Properties

Label 3630.2.a.bj.1.1
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 3630.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -3.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -3.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +3.38197 q^{13} -3.23607 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.09017 q^{17} +1.00000 q^{18} +1.23607 q^{19} -1.00000 q^{20} +3.23607 q^{21} +6.85410 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.38197 q^{26} -1.00000 q^{27} -3.23607 q^{28} -2.61803 q^{29} +1.00000 q^{30} -1.38197 q^{31} +1.00000 q^{32} -6.09017 q^{34} +3.23607 q^{35} +1.00000 q^{36} -6.38197 q^{37} +1.23607 q^{38} -3.38197 q^{39} -1.00000 q^{40} +1.23607 q^{41} +3.23607 q^{42} -0.0901699 q^{43} -1.00000 q^{45} +6.85410 q^{46} -4.32624 q^{47} -1.00000 q^{48} +3.47214 q^{49} +1.00000 q^{50} +6.09017 q^{51} +3.38197 q^{52} +8.47214 q^{53} -1.00000 q^{54} -3.23607 q^{56} -1.23607 q^{57} -2.61803 q^{58} +14.0902 q^{59} +1.00000 q^{60} -1.38197 q^{62} -3.23607 q^{63} +1.00000 q^{64} -3.38197 q^{65} -5.09017 q^{67} -6.09017 q^{68} -6.85410 q^{69} +3.23607 q^{70} +10.4721 q^{71} +1.00000 q^{72} +13.4164 q^{73} -6.38197 q^{74} -1.00000 q^{75} +1.23607 q^{76} -3.38197 q^{78} -4.09017 q^{79} -1.00000 q^{80} +1.00000 q^{81} +1.23607 q^{82} +4.00000 q^{83} +3.23607 q^{84} +6.09017 q^{85} -0.0901699 q^{86} +2.61803 q^{87} +17.2361 q^{89} -1.00000 q^{90} -10.9443 q^{91} +6.85410 q^{92} +1.38197 q^{93} -4.32624 q^{94} -1.23607 q^{95} -1.00000 q^{96} +18.6525 q^{97} +3.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} - 2q^{10} - 2q^{12} + 9q^{13} - 2q^{14} + 2q^{15} + 2q^{16} - q^{17} + 2q^{18} - 2q^{19} - 2q^{20} + 2q^{21} + 7q^{23} - 2q^{24} + 2q^{25} + 9q^{26} - 2q^{27} - 2q^{28} - 3q^{29} + 2q^{30} - 5q^{31} + 2q^{32} - q^{34} + 2q^{35} + 2q^{36} - 15q^{37} - 2q^{38} - 9q^{39} - 2q^{40} - 2q^{41} + 2q^{42} + 11q^{43} - 2q^{45} + 7q^{46} + 7q^{47} - 2q^{48} - 2q^{49} + 2q^{50} + q^{51} + 9q^{52} + 8q^{53} - 2q^{54} - 2q^{56} + 2q^{57} - 3q^{58} + 17q^{59} + 2q^{60} - 5q^{62} - 2q^{63} + 2q^{64} - 9q^{65} + q^{67} - q^{68} - 7q^{69} + 2q^{70} + 12q^{71} + 2q^{72} - 15q^{74} - 2q^{75} - 2q^{76} - 9q^{78} + 3q^{79} - 2q^{80} + 2q^{81} - 2q^{82} + 8q^{83} + 2q^{84} + q^{85} + 11q^{86} + 3q^{87} + 30q^{89} - 2q^{90} - 4q^{91} + 7q^{92} + 5q^{93} + 7q^{94} + 2q^{95} - 2q^{96} + 6q^{97} - 2q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −3.23607 −1.22312 −0.611559 0.791199i \(-0.709457\pi\)
−0.611559 + 0.791199i \(0.709457\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) 3.38197 0.937989 0.468994 0.883201i \(-0.344616\pi\)
0.468994 + 0.883201i \(0.344616\pi\)
\(14\) −3.23607 −0.864876
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −6.09017 −1.47708 −0.738542 0.674208i \(-0.764485\pi\)
−0.738542 + 0.674208i \(0.764485\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.23607 0.283573 0.141787 0.989897i \(-0.454715\pi\)
0.141787 + 0.989897i \(0.454715\pi\)
\(20\) −1.00000 −0.223607
\(21\) 3.23607 0.706168
\(22\) 0 0
\(23\) 6.85410 1.42918 0.714590 0.699544i \(-0.246614\pi\)
0.714590 + 0.699544i \(0.246614\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 3.38197 0.663258
\(27\) −1.00000 −0.192450
\(28\) −3.23607 −0.611559
\(29\) −2.61803 −0.486157 −0.243078 0.970007i \(-0.578157\pi\)
−0.243078 + 0.970007i \(0.578157\pi\)
\(30\) 1.00000 0.182574
\(31\) −1.38197 −0.248208 −0.124104 0.992269i \(-0.539606\pi\)
−0.124104 + 0.992269i \(0.539606\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.09017 −1.04446
\(35\) 3.23607 0.546995
\(36\) 1.00000 0.166667
\(37\) −6.38197 −1.04919 −0.524594 0.851352i \(-0.675783\pi\)
−0.524594 + 0.851352i \(0.675783\pi\)
\(38\) 1.23607 0.200517
\(39\) −3.38197 −0.541548
\(40\) −1.00000 −0.158114
\(41\) 1.23607 0.193041 0.0965207 0.995331i \(-0.469229\pi\)
0.0965207 + 0.995331i \(0.469229\pi\)
\(42\) 3.23607 0.499336
\(43\) −0.0901699 −0.0137508 −0.00687539 0.999976i \(-0.502189\pi\)
−0.00687539 + 0.999976i \(0.502189\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 6.85410 1.01058
\(47\) −4.32624 −0.631047 −0.315523 0.948918i \(-0.602180\pi\)
−0.315523 + 0.948918i \(0.602180\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.47214 0.496019
\(50\) 1.00000 0.141421
\(51\) 6.09017 0.852794
\(52\) 3.38197 0.468994
\(53\) 8.47214 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −3.23607 −0.432438
\(57\) −1.23607 −0.163721
\(58\) −2.61803 −0.343765
\(59\) 14.0902 1.83438 0.917192 0.398446i \(-0.130450\pi\)
0.917192 + 0.398446i \(0.130450\pi\)
\(60\) 1.00000 0.129099
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −1.38197 −0.175510
\(63\) −3.23607 −0.407706
\(64\) 1.00000 0.125000
\(65\) −3.38197 −0.419481
\(66\) 0 0
\(67\) −5.09017 −0.621863 −0.310932 0.950432i \(-0.600641\pi\)
−0.310932 + 0.950432i \(0.600641\pi\)
\(68\) −6.09017 −0.738542
\(69\) −6.85410 −0.825137
\(70\) 3.23607 0.386784
\(71\) 10.4721 1.24281 0.621407 0.783488i \(-0.286561\pi\)
0.621407 + 0.783488i \(0.286561\pi\)
\(72\) 1.00000 0.117851
\(73\) 13.4164 1.57027 0.785136 0.619324i \(-0.212593\pi\)
0.785136 + 0.619324i \(0.212593\pi\)
\(74\) −6.38197 −0.741888
\(75\) −1.00000 −0.115470
\(76\) 1.23607 0.141787
\(77\) 0 0
\(78\) −3.38197 −0.382932
\(79\) −4.09017 −0.460180 −0.230090 0.973169i \(-0.573902\pi\)
−0.230090 + 0.973169i \(0.573902\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 1.23607 0.136501
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 3.23607 0.353084
\(85\) 6.09017 0.660572
\(86\) −0.0901699 −0.00972328
\(87\) 2.61803 0.280683
\(88\) 0 0
\(89\) 17.2361 1.82702 0.913510 0.406817i \(-0.133361\pi\)
0.913510 + 0.406817i \(0.133361\pi\)
\(90\) −1.00000 −0.105409
\(91\) −10.9443 −1.14727
\(92\) 6.85410 0.714590
\(93\) 1.38197 0.143303
\(94\) −4.32624 −0.446217
\(95\) −1.23607 −0.126818
\(96\) −1.00000 −0.102062
\(97\) 18.6525 1.89387 0.946936 0.321422i \(-0.104161\pi\)
0.946936 + 0.321422i \(0.104161\pi\)
\(98\) 3.47214 0.350739
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 15.6180 1.55405 0.777026 0.629468i \(-0.216727\pi\)
0.777026 + 0.629468i \(0.216727\pi\)
\(102\) 6.09017 0.603017
\(103\) 6.76393 0.666470 0.333235 0.942844i \(-0.391860\pi\)
0.333235 + 0.942844i \(0.391860\pi\)
\(104\) 3.38197 0.331629
\(105\) −3.23607 −0.315808
\(106\) 8.47214 0.822887
\(107\) 6.29180 0.608251 0.304125 0.952632i \(-0.401636\pi\)
0.304125 + 0.952632i \(0.401636\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.47214 −0.236788 −0.118394 0.992967i \(-0.537775\pi\)
−0.118394 + 0.992967i \(0.537775\pi\)
\(110\) 0 0
\(111\) 6.38197 0.605749
\(112\) −3.23607 −0.305780
\(113\) −19.5623 −1.84027 −0.920133 0.391605i \(-0.871920\pi\)
−0.920133 + 0.391605i \(0.871920\pi\)
\(114\) −1.23607 −0.115768
\(115\) −6.85410 −0.639148
\(116\) −2.61803 −0.243078
\(117\) 3.38197 0.312663
\(118\) 14.0902 1.29711
\(119\) 19.7082 1.80665
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) 0 0
\(123\) −1.23607 −0.111452
\(124\) −1.38197 −0.124104
\(125\) −1.00000 −0.0894427
\(126\) −3.23607 −0.288292
\(127\) 14.6525 1.30020 0.650098 0.759850i \(-0.274728\pi\)
0.650098 + 0.759850i \(0.274728\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0.0901699 0.00793902
\(130\) −3.38197 −0.296618
\(131\) −3.38197 −0.295484 −0.147742 0.989026i \(-0.547200\pi\)
−0.147742 + 0.989026i \(0.547200\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) −5.09017 −0.439724
\(135\) 1.00000 0.0860663
\(136\) −6.09017 −0.522228
\(137\) −12.7984 −1.09344 −0.546719 0.837316i \(-0.684124\pi\)
−0.546719 + 0.837316i \(0.684124\pi\)
\(138\) −6.85410 −0.583460
\(139\) 8.76393 0.743347 0.371674 0.928364i \(-0.378784\pi\)
0.371674 + 0.928364i \(0.378784\pi\)
\(140\) 3.23607 0.273498
\(141\) 4.32624 0.364335
\(142\) 10.4721 0.878802
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 2.61803 0.217416
\(146\) 13.4164 1.11035
\(147\) −3.47214 −0.286377
\(148\) −6.38197 −0.524594
\(149\) −7.61803 −0.624094 −0.312047 0.950067i \(-0.601015\pi\)
−0.312047 + 0.950067i \(0.601015\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 20.9443 1.70442 0.852210 0.523199i \(-0.175262\pi\)
0.852210 + 0.523199i \(0.175262\pi\)
\(152\) 1.23607 0.100258
\(153\) −6.09017 −0.492361
\(154\) 0 0
\(155\) 1.38197 0.111002
\(156\) −3.38197 −0.270774
\(157\) −9.27051 −0.739867 −0.369934 0.929058i \(-0.620620\pi\)
−0.369934 + 0.929058i \(0.620620\pi\)
\(158\) −4.09017 −0.325396
\(159\) −8.47214 −0.671884
\(160\) −1.00000 −0.0790569
\(161\) −22.1803 −1.74806
\(162\) 1.00000 0.0785674
\(163\) 0.854102 0.0668984 0.0334492 0.999440i \(-0.489351\pi\)
0.0334492 + 0.999440i \(0.489351\pi\)
\(164\) 1.23607 0.0965207
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −13.2705 −1.02690 −0.513451 0.858119i \(-0.671633\pi\)
−0.513451 + 0.858119i \(0.671633\pi\)
\(168\) 3.23607 0.249668
\(169\) −1.56231 −0.120177
\(170\) 6.09017 0.467095
\(171\) 1.23607 0.0945245
\(172\) −0.0901699 −0.00687539
\(173\) −9.52786 −0.724390 −0.362195 0.932102i \(-0.617973\pi\)
−0.362195 + 0.932102i \(0.617973\pi\)
\(174\) 2.61803 0.198473
\(175\) −3.23607 −0.244624
\(176\) 0 0
\(177\) −14.0902 −1.05908
\(178\) 17.2361 1.29190
\(179\) 18.1459 1.35629 0.678144 0.734929i \(-0.262785\pi\)
0.678144 + 0.734929i \(0.262785\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 21.8885 1.62696 0.813481 0.581591i \(-0.197570\pi\)
0.813481 + 0.581591i \(0.197570\pi\)
\(182\) −10.9443 −0.811243
\(183\) 0 0
\(184\) 6.85410 0.505291
\(185\) 6.38197 0.469211
\(186\) 1.38197 0.101331
\(187\) 0 0
\(188\) −4.32624 −0.315523
\(189\) 3.23607 0.235389
\(190\) −1.23607 −0.0896738
\(191\) 9.23607 0.668298 0.334149 0.942520i \(-0.391551\pi\)
0.334149 + 0.942520i \(0.391551\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −20.9443 −1.50760 −0.753801 0.657103i \(-0.771782\pi\)
−0.753801 + 0.657103i \(0.771782\pi\)
\(194\) 18.6525 1.33917
\(195\) 3.38197 0.242188
\(196\) 3.47214 0.248010
\(197\) 10.9443 0.779747 0.389874 0.920868i \(-0.372519\pi\)
0.389874 + 0.920868i \(0.372519\pi\)
\(198\) 0 0
\(199\) 25.9787 1.84158 0.920791 0.390056i \(-0.127544\pi\)
0.920791 + 0.390056i \(0.127544\pi\)
\(200\) 1.00000 0.0707107
\(201\) 5.09017 0.359033
\(202\) 15.6180 1.09888
\(203\) 8.47214 0.594627
\(204\) 6.09017 0.426397
\(205\) −1.23607 −0.0863307
\(206\) 6.76393 0.471265
\(207\) 6.85410 0.476393
\(208\) 3.38197 0.234497
\(209\) 0 0
\(210\) −3.23607 −0.223310
\(211\) −25.2361 −1.73732 −0.868661 0.495406i \(-0.835019\pi\)
−0.868661 + 0.495406i \(0.835019\pi\)
\(212\) 8.47214 0.581869
\(213\) −10.4721 −0.717539
\(214\) 6.29180 0.430098
\(215\) 0.0901699 0.00614954
\(216\) −1.00000 −0.0680414
\(217\) 4.47214 0.303588
\(218\) −2.47214 −0.167434
\(219\) −13.4164 −0.906597
\(220\) 0 0
\(221\) −20.5967 −1.38549
\(222\) 6.38197 0.428330
\(223\) 7.23607 0.484563 0.242281 0.970206i \(-0.422104\pi\)
0.242281 + 0.970206i \(0.422104\pi\)
\(224\) −3.23607 −0.216219
\(225\) 1.00000 0.0666667
\(226\) −19.5623 −1.30127
\(227\) 20.3607 1.35139 0.675693 0.737183i \(-0.263845\pi\)
0.675693 + 0.737183i \(0.263845\pi\)
\(228\) −1.23607 −0.0818606
\(229\) 0.763932 0.0504820 0.0252410 0.999681i \(-0.491965\pi\)
0.0252410 + 0.999681i \(0.491965\pi\)
\(230\) −6.85410 −0.451946
\(231\) 0 0
\(232\) −2.61803 −0.171882
\(233\) −4.43769 −0.290723 −0.145362 0.989379i \(-0.546435\pi\)
−0.145362 + 0.989379i \(0.546435\pi\)
\(234\) 3.38197 0.221086
\(235\) 4.32624 0.282213
\(236\) 14.0902 0.917192
\(237\) 4.09017 0.265685
\(238\) 19.7082 1.27749
\(239\) −3.23607 −0.209324 −0.104662 0.994508i \(-0.533376\pi\)
−0.104662 + 0.994508i \(0.533376\pi\)
\(240\) 1.00000 0.0645497
\(241\) −23.8885 −1.53880 −0.769398 0.638769i \(-0.779444\pi\)
−0.769398 + 0.638769i \(0.779444\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −3.47214 −0.221827
\(246\) −1.23607 −0.0788088
\(247\) 4.18034 0.265989
\(248\) −1.38197 −0.0877549
\(249\) −4.00000 −0.253490
\(250\) −1.00000 −0.0632456
\(251\) 8.38197 0.529065 0.264533 0.964377i \(-0.414782\pi\)
0.264533 + 0.964377i \(0.414782\pi\)
\(252\) −3.23607 −0.203853
\(253\) 0 0
\(254\) 14.6525 0.919378
\(255\) −6.09017 −0.381381
\(256\) 1.00000 0.0625000
\(257\) 30.3607 1.89385 0.946924 0.321459i \(-0.104173\pi\)
0.946924 + 0.321459i \(0.104173\pi\)
\(258\) 0.0901699 0.00561374
\(259\) 20.6525 1.28328
\(260\) −3.38197 −0.209741
\(261\) −2.61803 −0.162052
\(262\) −3.38197 −0.208939
\(263\) 6.85410 0.422642 0.211321 0.977417i \(-0.432223\pi\)
0.211321 + 0.977417i \(0.432223\pi\)
\(264\) 0 0
\(265\) −8.47214 −0.520439
\(266\) −4.00000 −0.245256
\(267\) −17.2361 −1.05483
\(268\) −5.09017 −0.310932
\(269\) 3.20163 0.195207 0.0976033 0.995225i \(-0.468882\pi\)
0.0976033 + 0.995225i \(0.468882\pi\)
\(270\) 1.00000 0.0608581
\(271\) −6.38197 −0.387677 −0.193838 0.981033i \(-0.562094\pi\)
−0.193838 + 0.981033i \(0.562094\pi\)
\(272\) −6.09017 −0.369271
\(273\) 10.9443 0.662377
\(274\) −12.7984 −0.773178
\(275\) 0 0
\(276\) −6.85410 −0.412568
\(277\) −23.5066 −1.41237 −0.706187 0.708026i \(-0.749586\pi\)
−0.706187 + 0.708026i \(0.749586\pi\)
\(278\) 8.76393 0.525626
\(279\) −1.38197 −0.0827361
\(280\) 3.23607 0.193392
\(281\) −7.52786 −0.449075 −0.224537 0.974465i \(-0.572087\pi\)
−0.224537 + 0.974465i \(0.572087\pi\)
\(282\) 4.32624 0.257624
\(283\) −22.3262 −1.32716 −0.663579 0.748107i \(-0.730963\pi\)
−0.663579 + 0.748107i \(0.730963\pi\)
\(284\) 10.4721 0.621407
\(285\) 1.23607 0.0732183
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 1.00000 0.0589256
\(289\) 20.0902 1.18177
\(290\) 2.61803 0.153736
\(291\) −18.6525 −1.09343
\(292\) 13.4164 0.785136
\(293\) −20.0000 −1.16841 −0.584206 0.811605i \(-0.698594\pi\)
−0.584206 + 0.811605i \(0.698594\pi\)
\(294\) −3.47214 −0.202499
\(295\) −14.0902 −0.820361
\(296\) −6.38197 −0.370944
\(297\) 0 0
\(298\) −7.61803 −0.441301
\(299\) 23.1803 1.34055
\(300\) −1.00000 −0.0577350
\(301\) 0.291796 0.0168188
\(302\) 20.9443 1.20521
\(303\) −15.6180 −0.897233
\(304\) 1.23607 0.0708934
\(305\) 0 0
\(306\) −6.09017 −0.348152
\(307\) −33.4508 −1.90914 −0.954570 0.297985i \(-0.903685\pi\)
−0.954570 + 0.297985i \(0.903685\pi\)
\(308\) 0 0
\(309\) −6.76393 −0.384787
\(310\) 1.38197 0.0784904
\(311\) 9.23607 0.523729 0.261865 0.965105i \(-0.415663\pi\)
0.261865 + 0.965105i \(0.415663\pi\)
\(312\) −3.38197 −0.191466
\(313\) 8.47214 0.478873 0.239437 0.970912i \(-0.423037\pi\)
0.239437 + 0.970912i \(0.423037\pi\)
\(314\) −9.27051 −0.523165
\(315\) 3.23607 0.182332
\(316\) −4.09017 −0.230090
\(317\) −27.4164 −1.53986 −0.769929 0.638129i \(-0.779709\pi\)
−0.769929 + 0.638129i \(0.779709\pi\)
\(318\) −8.47214 −0.475094
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −6.29180 −0.351174
\(322\) −22.1803 −1.23606
\(323\) −7.52786 −0.418862
\(324\) 1.00000 0.0555556
\(325\) 3.38197 0.187598
\(326\) 0.854102 0.0473043
\(327\) 2.47214 0.136709
\(328\) 1.23607 0.0682504
\(329\) 14.0000 0.771845
\(330\) 0 0
\(331\) 9.23607 0.507660 0.253830 0.967249i \(-0.418310\pi\)
0.253830 + 0.967249i \(0.418310\pi\)
\(332\) 4.00000 0.219529
\(333\) −6.38197 −0.349730
\(334\) −13.2705 −0.726130
\(335\) 5.09017 0.278106
\(336\) 3.23607 0.176542
\(337\) −15.2361 −0.829962 −0.414981 0.909830i \(-0.636212\pi\)
−0.414981 + 0.909830i \(0.636212\pi\)
\(338\) −1.56231 −0.0849782
\(339\) 19.5623 1.06248
\(340\) 6.09017 0.330286
\(341\) 0 0
\(342\) 1.23607 0.0668389
\(343\) 11.4164 0.616428
\(344\) −0.0901699 −0.00486164
\(345\) 6.85410 0.369012
\(346\) −9.52786 −0.512221
\(347\) 35.8885 1.92660 0.963299 0.268431i \(-0.0865050\pi\)
0.963299 + 0.268431i \(0.0865050\pi\)
\(348\) 2.61803 0.140341
\(349\) −11.4164 −0.611106 −0.305553 0.952175i \(-0.598841\pi\)
−0.305553 + 0.952175i \(0.598841\pi\)
\(350\) −3.23607 −0.172975
\(351\) −3.38197 −0.180516
\(352\) 0 0
\(353\) 23.6180 1.25706 0.628531 0.777785i \(-0.283657\pi\)
0.628531 + 0.777785i \(0.283657\pi\)
\(354\) −14.0902 −0.748884
\(355\) −10.4721 −0.555803
\(356\) 17.2361 0.913510
\(357\) −19.7082 −1.04307
\(358\) 18.1459 0.959041
\(359\) −12.1803 −0.642854 −0.321427 0.946934i \(-0.604162\pi\)
−0.321427 + 0.946934i \(0.604162\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −17.4721 −0.919586
\(362\) 21.8885 1.15044
\(363\) 0 0
\(364\) −10.9443 −0.573636
\(365\) −13.4164 −0.702247
\(366\) 0 0
\(367\) −26.7639 −1.39707 −0.698533 0.715578i \(-0.746163\pi\)
−0.698533 + 0.715578i \(0.746163\pi\)
\(368\) 6.85410 0.357295
\(369\) 1.23607 0.0643471
\(370\) 6.38197 0.331783
\(371\) −27.4164 −1.42339
\(372\) 1.38197 0.0716516
\(373\) −11.5279 −0.596890 −0.298445 0.954427i \(-0.596468\pi\)
−0.298445 + 0.954427i \(0.596468\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −4.32624 −0.223109
\(377\) −8.85410 −0.456009
\(378\) 3.23607 0.166445
\(379\) −12.3607 −0.634925 −0.317463 0.948271i \(-0.602831\pi\)
−0.317463 + 0.948271i \(0.602831\pi\)
\(380\) −1.23607 −0.0634089
\(381\) −14.6525 −0.750669
\(382\) 9.23607 0.472558
\(383\) −0.326238 −0.0166700 −0.00833499 0.999965i \(-0.502653\pi\)
−0.00833499 + 0.999965i \(0.502653\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −20.9443 −1.06604
\(387\) −0.0901699 −0.00458360
\(388\) 18.6525 0.946936
\(389\) 6.85410 0.347517 0.173758 0.984788i \(-0.444409\pi\)
0.173758 + 0.984788i \(0.444409\pi\)
\(390\) 3.38197 0.171253
\(391\) −41.7426 −2.11102
\(392\) 3.47214 0.175369
\(393\) 3.38197 0.170598
\(394\) 10.9443 0.551364
\(395\) 4.09017 0.205799
\(396\) 0 0
\(397\) 2.20163 0.110496 0.0552482 0.998473i \(-0.482405\pi\)
0.0552482 + 0.998473i \(0.482405\pi\)
\(398\) 25.9787 1.30220
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) 22.0689 1.10207 0.551034 0.834483i \(-0.314234\pi\)
0.551034 + 0.834483i \(0.314234\pi\)
\(402\) 5.09017 0.253875
\(403\) −4.67376 −0.232817
\(404\) 15.6180 0.777026
\(405\) −1.00000 −0.0496904
\(406\) 8.47214 0.420465
\(407\) 0 0
\(408\) 6.09017 0.301508
\(409\) 12.9787 0.641756 0.320878 0.947121i \(-0.396022\pi\)
0.320878 + 0.947121i \(0.396022\pi\)
\(410\) −1.23607 −0.0610450
\(411\) 12.7984 0.631297
\(412\) 6.76393 0.333235
\(413\) −45.5967 −2.24367
\(414\) 6.85410 0.336861
\(415\) −4.00000 −0.196352
\(416\) 3.38197 0.165815
\(417\) −8.76393 −0.429172
\(418\) 0 0
\(419\) 22.8541 1.11650 0.558248 0.829674i \(-0.311474\pi\)
0.558248 + 0.829674i \(0.311474\pi\)
\(420\) −3.23607 −0.157904
\(421\) −2.47214 −0.120485 −0.0602423 0.998184i \(-0.519187\pi\)
−0.0602423 + 0.998184i \(0.519187\pi\)
\(422\) −25.2361 −1.22847
\(423\) −4.32624 −0.210349
\(424\) 8.47214 0.411443
\(425\) −6.09017 −0.295417
\(426\) −10.4721 −0.507377
\(427\) 0 0
\(428\) 6.29180 0.304125
\(429\) 0 0
\(430\) 0.0901699 0.00434838
\(431\) 22.3607 1.07708 0.538538 0.842601i \(-0.318977\pi\)
0.538538 + 0.842601i \(0.318977\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −22.8328 −1.09728 −0.548638 0.836060i \(-0.684853\pi\)
−0.548638 + 0.836060i \(0.684853\pi\)
\(434\) 4.47214 0.214669
\(435\) −2.61803 −0.125525
\(436\) −2.47214 −0.118394
\(437\) 8.47214 0.405277
\(438\) −13.4164 −0.641061
\(439\) 1.67376 0.0798843 0.0399422 0.999202i \(-0.487283\pi\)
0.0399422 + 0.999202i \(0.487283\pi\)
\(440\) 0 0
\(441\) 3.47214 0.165340
\(442\) −20.5967 −0.979687
\(443\) −18.6525 −0.886206 −0.443103 0.896471i \(-0.646122\pi\)
−0.443103 + 0.896471i \(0.646122\pi\)
\(444\) 6.38197 0.302875
\(445\) −17.2361 −0.817068
\(446\) 7.23607 0.342638
\(447\) 7.61803 0.360321
\(448\) −3.23607 −0.152890
\(449\) −11.1246 −0.525003 −0.262501 0.964932i \(-0.584547\pi\)
−0.262501 + 0.964932i \(0.584547\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −19.5623 −0.920133
\(453\) −20.9443 −0.984048
\(454\) 20.3607 0.955574
\(455\) 10.9443 0.513075
\(456\) −1.23607 −0.0578842
\(457\) 4.65248 0.217634 0.108817 0.994062i \(-0.465294\pi\)
0.108817 + 0.994062i \(0.465294\pi\)
\(458\) 0.763932 0.0356962
\(459\) 6.09017 0.284265
\(460\) −6.85410 −0.319574
\(461\) 23.2148 1.08122 0.540610 0.841273i \(-0.318193\pi\)
0.540610 + 0.841273i \(0.318193\pi\)
\(462\) 0 0
\(463\) 11.8885 0.552507 0.276254 0.961085i \(-0.410907\pi\)
0.276254 + 0.961085i \(0.410907\pi\)
\(464\) −2.61803 −0.121539
\(465\) −1.38197 −0.0640871
\(466\) −4.43769 −0.205572
\(467\) 28.5410 1.32072 0.660360 0.750949i \(-0.270404\pi\)
0.660360 + 0.750949i \(0.270404\pi\)
\(468\) 3.38197 0.156331
\(469\) 16.4721 0.760613
\(470\) 4.32624 0.199554
\(471\) 9.27051 0.427163
\(472\) 14.0902 0.648553
\(473\) 0 0
\(474\) 4.09017 0.187868
\(475\) 1.23607 0.0567147
\(476\) 19.7082 0.903324
\(477\) 8.47214 0.387912
\(478\) −3.23607 −0.148014
\(479\) 12.7639 0.583199 0.291599 0.956541i \(-0.405813\pi\)
0.291599 + 0.956541i \(0.405813\pi\)
\(480\) 1.00000 0.0456435
\(481\) −21.5836 −0.984127
\(482\) −23.8885 −1.08809
\(483\) 22.1803 1.00924
\(484\) 0 0
\(485\) −18.6525 −0.846965
\(486\) −1.00000 −0.0453609
\(487\) 11.4164 0.517327 0.258663 0.965968i \(-0.416718\pi\)
0.258663 + 0.965968i \(0.416718\pi\)
\(488\) 0 0
\(489\) −0.854102 −0.0386238
\(490\) −3.47214 −0.156855
\(491\) 28.0344 1.26518 0.632588 0.774488i \(-0.281993\pi\)
0.632588 + 0.774488i \(0.281993\pi\)
\(492\) −1.23607 −0.0557262
\(493\) 15.9443 0.718094
\(494\) 4.18034 0.188082
\(495\) 0 0
\(496\) −1.38197 −0.0620521
\(497\) −33.8885 −1.52011
\(498\) −4.00000 −0.179244
\(499\) −32.4721 −1.45365 −0.726826 0.686821i \(-0.759005\pi\)
−0.726826 + 0.686821i \(0.759005\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 13.2705 0.592883
\(502\) 8.38197 0.374105
\(503\) −24.3262 −1.08465 −0.542327 0.840168i \(-0.682456\pi\)
−0.542327 + 0.840168i \(0.682456\pi\)
\(504\) −3.23607 −0.144146
\(505\) −15.6180 −0.694993
\(506\) 0 0
\(507\) 1.56231 0.0693844
\(508\) 14.6525 0.650098
\(509\) 8.79837 0.389981 0.194991 0.980805i \(-0.437532\pi\)
0.194991 + 0.980805i \(0.437532\pi\)
\(510\) −6.09017 −0.269677
\(511\) −43.4164 −1.92063
\(512\) 1.00000 0.0441942
\(513\) −1.23607 −0.0545737
\(514\) 30.3607 1.33915
\(515\) −6.76393 −0.298054
\(516\) 0.0901699 0.00396951
\(517\) 0 0
\(518\) 20.6525 0.907418
\(519\) 9.52786 0.418227
\(520\) −3.38197 −0.148309
\(521\) −2.36068 −0.103423 −0.0517116 0.998662i \(-0.516468\pi\)
−0.0517116 + 0.998662i \(0.516468\pi\)
\(522\) −2.61803 −0.114588
\(523\) 38.8328 1.69804 0.849020 0.528360i \(-0.177193\pi\)
0.849020 + 0.528360i \(0.177193\pi\)
\(524\) −3.38197 −0.147742
\(525\) 3.23607 0.141234
\(526\) 6.85410 0.298853
\(527\) 8.41641 0.366624
\(528\) 0 0
\(529\) 23.9787 1.04255
\(530\) −8.47214 −0.368006
\(531\) 14.0902 0.611461
\(532\) −4.00000 −0.173422
\(533\) 4.18034 0.181071
\(534\) −17.2361 −0.745878
\(535\) −6.29180 −0.272018
\(536\) −5.09017 −0.219862
\(537\) −18.1459 −0.783053
\(538\) 3.20163 0.138032
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) 26.0689 1.12079 0.560394 0.828226i \(-0.310650\pi\)
0.560394 + 0.828226i \(0.310650\pi\)
\(542\) −6.38197 −0.274129
\(543\) −21.8885 −0.939327
\(544\) −6.09017 −0.261114
\(545\) 2.47214 0.105895
\(546\) 10.9443 0.468372
\(547\) 34.3262 1.46768 0.733842 0.679320i \(-0.237725\pi\)
0.733842 + 0.679320i \(0.237725\pi\)
\(548\) −12.7984 −0.546719
\(549\) 0 0
\(550\) 0 0
\(551\) −3.23607 −0.137861
\(552\) −6.85410 −0.291730
\(553\) 13.2361 0.562855
\(554\) −23.5066 −0.998699
\(555\) −6.38197 −0.270899
\(556\) 8.76393 0.371674
\(557\) −35.1246 −1.48828 −0.744139 0.668025i \(-0.767140\pi\)
−0.744139 + 0.668025i \(0.767140\pi\)
\(558\) −1.38197 −0.0585033
\(559\) −0.304952 −0.0128981
\(560\) 3.23607 0.136749
\(561\) 0 0
\(562\) −7.52786 −0.317544
\(563\) 17.0557 0.718813 0.359407 0.933181i \(-0.382979\pi\)
0.359407 + 0.933181i \(0.382979\pi\)
\(564\) 4.32624 0.182167
\(565\) 19.5623 0.822992
\(566\) −22.3262 −0.938442
\(567\) −3.23607 −0.135902
\(568\) 10.4721 0.439401
\(569\) −25.4164 −1.06551 −0.532756 0.846269i \(-0.678844\pi\)
−0.532756 + 0.846269i \(0.678844\pi\)
\(570\) 1.23607 0.0517732
\(571\) −28.3607 −1.18686 −0.593429 0.804887i \(-0.702226\pi\)
−0.593429 + 0.804887i \(0.702226\pi\)
\(572\) 0 0
\(573\) −9.23607 −0.385842
\(574\) −4.00000 −0.166957
\(575\) 6.85410 0.285836
\(576\) 1.00000 0.0416667
\(577\) 17.0557 0.710039 0.355020 0.934859i \(-0.384474\pi\)
0.355020 + 0.934859i \(0.384474\pi\)
\(578\) 20.0902 0.835641
\(579\) 20.9443 0.870414
\(580\) 2.61803 0.108708
\(581\) −12.9443 −0.537019
\(582\) −18.6525 −0.773170
\(583\) 0 0
\(584\) 13.4164 0.555175
\(585\) −3.38197 −0.139827
\(586\) −20.0000 −0.826192
\(587\) −26.2918 −1.08518 −0.542589 0.839998i \(-0.682556\pi\)
−0.542589 + 0.839998i \(0.682556\pi\)
\(588\) −3.47214 −0.143188
\(589\) −1.70820 −0.0703853
\(590\) −14.0902 −0.580083
\(591\) −10.9443 −0.450187
\(592\) −6.38197 −0.262297
\(593\) −17.6738 −0.725774 −0.362887 0.931833i \(-0.618209\pi\)
−0.362887 + 0.931833i \(0.618209\pi\)
\(594\) 0 0
\(595\) −19.7082 −0.807958
\(596\) −7.61803 −0.312047
\(597\) −25.9787 −1.06324
\(598\) 23.1803 0.947915
\(599\) 18.1803 0.742829 0.371414 0.928467i \(-0.378873\pi\)
0.371414 + 0.928467i \(0.378873\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −18.9443 −0.772753 −0.386376 0.922341i \(-0.626273\pi\)
−0.386376 + 0.922341i \(0.626273\pi\)
\(602\) 0.291796 0.0118927
\(603\) −5.09017 −0.207288
\(604\) 20.9443 0.852210
\(605\) 0 0
\(606\) −15.6180 −0.634439
\(607\) −4.47214 −0.181518 −0.0907592 0.995873i \(-0.528929\pi\)
−0.0907592 + 0.995873i \(0.528929\pi\)
\(608\) 1.23607 0.0501292
\(609\) −8.47214 −0.343308
\(610\) 0 0
\(611\) −14.6312 −0.591915
\(612\) −6.09017 −0.246181
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) −33.4508 −1.34997
\(615\) 1.23607 0.0498431
\(616\) 0 0
\(617\) 26.3607 1.06124 0.530621 0.847610i \(-0.321959\pi\)
0.530621 + 0.847610i \(0.321959\pi\)
\(618\) −6.76393 −0.272085
\(619\) 9.52786 0.382957 0.191479 0.981497i \(-0.438672\pi\)
0.191479 + 0.981497i \(0.438672\pi\)
\(620\) 1.38197 0.0555011
\(621\) −6.85410 −0.275046
\(622\) 9.23607 0.370333
\(623\) −55.7771 −2.23466
\(624\) −3.38197 −0.135387
\(625\) 1.00000 0.0400000
\(626\) 8.47214 0.338615
\(627\) 0 0
\(628\) −9.27051 −0.369934
\(629\) 38.8673 1.54974
\(630\) 3.23607 0.128928
\(631\) −5.67376 −0.225869 −0.112934 0.993602i \(-0.536025\pi\)
−0.112934 + 0.993602i \(0.536025\pi\)
\(632\) −4.09017 −0.162698
\(633\) 25.2361 1.00304
\(634\) −27.4164 −1.08884
\(635\) −14.6525 −0.581466
\(636\) −8.47214 −0.335942
\(637\) 11.7426 0.465261
\(638\) 0 0
\(639\) 10.4721 0.414271
\(640\) −1.00000 −0.0395285
\(641\) 12.1803 0.481095 0.240547 0.970637i \(-0.422673\pi\)
0.240547 + 0.970637i \(0.422673\pi\)
\(642\) −6.29180 −0.248317
\(643\) −4.79837 −0.189229 −0.0946147 0.995514i \(-0.530162\pi\)
−0.0946147 + 0.995514i \(0.530162\pi\)
\(644\) −22.1803 −0.874028
\(645\) −0.0901699 −0.00355044
\(646\) −7.52786 −0.296180
\(647\) −12.3820 −0.486785 −0.243393 0.969928i \(-0.578260\pi\)
−0.243393 + 0.969928i \(0.578260\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 3.38197 0.132652
\(651\) −4.47214 −0.175277
\(652\) 0.854102 0.0334492
\(653\) 24.5410 0.960364 0.480182 0.877169i \(-0.340571\pi\)
0.480182 + 0.877169i \(0.340571\pi\)
\(654\) 2.47214 0.0966682
\(655\) 3.38197 0.132144
\(656\) 1.23607 0.0482603
\(657\) 13.4164 0.523424
\(658\) 14.0000 0.545777
\(659\) −15.4164 −0.600538 −0.300269 0.953855i \(-0.597076\pi\)
−0.300269 + 0.953855i \(0.597076\pi\)
\(660\) 0 0
\(661\) 8.36068 0.325193 0.162596 0.986693i \(-0.448013\pi\)
0.162596 + 0.986693i \(0.448013\pi\)
\(662\) 9.23607 0.358970
\(663\) 20.5967 0.799911
\(664\) 4.00000 0.155230
\(665\) 4.00000 0.155113
\(666\) −6.38197 −0.247296
\(667\) −17.9443 −0.694805
\(668\) −13.2705 −0.513451
\(669\) −7.23607 −0.279763
\(670\) 5.09017 0.196650
\(671\) 0 0
\(672\) 3.23607 0.124834
\(673\) −15.7082 −0.605507 −0.302753 0.953069i \(-0.597906\pi\)
−0.302753 + 0.953069i \(0.597906\pi\)
\(674\) −15.2361 −0.586871
\(675\) −1.00000 −0.0384900
\(676\) −1.56231 −0.0600887
\(677\) 27.2361 1.04677 0.523384 0.852097i \(-0.324670\pi\)
0.523384 + 0.852097i \(0.324670\pi\)
\(678\) 19.5623 0.751286
\(679\) −60.3607 −2.31643
\(680\) 6.09017 0.233547
\(681\) −20.3607 −0.780223
\(682\) 0 0
\(683\) 21.1246 0.808311 0.404155 0.914690i \(-0.367565\pi\)
0.404155 + 0.914690i \(0.367565\pi\)
\(684\) 1.23607 0.0472622
\(685\) 12.7984 0.489001
\(686\) 11.4164 0.435880
\(687\) −0.763932 −0.0291458
\(688\) −0.0901699 −0.00343770
\(689\) 28.6525 1.09157
\(690\) 6.85410 0.260931
\(691\) −20.6525 −0.785657 −0.392829 0.919612i \(-0.628503\pi\)
−0.392829 + 0.919612i \(0.628503\pi\)
\(692\) −9.52786 −0.362195
\(693\) 0 0
\(694\) 35.8885 1.36231
\(695\) −8.76393 −0.332435
\(696\) 2.61803 0.0992363
\(697\) −7.52786 −0.285138
\(698\) −11.4164 −0.432117
\(699\) 4.43769 0.167849
\(700\) −3.23607 −0.122312
\(701\) −3.52786 −0.133246 −0.0666228 0.997778i \(-0.521222\pi\)
−0.0666228 + 0.997778i \(0.521222\pi\)
\(702\) −3.38197 −0.127644
\(703\) −7.88854 −0.297522
\(704\) 0 0
\(705\) −4.32624 −0.162936
\(706\) 23.6180 0.888876
\(707\) −50.5410 −1.90079
\(708\) −14.0902 −0.529541
\(709\) −10.5836 −0.397475 −0.198738 0.980053i \(-0.563684\pi\)
−0.198738 + 0.980053i \(0.563684\pi\)
\(710\) −10.4721 −0.393012
\(711\) −4.09017 −0.153393
\(712\) 17.2361 0.645949
\(713\) −9.47214 −0.354734
\(714\) −19.7082 −0.737561
\(715\) 0 0
\(716\) 18.1459 0.678144
\(717\) 3.23607 0.120853
\(718\) −12.1803 −0.454566
\(719\) 40.6525 1.51608 0.758041 0.652207i \(-0.226157\pi\)
0.758041 + 0.652207i \(0.226157\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −21.8885 −0.815172
\(722\) −17.4721 −0.650246
\(723\) 23.8885 0.888425
\(724\) 21.8885 0.813481
\(725\) −2.61803 −0.0972313
\(726\) 0 0
\(727\) −50.1803 −1.86109 −0.930543 0.366183i \(-0.880664\pi\)
−0.930543 + 0.366183i \(0.880664\pi\)
\(728\) −10.9443 −0.405622
\(729\) 1.00000 0.0370370
\(730\) −13.4164 −0.496564
\(731\) 0.549150 0.0203111
\(732\) 0 0
\(733\) 51.3262 1.89578 0.947889 0.318601i \(-0.103213\pi\)
0.947889 + 0.318601i \(0.103213\pi\)
\(734\) −26.7639 −0.987875
\(735\) 3.47214 0.128072
\(736\) 6.85410 0.252646
\(737\) 0 0
\(738\) 1.23607 0.0455003
\(739\) 27.0132 0.993695 0.496847 0.867838i \(-0.334491\pi\)
0.496847 + 0.867838i \(0.334491\pi\)
\(740\) 6.38197 0.234606
\(741\) −4.18034 −0.153569
\(742\) −27.4164 −1.00649
\(743\) 10.7984 0.396154 0.198077 0.980186i \(-0.436530\pi\)
0.198077 + 0.980186i \(0.436530\pi\)
\(744\) 1.38197 0.0506653
\(745\) 7.61803 0.279103
\(746\) −11.5279 −0.422065
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) −20.3607 −0.743963
\(750\) 1.00000 0.0365148
\(751\) −29.0344 −1.05948 −0.529741 0.848160i \(-0.677711\pi\)
−0.529741 + 0.848160i \(0.677711\pi\)
\(752\) −4.32624 −0.157762
\(753\) −8.38197 −0.305456
\(754\) −8.85410 −0.322447
\(755\) −20.9443 −0.762240
\(756\) 3.23607 0.117695
\(757\) −53.8541 −1.95736 −0.978680 0.205389i \(-0.934154\pi\)
−0.978680 + 0.205389i \(0.934154\pi\)
\(758\) −12.3607 −0.448960
\(759\) 0 0
\(760\) −1.23607 −0.0448369
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) −14.6525 −0.530803
\(763\) 8.00000 0.289619
\(764\) 9.23607 0.334149
\(765\) 6.09017 0.220191
\(766\) −0.326238 −0.0117875
\(767\) 47.6525 1.72063
\(768\) −1.00000 −0.0360844
\(769\) −24.4508 −0.881720 −0.440860 0.897576i \(-0.645327\pi\)
−0.440860 + 0.897576i \(0.645327\pi\)
\(770\) 0 0
\(771\) −30.3607 −1.09341
\(772\) −20.9443 −0.753801
\(773\) 19.5967 0.704846 0.352423 0.935841i \(-0.385358\pi\)
0.352423 + 0.935841i \(0.385358\pi\)
\(774\) −0.0901699 −0.00324109
\(775\) −1.38197 −0.0496417
\(776\) 18.6525 0.669585
\(777\) −20.6525 −0.740903
\(778\) 6.85410 0.245731
\(779\) 1.52786 0.0547414
\(780\) 3.38197 0.121094
\(781\) 0 0
\(782\) −41.7426 −1.49271
\(783\) 2.61803 0.0935609
\(784\) 3.47214 0.124005
\(785\) 9.27051 0.330879
\(786\) 3.38197 0.120631
\(787\) 21.0344 0.749797 0.374898 0.927066i \(-0.377678\pi\)
0.374898 + 0.927066i \(0.377678\pi\)
\(788\) 10.9443 0.389874
\(789\) −6.85410 −0.244012
\(790\) 4.09017 0.145522
\(791\) 63.3050 2.25086
\(792\) 0 0
\(793\) 0 0
\(794\) 2.20163 0.0781328
\(795\) 8.47214 0.300476
\(796\) 25.9787 0.920791
\(797\) −21.7082 −0.768944 −0.384472 0.923137i \(-0.625616\pi\)
−0.384472 + 0.923137i \(0.625616\pi\)
\(798\) 4.00000 0.141598
\(799\) 26.3475 0.932108
\(800\) 1.00000 0.0353553
\(801\) 17.2361 0.609007
\(802\) 22.0689 0.779279
\(803\) 0 0
\(804\) 5.09017 0.179516
\(805\) 22.1803 0.781754
\(806\) −4.67376 −0.164626
\(807\) −3.20163 −0.112703
\(808\) 15.6180 0.549441
\(809\) −2.65248 −0.0932561 −0.0466280 0.998912i \(-0.514848\pi\)
−0.0466280 + 0.998912i \(0.514848\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −4.58359 −0.160952 −0.0804758 0.996757i \(-0.525644\pi\)
−0.0804758 + 0.996757i \(0.525644\pi\)
\(812\) 8.47214 0.297314
\(813\) 6.38197 0.223825
\(814\) 0 0
\(815\) −0.854102 −0.0299179
\(816\) 6.09017 0.213199
\(817\) −0.111456 −0.00389936
\(818\) 12.9787 0.453790
\(819\) −10.9443 −0.382424
\(820\) −1.23607 −0.0431654
\(821\) −43.3050 −1.51135 −0.755677 0.654945i \(-0.772692\pi\)
−0.755677 + 0.654945i \(0.772692\pi\)
\(822\) 12.7984 0.446395
\(823\) −12.7639 −0.444923 −0.222461 0.974942i \(-0.571409\pi\)
−0.222461 + 0.974942i \(0.571409\pi\)
\(824\) 6.76393 0.235633
\(825\) 0 0
\(826\) −45.5967 −1.58651
\(827\) 30.2918 1.05335 0.526674 0.850067i \(-0.323439\pi\)
0.526674 + 0.850067i \(0.323439\pi\)
\(828\) 6.85410 0.238197
\(829\) 35.5967 1.23633 0.618163 0.786050i \(-0.287877\pi\)
0.618163 + 0.786050i \(0.287877\pi\)
\(830\) −4.00000 −0.138842
\(831\) 23.5066 0.815434
\(832\) 3.38197 0.117249
\(833\) −21.1459 −0.732662
\(834\) −8.76393 −0.303470
\(835\) 13.2705 0.459245
\(836\) 0 0
\(837\) 1.38197 0.0477677
\(838\) 22.8541 0.789482
\(839\) −23.3475 −0.806046 −0.403023 0.915190i \(-0.632041\pi\)
−0.403023 + 0.915190i \(0.632041\pi\)
\(840\) −3.23607 −0.111655
\(841\) −22.1459 −0.763652
\(842\) −2.47214 −0.0851954
\(843\) 7.52786 0.259273
\(844\) −25.2361 −0.868661
\(845\) 1.56231 0.0537450
\(846\) −4.32624 −0.148739
\(847\) 0 0
\(848\) 8.47214 0.290934
\(849\) 22.3262 0.766235
\(850\) −6.09017 −0.208891
\(851\) −43.7426 −1.49948
\(852\) −10.4721 −0.358769
\(853\) −27.5279 −0.942536 −0.471268 0.881990i \(-0.656204\pi\)
−0.471268 + 0.881990i \(0.656204\pi\)
\(854\) 0 0
\(855\) −1.23607 −0.0422726
\(856\) 6.29180 0.215049
\(857\) 24.1591 0.825258 0.412629 0.910899i \(-0.364611\pi\)
0.412629 + 0.910899i \(0.364611\pi\)
\(858\) 0 0
\(859\) −19.1246 −0.652523 −0.326262 0.945279i \(-0.605789\pi\)
−0.326262 + 0.945279i \(0.605789\pi\)
\(860\) 0.0901699 0.00307477
\(861\) 4.00000 0.136320
\(862\) 22.3607 0.761608
\(863\) −44.3394 −1.50933 −0.754665 0.656110i \(-0.772201\pi\)
−0.754665 + 0.656110i \(0.772201\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 9.52786 0.323957
\(866\) −22.8328 −0.775891
\(867\) −20.0902 −0.682298
\(868\) 4.47214 0.151794
\(869\) 0 0
\(870\) −2.61803 −0.0887597
\(871\) −17.2148 −0.583301
\(872\) −2.47214 −0.0837171
\(873\) 18.6525 0.631291
\(874\) 8.47214 0.286574
\(875\) 3.23607 0.109399
\(876\) −13.4164 −0.453298
\(877\) 18.9656 0.640421 0.320211 0.947346i \(-0.396246\pi\)
0.320211 + 0.947346i \(0.396246\pi\)
\(878\) 1.67376 0.0564867
\(879\) 20.0000 0.674583
\(880\) 0 0
\(881\) 8.00000 0.269527 0.134763 0.990878i \(-0.456973\pi\)
0.134763 + 0.990878i \(0.456973\pi\)
\(882\) 3.47214 0.116913
\(883\) −4.67376 −0.157285 −0.0786423 0.996903i \(-0.525058\pi\)
−0.0786423 + 0.996903i \(0.525058\pi\)
\(884\) −20.5967 −0.692744
\(885\) 14.0902 0.473636
\(886\) −18.6525 −0.626642
\(887\) −42.4508 −1.42536 −0.712680 0.701489i \(-0.752519\pi\)
−0.712680 + 0.701489i \(0.752519\pi\)
\(888\) 6.38197 0.214165
\(889\) −47.4164 −1.59030
\(890\) −17.2361 −0.577754
\(891\) 0 0
\(892\) 7.23607 0.242281
\(893\) −5.34752 −0.178948
\(894\) 7.61803 0.254785
\(895\) −18.1459 −0.606550
\(896\) −3.23607 −0.108109
\(897\) −23.1803 −0.773969
\(898\) −11.1246 −0.371233
\(899\) 3.61803 0.120668
\(900\) 1.00000 0.0333333
\(901\) −51.5967 −1.71894
\(902\) 0 0
\(903\) −0.291796 −0.00971037
\(904\) −19.5623 −0.650633
\(905\) −21.8885 −0.727600
\(906\) −20.9443 −0.695827
\(907\) −23.5066 −0.780523 −0.390262 0.920704i \(-0.627615\pi\)
−0.390262 + 0.920704i \(0.627615\pi\)
\(908\) 20.3607 0.675693
\(909\) 15.6180 0.518017
\(910\) 10.9443 0.362799
\(911\) 3.52786 0.116883 0.0584417 0.998291i \(-0.481387\pi\)
0.0584417 + 0.998291i \(0.481387\pi\)
\(912\) −1.23607 −0.0409303
\(913\) 0 0
\(914\) 4.65248 0.153890
\(915\) 0 0
\(916\) 0.763932 0.0252410
\(917\) 10.9443 0.361412
\(918\) 6.09017 0.201006
\(919\) 46.3394 1.52860 0.764298 0.644863i \(-0.223086\pi\)
0.764298 + 0.644863i \(0.223086\pi\)
\(920\) −6.85410 −0.225973
\(921\) 33.4508 1.10224
\(922\) 23.2148 0.764538
\(923\) 35.4164 1.16575
\(924\) 0 0
\(925\) −6.38197 −0.209838
\(926\) 11.8885 0.390682
\(927\) 6.76393 0.222157
\(928\) −2.61803 −0.0859412
\(929\) −30.4721 −0.999758 −0.499879 0.866095i \(-0.666622\pi\)
−0.499879 + 0.866095i \(0.666622\pi\)
\(930\) −1.38197 −0.0453165
\(931\) 4.29180 0.140658
\(932\) −4.43769 −0.145362
\(933\) −9.23607 −0.302375
\(934\) 28.5410 0.933891
\(935\) 0 0
\(936\) 3.38197 0.110543
\(937\) −42.6525 −1.39340 −0.696698 0.717365i \(-0.745348\pi\)
−0.696698 + 0.717365i \(0.745348\pi\)
\(938\) 16.4721 0.537834
\(939\) −8.47214 −0.276478
\(940\) 4.32624 0.141106
\(941\) 1.02129 0.0332930 0.0166465 0.999861i \(-0.494701\pi\)
0.0166465 + 0.999861i \(0.494701\pi\)
\(942\) 9.27051 0.302050
\(943\) 8.47214 0.275891
\(944\) 14.0902 0.458596
\(945\) −3.23607 −0.105269
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 4.09017 0.132843
\(949\) 45.3738 1.47290
\(950\) 1.23607 0.0401033
\(951\) 27.4164 0.889038
\(952\) 19.7082 0.638747
\(953\) −59.2148 −1.91815 −0.959077 0.283144i \(-0.908623\pi\)
−0.959077 + 0.283144i \(0.908623\pi\)
\(954\) 8.47214 0.274296
\(955\) −9.23607 −0.298872
\(956\) −3.23607 −0.104662
\(957\) 0 0
\(958\) 12.7639 0.412384
\(959\) 41.4164 1.33741
\(960\) 1.00000 0.0322749
\(961\) −29.0902 −0.938393
\(962\) −21.5836 −0.695883
\(963\) 6.29180 0.202750
\(964\) −23.8885 −0.769398
\(965\) 20.9443 0.674220
\(966\) 22.1803 0.713641
\(967\) 53.3050 1.71417 0.857086 0.515174i \(-0.172273\pi\)
0.857086 + 0.515174i \(0.172273\pi\)
\(968\) 0 0
\(969\) 7.52786 0.241830
\(970\) −18.6525 −0.598895
\(971\) 8.36068 0.268307 0.134153 0.990961i \(-0.457168\pi\)
0.134153 + 0.990961i \(0.457168\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −28.3607 −0.909202
\(974\) 11.4164 0.365805
\(975\) −3.38197 −0.108310
\(976\) 0 0
\(977\) 28.8328 0.922443 0.461222 0.887285i \(-0.347411\pi\)
0.461222 + 0.887285i \(0.347411\pi\)
\(978\) −0.854102 −0.0273112
\(979\) 0 0
\(980\) −3.47214 −0.110913
\(981\) −2.47214 −0.0789292
\(982\) 28.0344 0.894615
\(983\) −51.1935 −1.63282 −0.816409 0.577473i \(-0.804039\pi\)
−0.816409 + 0.577473i \(0.804039\pi\)
\(984\) −1.23607 −0.0394044
\(985\) −10.9443 −0.348713
\(986\) 15.9443 0.507769
\(987\) −14.0000 −0.445625
\(988\) 4.18034 0.132994
\(989\) −0.618034 −0.0196523
\(990\) 0 0
\(991\) 54.5197 1.73188 0.865938 0.500151i \(-0.166722\pi\)
0.865938 + 0.500151i \(0.166722\pi\)
\(992\) −1.38197 −0.0438775
\(993\) −9.23607 −0.293098
\(994\) −33.8885 −1.07488
\(995\) −25.9787 −0.823581
\(996\) −4.00000 −0.126745
\(997\) −25.0344 −0.792849 −0.396424 0.918067i \(-0.629749\pi\)
−0.396424 + 0.918067i \(0.629749\pi\)
\(998\) −32.4721 −1.02789
\(999\) 6.38197 0.201916
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 3630.2.a.bj.1.1 2
11.5 even 5 330.2.m.b.91.1 4
11.9 even 5 330.2.m.b.301.1 yes 4
11.10 odd 2 3630.2.a.bb.1.2 2
33.5 odd 10 990.2.n.e.91.1 4
33.20 odd 10 990.2.n.e.631.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
330.2.m.b.91.1 4 11.5 even 5
330.2.m.b.301.1 yes 4 11.9 even 5
990.2.n.e.91.1 4 33.5 odd 10
990.2.n.e.631.1 4 33.20 odd 10
3630.2.a.bb.1.2 2 11.10 odd 2
3630.2.a.bj.1.1 2 1.1 even 1 trivial