677154 + 0.735841i \(0.736787\pi\)
\(788\) 6.73281 0.239847
\(789\) 8.46907 0.301507
\(790\) −8.36641 −0.297663
\(791\) 0 0
\(792\) 2.05863 0.0731503
\(793\) 8.57946 0.304665
\(794\) 17.6121 0.625030
\(795\) 10.6776 0.378696
\(796\) 13.2311 0.468964
\(797\) 20.3810 0.721933 0.360966 0.932579i \(-0.382447\pi\)
0.360966 + 0.932579i \(0.382447\pi\)
\(798\) 0 0
\(799\) 72.2208 2.55499
\(800\) 1.00000 0.0353553
\(801\) −25.4880 −0.900573
\(802\) −32.8172 −1.15882
\(803\) 10.4983 0.370476
\(804\) 29.2311 1.03090
\(805\) 0 0
\(806\) −8.99656 −0.316890
\(807\) −20.1104 −0.707919
\(808\) 16.8793 0.593812
\(809\) −31.7294 −1.11555 −0.557773 0.829994i \(-0.688344\pi\)
−0.557773 + 0.829994i \(0.688344\pi\)
\(810\) 10.9379 0.384320
\(811\) −43.5095 −1.52782 −0.763912 0.645321i \(-0.776724\pi\)
−0.763912 + 0.645321i \(0.776724\pi\)
\(812\) 0 0
\(813\) −48.3449 −1.69553
\(814\) 4.24914 0.148932
\(815\) 18.2277 0.638487
\(816\) −14.6155 −0.511646
\(817\) 33.5086 1.17232
\(818\) 33.3561 1.16627
\(819\) 0 0
\(820\) −2.13187 −0.0744483
\(821\) −24.7766 −0.864711 −0.432355 0.901703i \(-0.642317\pi\)
−0.432355 + 0.901703i \(0.642317\pi\)
\(822\) 22.4914 0.784478
\(823\) 28.8647 1.00616 0.503080 0.864240i \(-0.332200\pi\)
0.503080 + 0.864240i \(0.332200\pi\)
\(824\) 6.61555 0.230464
\(825\) 2.24914 0.0783050
\(826\) 0 0
\(827\) 34.2277 1.19021 0.595106 0.803647i \(-0.297110\pi\)
0.595106 + 0.803647i \(0.297110\pi\)
\(828\) 12.8647 0.447079
\(829\) −23.6381 −0.820985 −0.410492 0.911864i \(-0.634643\pi\)
−0.410492 + 0.911864i \(0.634643\pi\)
\(830\) −8.49828 −0.294980
\(831\) 17.2019 0.596727
\(832\) −0.941367 −0.0326360
\(833\) 0 0
\(834\) −24.7000 −0.855290
\(835\) −8.00000 −0.276851
\(836\) 4.36641 0.151015
\(837\) −20.2345 −0.699408
\(838\) −12.3449 −0.426448
\(839\) 24.9053 0.859826 0.429913 0.902870i \(-0.358544\pi\)
0.429913 + 0.902870i \(0.358544\pi\)
\(840\) 0 0
\(841\) 47.5174 1.63853
\(842\) −8.87930 −0.306001
\(843\) −64.3604 −2.21669
\(844\) 23.1138 0.795611
\(845\) 12.1138 0.416728
\(846\) −22.8793 −0.786606
\(847\) 0 0
\(848\) −4.74742 −0.163027
\(849\) 6.47594 0.222254
\(850\) −6.49828 −0.222889
\(851\) 26.5535 0.910241
\(852\) −32.8724 −1.12619
\(853\) −22.9966 −0.787387 −0.393694 0.919242i \(-0.628803\pi\)
−0.393694 + 0.919242i \(0.628803\pi\)
\(854\) 0 0
\(855\) −8.98883 −0.307411
\(856\) −14.5535 −0.497428
\(857\) 31.2603 1.06783 0.533916 0.845538i \(-0.320720\pi\)
0.533916 + 0.845538i \(0.320720\pi\)
\(858\) −2.11727 −0.0722823
\(859\) −20.3449 −0.694160 −0.347080 0.937836i \(-0.612827\pi\)
−0.347080 + 0.937836i \(0.612827\pi\)
\(860\) −7.67418 −0.261687
\(861\) 0 0
\(862\) 18.2784 0.622563
\(863\) −58.3595 −1.98658 −0.993291 0.115644i \(-0.963107\pi\)
−0.993291 + 0.115644i \(0.963107\pi\)
\(864\) −2.11727 −0.0720309
\(865\) 0.117266 0.00398717
\(866\) −6.86469 −0.233272
\(867\) 56.7405 1.92701
\(868\) 0 0
\(869\) 8.36641 0.283811
\(870\) −19.6742 −0.667017
\(871\) −12.2345 −0.414551
\(872\) −0.249141 −0.00843696
\(873\) 31.6267 1.07040
\(874\) 27.2863 0.922973
\(875\) 0 0
\(876\) 23.6121 0.797779
\(877\) 11.9639 0.403992 0.201996 0.979386i \(-0.435257\pi\)
0.201996 + 0.979386i \(0.435257\pi\)
\(878\) 11.8466 0.399805
\(879\) 29.1070 0.981753
\(880\) −1.00000 −0.0337100
\(881\) −5.88961 −0.198426 −0.0992130 0.995066i \(-0.531633\pi\)
−0.0992130 + 0.995066i \(0.531633\pi\)
\(882\) 0 0
\(883\) −19.4880 −0.655822 −0.327911 0.944709i \(-0.606345\pi\)
−0.327911 + 0.944709i \(0.606345\pi\)
\(884\) 6.11727 0.205746
\(885\) −4.23453 −0.142342
\(886\) 17.8827 0.600782
\(887\) −50.4622 −1.69435 −0.847177 0.531310i \(-0.821700\pi\)
−0.847177 + 0.531310i \(0.821700\pi\)
\(888\) 9.55691 0.320709
\(889\) 0 0
\(890\) 12.3810 0.415013
\(891\) −10.9379 −0.366434
\(892\) −10.3810 −0.347582
\(893\) −48.5275 −1.62391
\(894\) 0.0328552 0.00109884
\(895\) 22.5535 0.753880
\(896\) 0 0
\(897\) −13.2311 −0.441773
\(898\) −16.8172 −0.561198
\(899\) 83.5984 2.78816
\(900\) 2.05863 0.0686211
\(901\) 30.8501 1.02777
\(902\) 2.13187 0.0709836
\(903\) 0 0
\(904\) 10.9966 0.365740
\(905\) 20.8793 0.694051
\(906\) −34.2897 −1.13920
\(907\) −16.8432 −0.559269 −0.279635 0.960106i \(-0.590213\pi\)
−0.279635 + 0.960106i \(0.590213\pi\)
\(908\) −16.1104 −0.534642
\(909\) 34.7483 1.15253
\(910\) 0 0
\(911\) −38.6155 −1.27939 −0.639695 0.768629i \(-0.720939\pi\)
−0.639695 + 0.768629i \(0.720939\pi\)
\(912\) 9.82066 0.325195
\(913\) 8.49828 0.281252
\(914\) −12.2277 −0.404455
\(915\) 20.4983 0.677652
\(916\) −24.2897 −0.802555
\(917\) 0 0
\(918\) 13.7586 0.454101
\(919\) 17.2818 0.570074 0.285037 0.958517i \(-0.407994\pi\)
0.285037 + 0.958517i \(0.407994\pi\)
\(920\) −6.24914 −0.206028
\(921\) −1.12070 −0.0369285
\(922\) 16.6155 0.547204
\(923\) 13.7586 0.452870
\(924\) 0 0
\(925\) 4.24914 0.139711
\(926\) 18.4837 0.607411
\(927\) 13.6190 0.447306
\(928\) 8.74742 0.287148
\(929\) −2.91539 −0.0956508 −0.0478254 0.998856i \(-0.515229\pi\)
−0.0478254 + 0.998856i \(0.515229\pi\)
\(930\) −21.4948 −0.704844
\(931\) 0 0
\(932\) 6.88617 0.225564
\(933\) 52.7191 1.72594
\(934\) −24.3956 −0.798249
\(935\) 6.49828 0.212517
\(936\) −1.93793 −0.0633432
\(937\) −22.5795 −0.737639 −0.368819 0.929501i \(-0.620238\pi\)
−0.368819 + 0.929501i \(0.620238\pi\)
\(938\) 0 0
\(939\) −21.0741 −0.687727
\(940\) 11.1138 0.362493
\(941\) 21.7655 0.709534 0.354767 0.934955i \(-0.384560\pi\)
0.354767 + 0.934955i \(0.384560\pi\)
\(942\) 12.3741 0.403171
\(943\) 13.3224 0.433836
\(944\) 1.88273 0.0612778
\(945\) 0 0
\(946\) 7.67418 0.249509
\(947\) −1.49484 −0.0485759 −0.0242879 0.999705i \(-0.507732\pi\)
−0.0242879 + 0.999705i \(0.507732\pi\)
\(948\) 18.8172 0.611155
\(949\) −9.88273 −0.320807
\(950\) 4.36641 0.141665
\(951\) −22.4431 −0.727767
\(952\) 0 0
\(953\) −4.83098 −0.156491 −0.0782453 0.996934i \(-0.524932\pi\)
−0.0782453 + 0.996934i \(0.524932\pi\)
\(954\) −9.77320 −0.316419
\(955\) −3.11383 −0.100761
\(956\) −11.1353 −0.360142
\(957\) 19.6742 0.635976
\(958\) 27.7655 0.897062
\(959\) 0 0
\(960\) −2.24914 −0.0725907
\(961\) 60.3346 1.94628
\(962\) −4.00000 −0.128965
\(963\) −29.9603 −0.965456
\(964\) −2.36641 −0.0762168
\(965\) 6.17246 0.198699
\(966\) 0 0
\(967\) −51.9278 −1.66989 −0.834943 0.550336i \(-0.814499\pi\)
−0.834943 + 0.550336i \(0.814499\pi\)
\(968\) 1.00000 0.0321412
\(969\) −63.8174 −2.05011
\(970\) −15.3630 −0.493275
\(971\) −59.6933 −1.91565 −0.957824 0.287354i \(-0.907224\pi\)
−0.957824 + 0.287354i \(0.907224\pi\)
\(972\) −18.2491 −0.585341
\(973\) 0 0
\(974\) 10.0958 0.323490
\(975\) −2.11727 −0.0678068
\(976\) −9.11383 −0.291727
\(977\) 31.2311 0.999171 0.499586 0.866265i \(-0.333486\pi\)
0.499586 + 0.866265i \(0.333486\pi\)
\(978\) −40.9966 −1.31093
\(979\) −12.3810 −0.395699
\(980\) 0 0
\(981\) −0.512889 −0.0163753
\(982\) −32.1104 −1.02468
\(983\) 22.6155 0.721324 0.360662 0.932697i \(-0.382551\pi\)
0.360662 + 0.932697i \(0.382551\pi\)
\(984\) 4.79488 0.152855
\(985\) −6.73281 −0.214525
\(986\) −56.8432 −1.81026
\(987\) 0 0
\(988\) −4.11039 −0.130769
\(989\) 47.9570 1.52494
\(990\) −2.05863 −0.0654277
\(991\) 22.4102 0.711884 0.355942 0.934508i \(-0.384160\pi\)
0.355942 + 0.934508i \(0.384160\pi\)
\(992\) 9.55691 0.303432
\(993\) −45.9639 −1.45862
\(994\) 0 0
\(995\) −13.2311 −0.419454
\(996\) 19.1138 0.605645
\(997\) 27.3415 0.865914 0.432957 0.901415i \(-0.357470\pi\)
0.432957 + 0.901415i \(0.357470\pi\)
\(998\) −8.79488 −0.278397
\(999\) −8.99656 −0.284639
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 5390.2.a.ca.1.3 3
7.6 odd 2 770.2.a.m.1.1 3
21.20 even 2 6930.2.a.ce.1.2 3
28.27 even 2 6160.2.a.bf.1.3 3
35.13 even 4 3850.2.c.ba.1849.1 6
35.27 even 4 3850.2.c.ba.1849.6 6
35.34 odd 2 3850.2.a.bt.1.3 3
77.76 even 2 8470.2.a.ci.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.1 3 7.6 odd 2
3850.2.a.bt.1.3 3 35.34 odd 2
3850.2.c.ba.1849.1 6 35.13 even 4
3850.2.c.ba.1849.6 6 35.27 even 4
5390.2.a.ca.1.3 3 1.1 even 1 trivial
6160.2.a.bf.1.3 3 28.27 even 2
6930.2.a.ce.1.2 3 21.20 even 2
8470.2.a.ci.1.1 3 77.76 even 2