Properties

Label 4560.2.a.bs.1.1
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1772.1
Defining polynomial: \(x^{3} - x^{2} - 12 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.32803\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.20191 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.20191 q^{7} +1.00000 q^{9} -5.20191 q^{11} +5.45415 q^{13} -1.00000 q^{15} +4.00000 q^{17} -1.00000 q^{19} +3.20191 q^{21} +8.65606 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.20191 q^{29} +2.00000 q^{31} +5.20191 q^{33} -3.20191 q^{35} -11.8580 q^{37} -5.45415 q^{39} -9.85797 q^{41} +3.45415 q^{43} +1.00000 q^{45} -8.65606 q^{47} +3.25224 q^{49} -4.00000 q^{51} -10.6561 q^{53} -5.20191 q^{55} +1.00000 q^{57} +13.0599 q^{59} +3.74776 q^{61} -3.20191 q^{63} +5.45415 q^{65} -4.00000 q^{67} -8.65606 q^{69} -6.90830 q^{71} +6.00000 q^{73} -1.00000 q^{75} +16.6561 q^{77} +6.15158 q^{79} +1.00000 q^{81} +8.40382 q^{83} +4.00000 q^{85} +3.20191 q^{87} +13.8580 q^{89} -17.4637 q^{91} -2.00000 q^{93} -1.00000 q^{95} +0.142026 q^{97} -5.20191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 3q^{5} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 3q^{5} + 3q^{9} - 6q^{11} + 4q^{13} - 3q^{15} + 12q^{17} - 3q^{19} + 4q^{23} + 3q^{25} - 3q^{27} + 6q^{31} + 6q^{33} - 4q^{37} - 4q^{39} + 2q^{41} - 2q^{43} + 3q^{45} - 4q^{47} + 7q^{49} - 12q^{51} - 10q^{53} - 6q^{55} + 3q^{57} - 2q^{59} + 14q^{61} + 4q^{65} - 12q^{67} - 4q^{69} + 4q^{71} + 18q^{73} - 3q^{75} + 28q^{77} + 2q^{79} + 3q^{81} + 6q^{83} + 12q^{85} + 10q^{89} + 8q^{91} - 6q^{93} - 3q^{95} + 32q^{97} - 6q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.20191 −1.21021 −0.605104 0.796146i \(-0.706869\pi\)
−0.605104 + 0.796146i \(0.706869\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.20191 −1.56844 −0.784218 0.620486i \(-0.786935\pi\)
−0.784218 + 0.620486i \(0.786935\pi\)
\(12\) 0 0
\(13\) 5.45415 1.51271 0.756355 0.654162i \(-0.226978\pi\)
0.756355 + 0.654162i \(0.226978\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 3.20191 0.698714
\(22\) 0 0
\(23\) 8.65606 1.80491 0.902457 0.430780i \(-0.141762\pi\)
0.902457 + 0.430780i \(0.141762\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.20191 −0.594580 −0.297290 0.954787i \(-0.596083\pi\)
−0.297290 + 0.954787i \(0.596083\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 5.20191 0.905537
\(34\) 0 0
\(35\) −3.20191 −0.541222
\(36\) 0 0
\(37\) −11.8580 −1.94944 −0.974719 0.223432i \(-0.928274\pi\)
−0.974719 + 0.223432i \(0.928274\pi\)
\(38\) 0 0
\(39\) −5.45415 −0.873363
\(40\) 0 0
\(41\) −9.85797 −1.53956 −0.769778 0.638311i \(-0.779633\pi\)
−0.769778 + 0.638311i \(0.779633\pi\)
\(42\) 0 0
\(43\) 3.45415 0.526753 0.263377 0.964693i \(-0.415164\pi\)
0.263377 + 0.964693i \(0.415164\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −8.65606 −1.26262 −0.631308 0.775532i \(-0.717482\pi\)
−0.631308 + 0.775532i \(0.717482\pi\)
\(48\) 0 0
\(49\) 3.25224 0.464606
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −10.6561 −1.46372 −0.731861 0.681454i \(-0.761348\pi\)
−0.731861 + 0.681454i \(0.761348\pi\)
\(54\) 0 0
\(55\) −5.20191 −0.701426
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 13.0599 1.70025 0.850126 0.526579i \(-0.176526\pi\)
0.850126 + 0.526579i \(0.176526\pi\)
\(60\) 0 0
\(61\) 3.74776 0.479852 0.239926 0.970791i \(-0.422877\pi\)
0.239926 + 0.970791i \(0.422877\pi\)
\(62\) 0 0
\(63\) −3.20191 −0.403403
\(64\) 0 0
\(65\) 5.45415 0.676504
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −8.65606 −1.04207
\(70\) 0 0
\(71\) −6.90830 −0.819865 −0.409932 0.912116i \(-0.634448\pi\)
−0.409932 + 0.912116i \(0.634448\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 16.6561 1.89813
\(78\) 0 0
\(79\) 6.15158 0.692107 0.346054 0.938215i \(-0.387522\pi\)
0.346054 + 0.938215i \(0.387522\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.40382 0.922439 0.461220 0.887286i \(-0.347412\pi\)
0.461220 + 0.887286i \(0.347412\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 3.20191 0.343281
\(88\) 0 0
\(89\) 13.8580 1.46894 0.734471 0.678640i \(-0.237430\pi\)
0.734471 + 0.678640i \(0.237430\pi\)
\(90\) 0 0
\(91\) −17.4637 −1.83069
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 0.142026 0.0144205 0.00721026 0.999974i \(-0.497705\pi\)
0.00721026 + 0.999974i \(0.497705\pi\)
\(98\) 0 0
\(99\) −5.20191 −0.522812
\(100\) 0 0
\(101\) 4.40382 0.438197 0.219098 0.975703i \(-0.429688\pi\)
0.219098 + 0.975703i \(0.429688\pi\)
\(102\) 0 0
\(103\) 12.8076 1.26197 0.630987 0.775793i \(-0.282650\pi\)
0.630987 + 0.775793i \(0.282650\pi\)
\(104\) 0 0
\(105\) 3.20191 0.312475
\(106\) 0 0
\(107\) 12.8076 1.23816 0.619081 0.785327i \(-0.287505\pi\)
0.619081 + 0.785327i \(0.287505\pi\)
\(108\) 0 0
\(109\) 15.0599 1.44248 0.721238 0.692688i \(-0.243574\pi\)
0.721238 + 0.692688i \(0.243574\pi\)
\(110\) 0 0
\(111\) 11.8580 1.12551
\(112\) 0 0
\(113\) 5.05989 0.475994 0.237997 0.971266i \(-0.423509\pi\)
0.237997 + 0.971266i \(0.423509\pi\)
\(114\) 0 0
\(115\) 8.65606 0.807182
\(116\) 0 0
\(117\) 5.45415 0.504236
\(118\) 0 0
\(119\) −12.8076 −1.17408
\(120\) 0 0
\(121\) 16.0599 1.45999
\(122\) 0 0
\(123\) 9.85797 0.888864
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −3.49552 −0.310177 −0.155089 0.987901i \(-0.549566\pi\)
−0.155089 + 0.987901i \(0.549566\pi\)
\(128\) 0 0
\(129\) −3.45415 −0.304121
\(130\) 0 0
\(131\) 15.6057 1.36348 0.681740 0.731595i \(-0.261224\pi\)
0.681740 + 0.731595i \(0.261224\pi\)
\(132\) 0 0
\(133\) 3.20191 0.277641
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −0.656063 −0.0560512 −0.0280256 0.999607i \(-0.508922\pi\)
−0.0280256 + 0.999607i \(0.508922\pi\)
\(138\) 0 0
\(139\) −6.40382 −0.543165 −0.271583 0.962415i \(-0.587547\pi\)
−0.271583 + 0.962415i \(0.587547\pi\)
\(140\) 0 0
\(141\) 8.65606 0.728972
\(142\) 0 0
\(143\) −28.3720 −2.37259
\(144\) 0 0
\(145\) −3.20191 −0.265904
\(146\) 0 0
\(147\) −3.25224 −0.268240
\(148\) 0 0
\(149\) −23.3121 −1.90980 −0.954902 0.296922i \(-0.904040\pi\)
−0.954902 + 0.296922i \(0.904040\pi\)
\(150\) 0 0
\(151\) 23.3121 1.89711 0.948557 0.316607i \(-0.102544\pi\)
0.948557 + 0.316607i \(0.102544\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 6.25224 0.498983 0.249491 0.968377i \(-0.419737\pi\)
0.249491 + 0.968377i \(0.419737\pi\)
\(158\) 0 0
\(159\) 10.6561 0.845081
\(160\) 0 0
\(161\) −27.7159 −2.18432
\(162\) 0 0
\(163\) −9.85797 −0.772136 −0.386068 0.922470i \(-0.626167\pi\)
−0.386068 + 0.922470i \(0.626167\pi\)
\(164\) 0 0
\(165\) 5.20191 0.404968
\(166\) 0 0
\(167\) 6.25224 0.483813 0.241906 0.970300i \(-0.422227\pi\)
0.241906 + 0.970300i \(0.422227\pi\)
\(168\) 0 0
\(169\) 16.7478 1.28829
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 9.05989 0.688810 0.344405 0.938821i \(-0.388081\pi\)
0.344405 + 0.938821i \(0.388081\pi\)
\(174\) 0 0
\(175\) −3.20191 −0.242042
\(176\) 0 0
\(177\) −13.0599 −0.981641
\(178\) 0 0
\(179\) 4.25224 0.317827 0.158914 0.987292i \(-0.449201\pi\)
0.158914 + 0.987292i \(0.449201\pi\)
\(180\) 0 0
\(181\) 18.5554 1.37921 0.689606 0.724184i \(-0.257784\pi\)
0.689606 + 0.724184i \(0.257784\pi\)
\(182\) 0 0
\(183\) −3.74776 −0.277042
\(184\) 0 0
\(185\) −11.8580 −0.871816
\(186\) 0 0
\(187\) −20.8076 −1.52161
\(188\) 0 0
\(189\) 3.20191 0.232905
\(190\) 0 0
\(191\) 22.5140 1.62906 0.814529 0.580122i \(-0.196995\pi\)
0.814529 + 0.580122i \(0.196995\pi\)
\(192\) 0 0
\(193\) −17.7573 −1.27820 −0.639100 0.769124i \(-0.720693\pi\)
−0.639100 + 0.769124i \(0.720693\pi\)
\(194\) 0 0
\(195\) −5.45415 −0.390580
\(196\) 0 0
\(197\) 15.4637 1.10174 0.550872 0.834590i \(-0.314295\pi\)
0.550872 + 0.834590i \(0.314295\pi\)
\(198\) 0 0
\(199\) −2.90830 −0.206164 −0.103082 0.994673i \(-0.532870\pi\)
−0.103082 + 0.994673i \(0.532870\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 10.2522 0.719566
\(204\) 0 0
\(205\) −9.85797 −0.688511
\(206\) 0 0
\(207\) 8.65606 0.601638
\(208\) 0 0
\(209\) 5.20191 0.359824
\(210\) 0 0
\(211\) 3.49552 0.240642 0.120321 0.992735i \(-0.461608\pi\)
0.120321 + 0.992735i \(0.461608\pi\)
\(212\) 0 0
\(213\) 6.90830 0.473349
\(214\) 0 0
\(215\) 3.45415 0.235571
\(216\) 0 0
\(217\) −6.40382 −0.434720
\(218\) 0 0
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 21.8166 1.46754
\(222\) 0 0
\(223\) −0.504478 −0.0337823 −0.0168912 0.999857i \(-0.505377\pi\)
−0.0168912 + 0.999857i \(0.505377\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −19.5644 −1.29853 −0.649266 0.760561i \(-0.724924\pi\)
−0.649266 + 0.760561i \(0.724924\pi\)
\(228\) 0 0
\(229\) 15.7478 1.04064 0.520321 0.853971i \(-0.325812\pi\)
0.520321 + 0.853971i \(0.325812\pi\)
\(230\) 0 0
\(231\) −16.6561 −1.09589
\(232\) 0 0
\(233\) −17.1605 −1.12422 −0.562112 0.827061i \(-0.690011\pi\)
−0.562112 + 0.827061i \(0.690011\pi\)
\(234\) 0 0
\(235\) −8.65606 −0.564659
\(236\) 0 0
\(237\) −6.15158 −0.399588
\(238\) 0 0
\(239\) 17.7064 1.14533 0.572666 0.819789i \(-0.305909\pi\)
0.572666 + 0.819789i \(0.305909\pi\)
\(240\) 0 0
\(241\) 23.8166 1.53416 0.767081 0.641550i \(-0.221708\pi\)
0.767081 + 0.641550i \(0.221708\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.25224 0.207778
\(246\) 0 0
\(247\) −5.45415 −0.347039
\(248\) 0 0
\(249\) −8.40382 −0.532571
\(250\) 0 0
\(251\) −7.10126 −0.448227 −0.224114 0.974563i \(-0.571949\pi\)
−0.224114 + 0.974563i \(0.571949\pi\)
\(252\) 0 0
\(253\) −45.0281 −2.83089
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 0 0
\(257\) 21.5644 1.34515 0.672574 0.740030i \(-0.265189\pi\)
0.672574 + 0.740030i \(0.265189\pi\)
\(258\) 0 0
\(259\) 37.9682 2.35923
\(260\) 0 0
\(261\) −3.20191 −0.198193
\(262\) 0 0
\(263\) −6.25224 −0.385530 −0.192765 0.981245i \(-0.561745\pi\)
−0.192765 + 0.981245i \(0.561745\pi\)
\(264\) 0 0
\(265\) −10.6561 −0.654597
\(266\) 0 0
\(267\) −13.8580 −0.848094
\(268\) 0 0
\(269\) 16.5140 1.00688 0.503439 0.864031i \(-0.332068\pi\)
0.503439 + 0.864031i \(0.332068\pi\)
\(270\) 0 0
\(271\) 22.4038 1.36094 0.680468 0.732778i \(-0.261777\pi\)
0.680468 + 0.732778i \(0.261777\pi\)
\(272\) 0 0
\(273\) 17.4637 1.05695
\(274\) 0 0
\(275\) −5.20191 −0.313687
\(276\) 0 0
\(277\) 7.34394 0.441254 0.220627 0.975358i \(-0.429190\pi\)
0.220627 + 0.975358i \(0.429190\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −1.85797 −0.110837 −0.0554187 0.998463i \(-0.517649\pi\)
−0.0554187 + 0.998463i \(0.517649\pi\)
\(282\) 0 0
\(283\) −1.05033 −0.0624355 −0.0312177 0.999513i \(-0.509939\pi\)
−0.0312177 + 0.999513i \(0.509939\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 31.5644 1.86319
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −0.142026 −0.00832569
\(292\) 0 0
\(293\) −15.7478 −0.919994 −0.459997 0.887920i \(-0.652150\pi\)
−0.459997 + 0.887920i \(0.652150\pi\)
\(294\) 0 0
\(295\) 13.0599 0.760376
\(296\) 0 0
\(297\) 5.20191 0.301846
\(298\) 0 0
\(299\) 47.2115 2.73031
\(300\) 0 0
\(301\) −11.0599 −0.637481
\(302\) 0 0
\(303\) −4.40382 −0.252993
\(304\) 0 0
\(305\) 3.74776 0.214596
\(306\) 0 0
\(307\) −17.5962 −1.00427 −0.502133 0.864790i \(-0.667451\pi\)
−0.502133 + 0.864790i \(0.667451\pi\)
\(308\) 0 0
\(309\) −12.8076 −0.728602
\(310\) 0 0
\(311\) −6.51404 −0.369377 −0.184689 0.982797i \(-0.559128\pi\)
−0.184689 + 0.982797i \(0.559128\pi\)
\(312\) 0 0
\(313\) 29.2115 1.65113 0.825565 0.564307i \(-0.190857\pi\)
0.825565 + 0.564307i \(0.190857\pi\)
\(314\) 0 0
\(315\) −3.20191 −0.180407
\(316\) 0 0
\(317\) −28.4727 −1.59918 −0.799592 0.600543i \(-0.794951\pi\)
−0.799592 + 0.600543i \(0.794951\pi\)
\(318\) 0 0
\(319\) 16.6561 0.932560
\(320\) 0 0
\(321\) −12.8076 −0.714853
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) 5.45415 0.302542
\(326\) 0 0
\(327\) −15.0599 −0.832814
\(328\) 0 0
\(329\) 27.7159 1.52803
\(330\) 0 0
\(331\) −21.9682 −1.20748 −0.603740 0.797181i \(-0.706324\pi\)
−0.603740 + 0.797181i \(0.706324\pi\)
\(332\) 0 0
\(333\) −11.8580 −0.649813
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 11.3535 0.618464 0.309232 0.950987i \(-0.399928\pi\)
0.309232 + 0.950987i \(0.399928\pi\)
\(338\) 0 0
\(339\) −5.05989 −0.274815
\(340\) 0 0
\(341\) −10.4038 −0.563399
\(342\) 0 0
\(343\) 12.0000 0.647939
\(344\) 0 0
\(345\) −8.65606 −0.466027
\(346\) 0 0
\(347\) −21.2115 −1.13869 −0.569346 0.822098i \(-0.692803\pi\)
−0.569346 + 0.822098i \(0.692803\pi\)
\(348\) 0 0
\(349\) −11.3121 −0.605524 −0.302762 0.953066i \(-0.597909\pi\)
−0.302762 + 0.953066i \(0.597909\pi\)
\(350\) 0 0
\(351\) −5.45415 −0.291121
\(352\) 0 0
\(353\) 13.4637 0.716601 0.358300 0.933606i \(-0.383356\pi\)
0.358300 + 0.933606i \(0.383356\pi\)
\(354\) 0 0
\(355\) −6.90830 −0.366655
\(356\) 0 0
\(357\) 12.8076 0.677853
\(358\) 0 0
\(359\) 3.88979 0.205295 0.102648 0.994718i \(-0.467269\pi\)
0.102648 + 0.994718i \(0.467269\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −16.0599 −0.842925
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) −16.5140 −0.862026 −0.431013 0.902346i \(-0.641844\pi\)
−0.431013 + 0.902346i \(0.641844\pi\)
\(368\) 0 0
\(369\) −9.85797 −0.513186
\(370\) 0 0
\(371\) 34.1198 1.77141
\(372\) 0 0
\(373\) 9.45415 0.489517 0.244759 0.969584i \(-0.421291\pi\)
0.244759 + 0.969584i \(0.421291\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −17.4637 −0.899427
\(378\) 0 0
\(379\) −5.46371 −0.280652 −0.140326 0.990105i \(-0.544815\pi\)
−0.140326 + 0.990105i \(0.544815\pi\)
\(380\) 0 0
\(381\) 3.49552 0.179081
\(382\) 0 0
\(383\) 22.1198 1.13027 0.565134 0.824999i \(-0.308825\pi\)
0.565134 + 0.824999i \(0.308825\pi\)
\(384\) 0 0
\(385\) 16.6561 0.848872
\(386\) 0 0
\(387\) 3.45415 0.175584
\(388\) 0 0
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 34.6243 1.75102
\(392\) 0 0
\(393\) −15.6057 −0.787205
\(394\) 0 0
\(395\) 6.15158 0.309520
\(396\) 0 0
\(397\) 33.1605 1.66428 0.832140 0.554566i \(-0.187116\pi\)
0.832140 + 0.554566i \(0.187116\pi\)
\(398\) 0 0
\(399\) −3.20191 −0.160296
\(400\) 0 0
\(401\) −37.0694 −1.85116 −0.925580 0.378552i \(-0.876422\pi\)
−0.925580 + 0.378552i \(0.876422\pi\)
\(402\) 0 0
\(403\) 10.9083 0.543381
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 61.6841 3.05757
\(408\) 0 0
\(409\) 15.5962 0.771181 0.385591 0.922670i \(-0.373998\pi\)
0.385591 + 0.922670i \(0.373998\pi\)
\(410\) 0 0
\(411\) 0.656063 0.0323612
\(412\) 0 0
\(413\) −41.8166 −2.05766
\(414\) 0 0
\(415\) 8.40382 0.412527
\(416\) 0 0
\(417\) 6.40382 0.313597
\(418\) 0 0
\(419\) −5.70639 −0.278775 −0.139388 0.990238i \(-0.544513\pi\)
−0.139388 + 0.990238i \(0.544513\pi\)
\(420\) 0 0
\(421\) 3.64711 0.177749 0.0888745 0.996043i \(-0.471673\pi\)
0.0888745 + 0.996043i \(0.471673\pi\)
\(422\) 0 0
\(423\) −8.65606 −0.420872
\(424\) 0 0
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) −12.0000 −0.580721
\(428\) 0 0
\(429\) 28.3720 1.36981
\(430\) 0 0
\(431\) −40.5236 −1.95195 −0.975976 0.217876i \(-0.930087\pi\)
−0.975976 + 0.217876i \(0.930087\pi\)
\(432\) 0 0
\(433\) −8.14203 −0.391281 −0.195640 0.980676i \(-0.562679\pi\)
−0.195640 + 0.980676i \(0.562679\pi\)
\(434\) 0 0
\(435\) 3.20191 0.153520
\(436\) 0 0
\(437\) −8.65606 −0.414076
\(438\) 0 0
\(439\) 30.1516 1.43906 0.719528 0.694463i \(-0.244358\pi\)
0.719528 + 0.694463i \(0.244358\pi\)
\(440\) 0 0
\(441\) 3.25224 0.154869
\(442\) 0 0
\(443\) −24.6243 −1.16993 −0.584967 0.811057i \(-0.698892\pi\)
−0.584967 + 0.811057i \(0.698892\pi\)
\(444\) 0 0
\(445\) 13.8580 0.656931
\(446\) 0 0
\(447\) 23.3121 1.10263
\(448\) 0 0
\(449\) −0.545849 −0.0257602 −0.0128801 0.999917i \(-0.504100\pi\)
−0.0128801 + 0.999917i \(0.504100\pi\)
\(450\) 0 0
\(451\) 51.2803 2.41470
\(452\) 0 0
\(453\) −23.3121 −1.09530
\(454\) 0 0
\(455\) −17.4637 −0.818711
\(456\) 0 0
\(457\) 34.2204 1.60076 0.800382 0.599490i \(-0.204630\pi\)
0.800382 + 0.599490i \(0.204630\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −0.687875 −0.0320375 −0.0160188 0.999872i \(-0.505099\pi\)
−0.0160188 + 0.999872i \(0.505099\pi\)
\(462\) 0 0
\(463\) 4.79809 0.222986 0.111493 0.993765i \(-0.464437\pi\)
0.111493 + 0.993765i \(0.464437\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 0 0
\(467\) −11.0090 −0.509434 −0.254717 0.967016i \(-0.581982\pi\)
−0.254717 + 0.967016i \(0.581982\pi\)
\(468\) 0 0
\(469\) 12.8076 0.591402
\(470\) 0 0
\(471\) −6.25224 −0.288088
\(472\) 0 0
\(473\) −17.9682 −0.826178
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −10.6561 −0.487908
\(478\) 0 0
\(479\) 26.7981 1.22444 0.612218 0.790689i \(-0.290278\pi\)
0.612218 + 0.790689i \(0.290278\pi\)
\(480\) 0 0
\(481\) −64.6752 −2.94893
\(482\) 0 0
\(483\) 27.7159 1.26112
\(484\) 0 0
\(485\) 0.142026 0.00644905
\(486\) 0 0
\(487\) −18.7070 −0.847695 −0.423847 0.905734i \(-0.639321\pi\)
−0.423847 + 0.905734i \(0.639321\pi\)
\(488\) 0 0
\(489\) 9.85797 0.445793
\(490\) 0 0
\(491\) −12.1102 −0.546526 −0.273263 0.961939i \(-0.588103\pi\)
−0.273263 + 0.961939i \(0.588103\pi\)
\(492\) 0 0
\(493\) −12.8076 −0.576827
\(494\) 0 0
\(495\) −5.20191 −0.233809
\(496\) 0 0
\(497\) 22.1198 0.992207
\(498\) 0 0
\(499\) 1.59618 0.0714547 0.0357273 0.999362i \(-0.488625\pi\)
0.0357273 + 0.999362i \(0.488625\pi\)
\(500\) 0 0
\(501\) −6.25224 −0.279329
\(502\) 0 0
\(503\) −2.55541 −0.113940 −0.0569700 0.998376i \(-0.518144\pi\)
−0.0569700 + 0.998376i \(0.518144\pi\)
\(504\) 0 0
\(505\) 4.40382 0.195968
\(506\) 0 0
\(507\) −16.7478 −0.743794
\(508\) 0 0
\(509\) 0.210868 0.00934655 0.00467328 0.999989i \(-0.498512\pi\)
0.00467328 + 0.999989i \(0.498512\pi\)
\(510\) 0 0
\(511\) −19.2115 −0.849865
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 12.8076 0.564372
\(516\) 0 0
\(517\) 45.0281 1.98033
\(518\) 0 0
\(519\) −9.05989 −0.397685
\(520\) 0 0
\(521\) −8.04137 −0.352299 −0.176149 0.984363i \(-0.556364\pi\)
−0.176149 + 0.984363i \(0.556364\pi\)
\(522\) 0 0
\(523\) −5.39487 −0.235901 −0.117951 0.993019i \(-0.537632\pi\)
−0.117951 + 0.993019i \(0.537632\pi\)
\(524\) 0 0
\(525\) 3.20191 0.139743
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 51.9274 2.25771
\(530\) 0 0
\(531\) 13.0599 0.566751
\(532\) 0 0
\(533\) −53.7669 −2.32890
\(534\) 0 0
\(535\) 12.8076 0.553723
\(536\) 0 0
\(537\) −4.25224 −0.183498
\(538\) 0 0
\(539\) −16.9179 −0.728704
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) −18.5554 −0.796289
\(544\) 0 0
\(545\) 15.0599 0.645095
\(546\) 0 0
\(547\) −9.59618 −0.410303 −0.205151 0.978730i \(-0.565769\pi\)
−0.205151 + 0.978730i \(0.565769\pi\)
\(548\) 0 0
\(549\) 3.74776 0.159951
\(550\) 0 0
\(551\) 3.20191 0.136406
\(552\) 0 0
\(553\) −19.6968 −0.837594
\(554\) 0 0
\(555\) 11.8580 0.503343
\(556\) 0 0
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 18.8395 0.796824
\(560\) 0 0
\(561\) 20.8076 0.878500
\(562\) 0 0
\(563\) −38.2522 −1.61214 −0.806070 0.591820i \(-0.798409\pi\)
−0.806070 + 0.591820i \(0.798409\pi\)
\(564\) 0 0
\(565\) 5.05989 0.212871
\(566\) 0 0
\(567\) −3.20191 −0.134468
\(568\) 0 0
\(569\) −16.5458 −0.693638 −0.346819 0.937932i \(-0.612738\pi\)
−0.346819 + 0.937932i \(0.612738\pi\)
\(570\) 0 0
\(571\) 31.7159 1.32727 0.663636 0.748056i \(-0.269013\pi\)
0.663636 + 0.748056i \(0.269013\pi\)
\(572\) 0 0
\(573\) −22.5140 −0.940537
\(574\) 0 0
\(575\) 8.65606 0.360983
\(576\) 0 0
\(577\) 20.7070 0.862043 0.431022 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(578\) 0 0
\(579\) 17.7573 0.737969
\(580\) 0 0
\(581\) −26.9083 −1.11634
\(582\) 0 0
\(583\) 55.4319 2.29575
\(584\) 0 0
\(585\) 5.45415 0.225501
\(586\) 0 0
\(587\) 11.3121 0.466901 0.233451 0.972369i \(-0.424998\pi\)
0.233451 + 0.972369i \(0.424998\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) −15.4637 −0.636092
\(592\) 0 0
\(593\) −24.9592 −1.02495 −0.512476 0.858701i \(-0.671272\pi\)
−0.512476 + 0.858701i \(0.671272\pi\)
\(594\) 0 0
\(595\) −12.8076 −0.525062
\(596\) 0 0
\(597\) 2.90830 0.119029
\(598\) 0 0
\(599\) 42.4229 1.73335 0.866677 0.498869i \(-0.166251\pi\)
0.866677 + 0.498869i \(0.166251\pi\)
\(600\) 0 0
\(601\) 16.9083 0.689704 0.344852 0.938657i \(-0.387929\pi\)
0.344852 + 0.938657i \(0.387929\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 16.0599 0.652927
\(606\) 0 0
\(607\) 1.39487 0.0566159 0.0283080 0.999599i \(-0.490988\pi\)
0.0283080 + 0.999599i \(0.490988\pi\)
\(608\) 0 0
\(609\) −10.2522 −0.415442
\(610\) 0 0
\(611\) −47.2115 −1.90997
\(612\) 0 0
\(613\) 36.8765 1.48943 0.744714 0.667384i \(-0.232586\pi\)
0.744714 + 0.667384i \(0.232586\pi\)
\(614\) 0 0
\(615\) 9.85797 0.397512
\(616\) 0 0
\(617\) 16.3032 0.656341 0.328170 0.944619i \(-0.393568\pi\)
0.328170 + 0.944619i \(0.393568\pi\)
\(618\) 0 0
\(619\) 36.5236 1.46801 0.734004 0.679146i \(-0.237649\pi\)
0.734004 + 0.679146i \(0.237649\pi\)
\(620\) 0 0
\(621\) −8.65606 −0.347356
\(622\) 0 0
\(623\) −44.3720 −1.77773
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −5.20191 −0.207744
\(628\) 0 0
\(629\) −47.4319 −1.89123
\(630\) 0 0
\(631\) −10.6243 −0.422945 −0.211472 0.977384i \(-0.567826\pi\)
−0.211472 + 0.977384i \(0.567826\pi\)
\(632\) 0 0
\(633\) −3.49552 −0.138935
\(634\) 0 0
\(635\) −3.49552 −0.138716
\(636\) 0 0
\(637\) 17.7382 0.702813
\(638\) 0 0
\(639\) −6.90830 −0.273288
\(640\) 0 0
\(641\) 18.1611 0.717322 0.358661 0.933468i \(-0.383233\pi\)
0.358661 + 0.933468i \(0.383233\pi\)
\(642\) 0 0
\(643\) −27.4542 −1.08269 −0.541343 0.840802i \(-0.682084\pi\)
−0.541343 + 0.840802i \(0.682084\pi\)
\(644\) 0 0
\(645\) −3.45415 −0.136007
\(646\) 0 0
\(647\) −19.5644 −0.769155 −0.384577 0.923093i \(-0.625653\pi\)
−0.384577 + 0.923093i \(0.625653\pi\)
\(648\) 0 0
\(649\) −67.9364 −2.66674
\(650\) 0 0
\(651\) 6.40382 0.250986
\(652\) 0 0
\(653\) −38.1516 −1.49299 −0.746493 0.665393i \(-0.768264\pi\)
−0.746493 + 0.665393i \(0.768264\pi\)
\(654\) 0 0
\(655\) 15.6057 0.609767
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 12.4727 0.485866 0.242933 0.970043i \(-0.421890\pi\)
0.242933 + 0.970043i \(0.421890\pi\)
\(660\) 0 0
\(661\) 11.5644 0.449802 0.224901 0.974382i \(-0.427794\pi\)
0.224901 + 0.974382i \(0.427794\pi\)
\(662\) 0 0
\(663\) −21.8166 −0.847287
\(664\) 0 0
\(665\) 3.20191 0.124165
\(666\) 0 0
\(667\) −27.7159 −1.07317
\(668\) 0 0
\(669\) 0.504478 0.0195042
\(670\) 0 0
\(671\) −19.4955 −0.752616
\(672\) 0 0
\(673\) −6.46311 −0.249134 −0.124567 0.992211i \(-0.539754\pi\)
−0.124567 + 0.992211i \(0.539754\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −31.9682 −1.22864 −0.614319 0.789058i \(-0.710569\pi\)
−0.614319 + 0.789058i \(0.710569\pi\)
\(678\) 0 0
\(679\) −0.454754 −0.0174518
\(680\) 0 0
\(681\) 19.5644 0.749708
\(682\) 0 0
\(683\) −2.68787 −0.102849 −0.0514243 0.998677i \(-0.516376\pi\)
−0.0514243 + 0.998677i \(0.516376\pi\)
\(684\) 0 0
\(685\) −0.656063 −0.0250669
\(686\) 0 0
\(687\) −15.7478 −0.600815
\(688\) 0 0
\(689\) −58.1198 −2.21419
\(690\) 0 0
\(691\) 26.9083 1.02364 0.511820 0.859093i \(-0.328971\pi\)
0.511820 + 0.859093i \(0.328971\pi\)
\(692\) 0 0
\(693\) 16.6561 0.632711
\(694\) 0 0
\(695\) −6.40382 −0.242911
\(696\) 0 0
\(697\) −39.4319 −1.49359
\(698\) 0 0
\(699\) 17.1605 0.649071
\(700\) 0 0
\(701\) 32.6243 1.23220 0.616100 0.787668i \(-0.288712\pi\)
0.616100 + 0.787668i \(0.288712\pi\)
\(702\) 0 0
\(703\) 11.8580 0.447232
\(704\) 0 0
\(705\) 8.65606 0.326006
\(706\) 0 0
\(707\) −14.1007 −0.530310
\(708\) 0 0
\(709\) −29.3631 −1.10275 −0.551376 0.834257i \(-0.685897\pi\)
−0.551376 + 0.834257i \(0.685897\pi\)
\(710\) 0 0
\(711\) 6.15158 0.230702
\(712\) 0 0
\(713\) 17.3121 0.648344
\(714\) 0 0
\(715\) −28.3720 −1.06105
\(716\) 0 0
\(717\) −17.7064 −0.661257
\(718\) 0 0
\(719\) −28.4134 −1.05964 −0.529820 0.848110i \(-0.677741\pi\)
−0.529820 + 0.848110i \(0.677741\pi\)
\(720\) 0 0
\(721\) −41.0090 −1.52725
\(722\) 0 0
\(723\) −23.8166 −0.885749
\(724\) 0 0
\(725\) −3.20191 −0.118916
\(726\) 0 0
\(727\) 46.1293 1.71084 0.855421 0.517933i \(-0.173298\pi\)
0.855421 + 0.517933i \(0.173298\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 13.8166 0.511026
\(732\) 0 0
\(733\) 28.6561 1.05844 0.529218 0.848486i \(-0.322485\pi\)
0.529218 + 0.848486i \(0.322485\pi\)
\(734\) 0 0
\(735\) −3.25224 −0.119961
\(736\) 0 0
\(737\) 20.8076 0.766460
\(738\) 0 0
\(739\) 25.8166 0.949679 0.474840 0.880072i \(-0.342506\pi\)
0.474840 + 0.880072i \(0.342506\pi\)
\(740\) 0 0
\(741\) 5.45415 0.200363
\(742\) 0 0
\(743\) 38.1198 1.39848 0.699239 0.714888i \(-0.253522\pi\)
0.699239 + 0.714888i \(0.253522\pi\)
\(744\) 0 0
\(745\) −23.3121 −0.854090
\(746\) 0 0
\(747\) 8.40382 0.307480
\(748\) 0 0
\(749\) −41.0090 −1.49843
\(750\) 0 0
\(751\) −7.31213 −0.266823 −0.133412 0.991061i \(-0.542593\pi\)
−0.133412 + 0.991061i \(0.542593\pi\)
\(752\) 0 0
\(753\) 7.10126 0.258784
\(754\) 0 0
\(755\) 23.3121 0.848415
\(756\) 0 0
\(757\) −3.86753 −0.140568 −0.0702839 0.997527i \(-0.522391\pi\)
−0.0702839 + 0.997527i \(0.522391\pi\)
\(758\) 0 0
\(759\) 45.0281 1.63442
\(760\) 0 0
\(761\) −12.3211 −0.446639 −0.223319 0.974745i \(-0.571689\pi\)
−0.223319 + 0.974745i \(0.571689\pi\)
\(762\) 0 0
\(763\) −48.2204 −1.74570
\(764\) 0 0
\(765\) 4.00000 0.144620
\(766\) 0 0
\(767\) 71.2306 2.57199
\(768\) 0 0
\(769\) −8.11977 −0.292806 −0.146403 0.989225i \(-0.546770\pi\)
−0.146403 + 0.989225i \(0.546770\pi\)
\(770\) 0 0
\(771\) −21.5644 −0.776622
\(772\) 0 0
\(773\) −17.3439 −0.623818 −0.311909 0.950112i \(-0.600968\pi\)
−0.311909 + 0.950112i \(0.600968\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) −37.9682 −1.36210
\(778\) 0 0
\(779\) 9.85797 0.353199
\(780\) 0 0
\(781\) 35.9364 1.28590
\(782\) 0 0
\(783\) 3.20191 0.114427
\(784\) 0 0
\(785\) 6.25224 0.223152
\(786\) 0 0
\(787\) 28.2204 1.00595 0.502975 0.864301i \(-0.332239\pi\)
0.502975 + 0.864301i \(0.332239\pi\)
\(788\) 0 0
\(789\) 6.25224 0.222586
\(790\) 0 0
\(791\) −16.2013 −0.576052
\(792\) 0 0
\(793\) 20.4409 0.725876
\(794\) 0 0
\(795\) 10.6561 0.377932
\(796\) 0 0
\(797\) 1.56436 0.0554126 0.0277063 0.999616i \(-0.491180\pi\)
0.0277063 + 0.999616i \(0.491180\pi\)
\(798\) 0 0
\(799\) −34.6243 −1.22492
\(800\) 0 0
\(801\) 13.8580 0.489647
\(802\) 0 0
\(803\) −31.2115 −1.10143
\(804\) 0 0
\(805\) −27.7159 −0.976859
\(806\) 0 0
\(807\) −16.5140 −0.581322
\(808\) 0 0
\(809\) −6.80765 −0.239344 −0.119672 0.992813i \(-0.538184\pi\)
−0.119672 + 0.992813i \(0.538184\pi\)
\(810\) 0 0
\(811\) −13.5134 −0.474521 −0.237260 0.971446i \(-0.576249\pi\)
−0.237260 + 0.971446i \(0.576249\pi\)
\(812\) 0 0
\(813\) −22.4038 −0.785736
\(814\) 0 0
\(815\) −9.85797 −0.345310
\(816\) 0 0
\(817\) −3.45415 −0.120845
\(818\) 0 0
\(819\) −17.4637 −0.610231
\(820\) 0 0
\(821\) −31.5326 −1.10049 −0.550247 0.835002i \(-0.685466\pi\)
−0.550247 + 0.835002i \(0.685466\pi\)
\(822\) 0 0
\(823\) 38.1102 1.32844 0.664219 0.747538i \(-0.268764\pi\)
0.664219 + 0.747538i \(0.268764\pi\)
\(824\) 0 0
\(825\) 5.20191 0.181107
\(826\) 0 0
\(827\) −1.74776 −0.0607756 −0.0303878 0.999538i \(-0.509674\pi\)
−0.0303878 + 0.999538i \(0.509674\pi\)
\(828\) 0 0
\(829\) −10.2522 −0.356075 −0.178037 0.984024i \(-0.556975\pi\)
−0.178037 + 0.984024i \(0.556975\pi\)
\(830\) 0 0
\(831\) −7.34394 −0.254758
\(832\) 0 0
\(833\) 13.0090 0.450734
\(834\) 0 0
\(835\) 6.25224 0.216368
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) −19.7159 −0.680670 −0.340335 0.940304i \(-0.610540\pi\)
−0.340335 + 0.940304i \(0.610540\pi\)
\(840\) 0 0
\(841\) −18.7478 −0.646475
\(842\) 0 0
\(843\) 1.85797 0.0639920
\(844\) 0 0
\(845\) 16.7478 0.576140
\(846\) 0 0
\(847\) −51.4223 −1.76689
\(848\) 0 0
\(849\) 1.05033 0.0360471
\(850\) 0 0
\(851\) −102.643 −3.51857
\(852\) 0 0
\(853\) 22.2522 0.761902 0.380951 0.924595i \(-0.375597\pi\)
0.380951 + 0.924595i \(0.375597\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 53.7848 1.83725 0.918627 0.395126i \(-0.129299\pi\)
0.918627 + 0.395126i \(0.129299\pi\)
\(858\) 0 0
\(859\) −5.31213 −0.181247 −0.0906237 0.995885i \(-0.528886\pi\)
−0.0906237 + 0.995885i \(0.528886\pi\)
\(860\) 0 0
\(861\) −31.5644 −1.07571
\(862\) 0 0
\(863\) −9.31213 −0.316988 −0.158494 0.987360i \(-0.550664\pi\)
−0.158494 + 0.987360i \(0.550664\pi\)
\(864\) 0 0
\(865\) 9.05989 0.308045
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −21.8166 −0.739227
\(872\) 0 0
\(873\) 0.142026 0.00480684
\(874\) 0 0
\(875\) −3.20191 −0.108244
\(876\) 0 0
\(877\) 24.3625 0.822662 0.411331 0.911486i \(-0.365064\pi\)
0.411331 + 0.911486i \(0.365064\pi\)
\(878\) 0 0
\(879\) 15.7478 0.531159
\(880\) 0 0
\(881\) −59.0281 −1.98871 −0.994353 0.106122i \(-0.966157\pi\)
−0.994353 + 0.106122i \(0.966157\pi\)
\(882\) 0 0
\(883\) −39.1701 −1.31818 −0.659089 0.752065i \(-0.729058\pi\)
−0.659089 + 0.752065i \(0.729058\pi\)
\(884\) 0 0
\(885\) −13.0599 −0.439003
\(886\) 0 0
\(887\) −55.4319 −1.86122 −0.930610 0.366011i \(-0.880723\pi\)
−0.930610 + 0.366011i \(0.880723\pi\)
\(888\) 0 0
\(889\) 11.1924 0.375379
\(890\) 0 0
\(891\) −5.20191 −0.174271
\(892\) 0 0
\(893\) 8.65606 0.289664
\(894\) 0 0
\(895\) 4.25224 0.142137
\(896\) 0 0
\(897\) −47.2115 −1.57635
\(898\) 0 0
\(899\) −6.40382 −0.213579
\(900\) 0 0
\(901\) −42.6243 −1.42002
\(902\) 0 0
\(903\) 11.0599 0.368050
\(904\) 0 0
\(905\) 18.5554 0.616803
\(906\) 0 0
\(907\) −18.9083 −0.627840 −0.313920 0.949449i \(-0.601642\pi\)
−0.313920 + 0.949449i \(0.601642\pi\)
\(908\) 0 0
\(909\) 4.40382 0.146066
\(910\) 0 0
\(911\) 25.8166 0.855342 0.427671 0.903934i \(-0.359334\pi\)
0.427671 + 0.903934i \(0.359334\pi\)
\(912\) 0 0
\(913\) −43.7159 −1.44679
\(914\) 0 0
\(915\) −3.74776 −0.123897
\(916\) 0 0
\(917\) −49.9682 −1.65009
\(918\) 0 0
\(919\) 38.1198 1.25746 0.628728 0.777626i \(-0.283576\pi\)
0.628728 + 0.777626i \(0.283576\pi\)
\(920\) 0 0
\(921\) 17.5962 0.579814
\(922\) 0 0
\(923\) −37.6789 −1.24022
\(924\) 0 0
\(925\) −11.8580 −0.389888
\(926\) 0 0
\(927\) 12.8076 0.420658
\(928\) 0 0
\(929\) 33.5146 1.09958 0.549790 0.835303i \(-0.314708\pi\)
0.549790 + 0.835303i \(0.314708\pi\)
\(930\) 0 0
\(931\) −3.25224 −0.106588
\(932\) 0 0
\(933\) 6.51404 0.213260
\(934\) 0 0
\(935\) −20.8076 −0.680483
\(936\) 0 0
\(937\) 52.8447 1.72636 0.863180 0.504896i \(-0.168469\pi\)
0.863180 + 0.504896i \(0.168469\pi\)
\(938\) 0 0
\(939\) −29.2115 −0.953280
\(940\) 0 0
\(941\) 52.8172 1.72179 0.860896 0.508781i \(-0.169904\pi\)
0.860896 + 0.508781i \(0.169904\pi\)
\(942\) 0 0
\(943\) −85.3312 −2.77877
\(944\) 0 0
\(945\) 3.20191 0.104158
\(946\) 0 0
\(947\) −19.0917 −0.620397 −0.310198 0.950672i \(-0.600395\pi\)
−0.310198 + 0.950672i \(0.600395\pi\)
\(948\) 0 0
\(949\) 32.7249 1.06230
\(950\) 0 0
\(951\) 28.4727 0.923289
\(952\) 0 0
\(953\) 46.1707 1.49562 0.747808 0.663915i \(-0.231106\pi\)
0.747808 + 0.663915i \(0.231106\pi\)
\(954\) 0 0
\(955\) 22.5140 0.728537
\(956\) 0 0
\(957\) −16.6561 −0.538414
\(958\) 0 0
\(959\) 2.10065 0.0678337
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 12.8076 0.412721
\(964\) 0 0
\(965\) −17.7573 −0.571628
\(966\) 0 0
\(967\) 25.5230 0.820764 0.410382 0.911914i \(-0.365395\pi\)
0.410382 + 0.911914i \(0.365395\pi\)
\(968\) 0 0
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) −19.9682 −0.640810 −0.320405 0.947281i \(-0.603819\pi\)
−0.320405 + 0.947281i \(0.603819\pi\)
\(972\) 0 0
\(973\) 20.5045 0.657343
\(974\) 0 0
\(975\) −5.45415 −0.174673
\(976\) 0 0
\(977\) −1.34394 −0.0429964 −0.0214982 0.999769i \(-0.506844\pi\)
−0.0214982 + 0.999769i \(0.506844\pi\)
\(978\) 0 0
\(979\) −72.0880 −2.30394
\(980\) 0 0
\(981\) 15.0599 0.480825
\(982\) 0 0
\(983\) 52.3720 1.67041 0.835204 0.549940i \(-0.185350\pi\)
0.835204 + 0.549940i \(0.185350\pi\)
\(984\) 0 0
\(985\) 15.4637 0.492715
\(986\) 0 0
\(987\) −27.7159 −0.882208
\(988\) 0 0
\(989\) 29.8993 0.950744
\(990\) 0 0
\(991\) 9.28031 0.294799 0.147399 0.989077i \(-0.452910\pi\)
0.147399 + 0.989077i \(0.452910\pi\)
\(992\) 0 0
\(993\) 21.9682 0.697139
\(994\) 0 0
\(995\) −2.90830 −0.0921994
\(996\) 0 0
\(997\) 13.2433 0.419419 0.209709 0.977764i \(-0.432748\pi\)
0.209709 + 0.977764i \(0.432748\pi\)
\(998\) 0 0
\(999\) 11.8580 0.375170
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 4560.2.a.bs.1.1 3
4.3 odd 2 2280.2.a.u.1.3 3
12.11 even 2 6840.2.a.bh.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.u.1.3 3 4.3 odd 2
4560.2.a.bs.1.1 3 1.1 even 1 trivial
6840.2.a.bh.1.3 3 12.11 even 2