Properties

Label 4730.2.a.bf.1.9
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-1.13397\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.13397 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.13397 q^{6} -0.848165 q^{7} +1.00000 q^{8} -1.71412 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.13397 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.13397 q^{6} -0.848165 q^{7} +1.00000 q^{8} -1.71412 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.13397 q^{12} +0.342344 q^{13} -0.848165 q^{14} -1.13397 q^{15} +1.00000 q^{16} +0.0704024 q^{17} -1.71412 q^{18} +5.57111 q^{19} -1.00000 q^{20} -0.961790 q^{21} +1.00000 q^{22} +2.81812 q^{23} +1.13397 q^{24} +1.00000 q^{25} +0.342344 q^{26} -5.34565 q^{27} -0.848165 q^{28} +4.83392 q^{29} -1.13397 q^{30} +9.40344 q^{31} +1.00000 q^{32} +1.13397 q^{33} +0.0704024 q^{34} +0.848165 q^{35} -1.71412 q^{36} -3.61148 q^{37} +5.57111 q^{38} +0.388206 q^{39} -1.00000 q^{40} +0.000180909 q^{41} -0.961790 q^{42} +1.00000 q^{43} +1.00000 q^{44} +1.71412 q^{45} +2.81812 q^{46} -3.92012 q^{47} +1.13397 q^{48} -6.28062 q^{49} +1.00000 q^{50} +0.0798340 q^{51} +0.342344 q^{52} -11.1383 q^{53} -5.34565 q^{54} -1.00000 q^{55} -0.848165 q^{56} +6.31745 q^{57} +4.83392 q^{58} +8.16088 q^{59} -1.13397 q^{60} +10.5635 q^{61} +9.40344 q^{62} +1.45386 q^{63} +1.00000 q^{64} -0.342344 q^{65} +1.13397 q^{66} -10.2375 q^{67} +0.0704024 q^{68} +3.19565 q^{69} +0.848165 q^{70} +3.85197 q^{71} -1.71412 q^{72} +1.31364 q^{73} -3.61148 q^{74} +1.13397 q^{75} +5.57111 q^{76} -0.848165 q^{77} +0.388206 q^{78} +11.2207 q^{79} -1.00000 q^{80} -0.919431 q^{81} +0.000180909 q^{82} +13.9437 q^{83} -0.961790 q^{84} -0.0704024 q^{85} +1.00000 q^{86} +5.48150 q^{87} +1.00000 q^{88} +12.5639 q^{89} +1.71412 q^{90} -0.290364 q^{91} +2.81812 q^{92} +10.6632 q^{93} -3.92012 q^{94} -5.57111 q^{95} +1.13397 q^{96} +16.4417 q^{97} -6.28062 q^{98} -1.71412 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} + O(q^{10}) \) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} - 13q^{10} + 13q^{11} + 11q^{13} + 7q^{14} + 13q^{16} - 2q^{17} + 25q^{18} + 7q^{19} - 13q^{20} + 12q^{21} + 13q^{22} + 12q^{23} + 13q^{25} + 11q^{26} - 15q^{27} + 7q^{28} + 14q^{29} + 20q^{31} + 13q^{32} - 2q^{34} - 7q^{35} + 25q^{36} + 17q^{37} + 7q^{38} - 4q^{39} - 13q^{40} + 9q^{41} + 12q^{42} + 13q^{43} + 13q^{44} - 25q^{45} + 12q^{46} - 9q^{47} + 30q^{49} + 13q^{50} - 3q^{51} + 11q^{52} + 22q^{53} - 15q^{54} - 13q^{55} + 7q^{56} + 17q^{57} + 14q^{58} + 19q^{59} + 2q^{61} + 20q^{62} + 12q^{63} + 13q^{64} - 11q^{65} + 9q^{67} - 2q^{68} - 6q^{69} - 7q^{70} + 6q^{71} + 25q^{72} + 7q^{73} + 17q^{74} + 7q^{76} + 7q^{77} - 4q^{78} + 50q^{79} - 13q^{80} + 85q^{81} + 9q^{82} + q^{83} + 12q^{84} + 2q^{85} + 13q^{86} - 21q^{87} + 13q^{88} + 5q^{89} - 25q^{90} + 5q^{91} + 12q^{92} + 3q^{93} - 9q^{94} - 7q^{95} + 20q^{97} + 30q^{98} + 25q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.13397 0.654696 0.327348 0.944904i \(-0.393845\pi\)
0.327348 + 0.944904i \(0.393845\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.13397 0.462940
\(7\) −0.848165 −0.320576 −0.160288 0.987070i \(-0.551242\pi\)
−0.160288 + 0.987070i \(0.551242\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.71412 −0.571373
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 1.13397 0.327348
\(13\) 0.342344 0.0949491 0.0474745 0.998872i \(-0.484883\pi\)
0.0474745 + 0.998872i \(0.484883\pi\)
\(14\) −0.848165 −0.226682
\(15\) −1.13397 −0.292789
\(16\) 1.00000 0.250000
\(17\) 0.0704024 0.0170751 0.00853755 0.999964i \(-0.497282\pi\)
0.00853755 + 0.999964i \(0.497282\pi\)
\(18\) −1.71412 −0.404022
\(19\) 5.57111 1.27810 0.639050 0.769165i \(-0.279328\pi\)
0.639050 + 0.769165i \(0.279328\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.961790 −0.209880
\(22\) 1.00000 0.213201
\(23\) 2.81812 0.587619 0.293809 0.955864i \(-0.405077\pi\)
0.293809 + 0.955864i \(0.405077\pi\)
\(24\) 1.13397 0.231470
\(25\) 1.00000 0.200000
\(26\) 0.342344 0.0671391
\(27\) −5.34565 −1.02877
\(28\) −0.848165 −0.160288
\(29\) 4.83392 0.897636 0.448818 0.893623i \(-0.351845\pi\)
0.448818 + 0.893623i \(0.351845\pi\)
\(30\) −1.13397 −0.207033
\(31\) 9.40344 1.68891 0.844454 0.535629i \(-0.179925\pi\)
0.844454 + 0.535629i \(0.179925\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.13397 0.197398
\(34\) 0.0704024 0.0120739
\(35\) 0.848165 0.143366
\(36\) −1.71412 −0.285687
\(37\) −3.61148 −0.593724 −0.296862 0.954920i \(-0.595940\pi\)
−0.296862 + 0.954920i \(0.595940\pi\)
\(38\) 5.57111 0.903753
\(39\) 0.388206 0.0621628
\(40\) −1.00000 −0.158114
\(41\) 0.000180909 0 2.82533e−5 0 1.41267e−5 1.00000i \(-0.499996\pi\)
1.41267e−5 1.00000i \(0.499996\pi\)
\(42\) −0.961790 −0.148407
\(43\) 1.00000 0.152499
\(44\) 1.00000 0.150756
\(45\) 1.71412 0.255526
\(46\) 2.81812 0.415509
\(47\) −3.92012 −0.571809 −0.285904 0.958258i \(-0.592294\pi\)
−0.285904 + 0.958258i \(0.592294\pi\)
\(48\) 1.13397 0.163674
\(49\) −6.28062 −0.897231
\(50\) 1.00000 0.141421
\(51\) 0.0798340 0.0111790
\(52\) 0.342344 0.0474745
\(53\) −11.1383 −1.52996 −0.764982 0.644052i \(-0.777252\pi\)
−0.764982 + 0.644052i \(0.777252\pi\)
\(54\) −5.34565 −0.727451
\(55\) −1.00000 −0.134840
\(56\) −0.848165 −0.113341
\(57\) 6.31745 0.836767
\(58\) 4.83392 0.634725
\(59\) 8.16088 1.06246 0.531228 0.847229i \(-0.321731\pi\)
0.531228 + 0.847229i \(0.321731\pi\)
\(60\) −1.13397 −0.146394
\(61\) 10.5635 1.35251 0.676257 0.736666i \(-0.263601\pi\)
0.676257 + 0.736666i \(0.263601\pi\)
\(62\) 9.40344 1.19424
\(63\) 1.45386 0.183169
\(64\) 1.00000 0.125000
\(65\) −0.342344 −0.0424625
\(66\) 1.13397 0.139582
\(67\) −10.2375 −1.25071 −0.625357 0.780339i \(-0.715047\pi\)
−0.625357 + 0.780339i \(0.715047\pi\)
\(68\) 0.0704024 0.00853755
\(69\) 3.19565 0.384711
\(70\) 0.848165 0.101375
\(71\) 3.85197 0.457145 0.228572 0.973527i \(-0.426594\pi\)
0.228572 + 0.973527i \(0.426594\pi\)
\(72\) −1.71412 −0.202011
\(73\) 1.31364 0.153750 0.0768749 0.997041i \(-0.475506\pi\)
0.0768749 + 0.997041i \(0.475506\pi\)
\(74\) −3.61148 −0.419827
\(75\) 1.13397 0.130939
\(76\) 5.57111 0.639050
\(77\) −0.848165 −0.0966573
\(78\) 0.388206 0.0439557
\(79\) 11.2207 1.26242 0.631211 0.775611i \(-0.282558\pi\)
0.631211 + 0.775611i \(0.282558\pi\)
\(80\) −1.00000 −0.111803
\(81\) −0.919431 −0.102159
\(82\) 0.000180909 0 1.99781e−5 0
\(83\) 13.9437 1.53052 0.765262 0.643719i \(-0.222609\pi\)
0.765262 + 0.643719i \(0.222609\pi\)
\(84\) −0.961790 −0.104940
\(85\) −0.0704024 −0.00763621
\(86\) 1.00000 0.107833
\(87\) 5.48150 0.587679
\(88\) 1.00000 0.106600
\(89\) 12.5639 1.33177 0.665887 0.746053i \(-0.268053\pi\)
0.665887 + 0.746053i \(0.268053\pi\)
\(90\) 1.71412 0.180684
\(91\) −0.290364 −0.0304384
\(92\) 2.81812 0.293809
\(93\) 10.6632 1.10572
\(94\) −3.92012 −0.404330
\(95\) −5.57111 −0.571584
\(96\) 1.13397 0.115735
\(97\) 16.4417 1.66940 0.834699 0.550707i \(-0.185642\pi\)
0.834699 + 0.550707i \(0.185642\pi\)
\(98\) −6.28062 −0.634438
\(99\) −1.71412 −0.172276
\(100\) 1.00000 0.100000
\(101\) −1.51515 −0.150763 −0.0753816 0.997155i \(-0.524018\pi\)
−0.0753816 + 0.997155i \(0.524018\pi\)
\(102\) 0.0798340 0.00790474
\(103\) 15.5004 1.52730 0.763648 0.645633i \(-0.223406\pi\)
0.763648 + 0.645633i \(0.223406\pi\)
\(104\) 0.342344 0.0335696
\(105\) 0.961790 0.0938611
\(106\) −11.1383 −1.08185
\(107\) 6.90709 0.667733 0.333867 0.942620i \(-0.391646\pi\)
0.333867 + 0.942620i \(0.391646\pi\)
\(108\) −5.34565 −0.514386
\(109\) −4.88144 −0.467557 −0.233779 0.972290i \(-0.575109\pi\)
−0.233779 + 0.972290i \(0.575109\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −4.09530 −0.388709
\(112\) −0.848165 −0.0801440
\(113\) −9.98848 −0.939638 −0.469819 0.882763i \(-0.655681\pi\)
−0.469819 + 0.882763i \(0.655681\pi\)
\(114\) 6.31745 0.591683
\(115\) −2.81812 −0.262791
\(116\) 4.83392 0.448818
\(117\) −0.586818 −0.0542514
\(118\) 8.16088 0.751270
\(119\) −0.0597128 −0.00547387
\(120\) −1.13397 −0.103516
\(121\) 1.00000 0.0909091
\(122\) 10.5635 0.956371
\(123\) 0.000205145 0 1.84973e−5 0
\(124\) 9.40344 0.844454
\(125\) −1.00000 −0.0894427
\(126\) 1.45386 0.129520
\(127\) 10.8227 0.960357 0.480178 0.877171i \(-0.340572\pi\)
0.480178 + 0.877171i \(0.340572\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.13397 0.0998402
\(130\) −0.342344 −0.0300255
\(131\) 3.12492 0.273025 0.136513 0.990638i \(-0.456411\pi\)
0.136513 + 0.990638i \(0.456411\pi\)
\(132\) 1.13397 0.0986991
\(133\) −4.72522 −0.409728
\(134\) −10.2375 −0.884388
\(135\) 5.34565 0.460081
\(136\) 0.0704024 0.00603696
\(137\) −0.152063 −0.0129916 −0.00649580 0.999979i \(-0.502068\pi\)
−0.00649580 + 0.999979i \(0.502068\pi\)
\(138\) 3.19565 0.272032
\(139\) 11.2006 0.950019 0.475010 0.879981i \(-0.342445\pi\)
0.475010 + 0.879981i \(0.342445\pi\)
\(140\) 0.848165 0.0716830
\(141\) −4.44529 −0.374361
\(142\) 3.85197 0.323250
\(143\) 0.342344 0.0286282
\(144\) −1.71412 −0.142843
\(145\) −4.83392 −0.401435
\(146\) 1.31364 0.108718
\(147\) −7.12201 −0.587413
\(148\) −3.61148 −0.296862
\(149\) 9.57487 0.784404 0.392202 0.919879i \(-0.371713\pi\)
0.392202 + 0.919879i \(0.371713\pi\)
\(150\) 1.13397 0.0925880
\(151\) −14.6795 −1.19460 −0.597300 0.802018i \(-0.703760\pi\)
−0.597300 + 0.802018i \(0.703760\pi\)
\(152\) 5.57111 0.451877
\(153\) −0.120678 −0.00975626
\(154\) −0.848165 −0.0683471
\(155\) −9.40344 −0.755302
\(156\) 0.388206 0.0310814
\(157\) −20.0963 −1.60386 −0.801930 0.597418i \(-0.796193\pi\)
−0.801930 + 0.597418i \(0.796193\pi\)
\(158\) 11.2207 0.892668
\(159\) −12.6305 −1.00166
\(160\) −1.00000 −0.0790569
\(161\) −2.39023 −0.188376
\(162\) −0.919431 −0.0722374
\(163\) 8.72493 0.683390 0.341695 0.939811i \(-0.388999\pi\)
0.341695 + 0.939811i \(0.388999\pi\)
\(164\) 0.000180909 0 1.41267e−5 0
\(165\) −1.13397 −0.0882792
\(166\) 13.9437 1.08224
\(167\) 3.33471 0.258047 0.129024 0.991642i \(-0.458816\pi\)
0.129024 + 0.991642i \(0.458816\pi\)
\(168\) −0.961790 −0.0742037
\(169\) −12.8828 −0.990985
\(170\) −0.0704024 −0.00539962
\(171\) −9.54955 −0.730272
\(172\) 1.00000 0.0762493
\(173\) −5.02943 −0.382381 −0.191190 0.981553i \(-0.561235\pi\)
−0.191190 + 0.981553i \(0.561235\pi\)
\(174\) 5.48150 0.415552
\(175\) −0.848165 −0.0641152
\(176\) 1.00000 0.0753778
\(177\) 9.25416 0.695586
\(178\) 12.5639 0.941706
\(179\) 4.55404 0.340385 0.170193 0.985411i \(-0.445561\pi\)
0.170193 + 0.985411i \(0.445561\pi\)
\(180\) 1.71412 0.127763
\(181\) 22.2730 1.65554 0.827770 0.561068i \(-0.189609\pi\)
0.827770 + 0.561068i \(0.189609\pi\)
\(182\) −0.290364 −0.0215232
\(183\) 11.9786 0.885485
\(184\) 2.81812 0.207755
\(185\) 3.61148 0.265522
\(186\) 10.6632 0.781862
\(187\) 0.0704024 0.00514833
\(188\) −3.92012 −0.285904
\(189\) 4.53399 0.329800
\(190\) −5.57111 −0.404171
\(191\) 9.24227 0.668747 0.334374 0.942441i \(-0.391475\pi\)
0.334374 + 0.942441i \(0.391475\pi\)
\(192\) 1.13397 0.0818370
\(193\) 15.2378 1.09684 0.548419 0.836204i \(-0.315230\pi\)
0.548419 + 0.836204i \(0.315230\pi\)
\(194\) 16.4417 1.18044
\(195\) −0.388206 −0.0278000
\(196\) −6.28062 −0.448615
\(197\) −17.6609 −1.25829 −0.629143 0.777290i \(-0.716594\pi\)
−0.629143 + 0.777290i \(0.716594\pi\)
\(198\) −1.71412 −0.121817
\(199\) −7.99206 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(200\) 1.00000 0.0707107
\(201\) −11.6090 −0.818837
\(202\) −1.51515 −0.106606
\(203\) −4.09996 −0.287761
\(204\) 0.0798340 0.00558950
\(205\) −0.000180909 0 −1.26353e−5 0
\(206\) 15.5004 1.07996
\(207\) −4.83060 −0.335750
\(208\) 0.342344 0.0237373
\(209\) 5.57111 0.385362
\(210\) 0.961790 0.0663698
\(211\) −7.85637 −0.540855 −0.270427 0.962740i \(-0.587165\pi\)
−0.270427 + 0.962740i \(0.587165\pi\)
\(212\) −11.1383 −0.764982
\(213\) 4.36800 0.299291
\(214\) 6.90709 0.472159
\(215\) −1.00000 −0.0681994
\(216\) −5.34565 −0.363726
\(217\) −7.97566 −0.541423
\(218\) −4.88144 −0.330613
\(219\) 1.48962 0.100659
\(220\) −1.00000 −0.0674200
\(221\) 0.0241018 0.00162126
\(222\) −4.09530 −0.274859
\(223\) 21.5912 1.44585 0.722926 0.690925i \(-0.242797\pi\)
0.722926 + 0.690925i \(0.242797\pi\)
\(224\) −0.848165 −0.0566704
\(225\) −1.71412 −0.114275
\(226\) −9.98848 −0.664424
\(227\) −26.2071 −1.73943 −0.869715 0.493555i \(-0.835697\pi\)
−0.869715 + 0.493555i \(0.835697\pi\)
\(228\) 6.31745 0.418383
\(229\) −0.315652 −0.0208589 −0.0104294 0.999946i \(-0.503320\pi\)
−0.0104294 + 0.999946i \(0.503320\pi\)
\(230\) −2.81812 −0.185821
\(231\) −0.961790 −0.0632812
\(232\) 4.83392 0.317362
\(233\) −19.3106 −1.26508 −0.632540 0.774528i \(-0.717988\pi\)
−0.632540 + 0.774528i \(0.717988\pi\)
\(234\) −0.586818 −0.0383615
\(235\) 3.92012 0.255721
\(236\) 8.16088 0.531228
\(237\) 12.7238 0.826503
\(238\) −0.0597128 −0.00387061
\(239\) −10.5435 −0.682005 −0.341002 0.940062i \(-0.610766\pi\)
−0.341002 + 0.940062i \(0.610766\pi\)
\(240\) −1.13397 −0.0731972
\(241\) −22.6988 −1.46216 −0.731078 0.682294i \(-0.760983\pi\)
−0.731078 + 0.682294i \(0.760983\pi\)
\(242\) 1.00000 0.0642824
\(243\) 14.9944 0.961888
\(244\) 10.5635 0.676257
\(245\) 6.28062 0.401254
\(246\) 0.000205145 0 1.30796e−5 0
\(247\) 1.90723 0.121354
\(248\) 9.40344 0.597119
\(249\) 15.8117 1.00203
\(250\) −1.00000 −0.0632456
\(251\) −1.08995 −0.0687972 −0.0343986 0.999408i \(-0.510952\pi\)
−0.0343986 + 0.999408i \(0.510952\pi\)
\(252\) 1.45386 0.0915843
\(253\) 2.81812 0.177174
\(254\) 10.8227 0.679075
\(255\) −0.0798340 −0.00499940
\(256\) 1.00000 0.0625000
\(257\) 5.90193 0.368152 0.184076 0.982912i \(-0.441071\pi\)
0.184076 + 0.982912i \(0.441071\pi\)
\(258\) 1.13397 0.0705977
\(259\) 3.06313 0.190334
\(260\) −0.342344 −0.0212313
\(261\) −8.28592 −0.512885
\(262\) 3.12492 0.193058
\(263\) −5.57837 −0.343977 −0.171989 0.985099i \(-0.555019\pi\)
−0.171989 + 0.985099i \(0.555019\pi\)
\(264\) 1.13397 0.0697908
\(265\) 11.1383 0.684221
\(266\) −4.72522 −0.289722
\(267\) 14.2471 0.871906
\(268\) −10.2375 −0.625357
\(269\) −7.51121 −0.457966 −0.228983 0.973430i \(-0.573540\pi\)
−0.228983 + 0.973430i \(0.573540\pi\)
\(270\) 5.34565 0.325326
\(271\) −21.0433 −1.27829 −0.639145 0.769086i \(-0.720712\pi\)
−0.639145 + 0.769086i \(0.720712\pi\)
\(272\) 0.0704024 0.00426877
\(273\) −0.329263 −0.0199279
\(274\) −0.152063 −0.00918644
\(275\) 1.00000 0.0603023
\(276\) 3.19565 0.192356
\(277\) 31.3505 1.88367 0.941834 0.336079i \(-0.109101\pi\)
0.941834 + 0.336079i \(0.109101\pi\)
\(278\) 11.2006 0.671765
\(279\) −16.1186 −0.964997
\(280\) 0.848165 0.0506875
\(281\) 4.44791 0.265340 0.132670 0.991160i \(-0.457645\pi\)
0.132670 + 0.991160i \(0.457645\pi\)
\(282\) −4.44529 −0.264713
\(283\) −5.42621 −0.322555 −0.161277 0.986909i \(-0.551561\pi\)
−0.161277 + 0.986909i \(0.551561\pi\)
\(284\) 3.85197 0.228572
\(285\) −6.31745 −0.374213
\(286\) 0.342344 0.0202432
\(287\) −0.000153441 0 −9.05733e−6 0
\(288\) −1.71412 −0.101006
\(289\) −16.9950 −0.999708
\(290\) −4.83392 −0.283858
\(291\) 18.6443 1.09295
\(292\) 1.31364 0.0768749
\(293\) −9.47735 −0.553673 −0.276836 0.960917i \(-0.589286\pi\)
−0.276836 + 0.960917i \(0.589286\pi\)
\(294\) −7.12201 −0.415364
\(295\) −8.16088 −0.475145
\(296\) −3.61148 −0.209913
\(297\) −5.34565 −0.310186
\(298\) 9.57487 0.554657
\(299\) 0.964766 0.0557938
\(300\) 1.13397 0.0654696
\(301\) −0.848165 −0.0488874
\(302\) −14.6795 −0.844709
\(303\) −1.71813 −0.0987041
\(304\) 5.57111 0.319525
\(305\) −10.5635 −0.604862
\(306\) −0.120678 −0.00689871
\(307\) −11.4070 −0.651033 −0.325516 0.945536i \(-0.605538\pi\)
−0.325516 + 0.945536i \(0.605538\pi\)
\(308\) −0.848165 −0.0483287
\(309\) 17.5769 0.999914
\(310\) −9.40344 −0.534079
\(311\) −28.3111 −1.60538 −0.802688 0.596399i \(-0.796598\pi\)
−0.802688 + 0.596399i \(0.796598\pi\)
\(312\) 0.388206 0.0219779
\(313\) −9.95623 −0.562760 −0.281380 0.959596i \(-0.590792\pi\)
−0.281380 + 0.959596i \(0.590792\pi\)
\(314\) −20.0963 −1.13410
\(315\) −1.45386 −0.0819155
\(316\) 11.2207 0.631211
\(317\) −18.5138 −1.03984 −0.519918 0.854216i \(-0.674038\pi\)
−0.519918 + 0.854216i \(0.674038\pi\)
\(318\) −12.6305 −0.708281
\(319\) 4.83392 0.270648
\(320\) −1.00000 −0.0559017
\(321\) 7.83240 0.437162
\(322\) −2.39023 −0.133202
\(323\) 0.392220 0.0218237
\(324\) −0.919431 −0.0510795
\(325\) 0.342344 0.0189898
\(326\) 8.72493 0.483229
\(327\) −5.53539 −0.306108
\(328\) 0.000180909 0 9.98905e−6 0
\(329\) 3.32491 0.183308
\(330\) −1.13397 −0.0624228
\(331\) −13.6151 −0.748355 −0.374178 0.927357i \(-0.622075\pi\)
−0.374178 + 0.927357i \(0.622075\pi\)
\(332\) 13.9437 0.765262
\(333\) 6.19052 0.339238
\(334\) 3.33471 0.182467
\(335\) 10.2375 0.559336
\(336\) −0.961790 −0.0524700
\(337\) −35.5932 −1.93888 −0.969441 0.245323i \(-0.921106\pi\)
−0.969441 + 0.245323i \(0.921106\pi\)
\(338\) −12.8828 −0.700732
\(339\) −11.3266 −0.615177
\(340\) −0.0704024 −0.00381811
\(341\) 9.40344 0.509225
\(342\) −9.54955 −0.516381
\(343\) 11.2641 0.608207
\(344\) 1.00000 0.0539164
\(345\) −3.19565 −0.172048
\(346\) −5.02943 −0.270384
\(347\) 13.1776 0.707412 0.353706 0.935357i \(-0.384921\pi\)
0.353706 + 0.935357i \(0.384921\pi\)
\(348\) 5.48150 0.293839
\(349\) −29.3920 −1.57332 −0.786658 0.617389i \(-0.788190\pi\)
−0.786658 + 0.617389i \(0.788190\pi\)
\(350\) −0.848165 −0.0453363
\(351\) −1.83005 −0.0976809
\(352\) 1.00000 0.0533002
\(353\) 19.7084 1.04897 0.524485 0.851420i \(-0.324258\pi\)
0.524485 + 0.851420i \(0.324258\pi\)
\(354\) 9.25416 0.491853
\(355\) −3.85197 −0.204441
\(356\) 12.5639 0.665887
\(357\) −0.0677124 −0.00358372
\(358\) 4.55404 0.240689
\(359\) 6.74743 0.356116 0.178058 0.984020i \(-0.443019\pi\)
0.178058 + 0.984020i \(0.443019\pi\)
\(360\) 1.71412 0.0903421
\(361\) 12.0373 0.633540
\(362\) 22.2730 1.17064
\(363\) 1.13397 0.0595178
\(364\) −0.290364 −0.0152192
\(365\) −1.31364 −0.0687590
\(366\) 11.9786 0.626132
\(367\) 7.26442 0.379200 0.189600 0.981861i \(-0.439281\pi\)
0.189600 + 0.981861i \(0.439281\pi\)
\(368\) 2.81812 0.146905
\(369\) −0.000310100 0 −1.61432e−5 0
\(370\) 3.61148 0.187752
\(371\) 9.44712 0.490470
\(372\) 10.6632 0.552860
\(373\) 12.5117 0.647829 0.323915 0.946086i \(-0.395001\pi\)
0.323915 + 0.946086i \(0.395001\pi\)
\(374\) 0.0704024 0.00364042
\(375\) −1.13397 −0.0585578
\(376\) −3.92012 −0.202165
\(377\) 1.65486 0.0852297
\(378\) 4.53399 0.233204
\(379\) −0.0643888 −0.00330743 −0.00165372 0.999999i \(-0.500526\pi\)
−0.00165372 + 0.999999i \(0.500526\pi\)
\(380\) −5.57111 −0.285792
\(381\) 12.2725 0.628742
\(382\) 9.24227 0.472876
\(383\) −18.6129 −0.951074 −0.475537 0.879696i \(-0.657746\pi\)
−0.475537 + 0.879696i \(0.657746\pi\)
\(384\) 1.13397 0.0578675
\(385\) 0.848165 0.0432265
\(386\) 15.2378 0.775582
\(387\) −1.71412 −0.0871336
\(388\) 16.4417 0.834699
\(389\) 10.1939 0.516851 0.258426 0.966031i \(-0.416796\pi\)
0.258426 + 0.966031i \(0.416796\pi\)
\(390\) −0.388206 −0.0196576
\(391\) 0.198402 0.0100336
\(392\) −6.28062 −0.317219
\(393\) 3.54355 0.178749
\(394\) −17.6609 −0.889743
\(395\) −11.2207 −0.564573
\(396\) −1.71412 −0.0861378
\(397\) −36.5458 −1.83418 −0.917090 0.398681i \(-0.869468\pi\)
−0.917090 + 0.398681i \(0.869468\pi\)
\(398\) −7.99206 −0.400606
\(399\) −5.35824 −0.268247
\(400\) 1.00000 0.0500000
\(401\) −26.2421 −1.31047 −0.655235 0.755425i \(-0.727430\pi\)
−0.655235 + 0.755425i \(0.727430\pi\)
\(402\) −11.6090 −0.579005
\(403\) 3.21921 0.160360
\(404\) −1.51515 −0.0753816
\(405\) 0.919431 0.0456869
\(406\) −4.09996 −0.203478
\(407\) −3.61148 −0.179015
\(408\) 0.0798340 0.00395237
\(409\) 33.6212 1.66246 0.831230 0.555928i \(-0.187637\pi\)
0.831230 + 0.555928i \(0.187637\pi\)
\(410\) −0.000180909 0 −8.93448e−6 0
\(411\) −0.172434 −0.00850554
\(412\) 15.5004 0.763648
\(413\) −6.92177 −0.340598
\(414\) −4.83060 −0.237411
\(415\) −13.9437 −0.684471
\(416\) 0.342344 0.0167848
\(417\) 12.7011 0.621974
\(418\) 5.57111 0.272492
\(419\) 18.6822 0.912683 0.456342 0.889805i \(-0.349159\pi\)
0.456342 + 0.889805i \(0.349159\pi\)
\(420\) 0.961790 0.0469306
\(421\) 21.0598 1.02639 0.513195 0.858272i \(-0.328462\pi\)
0.513195 + 0.858272i \(0.328462\pi\)
\(422\) −7.85637 −0.382442
\(423\) 6.71956 0.326716
\(424\) −11.1383 −0.540924
\(425\) 0.0704024 0.00341502
\(426\) 4.36800 0.211630
\(427\) −8.95956 −0.433583
\(428\) 6.90709 0.333867
\(429\) 0.388206 0.0187428
\(430\) −1.00000 −0.0482243
\(431\) 14.0491 0.676723 0.338361 0.941016i \(-0.390127\pi\)
0.338361 + 0.941016i \(0.390127\pi\)
\(432\) −5.34565 −0.257193
\(433\) −31.1900 −1.49890 −0.749449 0.662062i \(-0.769681\pi\)
−0.749449 + 0.662062i \(0.769681\pi\)
\(434\) −7.97566 −0.382844
\(435\) −5.48150 −0.262818
\(436\) −4.88144 −0.233779
\(437\) 15.7001 0.751035
\(438\) 1.48962 0.0711769
\(439\) 19.2785 0.920112 0.460056 0.887890i \(-0.347829\pi\)
0.460056 + 0.887890i \(0.347829\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 10.7657 0.512654
\(442\) 0.0241018 0.00114641
\(443\) 5.15368 0.244859 0.122429 0.992477i \(-0.460932\pi\)
0.122429 + 0.992477i \(0.460932\pi\)
\(444\) −4.09530 −0.194354
\(445\) −12.5639 −0.595587
\(446\) 21.5912 1.02237
\(447\) 10.8576 0.513546
\(448\) −0.848165 −0.0400720
\(449\) −23.5580 −1.11177 −0.555885 0.831259i \(-0.687621\pi\)
−0.555885 + 0.831259i \(0.687621\pi\)
\(450\) −1.71412 −0.0808044
\(451\) 0.000180909 0 8.51869e−6 0
\(452\) −9.98848 −0.469819
\(453\) −16.6460 −0.782099
\(454\) −26.2071 −1.22996
\(455\) 0.290364 0.0136125
\(456\) 6.31745 0.295842
\(457\) −11.1303 −0.520655 −0.260328 0.965520i \(-0.583831\pi\)
−0.260328 + 0.965520i \(0.583831\pi\)
\(458\) −0.315652 −0.0147495
\(459\) −0.376347 −0.0175664
\(460\) −2.81812 −0.131396
\(461\) −32.8012 −1.52771 −0.763853 0.645390i \(-0.776695\pi\)
−0.763853 + 0.645390i \(0.776695\pi\)
\(462\) −0.961790 −0.0447465
\(463\) −25.8484 −1.20128 −0.600640 0.799520i \(-0.705087\pi\)
−0.600640 + 0.799520i \(0.705087\pi\)
\(464\) 4.83392 0.224409
\(465\) −10.6632 −0.494493
\(466\) −19.3106 −0.894547
\(467\) −23.7532 −1.09917 −0.549584 0.835439i \(-0.685214\pi\)
−0.549584 + 0.835439i \(0.685214\pi\)
\(468\) −0.586818 −0.0271257
\(469\) 8.68312 0.400949
\(470\) 3.92012 0.180822
\(471\) −22.7885 −1.05004
\(472\) 8.16088 0.375635
\(473\) 1.00000 0.0459800
\(474\) 12.7238 0.584426
\(475\) 5.57111 0.255620
\(476\) −0.0597128 −0.00273693
\(477\) 19.0924 0.874181
\(478\) −10.5435 −0.482250
\(479\) −26.0329 −1.18947 −0.594737 0.803920i \(-0.702744\pi\)
−0.594737 + 0.803920i \(0.702744\pi\)
\(480\) −1.13397 −0.0517582
\(481\) −1.23637 −0.0563736
\(482\) −22.6988 −1.03390
\(483\) −2.71044 −0.123329
\(484\) 1.00000 0.0454545
\(485\) −16.4417 −0.746577
\(486\) 14.9944 0.680158
\(487\) 31.4385 1.42461 0.712306 0.701869i \(-0.247651\pi\)
0.712306 + 0.701869i \(0.247651\pi\)
\(488\) 10.5635 0.478186
\(489\) 9.89378 0.447412
\(490\) 6.28062 0.283729
\(491\) 7.08558 0.319768 0.159884 0.987136i \(-0.448888\pi\)
0.159884 + 0.987136i \(0.448888\pi\)
\(492\) 0.000205145 0 9.24866e−6 0
\(493\) 0.340320 0.0153272
\(494\) 1.90723 0.0858105
\(495\) 1.71412 0.0770440
\(496\) 9.40344 0.422227
\(497\) −3.26710 −0.146550
\(498\) 15.8117 0.708541
\(499\) 18.5102 0.828630 0.414315 0.910133i \(-0.364021\pi\)
0.414315 + 0.910133i \(0.364021\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 3.78144 0.168943
\(502\) −1.08995 −0.0486470
\(503\) 8.97093 0.399994 0.199997 0.979797i \(-0.435907\pi\)
0.199997 + 0.979797i \(0.435907\pi\)
\(504\) 1.45386 0.0647599
\(505\) 1.51515 0.0674234
\(506\) 2.81812 0.125281
\(507\) −14.6087 −0.648794
\(508\) 10.8227 0.480178
\(509\) −11.1247 −0.493094 −0.246547 0.969131i \(-0.579296\pi\)
−0.246547 + 0.969131i \(0.579296\pi\)
\(510\) −0.0798340 −0.00353511
\(511\) −1.11418 −0.0492885
\(512\) 1.00000 0.0441942
\(513\) −29.7812 −1.31487
\(514\) 5.90193 0.260323
\(515\) −15.5004 −0.683027
\(516\) 1.13397 0.0499201
\(517\) −3.92012 −0.172407
\(518\) 3.06313 0.134586
\(519\) −5.70321 −0.250343
\(520\) −0.342344 −0.0150128
\(521\) 21.7350 0.952226 0.476113 0.879384i \(-0.342045\pi\)
0.476113 + 0.879384i \(0.342045\pi\)
\(522\) −8.28592 −0.362665
\(523\) −44.6326 −1.95165 −0.975823 0.218560i \(-0.929864\pi\)
−0.975823 + 0.218560i \(0.929864\pi\)
\(524\) 3.12492 0.136513
\(525\) −0.961790 −0.0419760
\(526\) −5.57837 −0.243229
\(527\) 0.662025 0.0288382
\(528\) 1.13397 0.0493496
\(529\) −15.0582 −0.654704
\(530\) 11.1383 0.483817
\(531\) −13.9887 −0.607059
\(532\) −4.72522 −0.204864
\(533\) 6.19332e−5 0 2.68262e−6 0
\(534\) 14.2471 0.616531
\(535\) −6.90709 −0.298619
\(536\) −10.2375 −0.442194
\(537\) 5.16413 0.222849
\(538\) −7.51121 −0.323831
\(539\) −6.28062 −0.270525
\(540\) 5.34565 0.230040
\(541\) 18.2060 0.782737 0.391369 0.920234i \(-0.372002\pi\)
0.391369 + 0.920234i \(0.372002\pi\)
\(542\) −21.0433 −0.903887
\(543\) 25.2568 1.08387
\(544\) 0.0704024 0.00301848
\(545\) 4.88144 0.209098
\(546\) −0.329263 −0.0140912
\(547\) −2.00044 −0.0855327 −0.0427663 0.999085i \(-0.513617\pi\)
−0.0427663 + 0.999085i \(0.513617\pi\)
\(548\) −0.152063 −0.00649580
\(549\) −18.1070 −0.772790
\(550\) 1.00000 0.0426401
\(551\) 26.9303 1.14727
\(552\) 3.19565 0.136016
\(553\) −9.51697 −0.404703
\(554\) 31.3505 1.33195
\(555\) 4.09530 0.173836
\(556\) 11.2006 0.475010
\(557\) 1.09153 0.0462498 0.0231249 0.999733i \(-0.492638\pi\)
0.0231249 + 0.999733i \(0.492638\pi\)
\(558\) −16.1186 −0.682356
\(559\) 0.342344 0.0144796
\(560\) 0.848165 0.0358415
\(561\) 0.0798340 0.00337059
\(562\) 4.44791 0.187624
\(563\) −25.4926 −1.07438 −0.537192 0.843460i \(-0.680515\pi\)
−0.537192 + 0.843460i \(0.680515\pi\)
\(564\) −4.44529 −0.187180
\(565\) 9.98848 0.420219
\(566\) −5.42621 −0.228081
\(567\) 0.779829 0.0327498
\(568\) 3.85197 0.161625
\(569\) 44.0232 1.84555 0.922774 0.385342i \(-0.125917\pi\)
0.922774 + 0.385342i \(0.125917\pi\)
\(570\) −6.31745 −0.264609
\(571\) 26.8498 1.12363 0.561814 0.827264i \(-0.310104\pi\)
0.561814 + 0.827264i \(0.310104\pi\)
\(572\) 0.342344 0.0143141
\(573\) 10.4804 0.437826
\(574\) −0.000153441 0 −6.40450e−6 0
\(575\) 2.81812 0.117524
\(576\) −1.71412 −0.0714217
\(577\) 25.0342 1.04219 0.521094 0.853499i \(-0.325524\pi\)
0.521094 + 0.853499i \(0.325524\pi\)
\(578\) −16.9950 −0.706901
\(579\) 17.2791 0.718095
\(580\) −4.83392 −0.200718
\(581\) −11.8266 −0.490649
\(582\) 18.6443 0.772831
\(583\) −11.1383 −0.461301
\(584\) 1.31364 0.0543588
\(585\) 0.586818 0.0242620
\(586\) −9.47735 −0.391506
\(587\) −30.4262 −1.25582 −0.627912 0.778284i \(-0.716090\pi\)
−0.627912 + 0.778284i \(0.716090\pi\)
\(588\) −7.12201 −0.293707
\(589\) 52.3876 2.15859
\(590\) −8.16088 −0.335978
\(591\) −20.0269 −0.823795
\(592\) −3.61148 −0.148431
\(593\) −20.6653 −0.848622 −0.424311 0.905517i \(-0.639484\pi\)
−0.424311 + 0.905517i \(0.639484\pi\)
\(594\) −5.34565 −0.219335
\(595\) 0.0597128 0.00244799
\(596\) 9.57487 0.392202
\(597\) −9.06273 −0.370913
\(598\) 0.964766 0.0394522
\(599\) 47.9050 1.95734 0.978672 0.205430i \(-0.0658594\pi\)
0.978672 + 0.205430i \(0.0658594\pi\)
\(600\) 1.13397 0.0462940
\(601\) −14.6296 −0.596754 −0.298377 0.954448i \(-0.596445\pi\)
−0.298377 + 0.954448i \(0.596445\pi\)
\(602\) −0.848165 −0.0345686
\(603\) 17.5484 0.714625
\(604\) −14.6795 −0.597300
\(605\) −1.00000 −0.0406558
\(606\) −1.71813 −0.0697943
\(607\) 7.67710 0.311604 0.155802 0.987788i \(-0.450204\pi\)
0.155802 + 0.987788i \(0.450204\pi\)
\(608\) 5.57111 0.225938
\(609\) −4.64922 −0.188396
\(610\) −10.5635 −0.427702
\(611\) −1.34203 −0.0542927
\(612\) −0.120678 −0.00487813
\(613\) 11.1679 0.451068 0.225534 0.974235i \(-0.427587\pi\)
0.225534 + 0.974235i \(0.427587\pi\)
\(614\) −11.4070 −0.460350
\(615\) −0.000205145 0 −8.27225e−6 0
\(616\) −0.848165 −0.0341735
\(617\) 7.35510 0.296105 0.148052 0.988980i \(-0.452700\pi\)
0.148052 + 0.988980i \(0.452700\pi\)
\(618\) 17.5769 0.707046
\(619\) 21.9306 0.881466 0.440733 0.897638i \(-0.354719\pi\)
0.440733 + 0.897638i \(0.354719\pi\)
\(620\) −9.40344 −0.377651
\(621\) −15.0647 −0.604525
\(622\) −28.3111 −1.13517
\(623\) −10.6563 −0.426935
\(624\) 0.388206 0.0155407
\(625\) 1.00000 0.0400000
\(626\) −9.95623 −0.397931
\(627\) 6.31745 0.252295
\(628\) −20.0963 −0.801930
\(629\) −0.254257 −0.0101379
\(630\) −1.45386 −0.0579230
\(631\) −45.2462 −1.80122 −0.900612 0.434624i \(-0.856881\pi\)
−0.900612 + 0.434624i \(0.856881\pi\)
\(632\) 11.2207 0.446334
\(633\) −8.90886 −0.354095
\(634\) −18.5138 −0.735275
\(635\) −10.8227 −0.429485
\(636\) −12.6305 −0.500830
\(637\) −2.15013 −0.0851913
\(638\) 4.83392 0.191377
\(639\) −6.60274 −0.261200
\(640\) −1.00000 −0.0395285
\(641\) −30.4273 −1.20181 −0.600903 0.799322i \(-0.705192\pi\)
−0.600903 + 0.799322i \(0.705192\pi\)
\(642\) 7.83240 0.309120
\(643\) 31.2404 1.23200 0.616001 0.787746i \(-0.288752\pi\)
0.616001 + 0.787746i \(0.288752\pi\)
\(644\) −2.39023 −0.0941882
\(645\) −1.13397 −0.0446499
\(646\) 0.392220 0.0154317
\(647\) 8.85182 0.348001 0.174001 0.984746i \(-0.444331\pi\)
0.174001 + 0.984746i \(0.444331\pi\)
\(648\) −0.919431 −0.0361187
\(649\) 8.16088 0.320343
\(650\) 0.342344 0.0134278
\(651\) −9.04413 −0.354468
\(652\) 8.72493 0.341695
\(653\) 35.0578 1.37192 0.685959 0.727640i \(-0.259383\pi\)
0.685959 + 0.727640i \(0.259383\pi\)
\(654\) −5.53539 −0.216451
\(655\) −3.12492 −0.122101
\(656\) 0.000180909 0 7.06333e−6 0
\(657\) −2.25173 −0.0878485
\(658\) 3.32491 0.129618
\(659\) −5.76793 −0.224686 −0.112343 0.993669i \(-0.535836\pi\)
−0.112343 + 0.993669i \(0.535836\pi\)
\(660\) −1.13397 −0.0441396
\(661\) 11.8226 0.459846 0.229923 0.973209i \(-0.426153\pi\)
0.229923 + 0.973209i \(0.426153\pi\)
\(662\) −13.6151 −0.529167
\(663\) 0.0273307 0.00106144
\(664\) 13.9437 0.541122
\(665\) 4.72522 0.183236
\(666\) 6.19052 0.239878
\(667\) 13.6226 0.527468
\(668\) 3.33471 0.129024
\(669\) 24.4837 0.946593
\(670\) 10.2375 0.395511
\(671\) 10.5635 0.407798
\(672\) −0.961790 −0.0371019
\(673\) −6.74822 −0.260125 −0.130062 0.991506i \(-0.541518\pi\)
−0.130062 + 0.991506i \(0.541518\pi\)
\(674\) −35.5932 −1.37100
\(675\) −5.34565 −0.205754
\(676\) −12.8828 −0.495492
\(677\) 24.9848 0.960244 0.480122 0.877202i \(-0.340592\pi\)
0.480122 + 0.877202i \(0.340592\pi\)
\(678\) −11.3266 −0.434996
\(679\) −13.9452 −0.535169
\(680\) −0.0704024 −0.00269981
\(681\) −29.7180 −1.13880
\(682\) 9.40344 0.360076
\(683\) 7.50610 0.287213 0.143606 0.989635i \(-0.454130\pi\)
0.143606 + 0.989635i \(0.454130\pi\)
\(684\) −9.54955 −0.365136
\(685\) 0.152063 0.00581002
\(686\) 11.2641 0.430067
\(687\) −0.357939 −0.0136562
\(688\) 1.00000 0.0381246
\(689\) −3.81313 −0.145269
\(690\) −3.19565 −0.121656
\(691\) −49.3985 −1.87921 −0.939603 0.342266i \(-0.888806\pi\)
−0.939603 + 0.342266i \(0.888806\pi\)
\(692\) −5.02943 −0.191190
\(693\) 1.45386 0.0552274
\(694\) 13.1776 0.500216
\(695\) −11.2006 −0.424861
\(696\) 5.48150 0.207776
\(697\) 1.27365e−5 0 4.82428e−7 0
\(698\) −29.3920 −1.11250
\(699\) −21.8976 −0.828243
\(700\) −0.848165 −0.0320576
\(701\) −30.8242 −1.16421 −0.582107 0.813112i \(-0.697771\pi\)
−0.582107 + 0.813112i \(0.697771\pi\)
\(702\) −1.83005 −0.0690708
\(703\) −20.1200 −0.758839
\(704\) 1.00000 0.0376889
\(705\) 4.44529 0.167419
\(706\) 19.7084 0.741734
\(707\) 1.28510 0.0483311
\(708\) 9.25416 0.347793
\(709\) 3.99792 0.150145 0.0750725 0.997178i \(-0.476081\pi\)
0.0750725 + 0.997178i \(0.476081\pi\)
\(710\) −3.85197 −0.144562
\(711\) −19.2336 −0.721315
\(712\) 12.5639 0.470853
\(713\) 26.5000 0.992433
\(714\) −0.0677124 −0.00253407
\(715\) −0.342344 −0.0128029
\(716\) 4.55404 0.170193
\(717\) −11.9560 −0.446506
\(718\) 6.74743 0.251812
\(719\) 4.61066 0.171949 0.0859743 0.996297i \(-0.472600\pi\)
0.0859743 + 0.996297i \(0.472600\pi\)
\(720\) 1.71412 0.0638815
\(721\) −13.1469 −0.489614
\(722\) 12.0373 0.447980
\(723\) −25.7396 −0.957268
\(724\) 22.2730 0.827770
\(725\) 4.83392 0.179527
\(726\) 1.13397 0.0420854
\(727\) 40.3383 1.49607 0.748033 0.663662i \(-0.230998\pi\)
0.748033 + 0.663662i \(0.230998\pi\)
\(728\) −0.290364 −0.0107616
\(729\) 19.7614 0.731903
\(730\) −1.31364 −0.0486200
\(731\) 0.0704024 0.00260393
\(732\) 11.9786 0.442742
\(733\) 43.0755 1.59103 0.795515 0.605934i \(-0.207200\pi\)
0.795515 + 0.605934i \(0.207200\pi\)
\(734\) 7.26442 0.268135
\(735\) 7.12201 0.262699
\(736\) 2.81812 0.103877
\(737\) −10.2375 −0.377105
\(738\) −0.000310100 0 −1.14150e−5 0
\(739\) −15.8148 −0.581758 −0.290879 0.956760i \(-0.593948\pi\)
−0.290879 + 0.956760i \(0.593948\pi\)
\(740\) 3.61148 0.132761
\(741\) 2.16274 0.0794502
\(742\) 9.44712 0.346815
\(743\) −34.2360 −1.25600 −0.627999 0.778214i \(-0.716126\pi\)
−0.627999 + 0.778214i \(0.716126\pi\)
\(744\) 10.6632 0.390931
\(745\) −9.57487 −0.350796
\(746\) 12.5117 0.458084
\(747\) −23.9012 −0.874501
\(748\) 0.0704024 0.00257417
\(749\) −5.85835 −0.214059
\(750\) −1.13397 −0.0414066
\(751\) 3.12619 0.114076 0.0570382 0.998372i \(-0.481834\pi\)
0.0570382 + 0.998372i \(0.481834\pi\)
\(752\) −3.92012 −0.142952
\(753\) −1.23597 −0.0450412
\(754\) 1.65486 0.0602665
\(755\) 14.6795 0.534241
\(756\) 4.53399 0.164900
\(757\) −16.6448 −0.604966 −0.302483 0.953155i \(-0.597815\pi\)
−0.302483 + 0.953155i \(0.597815\pi\)
\(758\) −0.0643888 −0.00233871
\(759\) 3.19565 0.115995
\(760\) −5.57111 −0.202085
\(761\) −15.3065 −0.554862 −0.277431 0.960746i \(-0.589483\pi\)
−0.277431 + 0.960746i \(0.589483\pi\)
\(762\) 12.2725 0.444587
\(763\) 4.14027 0.149888
\(764\) 9.24227 0.334374
\(765\) 0.120678 0.00436313
\(766\) −18.6129 −0.672511
\(767\) 2.79383 0.100879
\(768\) 1.13397 0.0409185
\(769\) −40.4137 −1.45736 −0.728678 0.684856i \(-0.759865\pi\)
−0.728678 + 0.684856i \(0.759865\pi\)
\(770\) 0.848165 0.0305657
\(771\) 6.69259 0.241028
\(772\) 15.2378 0.548419
\(773\) 44.4971 1.60045 0.800225 0.599699i \(-0.204713\pi\)
0.800225 + 0.599699i \(0.204713\pi\)
\(774\) −1.71412 −0.0616128
\(775\) 9.40344 0.337781
\(776\) 16.4417 0.590221
\(777\) 3.47349 0.124611
\(778\) 10.1939 0.365469
\(779\) 0.00100787 3.61105e−5 0
\(780\) −0.388206 −0.0139000
\(781\) 3.85197 0.137834
\(782\) 0.198402 0.00709486
\(783\) −25.8405 −0.923463
\(784\) −6.28062 −0.224308
\(785\) 20.0963 0.717268
\(786\) 3.54355 0.126394
\(787\) −15.8840 −0.566202 −0.283101 0.959090i \(-0.591363\pi\)
−0.283101 + 0.959090i \(0.591363\pi\)
\(788\) −17.6609 −0.629143
\(789\) −6.32569 −0.225200
\(790\) −11.2207 −0.399213
\(791\) 8.47188 0.301225
\(792\) −1.71412 −0.0609086
\(793\) 3.61634 0.128420
\(794\) −36.5458 −1.29696
\(795\) 12.6305 0.447956
\(796\) −7.99206 −0.283271
\(797\) −24.0868 −0.853198 −0.426599 0.904441i \(-0.640289\pi\)
−0.426599 + 0.904441i \(0.640289\pi\)
\(798\) −5.35824 −0.189680
\(799\) −0.275986 −0.00976369
\(800\) 1.00000 0.0353553
\(801\) −21.5361 −0.760940
\(802\) −26.2421 −0.926642
\(803\) 1.31364 0.0463573
\(804\) −11.6090 −0.409419
\(805\) 2.39023 0.0842445
\(806\) 3.21921 0.113392
\(807\) −8.51746 −0.299829
\(808\) −1.51515 −0.0533029
\(809\) 24.9824 0.878335 0.439168 0.898405i \(-0.355273\pi\)
0.439168 + 0.898405i \(0.355273\pi\)
\(810\) 0.919431 0.0323055
\(811\) 48.5670 1.70542 0.852709 0.522386i \(-0.174958\pi\)
0.852709 + 0.522386i \(0.174958\pi\)
\(812\) −4.09996 −0.143880
\(813\) −23.8624 −0.836891
\(814\) −3.61148 −0.126582
\(815\) −8.72493 −0.305621
\(816\) 0.0798340 0.00279475
\(817\) 5.57111 0.194908
\(818\) 33.6212 1.17554
\(819\) 0.497719 0.0173917
\(820\) −0.000180909 0 −6.31763e−6 0
\(821\) 16.1280 0.562872 0.281436 0.959580i \(-0.409189\pi\)
0.281436 + 0.959580i \(0.409189\pi\)
\(822\) −0.172434 −0.00601432
\(823\) 24.3566 0.849019 0.424509 0.905423i \(-0.360447\pi\)
0.424509 + 0.905423i \(0.360447\pi\)
\(824\) 15.5004 0.539980
\(825\) 1.13397 0.0394796
\(826\) −6.92177 −0.240839
\(827\) 12.4042 0.431335 0.215668 0.976467i \(-0.430807\pi\)
0.215668 + 0.976467i \(0.430807\pi\)
\(828\) −4.83060 −0.167875
\(829\) 22.3728 0.777041 0.388520 0.921440i \(-0.372986\pi\)
0.388520 + 0.921440i \(0.372986\pi\)
\(830\) −13.9437 −0.483994
\(831\) 35.5504 1.23323
\(832\) 0.342344 0.0118686
\(833\) −0.442171 −0.0153203
\(834\) 12.7011 0.439802
\(835\) −3.33471 −0.115402
\(836\) 5.57111 0.192681
\(837\) −50.2675 −1.73750
\(838\) 18.6822 0.645365
\(839\) 32.2696 1.11407 0.557036 0.830488i \(-0.311939\pi\)
0.557036 + 0.830488i \(0.311939\pi\)
\(840\) 0.961790 0.0331849
\(841\) −5.63323 −0.194249
\(842\) 21.0598 0.725767
\(843\) 5.04378 0.173717
\(844\) −7.85637 −0.270427
\(845\) 12.8828 0.443182
\(846\) 6.71956 0.231023
\(847\) −0.848165 −0.0291433
\(848\) −11.1383 −0.382491
\(849\) −6.15314 −0.211175
\(850\) 0.0704024 0.00241478
\(851\) −10.1776 −0.348883
\(852\) 4.36800 0.149645
\(853\) 27.3554 0.936630 0.468315 0.883562i \(-0.344861\pi\)
0.468315 + 0.883562i \(0.344861\pi\)
\(854\) −8.95956 −0.306590
\(855\) 9.54955 0.326588
\(856\) 6.90709 0.236079
\(857\) 2.14390 0.0732342 0.0366171 0.999329i \(-0.488342\pi\)
0.0366171 + 0.999329i \(0.488342\pi\)
\(858\) 0.388206 0.0132531
\(859\) −52.9184 −1.80555 −0.902777 0.430110i \(-0.858475\pi\)
−0.902777 + 0.430110i \(0.858475\pi\)
\(860\) −1.00000 −0.0340997
\(861\) −0.000173997 0 −5.92980e−6 0
\(862\) 14.0491 0.478515
\(863\) 11.7062 0.398485 0.199243 0.979950i \(-0.436152\pi\)
0.199243 + 0.979950i \(0.436152\pi\)
\(864\) −5.34565 −0.181863
\(865\) 5.02943 0.171006
\(866\) −31.1900 −1.05988
\(867\) −19.2718 −0.654505
\(868\) −7.97566 −0.270712
\(869\) 11.2207 0.380635
\(870\) −5.48150 −0.185840
\(871\) −3.50476 −0.118754
\(872\) −4.88144 −0.165307
\(873\) −28.1830 −0.953849
\(874\) 15.7001 0.531062
\(875\) 0.848165 0.0286732
\(876\) 1.48962 0.0503297
\(877\) 46.9501 1.58539 0.792696 0.609618i \(-0.208677\pi\)
0.792696 + 0.609618i \(0.208677\pi\)
\(878\) 19.2785 0.650617
\(879\) −10.7470 −0.362487
\(880\) −1.00000 −0.0337100
\(881\) −35.7035 −1.20288 −0.601440 0.798918i \(-0.705406\pi\)
−0.601440 + 0.798918i \(0.705406\pi\)
\(882\) 10.7657 0.362501
\(883\) 7.33661 0.246897 0.123448 0.992351i \(-0.460605\pi\)
0.123448 + 0.992351i \(0.460605\pi\)
\(884\) 0.0241018 0.000810632 0
\(885\) −9.25416 −0.311075
\(886\) 5.15368 0.173141
\(887\) −43.2495 −1.45218 −0.726088 0.687601i \(-0.758664\pi\)
−0.726088 + 0.687601i \(0.758664\pi\)
\(888\) −4.09530 −0.137429
\(889\) −9.17941 −0.307867
\(890\) −12.5639 −0.421144
\(891\) −0.919431 −0.0308021
\(892\) 21.5912 0.722926
\(893\) −21.8394 −0.730829
\(894\) 10.8576 0.363132
\(895\) −4.55404 −0.152225
\(896\) −0.848165 −0.0283352
\(897\) 1.09401 0.0365280
\(898\) −23.5580 −0.786141
\(899\) 45.4555 1.51602
\(900\) −1.71412 −0.0571373
\(901\) −0.784164 −0.0261243
\(902\) 0.000180909 0 6.02362e−6 0
\(903\) −0.961790 −0.0320064
\(904\) −9.98848 −0.332212
\(905\) −22.2730 −0.740380
\(906\) −16.6460 −0.553028
\(907\) 23.3658 0.775849 0.387925 0.921691i \(-0.373192\pi\)
0.387925 + 0.921691i \(0.373192\pi\)
\(908\) −26.2071 −0.869715
\(909\) 2.59715 0.0861421
\(910\) 0.290364 0.00962547
\(911\) 9.06714 0.300408 0.150204 0.988655i \(-0.452007\pi\)
0.150204 + 0.988655i \(0.452007\pi\)
\(912\) 6.31745 0.209192
\(913\) 13.9437 0.461470
\(914\) −11.1303 −0.368159
\(915\) −11.9786 −0.396001
\(916\) −0.315652 −0.0104294
\(917\) −2.65045 −0.0875254
\(918\) −0.376347 −0.0124213
\(919\) −14.1835 −0.467872 −0.233936 0.972252i \(-0.575161\pi\)
−0.233936 + 0.972252i \(0.575161\pi\)
\(920\) −2.81812 −0.0929107
\(921\) −12.9352 −0.426228
\(922\) −32.8012 −1.08025
\(923\) 1.31870 0.0434055
\(924\) −0.961790 −0.0316406
\(925\) −3.61148 −0.118745
\(926\) −25.8484 −0.849433
\(927\) −26.5695 −0.872656
\(928\) 4.83392 0.158681
\(929\) 34.2554 1.12388 0.561942 0.827177i \(-0.310054\pi\)
0.561942 + 0.827177i \(0.310054\pi\)
\(930\) −10.6632 −0.349659
\(931\) −34.9900 −1.14675
\(932\) −19.3106 −0.632540
\(933\) −32.1039 −1.05103
\(934\) −23.7532 −0.777229
\(935\) −0.0704024 −0.00230241
\(936\) −0.586818 −0.0191808
\(937\) −39.8709 −1.30253 −0.651263 0.758852i \(-0.725760\pi\)
−0.651263 + 0.758852i \(0.725760\pi\)
\(938\) 8.68312 0.283514
\(939\) −11.2900 −0.368436
\(940\) 3.92012 0.127860
\(941\) −51.0812 −1.66520 −0.832600 0.553874i \(-0.813149\pi\)
−0.832600 + 0.553874i \(0.813149\pi\)
\(942\) −22.7885 −0.742491
\(943\) 0.000509824 0 1.66022e−5 0
\(944\) 8.16088 0.265614
\(945\) −4.53399 −0.147491
\(946\) 1.00000 0.0325128
\(947\) −58.9423 −1.91537 −0.957684 0.287822i \(-0.907069\pi\)
−0.957684 + 0.287822i \(0.907069\pi\)
\(948\) 12.7238 0.413251
\(949\) 0.449716 0.0145984
\(950\) 5.57111 0.180751
\(951\) −20.9940 −0.680776
\(952\) −0.0597128 −0.00193530
\(953\) 35.1078 1.13725 0.568627 0.822595i \(-0.307475\pi\)
0.568627 + 0.822595i \(0.307475\pi\)
\(954\) 19.0924 0.618139
\(955\) −9.24227 −0.299073
\(956\) −10.5435 −0.341002
\(957\) 5.48150 0.177192
\(958\) −26.0329 −0.841085
\(959\) 0.128974 0.00416479
\(960\) −1.13397 −0.0365986
\(961\) 57.4246 1.85241
\(962\) −1.23637 −0.0398621
\(963\) −11.8396 −0.381525
\(964\) −22.6988 −0.731078
\(965\) −15.2378 −0.490521
\(966\) −2.71044 −0.0872070
\(967\) −37.5885 −1.20877 −0.604383 0.796694i \(-0.706580\pi\)
−0.604383 + 0.796694i \(0.706580\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0.444764 0.0142879
\(970\) −16.4417 −0.527910
\(971\) 16.5199 0.530149 0.265075 0.964228i \(-0.414603\pi\)
0.265075 + 0.964228i \(0.414603\pi\)
\(972\) 14.9944 0.480944
\(973\) −9.49992 −0.304553
\(974\) 31.4385 1.00735
\(975\) 0.388206 0.0124326
\(976\) 10.5635 0.338128
\(977\) −37.0392 −1.18499 −0.592495 0.805574i \(-0.701857\pi\)
−0.592495 + 0.805574i \(0.701857\pi\)
\(978\) 9.89378 0.316368
\(979\) 12.5639 0.401545
\(980\) 6.28062 0.200627
\(981\) 8.36738 0.267150
\(982\) 7.08558 0.226110
\(983\) 45.9835 1.46665 0.733323 0.679880i \(-0.237968\pi\)
0.733323 + 0.679880i \(0.237968\pi\)
\(984\) 0.000205145 0 6.53979e−6 0
\(985\) 17.6609 0.562723
\(986\) 0.340320 0.0108380
\(987\) 3.77034 0.120011
\(988\) 1.90723 0.0606772
\(989\) 2.81812 0.0896110
\(990\) 1.71412 0.0544783
\(991\) 8.48927 0.269671 0.134835 0.990868i \(-0.456949\pi\)
0.134835 + 0.990868i \(0.456949\pi\)
\(992\) 9.40344 0.298559
\(993\) −15.4391 −0.489945
\(994\) −3.26710 −0.103626
\(995\) 7.99206 0.253365
\(996\) 15.8117 0.501014
\(997\) 38.1707 1.20888 0.604439 0.796651i \(-0.293397\pi\)
0.604439 + 0.796651i \(0.293397\pi\)
\(998\) 18.5102 0.585930
\(999\) 19.3057 0.610807
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 4730.2.a.bf.1.9 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.bf.1.9 13 1.1 even 1 trivial