Properties

Label 585.2.a.l.1.1
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{5} -4.56155 q^{7} +2.43845 q^{8} +O(q^{10})\) \(q-1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{5} -4.56155 q^{7} +2.43845 q^{8} -1.56155 q^{10} +2.56155 q^{11} -1.00000 q^{13} +7.12311 q^{14} -4.68466 q^{16} -2.56155 q^{17} +3.12311 q^{19} +0.438447 q^{20} -4.00000 q^{22} +6.56155 q^{23} +1.00000 q^{25} +1.56155 q^{26} -2.00000 q^{28} +1.12311 q^{29} +6.00000 q^{31} +2.43845 q^{32} +4.00000 q^{34} -4.56155 q^{35} +1.68466 q^{37} -4.87689 q^{38} +2.43845 q^{40} -0.561553 q^{41} -5.12311 q^{43} +1.12311 q^{44} -10.2462 q^{46} +2.87689 q^{47} +13.8078 q^{49} -1.56155 q^{50} -0.438447 q^{52} +7.68466 q^{53} +2.56155 q^{55} -11.1231 q^{56} -1.75379 q^{58} +12.0000 q^{59} +5.68466 q^{61} -9.36932 q^{62} +5.56155 q^{64} -1.00000 q^{65} +13.3693 q^{67} -1.12311 q^{68} +7.12311 q^{70} +14.5616 q^{71} -6.00000 q^{73} -2.63068 q^{74} +1.36932 q^{76} -11.6847 q^{77} -7.68466 q^{79} -4.68466 q^{80} +0.876894 q^{82} +16.4924 q^{83} -2.56155 q^{85} +8.00000 q^{86} +6.24621 q^{88} -1.68466 q^{89} +4.56155 q^{91} +2.87689 q^{92} -4.49242 q^{94} +3.12311 q^{95} -11.9309 q^{97} -21.5616 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8} + q^{10} + q^{11} - 2 q^{13} + 6 q^{14} + 3 q^{16} - q^{17} - 2 q^{19} + 5 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} - q^{26} - 4 q^{28} - 6 q^{29} + 12 q^{31} + 9 q^{32} + 8 q^{34} - 5 q^{35} - 9 q^{37} - 18 q^{38} + 9 q^{40} + 3 q^{41} - 2 q^{43} - 6 q^{44} - 4 q^{46} + 14 q^{47} + 7 q^{49} + q^{50} - 5 q^{52} + 3 q^{53} + q^{55} - 14 q^{56} - 20 q^{58} + 24 q^{59} - q^{61} + 6 q^{62} + 7 q^{64} - 2 q^{65} + 2 q^{67} + 6 q^{68} + 6 q^{70} + 25 q^{71} - 12 q^{73} - 30 q^{74} - 22 q^{76} - 11 q^{77} - 3 q^{79} + 3 q^{80} + 10 q^{82} - q^{85} + 16 q^{86} - 4 q^{88} + 9 q^{89} + 5 q^{91} + 14 q^{92} + 24 q^{94} - 2 q^{95} + 5 q^{97} - 39 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −1.10418 −0.552092 0.833783i \(-0.686170\pi\)
−0.552092 + 0.833783i \(0.686170\pi\)
\(3\) 0 0
\(4\) 0.438447 0.219224
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.56155 −1.72410 −0.862052 0.506819i \(-0.830821\pi\)
−0.862052 + 0.506819i \(0.830821\pi\)
\(8\) 2.43845 0.862121
\(9\) 0 0
\(10\) −1.56155 −0.493806
\(11\) 2.56155 0.772337 0.386169 0.922428i \(-0.373798\pi\)
0.386169 + 0.922428i \(0.373798\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 7.12311 1.90373
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) −2.56155 −0.621268 −0.310634 0.950530i \(-0.600541\pi\)
−0.310634 + 0.950530i \(0.600541\pi\)
\(18\) 0 0
\(19\) 3.12311 0.716490 0.358245 0.933628i \(-0.383375\pi\)
0.358245 + 0.933628i \(0.383375\pi\)
\(20\) 0.438447 0.0980398
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 6.56155 1.36818 0.684089 0.729398i \(-0.260200\pi\)
0.684089 + 0.729398i \(0.260200\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.56155 0.306246
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 1.12311 0.208555 0.104278 0.994548i \(-0.466747\pi\)
0.104278 + 0.994548i \(0.466747\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 2.43845 0.431061
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −4.56155 −0.771043
\(36\) 0 0
\(37\) 1.68466 0.276956 0.138478 0.990366i \(-0.455779\pi\)
0.138478 + 0.990366i \(0.455779\pi\)
\(38\) −4.87689 −0.791137
\(39\) 0 0
\(40\) 2.43845 0.385552
\(41\) −0.561553 −0.0876998 −0.0438499 0.999038i \(-0.513962\pi\)
−0.0438499 + 0.999038i \(0.513962\pi\)
\(42\) 0 0
\(43\) −5.12311 −0.781266 −0.390633 0.920546i \(-0.627744\pi\)
−0.390633 + 0.920546i \(0.627744\pi\)
\(44\) 1.12311 0.169315
\(45\) 0 0
\(46\) −10.2462 −1.51072
\(47\) 2.87689 0.419638 0.209819 0.977740i \(-0.432712\pi\)
0.209819 + 0.977740i \(0.432712\pi\)
\(48\) 0 0
\(49\) 13.8078 1.97254
\(50\) −1.56155 −0.220837
\(51\) 0 0
\(52\) −0.438447 −0.0608017
\(53\) 7.68466 1.05557 0.527785 0.849378i \(-0.323023\pi\)
0.527785 + 0.849378i \(0.323023\pi\)
\(54\) 0 0
\(55\) 2.56155 0.345400
\(56\) −11.1231 −1.48639
\(57\) 0 0
\(58\) −1.75379 −0.230284
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 5.68466 0.727846 0.363923 0.931429i \(-0.381437\pi\)
0.363923 + 0.931429i \(0.381437\pi\)
\(62\) −9.36932 −1.18990
\(63\) 0 0
\(64\) 5.56155 0.695194
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 13.3693 1.63332 0.816661 0.577118i \(-0.195823\pi\)
0.816661 + 0.577118i \(0.195823\pi\)
\(68\) −1.12311 −0.136197
\(69\) 0 0
\(70\) 7.12311 0.851374
\(71\) 14.5616 1.72814 0.864069 0.503373i \(-0.167908\pi\)
0.864069 + 0.503373i \(0.167908\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −2.63068 −0.305811
\(75\) 0 0
\(76\) 1.36932 0.157071
\(77\) −11.6847 −1.33159
\(78\) 0 0
\(79\) −7.68466 −0.864592 −0.432296 0.901732i \(-0.642296\pi\)
−0.432296 + 0.901732i \(0.642296\pi\)
\(80\) −4.68466 −0.523761
\(81\) 0 0
\(82\) 0.876894 0.0968368
\(83\) 16.4924 1.81028 0.905139 0.425115i \(-0.139766\pi\)
0.905139 + 0.425115i \(0.139766\pi\)
\(84\) 0 0
\(85\) −2.56155 −0.277839
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 6.24621 0.665848
\(89\) −1.68466 −0.178573 −0.0892867 0.996006i \(-0.528459\pi\)
−0.0892867 + 0.996006i \(0.528459\pi\)
\(90\) 0 0
\(91\) 4.56155 0.478181
\(92\) 2.87689 0.299937
\(93\) 0 0
\(94\) −4.49242 −0.463358
\(95\) 3.12311 0.320424
\(96\) 0 0
\(97\) −11.9309 −1.21140 −0.605698 0.795695i \(-0.707106\pi\)
−0.605698 + 0.795695i \(0.707106\pi\)
\(98\) −21.5616 −2.17805
\(99\) 0 0
\(100\) 0.438447 0.0438447
\(101\) −6.24621 −0.621521 −0.310761 0.950488i \(-0.600584\pi\)
−0.310761 + 0.950488i \(0.600584\pi\)
\(102\) 0 0
\(103\) −6.87689 −0.677601 −0.338800 0.940858i \(-0.610021\pi\)
−0.338800 + 0.940858i \(0.610021\pi\)
\(104\) −2.43845 −0.239109
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) −17.9309 −1.73344 −0.866721 0.498793i \(-0.833777\pi\)
−0.866721 + 0.498793i \(0.833777\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 21.3693 2.01921
\(113\) −13.1231 −1.23452 −0.617259 0.786760i \(-0.711757\pi\)
−0.617259 + 0.786760i \(0.711757\pi\)
\(114\) 0 0
\(115\) 6.56155 0.611868
\(116\) 0.492423 0.0457203
\(117\) 0 0
\(118\) −18.7386 −1.72503
\(119\) 11.6847 1.07113
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) −8.87689 −0.803676
\(123\) 0 0
\(124\) 2.63068 0.236242
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 18.2462 1.61909 0.809545 0.587058i \(-0.199714\pi\)
0.809545 + 0.587058i \(0.199714\pi\)
\(128\) −13.5616 −1.19868
\(129\) 0 0
\(130\) 1.56155 0.136957
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −14.2462 −1.23530
\(134\) −20.8769 −1.80349
\(135\) 0 0
\(136\) −6.24621 −0.535608
\(137\) −7.12311 −0.608568 −0.304284 0.952581i \(-0.598417\pi\)
−0.304284 + 0.952581i \(0.598417\pi\)
\(138\) 0 0
\(139\) 15.6847 1.33036 0.665178 0.746685i \(-0.268356\pi\)
0.665178 + 0.746685i \(0.268356\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −22.7386 −1.90818
\(143\) −2.56155 −0.214208
\(144\) 0 0
\(145\) 1.12311 0.0932688
\(146\) 9.36932 0.775410
\(147\) 0 0
\(148\) 0.738634 0.0607153
\(149\) 2.80776 0.230021 0.115010 0.993364i \(-0.463310\pi\)
0.115010 + 0.993364i \(0.463310\pi\)
\(150\) 0 0
\(151\) −13.3693 −1.08798 −0.543990 0.839092i \(-0.683087\pi\)
−0.543990 + 0.839092i \(0.683087\pi\)
\(152\) 7.61553 0.617701
\(153\) 0 0
\(154\) 18.2462 1.47032
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −21.3693 −1.70546 −0.852729 0.522354i \(-0.825054\pi\)
−0.852729 + 0.522354i \(0.825054\pi\)
\(158\) 12.0000 0.954669
\(159\) 0 0
\(160\) 2.43845 0.192776
\(161\) −29.9309 −2.35888
\(162\) 0 0
\(163\) −21.0540 −1.64907 −0.824537 0.565808i \(-0.808564\pi\)
−0.824537 + 0.565808i \(0.808564\pi\)
\(164\) −0.246211 −0.0192259
\(165\) 0 0
\(166\) −25.7538 −1.99888
\(167\) 13.1231 1.01550 0.507748 0.861506i \(-0.330478\pi\)
0.507748 + 0.861506i \(0.330478\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) −2.24621 −0.171272
\(173\) 21.1231 1.60596 0.802980 0.596006i \(-0.203247\pi\)
0.802980 + 0.596006i \(0.203247\pi\)
\(174\) 0 0
\(175\) −4.56155 −0.344821
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) 2.63068 0.197178
\(179\) −13.1231 −0.980867 −0.490433 0.871479i \(-0.663162\pi\)
−0.490433 + 0.871479i \(0.663162\pi\)
\(180\) 0 0
\(181\) 25.6847 1.90913 0.954563 0.298010i \(-0.0963228\pi\)
0.954563 + 0.298010i \(0.0963228\pi\)
\(182\) −7.12311 −0.528000
\(183\) 0 0
\(184\) 16.0000 1.17954
\(185\) 1.68466 0.123859
\(186\) 0 0
\(187\) −6.56155 −0.479828
\(188\) 1.26137 0.0919946
\(189\) 0 0
\(190\) −4.87689 −0.353807
\(191\) −21.6155 −1.56404 −0.782022 0.623250i \(-0.785812\pi\)
−0.782022 + 0.623250i \(0.785812\pi\)
\(192\) 0 0
\(193\) 15.4384 1.11128 0.555642 0.831422i \(-0.312473\pi\)
0.555642 + 0.831422i \(0.312473\pi\)
\(194\) 18.6307 1.33761
\(195\) 0 0
\(196\) 6.05398 0.432427
\(197\) 21.3693 1.52250 0.761250 0.648458i \(-0.224586\pi\)
0.761250 + 0.648458i \(0.224586\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 2.43845 0.172424
\(201\) 0 0
\(202\) 9.75379 0.686274
\(203\) −5.12311 −0.359572
\(204\) 0 0
\(205\) −0.561553 −0.0392205
\(206\) 10.7386 0.748196
\(207\) 0 0
\(208\) 4.68466 0.324823
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −10.2462 −0.705378 −0.352689 0.935741i \(-0.614733\pi\)
−0.352689 + 0.935741i \(0.614733\pi\)
\(212\) 3.36932 0.231406
\(213\) 0 0
\(214\) 28.0000 1.91404
\(215\) −5.12311 −0.349393
\(216\) 0 0
\(217\) −27.3693 −1.85795
\(218\) 3.12311 0.211523
\(219\) 0 0
\(220\) 1.12311 0.0757198
\(221\) 2.56155 0.172309
\(222\) 0 0
\(223\) 9.36932 0.627416 0.313708 0.949520i \(-0.398429\pi\)
0.313708 + 0.949520i \(0.398429\pi\)
\(224\) −11.1231 −0.743194
\(225\) 0 0
\(226\) 20.4924 1.36314
\(227\) 22.2462 1.47653 0.738266 0.674509i \(-0.235645\pi\)
0.738266 + 0.674509i \(0.235645\pi\)
\(228\) 0 0
\(229\) 8.87689 0.586602 0.293301 0.956020i \(-0.405246\pi\)
0.293301 + 0.956020i \(0.405246\pi\)
\(230\) −10.2462 −0.675615
\(231\) 0 0
\(232\) 2.73863 0.179800
\(233\) −7.19224 −0.471179 −0.235590 0.971853i \(-0.575702\pi\)
−0.235590 + 0.971853i \(0.575702\pi\)
\(234\) 0 0
\(235\) 2.87689 0.187668
\(236\) 5.26137 0.342486
\(237\) 0 0
\(238\) −18.2462 −1.18273
\(239\) −5.93087 −0.383636 −0.191818 0.981431i \(-0.561438\pi\)
−0.191818 + 0.981431i \(0.561438\pi\)
\(240\) 0 0
\(241\) −24.7386 −1.59356 −0.796778 0.604272i \(-0.793464\pi\)
−0.796778 + 0.604272i \(0.793464\pi\)
\(242\) 6.93087 0.445533
\(243\) 0 0
\(244\) 2.49242 0.159561
\(245\) 13.8078 0.882146
\(246\) 0 0
\(247\) −3.12311 −0.198718
\(248\) 14.6307 0.929049
\(249\) 0 0
\(250\) −1.56155 −0.0987613
\(251\) 9.75379 0.615654 0.307827 0.951442i \(-0.400398\pi\)
0.307827 + 0.951442i \(0.400398\pi\)
\(252\) 0 0
\(253\) 16.8078 1.05670
\(254\) −28.4924 −1.78777
\(255\) 0 0
\(256\) 10.0540 0.628373
\(257\) −21.1231 −1.31762 −0.658812 0.752308i \(-0.728941\pi\)
−0.658812 + 0.752308i \(0.728941\pi\)
\(258\) 0 0
\(259\) −7.68466 −0.477501
\(260\) −0.438447 −0.0271913
\(261\) 0 0
\(262\) 0 0
\(263\) 14.2462 0.878459 0.439230 0.898375i \(-0.355251\pi\)
0.439230 + 0.898375i \(0.355251\pi\)
\(264\) 0 0
\(265\) 7.68466 0.472065
\(266\) 22.2462 1.36400
\(267\) 0 0
\(268\) 5.86174 0.358063
\(269\) −9.12311 −0.556246 −0.278123 0.960546i \(-0.589712\pi\)
−0.278123 + 0.960546i \(0.589712\pi\)
\(270\) 0 0
\(271\) 5.36932 0.326163 0.163081 0.986613i \(-0.447857\pi\)
0.163081 + 0.986613i \(0.447857\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) 11.1231 0.671971
\(275\) 2.56155 0.154467
\(276\) 0 0
\(277\) 4.24621 0.255130 0.127565 0.991830i \(-0.459284\pi\)
0.127565 + 0.991830i \(0.459284\pi\)
\(278\) −24.4924 −1.46896
\(279\) 0 0
\(280\) −11.1231 −0.664733
\(281\) −12.2462 −0.730548 −0.365274 0.930900i \(-0.619025\pi\)
−0.365274 + 0.930900i \(0.619025\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 6.38447 0.378849
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 2.56155 0.151204
\(288\) 0 0
\(289\) −10.4384 −0.614026
\(290\) −1.75379 −0.102986
\(291\) 0 0
\(292\) −2.63068 −0.153949
\(293\) −3.75379 −0.219299 −0.109649 0.993970i \(-0.534973\pi\)
−0.109649 + 0.993970i \(0.534973\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 4.10795 0.238770
\(297\) 0 0
\(298\) −4.38447 −0.253986
\(299\) −6.56155 −0.379464
\(300\) 0 0
\(301\) 23.3693 1.34699
\(302\) 20.8769 1.20133
\(303\) 0 0
\(304\) −14.6307 −0.839127
\(305\) 5.68466 0.325503
\(306\) 0 0
\(307\) 3.43845 0.196243 0.0981213 0.995174i \(-0.468717\pi\)
0.0981213 + 0.995174i \(0.468717\pi\)
\(308\) −5.12311 −0.291916
\(309\) 0 0
\(310\) −9.36932 −0.532141
\(311\) 27.3693 1.55197 0.775986 0.630750i \(-0.217253\pi\)
0.775986 + 0.630750i \(0.217253\pi\)
\(312\) 0 0
\(313\) 20.8769 1.18003 0.590016 0.807392i \(-0.299121\pi\)
0.590016 + 0.807392i \(0.299121\pi\)
\(314\) 33.3693 1.88314
\(315\) 0 0
\(316\) −3.36932 −0.189539
\(317\) 34.4924 1.93729 0.968644 0.248454i \(-0.0799224\pi\)
0.968644 + 0.248454i \(0.0799224\pi\)
\(318\) 0 0
\(319\) 2.87689 0.161075
\(320\) 5.56155 0.310900
\(321\) 0 0
\(322\) 46.7386 2.60464
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 32.8769 1.82088
\(327\) 0 0
\(328\) −1.36932 −0.0756079
\(329\) −13.1231 −0.723500
\(330\) 0 0
\(331\) 20.8769 1.14750 0.573749 0.819031i \(-0.305489\pi\)
0.573749 + 0.819031i \(0.305489\pi\)
\(332\) 7.23106 0.396856
\(333\) 0 0
\(334\) −20.4924 −1.12130
\(335\) 13.3693 0.730444
\(336\) 0 0
\(337\) 2.49242 0.135771 0.0678855 0.997693i \(-0.478375\pi\)
0.0678855 + 0.997693i \(0.478375\pi\)
\(338\) −1.56155 −0.0849373
\(339\) 0 0
\(340\) −1.12311 −0.0609090
\(341\) 15.3693 0.832295
\(342\) 0 0
\(343\) −31.0540 −1.67676
\(344\) −12.4924 −0.673546
\(345\) 0 0
\(346\) −32.9848 −1.77328
\(347\) 11.0540 0.593408 0.296704 0.954969i \(-0.404112\pi\)
0.296704 + 0.954969i \(0.404112\pi\)
\(348\) 0 0
\(349\) 1.36932 0.0732979 0.0366489 0.999328i \(-0.488332\pi\)
0.0366489 + 0.999328i \(0.488332\pi\)
\(350\) 7.12311 0.380746
\(351\) 0 0
\(352\) 6.24621 0.332924
\(353\) 10.4924 0.558455 0.279228 0.960225i \(-0.409922\pi\)
0.279228 + 0.960225i \(0.409922\pi\)
\(354\) 0 0
\(355\) 14.5616 0.772847
\(356\) −0.738634 −0.0391475
\(357\) 0 0
\(358\) 20.4924 1.08306
\(359\) 2.24621 0.118550 0.0592752 0.998242i \(-0.481121\pi\)
0.0592752 + 0.998242i \(0.481121\pi\)
\(360\) 0 0
\(361\) −9.24621 −0.486643
\(362\) −40.1080 −2.10803
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −18.2462 −0.952444 −0.476222 0.879325i \(-0.657994\pi\)
−0.476222 + 0.879325i \(0.657994\pi\)
\(368\) −30.7386 −1.60236
\(369\) 0 0
\(370\) −2.63068 −0.136763
\(371\) −35.0540 −1.81991
\(372\) 0 0
\(373\) −4.24621 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(374\) 10.2462 0.529819
\(375\) 0 0
\(376\) 7.01515 0.361779
\(377\) −1.12311 −0.0578429
\(378\) 0 0
\(379\) 2.49242 0.128027 0.0640136 0.997949i \(-0.479610\pi\)
0.0640136 + 0.997949i \(0.479610\pi\)
\(380\) 1.36932 0.0702445
\(381\) 0 0
\(382\) 33.7538 1.72699
\(383\) 36.4924 1.86468 0.932338 0.361588i \(-0.117765\pi\)
0.932338 + 0.361588i \(0.117765\pi\)
\(384\) 0 0
\(385\) −11.6847 −0.595505
\(386\) −24.1080 −1.22706
\(387\) 0 0
\(388\) −5.23106 −0.265567
\(389\) −19.8617 −1.00703 −0.503515 0.863986i \(-0.667960\pi\)
−0.503515 + 0.863986i \(0.667960\pi\)
\(390\) 0 0
\(391\) −16.8078 −0.850005
\(392\) 33.6695 1.70057
\(393\) 0 0
\(394\) −33.3693 −1.68112
\(395\) −7.68466 −0.386657
\(396\) 0 0
\(397\) −8.56155 −0.429692 −0.214846 0.976648i \(-0.568925\pi\)
−0.214846 + 0.976648i \(0.568925\pi\)
\(398\) 12.4924 0.626189
\(399\) 0 0
\(400\) −4.68466 −0.234233
\(401\) 20.2462 1.01105 0.505524 0.862813i \(-0.331299\pi\)
0.505524 + 0.862813i \(0.331299\pi\)
\(402\) 0 0
\(403\) −6.00000 −0.298881
\(404\) −2.73863 −0.136252
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 4.31534 0.213904
\(408\) 0 0
\(409\) −15.1231 −0.747789 −0.373895 0.927471i \(-0.621978\pi\)
−0.373895 + 0.927471i \(0.621978\pi\)
\(410\) 0.876894 0.0433067
\(411\) 0 0
\(412\) −3.01515 −0.148546
\(413\) −54.7386 −2.69351
\(414\) 0 0
\(415\) 16.4924 0.809581
\(416\) −2.43845 −0.119555
\(417\) 0 0
\(418\) −12.4924 −0.611024
\(419\) −33.6155 −1.64223 −0.821113 0.570766i \(-0.806646\pi\)
−0.821113 + 0.570766i \(0.806646\pi\)
\(420\) 0 0
\(421\) 14.6307 0.713056 0.356528 0.934285i \(-0.383960\pi\)
0.356528 + 0.934285i \(0.383960\pi\)
\(422\) 16.0000 0.778868
\(423\) 0 0
\(424\) 18.7386 0.910029
\(425\) −2.56155 −0.124254
\(426\) 0 0
\(427\) −25.9309 −1.25488
\(428\) −7.86174 −0.380012
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) −36.4924 −1.75778 −0.878889 0.477026i \(-0.841715\pi\)
−0.878889 + 0.477026i \(0.841715\pi\)
\(432\) 0 0
\(433\) 31.6155 1.51935 0.759673 0.650306i \(-0.225359\pi\)
0.759673 + 0.650306i \(0.225359\pi\)
\(434\) 42.7386 2.05152
\(435\) 0 0
\(436\) −0.876894 −0.0419956
\(437\) 20.4924 0.980286
\(438\) 0 0
\(439\) −15.0540 −0.718487 −0.359244 0.933244i \(-0.616965\pi\)
−0.359244 + 0.933244i \(0.616965\pi\)
\(440\) 6.24621 0.297776
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 12.8078 0.608515 0.304258 0.952590i \(-0.401592\pi\)
0.304258 + 0.952590i \(0.401592\pi\)
\(444\) 0 0
\(445\) −1.68466 −0.0798605
\(446\) −14.6307 −0.692783
\(447\) 0 0
\(448\) −25.3693 −1.19859
\(449\) 10.1771 0.480286 0.240143 0.970738i \(-0.422806\pi\)
0.240143 + 0.970738i \(0.422806\pi\)
\(450\) 0 0
\(451\) −1.43845 −0.0677338
\(452\) −5.75379 −0.270635
\(453\) 0 0
\(454\) −34.7386 −1.63036
\(455\) 4.56155 0.213849
\(456\) 0 0
\(457\) −25.6847 −1.20148 −0.600739 0.799445i \(-0.705127\pi\)
−0.600739 + 0.799445i \(0.705127\pi\)
\(458\) −13.8617 −0.647717
\(459\) 0 0
\(460\) 2.87689 0.134136
\(461\) −29.0540 −1.35318 −0.676589 0.736361i \(-0.736543\pi\)
−0.676589 + 0.736361i \(0.736543\pi\)
\(462\) 0 0
\(463\) −0.0691303 −0.00321276 −0.00160638 0.999999i \(-0.500511\pi\)
−0.00160638 + 0.999999i \(0.500511\pi\)
\(464\) −5.26137 −0.244253
\(465\) 0 0
\(466\) 11.2311 0.520269
\(467\) −21.9309 −1.01484 −0.507420 0.861699i \(-0.669401\pi\)
−0.507420 + 0.861699i \(0.669401\pi\)
\(468\) 0 0
\(469\) −60.9848 −2.81602
\(470\) −4.49242 −0.207220
\(471\) 0 0
\(472\) 29.2614 1.34686
\(473\) −13.1231 −0.603401
\(474\) 0 0
\(475\) 3.12311 0.143298
\(476\) 5.12311 0.234817
\(477\) 0 0
\(478\) 9.26137 0.423605
\(479\) −11.0540 −0.505069 −0.252535 0.967588i \(-0.581264\pi\)
−0.252535 + 0.967588i \(0.581264\pi\)
\(480\) 0 0
\(481\) −1.68466 −0.0768138
\(482\) 38.6307 1.75958
\(483\) 0 0
\(484\) −1.94602 −0.0884557
\(485\) −11.9309 −0.541753
\(486\) 0 0
\(487\) 5.05398 0.229017 0.114509 0.993422i \(-0.463471\pi\)
0.114509 + 0.993422i \(0.463471\pi\)
\(488\) 13.8617 0.627491
\(489\) 0 0
\(490\) −21.5616 −0.974052
\(491\) 13.6155 0.614460 0.307230 0.951635i \(-0.400598\pi\)
0.307230 + 0.951635i \(0.400598\pi\)
\(492\) 0 0
\(493\) −2.87689 −0.129569
\(494\) 4.87689 0.219422
\(495\) 0 0
\(496\) −28.1080 −1.26208
\(497\) −66.4233 −2.97949
\(498\) 0 0
\(499\) −30.9848 −1.38707 −0.693536 0.720422i \(-0.743948\pi\)
−0.693536 + 0.720422i \(0.743948\pi\)
\(500\) 0.438447 0.0196080
\(501\) 0 0
\(502\) −15.2311 −0.679795
\(503\) 30.2462 1.34861 0.674306 0.738452i \(-0.264443\pi\)
0.674306 + 0.738452i \(0.264443\pi\)
\(504\) 0 0
\(505\) −6.24621 −0.277953
\(506\) −26.2462 −1.16679
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −19.4384 −0.861594 −0.430797 0.902449i \(-0.641768\pi\)
−0.430797 + 0.902449i \(0.641768\pi\)
\(510\) 0 0
\(511\) 27.3693 1.21075
\(512\) 11.4233 0.504843
\(513\) 0 0
\(514\) 32.9848 1.45490
\(515\) −6.87689 −0.303032
\(516\) 0 0
\(517\) 7.36932 0.324102
\(518\) 12.0000 0.527250
\(519\) 0 0
\(520\) −2.43845 −0.106933
\(521\) 31.8617 1.39589 0.697944 0.716152i \(-0.254098\pi\)
0.697944 + 0.716152i \(0.254098\pi\)
\(522\) 0 0
\(523\) 18.7386 0.819383 0.409692 0.912224i \(-0.365636\pi\)
0.409692 + 0.912224i \(0.365636\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −22.2462 −0.969981
\(527\) −15.3693 −0.669498
\(528\) 0 0
\(529\) 20.0540 0.871912
\(530\) −12.0000 −0.521247
\(531\) 0 0
\(532\) −6.24621 −0.270808
\(533\) 0.561553 0.0243236
\(534\) 0 0
\(535\) −17.9309 −0.775219
\(536\) 32.6004 1.40812
\(537\) 0 0
\(538\) 14.2462 0.614198
\(539\) 35.3693 1.52346
\(540\) 0 0
\(541\) 27.1231 1.16611 0.583057 0.812431i \(-0.301857\pi\)
0.583057 + 0.812431i \(0.301857\pi\)
\(542\) −8.38447 −0.360144
\(543\) 0 0
\(544\) −6.24621 −0.267804
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 19.3693 0.828172 0.414086 0.910238i \(-0.364101\pi\)
0.414086 + 0.910238i \(0.364101\pi\)
\(548\) −3.12311 −0.133412
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 3.50758 0.149428
\(552\) 0 0
\(553\) 35.0540 1.49065
\(554\) −6.63068 −0.281711
\(555\) 0 0
\(556\) 6.87689 0.291645
\(557\) −26.4924 −1.12252 −0.561260 0.827640i \(-0.689683\pi\)
−0.561260 + 0.827640i \(0.689683\pi\)
\(558\) 0 0
\(559\) 5.12311 0.216684
\(560\) 21.3693 0.903018
\(561\) 0 0
\(562\) 19.1231 0.806660
\(563\) −31.6847 −1.33535 −0.667675 0.744453i \(-0.732710\pi\)
−0.667675 + 0.744453i \(0.732710\pi\)
\(564\) 0 0
\(565\) −13.1231 −0.552093
\(566\) 6.24621 0.262548
\(567\) 0 0
\(568\) 35.5076 1.48986
\(569\) −41.1231 −1.72397 −0.861985 0.506934i \(-0.830779\pi\)
−0.861985 + 0.506934i \(0.830779\pi\)
\(570\) 0 0
\(571\) 15.0540 0.629989 0.314995 0.949093i \(-0.397997\pi\)
0.314995 + 0.949093i \(0.397997\pi\)
\(572\) −1.12311 −0.0469594
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 6.56155 0.273636
\(576\) 0 0
\(577\) 28.5616 1.18903 0.594517 0.804083i \(-0.297343\pi\)
0.594517 + 0.804083i \(0.297343\pi\)
\(578\) 16.3002 0.677998
\(579\) 0 0
\(580\) 0.492423 0.0204467
\(581\) −75.2311 −3.12111
\(582\) 0 0
\(583\) 19.6847 0.815255
\(584\) −14.6307 −0.605422
\(585\) 0 0
\(586\) 5.86174 0.242146
\(587\) 0.492423 0.0203245 0.0101622 0.999948i \(-0.496765\pi\)
0.0101622 + 0.999948i \(0.496765\pi\)
\(588\) 0 0
\(589\) 18.7386 0.772112
\(590\) −18.7386 −0.771457
\(591\) 0 0
\(592\) −7.89205 −0.324361
\(593\) 7.75379 0.318410 0.159205 0.987246i \(-0.449107\pi\)
0.159205 + 0.987246i \(0.449107\pi\)
\(594\) 0 0
\(595\) 11.6847 0.479024
\(596\) 1.23106 0.0504260
\(597\) 0 0
\(598\) 10.2462 0.418999
\(599\) 15.3693 0.627973 0.313987 0.949427i \(-0.398335\pi\)
0.313987 + 0.949427i \(0.398335\pi\)
\(600\) 0 0
\(601\) 41.5464 1.69471 0.847356 0.531025i \(-0.178193\pi\)
0.847356 + 0.531025i \(0.178193\pi\)
\(602\) −36.4924 −1.48732
\(603\) 0 0
\(604\) −5.86174 −0.238511
\(605\) −4.43845 −0.180449
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 7.61553 0.308850
\(609\) 0 0
\(610\) −8.87689 −0.359415
\(611\) −2.87689 −0.116387
\(612\) 0 0
\(613\) 37.5464 1.51648 0.758242 0.651973i \(-0.226058\pi\)
0.758242 + 0.651973i \(0.226058\pi\)
\(614\) −5.36932 −0.216688
\(615\) 0 0
\(616\) −28.4924 −1.14799
\(617\) −48.7386 −1.96214 −0.981072 0.193645i \(-0.937969\pi\)
−0.981072 + 0.193645i \(0.937969\pi\)
\(618\) 0 0
\(619\) −33.3693 −1.34123 −0.670613 0.741807i \(-0.733969\pi\)
−0.670613 + 0.741807i \(0.733969\pi\)
\(620\) 2.63068 0.105651
\(621\) 0 0
\(622\) −42.7386 −1.71366
\(623\) 7.68466 0.307879
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −32.6004 −1.30297
\(627\) 0 0
\(628\) −9.36932 −0.373876
\(629\) −4.31534 −0.172064
\(630\) 0 0
\(631\) 16.7386 0.666354 0.333177 0.942864i \(-0.391879\pi\)
0.333177 + 0.942864i \(0.391879\pi\)
\(632\) −18.7386 −0.745383
\(633\) 0 0
\(634\) −53.8617 −2.13912
\(635\) 18.2462 0.724079
\(636\) 0 0
\(637\) −13.8078 −0.547084
\(638\) −4.49242 −0.177857
\(639\) 0 0
\(640\) −13.5616 −0.536067
\(641\) 32.9848 1.30282 0.651412 0.758725i \(-0.274177\pi\)
0.651412 + 0.758725i \(0.274177\pi\)
\(642\) 0 0
\(643\) 1.68466 0.0664364 0.0332182 0.999448i \(-0.489424\pi\)
0.0332182 + 0.999448i \(0.489424\pi\)
\(644\) −13.1231 −0.517123
\(645\) 0 0
\(646\) 12.4924 0.491508
\(647\) 11.1922 0.440012 0.220006 0.975498i \(-0.429392\pi\)
0.220006 + 0.975498i \(0.429392\pi\)
\(648\) 0 0
\(649\) 30.7386 1.20660
\(650\) 1.56155 0.0612491
\(651\) 0 0
\(652\) −9.23106 −0.361516
\(653\) −8.63068 −0.337745 −0.168872 0.985638i \(-0.554013\pi\)
−0.168872 + 0.985638i \(0.554013\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.63068 0.102711
\(657\) 0 0
\(658\) 20.4924 0.798878
\(659\) 9.12311 0.355386 0.177693 0.984086i \(-0.443137\pi\)
0.177693 + 0.984086i \(0.443137\pi\)
\(660\) 0 0
\(661\) −26.4924 −1.03044 −0.515218 0.857059i \(-0.672289\pi\)
−0.515218 + 0.857059i \(0.672289\pi\)
\(662\) −32.6004 −1.26705
\(663\) 0 0
\(664\) 40.2159 1.56068
\(665\) −14.2462 −0.552444
\(666\) 0 0
\(667\) 7.36932 0.285341
\(668\) 5.75379 0.222621
\(669\) 0 0
\(670\) −20.8769 −0.806545
\(671\) 14.5616 0.562143
\(672\) 0 0
\(673\) −32.7386 −1.26198 −0.630991 0.775790i \(-0.717351\pi\)
−0.630991 + 0.775790i \(0.717351\pi\)
\(674\) −3.89205 −0.149916
\(675\) 0 0
\(676\) 0.438447 0.0168634
\(677\) −18.5616 −0.713378 −0.356689 0.934223i \(-0.616094\pi\)
−0.356689 + 0.934223i \(0.616094\pi\)
\(678\) 0 0
\(679\) 54.4233 2.08857
\(680\) −6.24621 −0.239531
\(681\) 0 0
\(682\) −24.0000 −0.919007
\(683\) −0.492423 −0.0188420 −0.00942101 0.999956i \(-0.502999\pi\)
−0.00942101 + 0.999956i \(0.502999\pi\)
\(684\) 0 0
\(685\) −7.12311 −0.272160
\(686\) 48.4924 1.85145
\(687\) 0 0
\(688\) 24.0000 0.914991
\(689\) −7.68466 −0.292762
\(690\) 0 0
\(691\) 19.6155 0.746210 0.373105 0.927789i \(-0.378293\pi\)
0.373105 + 0.927789i \(0.378293\pi\)
\(692\) 9.26137 0.352064
\(693\) 0 0
\(694\) −17.2614 −0.655233
\(695\) 15.6847 0.594953
\(696\) 0 0
\(697\) 1.43845 0.0544851
\(698\) −2.13826 −0.0809344
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) 16.9848 0.641509 0.320754 0.947162i \(-0.396064\pi\)
0.320754 + 0.947162i \(0.396064\pi\)
\(702\) 0 0
\(703\) 5.26137 0.198436
\(704\) 14.2462 0.536924
\(705\) 0 0
\(706\) −16.3845 −0.616638
\(707\) 28.4924 1.07157
\(708\) 0 0
\(709\) 0.876894 0.0329325 0.0164662 0.999864i \(-0.494758\pi\)
0.0164662 + 0.999864i \(0.494758\pi\)
\(710\) −22.7386 −0.853366
\(711\) 0 0
\(712\) −4.10795 −0.153952
\(713\) 39.3693 1.47439
\(714\) 0 0
\(715\) −2.56155 −0.0957966
\(716\) −5.75379 −0.215029
\(717\) 0 0
\(718\) −3.50758 −0.130902
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 31.3693 1.16825
\(722\) 14.4384 0.537343
\(723\) 0 0
\(724\) 11.2614 0.418525
\(725\) 1.12311 0.0417111
\(726\) 0 0
\(727\) 21.1231 0.783413 0.391706 0.920090i \(-0.371885\pi\)
0.391706 + 0.920090i \(0.371885\pi\)
\(728\) 11.1231 0.412250
\(729\) 0 0
\(730\) 9.36932 0.346774
\(731\) 13.1231 0.485376
\(732\) 0 0
\(733\) −20.5616 −0.759458 −0.379729 0.925098i \(-0.623983\pi\)
−0.379729 + 0.925098i \(0.623983\pi\)
\(734\) 28.4924 1.05167
\(735\) 0 0
\(736\) 16.0000 0.589768
\(737\) 34.2462 1.26148
\(738\) 0 0
\(739\) −0.738634 −0.0271711 −0.0135855 0.999908i \(-0.504325\pi\)
−0.0135855 + 0.999908i \(0.504325\pi\)
\(740\) 0.738634 0.0271527
\(741\) 0 0
\(742\) 54.7386 2.00952
\(743\) −30.7386 −1.12769 −0.563846 0.825880i \(-0.690679\pi\)
−0.563846 + 0.825880i \(0.690679\pi\)
\(744\) 0 0
\(745\) 2.80776 0.102869
\(746\) 6.63068 0.242767
\(747\) 0 0
\(748\) −2.87689 −0.105190
\(749\) 81.7926 2.98864
\(750\) 0 0
\(751\) −23.0540 −0.841252 −0.420626 0.907234i \(-0.638189\pi\)
−0.420626 + 0.907234i \(0.638189\pi\)
\(752\) −13.4773 −0.491465
\(753\) 0 0
\(754\) 1.75379 0.0638692
\(755\) −13.3693 −0.486559
\(756\) 0 0
\(757\) 18.4924 0.672119 0.336059 0.941841i \(-0.390906\pi\)
0.336059 + 0.941841i \(0.390906\pi\)
\(758\) −3.89205 −0.141366
\(759\) 0 0
\(760\) 7.61553 0.276244
\(761\) −37.2311 −1.34962 −0.674812 0.737989i \(-0.735775\pi\)
−0.674812 + 0.737989i \(0.735775\pi\)
\(762\) 0 0
\(763\) 9.12311 0.330279
\(764\) −9.47727 −0.342876
\(765\) 0 0
\(766\) −56.9848 −2.05895
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 18.2462 0.657548
\(771\) 0 0
\(772\) 6.76894 0.243620
\(773\) −0.876894 −0.0315397 −0.0157698 0.999876i \(-0.505020\pi\)
−0.0157698 + 0.999876i \(0.505020\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) −29.0928 −1.04437
\(777\) 0 0
\(778\) 31.0152 1.11195
\(779\) −1.75379 −0.0628360
\(780\) 0 0
\(781\) 37.3002 1.33471
\(782\) 26.2462 0.938563
\(783\) 0 0
\(784\) −64.6847 −2.31017
\(785\) −21.3693 −0.762704
\(786\) 0 0
\(787\) −13.3693 −0.476565 −0.238282 0.971196i \(-0.576584\pi\)
−0.238282 + 0.971196i \(0.576584\pi\)
\(788\) 9.36932 0.333768
\(789\) 0 0
\(790\) 12.0000 0.426941
\(791\) 59.8617 2.12844
\(792\) 0 0
\(793\) −5.68466 −0.201868
\(794\) 13.3693 0.474459
\(795\) 0 0
\(796\) −3.50758 −0.124323
\(797\) −7.68466 −0.272205 −0.136102 0.990695i \(-0.543458\pi\)
−0.136102 + 0.990695i \(0.543458\pi\)
\(798\) 0 0
\(799\) −7.36932 −0.260708
\(800\) 2.43845 0.0862121
\(801\) 0 0
\(802\) −31.6155 −1.11638
\(803\) −15.3693 −0.542371
\(804\) 0 0
\(805\) −29.9309 −1.05492
\(806\) 9.36932 0.330020
\(807\) 0 0
\(808\) −15.2311 −0.535827
\(809\) 40.4924 1.42364 0.711819 0.702363i \(-0.247872\pi\)
0.711819 + 0.702363i \(0.247872\pi\)
\(810\) 0 0
\(811\) −12.2462 −0.430023 −0.215011 0.976612i \(-0.568979\pi\)
−0.215011 + 0.976612i \(0.568979\pi\)
\(812\) −2.24621 −0.0788266
\(813\) 0 0
\(814\) −6.73863 −0.236189
\(815\) −21.0540 −0.737489
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 23.6155 0.825698
\(819\) 0 0
\(820\) −0.246211 −0.00859807
\(821\) −21.5464 −0.751974 −0.375987 0.926625i \(-0.622696\pi\)
−0.375987 + 0.926625i \(0.622696\pi\)
\(822\) 0 0
\(823\) −46.2462 −1.61204 −0.806021 0.591887i \(-0.798383\pi\)
−0.806021 + 0.591887i \(0.798383\pi\)
\(824\) −16.7689 −0.584174
\(825\) 0 0
\(826\) 85.4773 2.97413
\(827\) −1.12311 −0.0390542 −0.0195271 0.999809i \(-0.506216\pi\)
−0.0195271 + 0.999809i \(0.506216\pi\)
\(828\) 0 0
\(829\) −48.7386 −1.69276 −0.846381 0.532577i \(-0.821224\pi\)
−0.846381 + 0.532577i \(0.821224\pi\)
\(830\) −25.7538 −0.893927
\(831\) 0 0
\(832\) −5.56155 −0.192812
\(833\) −35.3693 −1.22547
\(834\) 0 0
\(835\) 13.1231 0.454144
\(836\) 3.50758 0.121312
\(837\) 0 0
\(838\) 52.4924 1.81332
\(839\) 41.7926 1.44284 0.721421 0.692497i \(-0.243490\pi\)
0.721421 + 0.692497i \(0.243490\pi\)
\(840\) 0 0
\(841\) −27.7386 −0.956505
\(842\) −22.8466 −0.787345
\(843\) 0 0
\(844\) −4.49242 −0.154636
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 20.2462 0.695668
\(848\) −36.0000 −1.23625
\(849\) 0 0
\(850\) 4.00000 0.137199
\(851\) 11.0540 0.378925
\(852\) 0 0
\(853\) −20.4233 −0.699280 −0.349640 0.936884i \(-0.613696\pi\)
−0.349640 + 0.936884i \(0.613696\pi\)
\(854\) 40.4924 1.38562
\(855\) 0 0
\(856\) −43.7235 −1.49444
\(857\) 6.56155 0.224138 0.112069 0.993700i \(-0.464252\pi\)
0.112069 + 0.993700i \(0.464252\pi\)
\(858\) 0 0
\(859\) −16.8078 −0.573474 −0.286737 0.958009i \(-0.592571\pi\)
−0.286737 + 0.958009i \(0.592571\pi\)
\(860\) −2.24621 −0.0765952
\(861\) 0 0
\(862\) 56.9848 1.94091
\(863\) 15.3693 0.523178 0.261589 0.965179i \(-0.415754\pi\)
0.261589 + 0.965179i \(0.415754\pi\)
\(864\) 0 0
\(865\) 21.1231 0.718207
\(866\) −49.3693 −1.67764
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) −19.6847 −0.667756
\(870\) 0 0
\(871\) −13.3693 −0.453002
\(872\) −4.87689 −0.165152
\(873\) 0 0
\(874\) −32.0000 −1.08242
\(875\) −4.56155 −0.154209
\(876\) 0 0
\(877\) −18.9848 −0.641073 −0.320536 0.947236i \(-0.603863\pi\)
−0.320536 + 0.947236i \(0.603863\pi\)
\(878\) 23.5076 0.793342
\(879\) 0 0
\(880\) −12.0000 −0.404520
\(881\) 3.36932 0.113515 0.0567576 0.998388i \(-0.481924\pi\)
0.0567576 + 0.998388i \(0.481924\pi\)
\(882\) 0 0
\(883\) 18.1080 0.609381 0.304691 0.952451i \(-0.401447\pi\)
0.304691 + 0.952451i \(0.401447\pi\)
\(884\) 1.12311 0.0377741
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 1.43845 0.0482983 0.0241492 0.999708i \(-0.492312\pi\)
0.0241492 + 0.999708i \(0.492312\pi\)
\(888\) 0 0
\(889\) −83.2311 −2.79148
\(890\) 2.63068 0.0881807
\(891\) 0 0
\(892\) 4.10795 0.137544
\(893\) 8.98485 0.300666
\(894\) 0 0
\(895\) −13.1231 −0.438657
\(896\) 61.8617 2.06666
\(897\) 0 0
\(898\) −15.8920 −0.530325
\(899\) 6.73863 0.224746
\(900\) 0 0
\(901\) −19.6847 −0.655791
\(902\) 2.24621 0.0747907
\(903\) 0 0
\(904\) −32.0000 −1.06430
\(905\) 25.6847 0.853787
\(906\) 0 0
\(907\) −27.3693 −0.908783 −0.454392 0.890802i \(-0.650143\pi\)
−0.454392 + 0.890802i \(0.650143\pi\)
\(908\) 9.75379 0.323691
\(909\) 0 0
\(910\) −7.12311 −0.236129
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 42.2462 1.39815
\(914\) 40.1080 1.32665
\(915\) 0 0
\(916\) 3.89205 0.128597
\(917\) 0 0
\(918\) 0 0
\(919\) 39.0540 1.28827 0.644136 0.764911i \(-0.277217\pi\)
0.644136 + 0.764911i \(0.277217\pi\)
\(920\) 16.0000 0.527504
\(921\) 0 0
\(922\) 45.3693 1.49416
\(923\) −14.5616 −0.479299
\(924\) 0 0
\(925\) 1.68466 0.0553912
\(926\) 0.107951 0.00354748
\(927\) 0 0
\(928\) 2.73863 0.0899001
\(929\) −38.6695 −1.26871 −0.634353 0.773044i \(-0.718733\pi\)
−0.634353 + 0.773044i \(0.718733\pi\)
\(930\) 0 0
\(931\) 43.1231 1.41330
\(932\) −3.15342 −0.103294
\(933\) 0 0
\(934\) 34.2462 1.12057
\(935\) −6.56155 −0.214586
\(936\) 0 0
\(937\) −2.63068 −0.0859407 −0.0429703 0.999076i \(-0.513682\pi\)
−0.0429703 + 0.999076i \(0.513682\pi\)
\(938\) 95.2311 3.10940
\(939\) 0 0
\(940\) 1.26137 0.0411412
\(941\) −21.1922 −0.690847 −0.345424 0.938447i \(-0.612265\pi\)
−0.345424 + 0.938447i \(0.612265\pi\)
\(942\) 0 0
\(943\) −3.68466 −0.119989
\(944\) −56.2159 −1.82967
\(945\) 0 0
\(946\) 20.4924 0.666266
\(947\) −46.6004 −1.51431 −0.757154 0.653236i \(-0.773411\pi\)
−0.757154 + 0.653236i \(0.773411\pi\)
\(948\) 0 0
\(949\) 6.00000 0.194768
\(950\) −4.87689 −0.158227
\(951\) 0 0
\(952\) 28.4924 0.923445
\(953\) 25.4384 0.824032 0.412016 0.911177i \(-0.364825\pi\)
0.412016 + 0.911177i \(0.364825\pi\)
\(954\) 0 0
\(955\) −21.6155 −0.699462
\(956\) −2.60037 −0.0841021
\(957\) 0 0
\(958\) 17.2614 0.557689
\(959\) 32.4924 1.04924
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 2.63068 0.0848166
\(963\) 0 0
\(964\) −10.8466 −0.349345
\(965\) 15.4384 0.496981
\(966\) 0 0
\(967\) 17.3693 0.558560 0.279280 0.960210i \(-0.409904\pi\)
0.279280 + 0.960210i \(0.409904\pi\)
\(968\) −10.8229 −0.347862
\(969\) 0 0
\(970\) 18.6307 0.598195
\(971\) 8.49242 0.272535 0.136267 0.990672i \(-0.456489\pi\)
0.136267 + 0.990672i \(0.456489\pi\)
\(972\) 0 0
\(973\) −71.5464 −2.29367
\(974\) −7.89205 −0.252878
\(975\) 0 0
\(976\) −26.6307 −0.852427
\(977\) 55.4773 1.77488 0.887438 0.460928i \(-0.152483\pi\)
0.887438 + 0.460928i \(0.152483\pi\)
\(978\) 0 0
\(979\) −4.31534 −0.137919
\(980\) 6.05398 0.193387
\(981\) 0 0
\(982\) −21.2614 −0.678477
\(983\) −6.73863 −0.214929 −0.107465 0.994209i \(-0.534273\pi\)
−0.107465 + 0.994209i \(0.534273\pi\)
\(984\) 0 0
\(985\) 21.3693 0.680883
\(986\) 4.49242 0.143068
\(987\) 0 0
\(988\) −1.36932 −0.0435638
\(989\) −33.6155 −1.06891
\(990\) 0 0
\(991\) −27.0540 −0.859398 −0.429699 0.902972i \(-0.641380\pi\)
−0.429699 + 0.902972i \(0.641380\pi\)
\(992\) 14.6307 0.464525
\(993\) 0 0
\(994\) 103.723 3.28991
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) 11.7538 0.372246 0.186123 0.982526i \(-0.440408\pi\)
0.186123 + 0.982526i \(0.440408\pi\)
\(998\) 48.3845 1.53158
\(999\) 0 0
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 585.2.a.l.1.1 yes 2
3.2 odd 2 585.2.a.j.1.2 2
4.3 odd 2 9360.2.a.cw.1.2 2
5.2 odd 4 2925.2.c.p.2224.2 4
5.3 odd 4 2925.2.c.p.2224.3 4
5.4 even 2 2925.2.a.x.1.2 2
12.11 even 2 9360.2.a.cl.1.2 2
13.12 even 2 7605.2.a.bd.1.2 2
15.2 even 4 2925.2.c.o.2224.3 4
15.8 even 4 2925.2.c.o.2224.2 4
15.14 odd 2 2925.2.a.bc.1.1 2
39.38 odd 2 7605.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.a.j.1.2 2 3.2 odd 2
585.2.a.l.1.1 yes 2 1.1 even 1 trivial
2925.2.a.x.1.2 2 5.4 even 2
2925.2.a.bc.1.1 2 15.14 odd 2
2925.2.c.o.2224.2 4 15.8 even 4
2925.2.c.o.2224.3 4 15.2 even 4
2925.2.c.p.2224.2 4 5.2 odd 4
2925.2.c.p.2224.3 4 5.3 odd 4
7605.2.a.bd.1.2 2 13.12 even 2
7605.2.a.bi.1.1 2 39.38 odd 2
9360.2.a.cl.1.2 2 12.11 even 2
9360.2.a.cw.1.2 2 4.3 odd 2