Properties

Label 2166.2.a.u.1.2
Level $2166$
Weight $2$
Character 2166.1
Self dual yes
Analytic conductor $17.296$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.2955970778\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.53209\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.18479 q^{5} +1.00000 q^{6} +0.532089 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.18479 q^{5} +1.00000 q^{6} +0.532089 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.18479 q^{10} +1.87939 q^{11} +1.00000 q^{12} +3.87939 q^{13} +0.532089 q^{14} +1.18479 q^{15} +1.00000 q^{16} +1.16250 q^{17} +1.00000 q^{18} +1.18479 q^{20} +0.532089 q^{21} +1.87939 q^{22} -6.70233 q^{23} +1.00000 q^{24} -3.59627 q^{25} +3.87939 q^{26} +1.00000 q^{27} +0.532089 q^{28} +4.02229 q^{29} +1.18479 q^{30} +1.95811 q^{31} +1.00000 q^{32} +1.87939 q^{33} +1.16250 q^{34} +0.630415 q^{35} +1.00000 q^{36} -6.88713 q^{37} +3.87939 q^{39} +1.18479 q^{40} +8.98545 q^{41} +0.532089 q^{42} +2.42602 q^{43} +1.87939 q^{44} +1.18479 q^{45} -6.70233 q^{46} +2.04189 q^{47} +1.00000 q^{48} -6.71688 q^{49} -3.59627 q^{50} +1.16250 q^{51} +3.87939 q^{52} -12.9709 q^{53} +1.00000 q^{54} +2.22668 q^{55} +0.532089 q^{56} +4.02229 q^{58} +2.68004 q^{59} +1.18479 q^{60} -11.3473 q^{61} +1.95811 q^{62} +0.532089 q^{63} +1.00000 q^{64} +4.59627 q^{65} +1.87939 q^{66} +11.1702 q^{67} +1.16250 q^{68} -6.70233 q^{69} +0.630415 q^{70} -6.07192 q^{71} +1.00000 q^{72} -0.327696 q^{73} -6.88713 q^{74} -3.59627 q^{75} +1.00000 q^{77} +3.87939 q^{78} +16.1334 q^{79} +1.18479 q^{80} +1.00000 q^{81} +8.98545 q^{82} +11.5740 q^{83} +0.532089 q^{84} +1.37733 q^{85} +2.42602 q^{86} +4.02229 q^{87} +1.87939 q^{88} +3.55943 q^{89} +1.18479 q^{90} +2.06418 q^{91} -6.70233 q^{92} +1.95811 q^{93} +2.04189 q^{94} +1.00000 q^{96} -5.87939 q^{97} -6.71688 q^{98} +1.87939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + 3q^{12} + 6q^{13} - 3q^{14} + 3q^{16} + 6q^{17} + 3q^{18} - 3q^{21} + 6q^{23} + 3q^{24} + 3q^{25} + 6q^{26} + 3q^{27} - 3q^{28} + 6q^{29} + 9q^{31} + 3q^{32} + 6q^{34} + 9q^{35} + 3q^{36} + 9q^{37} + 6q^{39} + 9q^{41} - 3q^{42} + 15q^{43} + 6q^{46} + 3q^{47} + 3q^{48} - 12q^{49} + 3q^{50} + 6q^{51} + 6q^{52} - 3q^{53} + 3q^{54} - 3q^{56} + 6q^{58} - 12q^{59} - 33q^{61} + 9q^{62} - 3q^{63} + 3q^{64} + 12q^{67} + 6q^{68} + 6q^{69} + 9q^{70} + 15q^{71} + 3q^{72} + 3q^{73} + 9q^{74} + 3q^{75} + 3q^{77} + 6q^{78} + 15q^{79} + 3q^{81} + 9q^{82} + 27q^{83} - 3q^{84} - 27q^{85} + 15q^{86} + 6q^{87} - 15q^{89} - 3q^{91} + 6q^{92} + 9q^{93} + 3q^{94} + 3q^{96} - 12q^{97} - 12q^{98} + O(q^{100}) \)

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
\(5\) 1.18479 0.529855 0.264928 0.964268i \(-0.414652\pi\)
0.264928 + 0.964268i \(0.414652\pi\)
\(6\) 1.00000 0.408248
\(7\) 0.532089 0.201111 0.100555 0.994931i \(-0.467938\pi\)
0.100555 + 0.994931i \(0.467938\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.18479 0.374664
\(11\) 1.87939 0.566656 0.283328 0.959023i \(-0.408561\pi\)
0.283328 + 0.959023i \(0.408561\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.87939 1.07595 0.537974 0.842961i \(-0.319190\pi\)
0.537974 + 0.842961i \(0.319190\pi\)
\(14\) 0.532089 0.142207
\(15\) 1.18479 0.305912
\(16\) 1.00000 0.250000
\(17\) 1.16250 0.281949 0.140974 0.990013i \(-0.454977\pi\)
0.140974 + 0.990013i \(0.454977\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0
\(20\) 1.18479 0.264928
\(21\) 0.532089 0.116111
\(22\) 1.87939 0.400686
\(23\) −6.70233 −1.39753 −0.698767 0.715350i \(-0.746267\pi\)
−0.698767 + 0.715350i \(0.746267\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.59627 −0.719253
\(26\) 3.87939 0.760810
\(27\) 1.00000 0.192450
\(28\) 0.532089 0.100555
\(29\) 4.02229 0.746920 0.373460 0.927646i \(-0.378171\pi\)
0.373460 + 0.927646i \(0.378171\pi\)
\(30\) 1.18479 0.216313
\(31\) 1.95811 0.351687 0.175844 0.984418i \(-0.443735\pi\)
0.175844 + 0.984418i \(0.443735\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.87939 0.327159
\(34\) 1.16250 0.199368
\(35\) 0.630415 0.106560
\(36\) 1.00000 0.166667
\(37\) −6.88713 −1.13224 −0.566118 0.824324i \(-0.691555\pi\)
−0.566118 + 0.824324i \(0.691555\pi\)
\(38\) 0 0
\(39\) 3.87939 0.621199
\(40\) 1.18479 0.187332
\(41\) 8.98545 1.40329 0.701646 0.712526i \(-0.252449\pi\)
0.701646 + 0.712526i \(0.252449\pi\)
\(42\) 0.532089 0.0821031
\(43\) 2.42602 0.369965 0.184982 0.982742i \(-0.440777\pi\)
0.184982 + 0.982742i \(0.440777\pi\)
\(44\) 1.87939 0.283328
\(45\) 1.18479 0.176618
\(46\) −6.70233 −0.988205
\(47\) 2.04189 0.297840 0.148920 0.988849i \(-0.452420\pi\)
0.148920 + 0.988849i \(0.452420\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.71688 −0.959554
\(50\) −3.59627 −0.508589
\(51\) 1.16250 0.162783
\(52\) 3.87939 0.537974
\(53\) −12.9709 −1.78169 −0.890845 0.454307i \(-0.849887\pi\)
−0.890845 + 0.454307i \(0.849887\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.22668 0.300246
\(56\) 0.532089 0.0711034
\(57\) 0 0
\(58\) 4.02229 0.528152
\(59\) 2.68004 0.348912 0.174456 0.984665i \(-0.444183\pi\)
0.174456 + 0.984665i \(0.444183\pi\)
\(60\) 1.18479 0.152956
\(61\) −11.3473 −1.45287 −0.726436 0.687234i \(-0.758825\pi\)
−0.726436 + 0.687234i \(0.758825\pi\)
\(62\) 1.95811 0.248680
\(63\) 0.532089 0.0670369
\(64\) 1.00000 0.125000
\(65\) 4.59627 0.570097
\(66\) 1.87939 0.231336
\(67\) 11.1702 1.36466 0.682331 0.731043i \(-0.260966\pi\)
0.682331 + 0.731043i \(0.260966\pi\)
\(68\) 1.16250 0.140974
\(69\) −6.70233 −0.806866
\(70\) 0.630415 0.0753490
\(71\) −6.07192 −0.720604 −0.360302 0.932836i \(-0.617326\pi\)
−0.360302 + 0.932836i \(0.617326\pi\)
\(72\) 1.00000 0.117851
\(73\) −0.327696 −0.0383539 −0.0191770 0.999816i \(-0.506105\pi\)
−0.0191770 + 0.999816i \(0.506105\pi\)
\(74\) −6.88713 −0.800612
\(75\) −3.59627 −0.415261
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 3.87939 0.439254
\(79\) 16.1334 1.81515 0.907575 0.419890i \(-0.137931\pi\)
0.907575 + 0.419890i \(0.137931\pi\)
\(80\) 1.18479 0.132464
\(81\) 1.00000 0.111111
\(82\) 8.98545 0.992277
\(83\) 11.5740 1.27041 0.635205 0.772344i \(-0.280916\pi\)
0.635205 + 0.772344i \(0.280916\pi\)
\(84\) 0.532089 0.0580557
\(85\) 1.37733 0.149392
\(86\) 2.42602 0.261605
\(87\) 4.02229 0.431235
\(88\) 1.87939 0.200343
\(89\) 3.55943 0.377299 0.188649 0.982044i \(-0.439589\pi\)
0.188649 + 0.982044i \(0.439589\pi\)
\(90\) 1.18479 0.124888
\(91\) 2.06418 0.216385
\(92\) −6.70233 −0.698767
\(93\) 1.95811 0.203047
\(94\) 2.04189 0.210605
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −5.87939 −0.596961 −0.298481 0.954416i \(-0.596480\pi\)
−0.298481 + 0.954416i \(0.596480\pi\)
\(98\) −6.71688 −0.678507
\(99\) 1.87939 0.188885
\(100\) −3.59627 −0.359627
\(101\) 12.6459 1.25831 0.629157 0.777278i \(-0.283400\pi\)
0.629157 + 0.777278i \(0.283400\pi\)
\(102\) 1.16250 0.115105
\(103\) −2.96316 −0.291969 −0.145985 0.989287i \(-0.546635\pi\)
−0.145985 + 0.989287i \(0.546635\pi\)
\(104\) 3.87939 0.380405
\(105\) 0.630415 0.0615222
\(106\) −12.9709 −1.25985
\(107\) −19.1138 −1.84780 −0.923901 0.382632i \(-0.875018\pi\)
−0.923901 + 0.382632i \(0.875018\pi\)
\(108\) 1.00000 0.0962250
\(109\) 16.4115 1.57193 0.785967 0.618268i \(-0.212166\pi\)
0.785967 + 0.618268i \(0.212166\pi\)
\(110\) 2.22668 0.212306
\(111\) −6.88713 −0.653697
\(112\) 0.532089 0.0502777
\(113\) 5.73648 0.539643 0.269821 0.962910i \(-0.413035\pi\)
0.269821 + 0.962910i \(0.413035\pi\)
\(114\) 0 0
\(115\) −7.94087 −0.740490
\(116\) 4.02229 0.373460
\(117\) 3.87939 0.358649
\(118\) 2.68004 0.246718
\(119\) 0.618555 0.0567029
\(120\) 1.18479 0.108156
\(121\) −7.46791 −0.678901
\(122\) −11.3473 −1.02734
\(123\) 8.98545 0.810191
\(124\) 1.95811 0.175844
\(125\) −10.1848 −0.910956
\(126\) 0.532089 0.0474022
\(127\) 1.88713 0.167455 0.0837277 0.996489i \(-0.473317\pi\)
0.0837277 + 0.996489i \(0.473317\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.42602 0.213599
\(130\) 4.59627 0.403119
\(131\) 10.7023 0.935067 0.467534 0.883975i \(-0.345143\pi\)
0.467534 + 0.883975i \(0.345143\pi\)
\(132\) 1.87939 0.163579
\(133\) 0 0
\(134\) 11.1702 0.964962
\(135\) 1.18479 0.101971
\(136\) 1.16250 0.0996839
\(137\) −11.2567 −0.961726 −0.480863 0.876796i \(-0.659677\pi\)
−0.480863 + 0.876796i \(0.659677\pi\)
\(138\) −6.70233 −0.570541
\(139\) −12.1702 −1.03227 −0.516133 0.856508i \(-0.672629\pi\)
−0.516133 + 0.856508i \(0.672629\pi\)
\(140\) 0.630415 0.0532798
\(141\) 2.04189 0.171958
\(142\) −6.07192 −0.509544
\(143\) 7.29086 0.609692
\(144\) 1.00000 0.0833333
\(145\) 4.76558 0.395760
\(146\) −0.327696 −0.0271203
\(147\) −6.71688 −0.553999
\(148\) −6.88713 −0.566118
\(149\) −16.3550 −1.33986 −0.669928 0.742426i \(-0.733675\pi\)
−0.669928 + 0.742426i \(0.733675\pi\)
\(150\) −3.59627 −0.293634
\(151\) −20.5226 −1.67010 −0.835052 0.550170i \(-0.814563\pi\)
−0.835052 + 0.550170i \(0.814563\pi\)
\(152\) 0 0
\(153\) 1.16250 0.0939829
\(154\) 1.00000 0.0805823
\(155\) 2.31996 0.186343
\(156\) 3.87939 0.310599
\(157\) −3.19934 −0.255335 −0.127668 0.991817i \(-0.540749\pi\)
−0.127668 + 0.991817i \(0.540749\pi\)
\(158\) 16.1334 1.28351
\(159\) −12.9709 −1.02866
\(160\) 1.18479 0.0936661
\(161\) −3.56624 −0.281059
\(162\) 1.00000 0.0785674
\(163\) −24.0077 −1.88043 −0.940216 0.340580i \(-0.889377\pi\)
−0.940216 + 0.340580i \(0.889377\pi\)
\(164\) 8.98545 0.701646
\(165\) 2.22668 0.173347
\(166\) 11.5740 0.898315
\(167\) −3.66044 −0.283254 −0.141627 0.989920i \(-0.545233\pi\)
−0.141627 + 0.989920i \(0.545233\pi\)
\(168\) 0.532089 0.0410515
\(169\) 2.04963 0.157664
\(170\) 1.37733 0.105636
\(171\) 0 0
\(172\) 2.42602 0.184982
\(173\) 5.33275 0.405441 0.202721 0.979237i \(-0.435022\pi\)
0.202721 + 0.979237i \(0.435022\pi\)
\(174\) 4.02229 0.304929
\(175\) −1.91353 −0.144650
\(176\) 1.87939 0.141664
\(177\) 2.68004 0.201445
\(178\) 3.55943 0.266791
\(179\) 25.2567 1.88778 0.943888 0.330267i \(-0.107139\pi\)
0.943888 + 0.330267i \(0.107139\pi\)
\(180\) 1.18479 0.0883092
\(181\) 17.3063 1.28637 0.643185 0.765711i \(-0.277613\pi\)
0.643185 + 0.765711i \(0.277613\pi\)
\(182\) 2.06418 0.153007
\(183\) −11.3473 −0.838816
\(184\) −6.70233 −0.494103
\(185\) −8.15982 −0.599922
\(186\) 1.95811 0.143576
\(187\) 2.18479 0.159768
\(188\) 2.04189 0.148920
\(189\) 0.532089 0.0387038
\(190\) 0 0
\(191\) −15.1780 −1.09824 −0.549120 0.835743i \(-0.685037\pi\)
−0.549120 + 0.835743i \(0.685037\pi\)
\(192\) 1.00000 0.0721688
\(193\) −13.0077 −0.936318 −0.468159 0.883644i \(-0.655082\pi\)
−0.468159 + 0.883644i \(0.655082\pi\)
\(194\) −5.87939 −0.422115
\(195\) 4.59627 0.329145
\(196\) −6.71688 −0.479777
\(197\) −8.94862 −0.637562 −0.318781 0.947828i \(-0.603274\pi\)
−0.318781 + 0.947828i \(0.603274\pi\)
\(198\) 1.87939 0.133562
\(199\) −4.50475 −0.319333 −0.159667 0.987171i \(-0.551042\pi\)
−0.159667 + 0.987171i \(0.551042\pi\)
\(200\) −3.59627 −0.254294
\(201\) 11.1702 0.787888
\(202\) 12.6459 0.889762
\(203\) 2.14022 0.150214
\(204\) 1.16250 0.0813915
\(205\) 10.6459 0.743542
\(206\) −2.96316 −0.206453
\(207\) −6.70233 −0.465844
\(208\) 3.87939 0.268987
\(209\) 0 0
\(210\) 0.630415 0.0435028
\(211\) 6.40373 0.440851 0.220426 0.975404i \(-0.429255\pi\)
0.220426 + 0.975404i \(0.429255\pi\)
\(212\) −12.9709 −0.890845
\(213\) −6.07192 −0.416041
\(214\) −19.1138 −1.30659
\(215\) 2.87433 0.196028
\(216\) 1.00000 0.0680414
\(217\) 1.04189 0.0707280
\(218\) 16.4115 1.11153
\(219\) −0.327696 −0.0221436
\(220\) 2.22668 0.150123
\(221\) 4.50980 0.303362
\(222\) −6.88713 −0.462234
\(223\) −3.24897 −0.217567 −0.108784 0.994065i \(-0.534696\pi\)
−0.108784 + 0.994065i \(0.534696\pi\)
\(224\) 0.532089 0.0355517
\(225\) −3.59627 −0.239751
\(226\) 5.73648 0.381585
\(227\) 11.7023 0.776711 0.388356 0.921510i \(-0.373043\pi\)
0.388356 + 0.921510i \(0.373043\pi\)
\(228\) 0 0
\(229\) −22.9067 −1.51372 −0.756860 0.653578i \(-0.773267\pi\)
−0.756860 + 0.653578i \(0.773267\pi\)
\(230\) −7.94087 −0.523606
\(231\) 1.00000 0.0657952
\(232\) 4.02229 0.264076
\(233\) −3.19934 −0.209596 −0.104798 0.994494i \(-0.533420\pi\)
−0.104798 + 0.994494i \(0.533420\pi\)
\(234\) 3.87939 0.253603
\(235\) 2.41921 0.157812
\(236\) 2.68004 0.174456
\(237\) 16.1334 1.04798
\(238\) 0.618555 0.0400950
\(239\) 6.17293 0.399294 0.199647 0.979868i \(-0.436020\pi\)
0.199647 + 0.979868i \(0.436020\pi\)
\(240\) 1.18479 0.0764780
\(241\) −1.53714 −0.0990160 −0.0495080 0.998774i \(-0.515765\pi\)
−0.0495080 + 0.998774i \(0.515765\pi\)
\(242\) −7.46791 −0.480056
\(243\) 1.00000 0.0641500
\(244\) −11.3473 −0.726436
\(245\) −7.95811 −0.508425
\(246\) 8.98545 0.572892
\(247\) 0 0
\(248\) 1.95811 0.124340
\(249\) 11.5740 0.733471
\(250\) −10.1848 −0.644143
\(251\) 17.9436 1.13259 0.566294 0.824203i \(-0.308377\pi\)
0.566294 + 0.824203i \(0.308377\pi\)
\(252\) 0.532089 0.0335184
\(253\) −12.5963 −0.791920
\(254\) 1.88713 0.118409
\(255\) 1.37733 0.0862515
\(256\) 1.00000 0.0625000
\(257\) −29.4124 −1.83470 −0.917348 0.398087i \(-0.869674\pi\)
−0.917348 + 0.398087i \(0.869674\pi\)
\(258\) 2.42602 0.151038
\(259\) −3.66456 −0.227705
\(260\) 4.59627 0.285048
\(261\) 4.02229 0.248973
\(262\) 10.7023 0.661192
\(263\) 10.6827 0.658726 0.329363 0.944203i \(-0.393166\pi\)
0.329363 + 0.944203i \(0.393166\pi\)
\(264\) 1.87939 0.115668
\(265\) −15.3678 −0.944038
\(266\) 0 0
\(267\) 3.55943 0.217834
\(268\) 11.1702 0.682331
\(269\) 20.5767 1.25458 0.627291 0.778785i \(-0.284164\pi\)
0.627291 + 0.778785i \(0.284164\pi\)
\(270\) 1.18479 0.0721042
\(271\) 1.87670 0.114001 0.0570006 0.998374i \(-0.481846\pi\)
0.0570006 + 0.998374i \(0.481846\pi\)
\(272\) 1.16250 0.0704871
\(273\) 2.06418 0.124930
\(274\) −11.2567 −0.680043
\(275\) −6.75877 −0.407569
\(276\) −6.70233 −0.403433
\(277\) 4.87258 0.292765 0.146382 0.989228i \(-0.453237\pi\)
0.146382 + 0.989228i \(0.453237\pi\)
\(278\) −12.1702 −0.729923
\(279\) 1.95811 0.117229
\(280\) 0.630415 0.0376745
\(281\) 0.403733 0.0240847 0.0120424 0.999927i \(-0.496167\pi\)
0.0120424 + 0.999927i \(0.496167\pi\)
\(282\) 2.04189 0.121593
\(283\) 17.5895 1.04558 0.522792 0.852460i \(-0.324890\pi\)
0.522792 + 0.852460i \(0.324890\pi\)
\(284\) −6.07192 −0.360302
\(285\) 0 0
\(286\) 7.29086 0.431118
\(287\) 4.78106 0.282217
\(288\) 1.00000 0.0589256
\(289\) −15.6486 −0.920505
\(290\) 4.76558 0.279844
\(291\) −5.87939 −0.344656
\(292\) −0.327696 −0.0191770
\(293\) 13.1753 0.769709 0.384855 0.922977i \(-0.374252\pi\)
0.384855 + 0.922977i \(0.374252\pi\)
\(294\) −6.71688 −0.391736
\(295\) 3.17530 0.184873
\(296\) −6.88713 −0.400306
\(297\) 1.87939 0.109053
\(298\) −16.3550 −0.947422
\(299\) −26.0009 −1.50367
\(300\) −3.59627 −0.207631
\(301\) 1.29086 0.0744039
\(302\) −20.5226 −1.18094
\(303\) 12.6459 0.726488
\(304\) 0 0
\(305\) −13.4442 −0.769812
\(306\) 1.16250 0.0664559
\(307\) −16.9017 −0.964629 −0.482315 0.875998i \(-0.660204\pi\)
−0.482315 + 0.875998i \(0.660204\pi\)
\(308\) 1.00000 0.0569803
\(309\) −2.96316 −0.168568
\(310\) 2.31996 0.131765
\(311\) −2.38144 −0.135039 −0.0675197 0.997718i \(-0.521509\pi\)
−0.0675197 + 0.997718i \(0.521509\pi\)
\(312\) 3.87939 0.219627
\(313\) −29.8631 −1.68796 −0.843981 0.536374i \(-0.819794\pi\)
−0.843981 + 0.536374i \(0.819794\pi\)
\(314\) −3.19934 −0.180549
\(315\) 0.630415 0.0355199
\(316\) 16.1334 0.907575
\(317\) 7.02229 0.394411 0.197206 0.980362i \(-0.436813\pi\)
0.197206 + 0.980362i \(0.436813\pi\)
\(318\) −12.9709 −0.727372
\(319\) 7.55943 0.423247
\(320\) 1.18479 0.0662319
\(321\) −19.1138 −1.06683
\(322\) −3.56624 −0.198739
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −13.9513 −0.773879
\(326\) −24.0077 −1.32967
\(327\) 16.4115 0.907557
\(328\) 8.98545 0.496139
\(329\) 1.08647 0.0598988
\(330\) 2.22668 0.122575
\(331\) −27.1985 −1.49497 −0.747483 0.664281i \(-0.768738\pi\)
−0.747483 + 0.664281i \(0.768738\pi\)
\(332\) 11.5740 0.635205
\(333\) −6.88713 −0.377412
\(334\) −3.66044 −0.200291
\(335\) 13.2344 0.723074
\(336\) 0.532089 0.0290278
\(337\) 11.2172 0.611039 0.305520 0.952186i \(-0.401170\pi\)
0.305520 + 0.952186i \(0.401170\pi\)
\(338\) 2.04963 0.111485
\(339\) 5.73648 0.311563
\(340\) 1.37733 0.0746960
\(341\) 3.68004 0.199286
\(342\) 0 0
\(343\) −7.29860 −0.394087
\(344\) 2.42602 0.130802
\(345\) −7.94087 −0.427522
\(346\) 5.33275 0.286690
\(347\) 1.93582 0.103920 0.0519602 0.998649i \(-0.483453\pi\)
0.0519602 + 0.998649i \(0.483453\pi\)
\(348\) 4.02229 0.215617
\(349\) 14.3969 0.770650 0.385325 0.922781i \(-0.374089\pi\)
0.385325 + 0.922781i \(0.374089\pi\)
\(350\) −1.91353 −0.102283
\(351\) 3.87939 0.207066
\(352\) 1.87939 0.100172
\(353\) 15.7237 0.836887 0.418444 0.908243i \(-0.362576\pi\)
0.418444 + 0.908243i \(0.362576\pi\)
\(354\) 2.68004 0.142443
\(355\) −7.19396 −0.381816
\(356\) 3.55943 0.188649
\(357\) 0.618555 0.0327374
\(358\) 25.2567 1.33486
\(359\) 15.5175 0.818984 0.409492 0.912314i \(-0.365706\pi\)
0.409492 + 0.912314i \(0.365706\pi\)
\(360\) 1.18479 0.0624440
\(361\) 0 0
\(362\) 17.3063 0.909601
\(363\) −7.46791 −0.391964
\(364\) 2.06418 0.108192
\(365\) −0.388252 −0.0203220
\(366\) −11.3473 −0.593133
\(367\) −15.7442 −0.821842 −0.410921 0.911671i \(-0.634793\pi\)
−0.410921 + 0.911671i \(0.634793\pi\)
\(368\) −6.70233 −0.349383
\(369\) 8.98545 0.467764
\(370\) −8.15982 −0.424209
\(371\) −6.90167 −0.358317
\(372\) 1.95811 0.101523
\(373\) 29.6287 1.53411 0.767057 0.641579i \(-0.221720\pi\)
0.767057 + 0.641579i \(0.221720\pi\)
\(374\) 2.18479 0.112973
\(375\) −10.1848 −0.525940
\(376\) 2.04189 0.105302
\(377\) 15.6040 0.803647
\(378\) 0.532089 0.0273677
\(379\) −4.72462 −0.242688 −0.121344 0.992611i \(-0.538720\pi\)
−0.121344 + 0.992611i \(0.538720\pi\)
\(380\) 0 0
\(381\) 1.88713 0.0966804
\(382\) −15.1780 −0.776573
\(383\) 5.24123 0.267814 0.133907 0.990994i \(-0.457248\pi\)
0.133907 + 0.990994i \(0.457248\pi\)
\(384\) 1.00000 0.0510310
\(385\) 1.18479 0.0603826
\(386\) −13.0077 −0.662077
\(387\) 2.42602 0.123322
\(388\) −5.87939 −0.298481
\(389\) −15.0770 −0.764433 −0.382216 0.924073i \(-0.624839\pi\)
−0.382216 + 0.924073i \(0.624839\pi\)
\(390\) 4.59627 0.232741
\(391\) −7.79149 −0.394033
\(392\) −6.71688 −0.339254
\(393\) 10.7023 0.539861
\(394\) −8.94862 −0.450825
\(395\) 19.1147 0.961767
\(396\) 1.87939 0.0944427
\(397\) −14.5449 −0.729987 −0.364993 0.931010i \(-0.618929\pi\)
−0.364993 + 0.931010i \(0.618929\pi\)
\(398\) −4.50475 −0.225803
\(399\) 0 0
\(400\) −3.59627 −0.179813
\(401\) −2.85473 −0.142559 −0.0712793 0.997456i \(-0.522708\pi\)
−0.0712793 + 0.997456i \(0.522708\pi\)
\(402\) 11.1702 0.557121
\(403\) 7.59627 0.378397
\(404\) 12.6459 0.629157
\(405\) 1.18479 0.0588728
\(406\) 2.14022 0.106217
\(407\) −12.9436 −0.641589
\(408\) 1.16250 0.0575525
\(409\) −34.2276 −1.69245 −0.846223 0.532828i \(-0.821129\pi\)
−0.846223 + 0.532828i \(0.821129\pi\)
\(410\) 10.6459 0.525763
\(411\) −11.2567 −0.555253
\(412\) −2.96316 −0.145985
\(413\) 1.42602 0.0701700
\(414\) −6.70233 −0.329402
\(415\) 13.7128 0.673133
\(416\) 3.87939 0.190203
\(417\) −12.1702 −0.595979
\(418\) 0 0
\(419\) −7.91891 −0.386864 −0.193432 0.981114i \(-0.561962\pi\)
−0.193432 + 0.981114i \(0.561962\pi\)
\(420\) 0.630415 0.0307611
\(421\) −16.2098 −0.790016 −0.395008 0.918678i \(-0.629258\pi\)
−0.395008 + 0.918678i \(0.629258\pi\)
\(422\) 6.40373 0.311729
\(423\) 2.04189 0.0992800
\(424\) −12.9709 −0.629923
\(425\) −4.18067 −0.202792
\(426\) −6.07192 −0.294185
\(427\) −6.03777 −0.292188
\(428\) −19.1138 −0.923901
\(429\) 7.29086 0.352006
\(430\) 2.87433 0.138613
\(431\) −25.4320 −1.22502 −0.612508 0.790464i \(-0.709839\pi\)
−0.612508 + 0.790464i \(0.709839\pi\)
\(432\) 1.00000 0.0481125
\(433\) −11.8479 −0.569375 −0.284687 0.958620i \(-0.591890\pi\)
−0.284687 + 0.958620i \(0.591890\pi\)
\(434\) 1.04189 0.0500123
\(435\) 4.76558 0.228492
\(436\) 16.4115 0.785967
\(437\) 0 0
\(438\) −0.327696 −0.0156579
\(439\) −25.8598 −1.23422 −0.617110 0.786877i \(-0.711697\pi\)
−0.617110 + 0.786877i \(0.711697\pi\)
\(440\) 2.22668 0.106153
\(441\) −6.71688 −0.319851
\(442\) 4.50980 0.214509
\(443\) 0.923029 0.0438544 0.0219272 0.999760i \(-0.493020\pi\)
0.0219272 + 0.999760i \(0.493020\pi\)
\(444\) −6.88713 −0.326849
\(445\) 4.21719 0.199914
\(446\) −3.24897 −0.153843
\(447\) −16.3550 −0.773567
\(448\) 0.532089 0.0251388
\(449\) −9.34461 −0.440999 −0.220500 0.975387i \(-0.570769\pi\)
−0.220500 + 0.975387i \(0.570769\pi\)
\(450\) −3.59627 −0.169530
\(451\) 16.8871 0.795184
\(452\) 5.73648 0.269821
\(453\) −20.5226 −0.964236
\(454\) 11.7023 0.549218
\(455\) 2.44562 0.114653
\(456\) 0 0
\(457\) 30.9009 1.44548 0.722740 0.691120i \(-0.242882\pi\)
0.722740 + 0.691120i \(0.242882\pi\)
\(458\) −22.9067 −1.07036
\(459\) 1.16250 0.0542610
\(460\) −7.94087 −0.370245
\(461\) −29.6878 −1.38270 −0.691349 0.722521i \(-0.742983\pi\)
−0.691349 + 0.722521i \(0.742983\pi\)
\(462\) 1.00000 0.0465242
\(463\) −11.8307 −0.549819 −0.274909 0.961470i \(-0.588648\pi\)
−0.274909 + 0.961470i \(0.588648\pi\)
\(464\) 4.02229 0.186730
\(465\) 2.31996 0.107585
\(466\) −3.19934 −0.148207
\(467\) 38.3182 1.77315 0.886577 0.462580i \(-0.153076\pi\)
0.886577 + 0.462580i \(0.153076\pi\)
\(468\) 3.87939 0.179325
\(469\) 5.94356 0.274448
\(470\) 2.41921 0.111590
\(471\) −3.19934 −0.147418
\(472\) 2.68004 0.123359
\(473\) 4.55943 0.209643
\(474\) 16.1334 0.741032
\(475\) 0 0
\(476\) 0.618555 0.0283514
\(477\) −12.9709 −0.593897
\(478\) 6.17293 0.282343
\(479\) 37.2841 1.70355 0.851776 0.523906i \(-0.175526\pi\)
0.851776 + 0.523906i \(0.175526\pi\)
\(480\) 1.18479 0.0540781
\(481\) −26.7178 −1.21823
\(482\) −1.53714 −0.0700149
\(483\) −3.56624 −0.162269
\(484\) −7.46791 −0.339451
\(485\) −6.96585 −0.316303
\(486\) 1.00000 0.0453609
\(487\) −32.9786 −1.49441 −0.747203 0.664596i \(-0.768603\pi\)
−0.747203 + 0.664596i \(0.768603\pi\)
\(488\) −11.3473 −0.513668
\(489\) −24.0077 −1.08567
\(490\) −7.95811 −0.359511
\(491\) 17.6810 0.797931 0.398966 0.916966i \(-0.369369\pi\)
0.398966 + 0.916966i \(0.369369\pi\)
\(492\) 8.98545 0.405095
\(493\) 4.67593 0.210593
\(494\) 0 0
\(495\) 2.22668 0.100082
\(496\) 1.95811 0.0879218
\(497\) −3.23080 −0.144921
\(498\) 11.5740 0.518642
\(499\) 27.1239 1.21423 0.607117 0.794613i \(-0.292326\pi\)
0.607117 + 0.794613i \(0.292326\pi\)
\(500\) −10.1848 −0.455478
\(501\) −3.66044 −0.163537
\(502\) 17.9436 0.800860
\(503\) 0.571290 0.0254725 0.0127363 0.999919i \(-0.495946\pi\)
0.0127363 + 0.999919i \(0.495946\pi\)
\(504\) 0.532089 0.0237011
\(505\) 14.9828 0.666724
\(506\) −12.5963 −0.559972
\(507\) 2.04963 0.0910273
\(508\) 1.88713 0.0837277
\(509\) −40.5509 −1.79739 −0.898693 0.438579i \(-0.855482\pi\)
−0.898693 + 0.438579i \(0.855482\pi\)
\(510\) 1.37733 0.0609890
\(511\) −0.174363 −0.00771338
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −29.4124 −1.29733
\(515\) −3.51073 −0.154701
\(516\) 2.42602 0.106800
\(517\) 3.83750 0.168773
\(518\) −3.66456 −0.161012
\(519\) 5.33275 0.234082
\(520\) 4.59627 0.201560
\(521\) −18.8999 −0.828020 −0.414010 0.910272i \(-0.635872\pi\)
−0.414010 + 0.910272i \(0.635872\pi\)
\(522\) 4.02229 0.176051
\(523\) 4.74186 0.207347 0.103673 0.994611i \(-0.466940\pi\)
0.103673 + 0.994611i \(0.466940\pi\)
\(524\) 10.7023 0.467534
\(525\) −1.91353 −0.0835134
\(526\) 10.6827 0.465789
\(527\) 2.27631 0.0991577
\(528\) 1.87939 0.0817897
\(529\) 21.9213 0.953099
\(530\) −15.3678 −0.667536
\(531\) 2.68004 0.116304
\(532\) 0 0
\(533\) 34.8580 1.50987
\(534\) 3.55943 0.154032
\(535\) −22.6459 −0.979067
\(536\) 11.1702 0.482481
\(537\) 25.2567 1.08991
\(538\) 20.5767 0.887123
\(539\) −12.6236 −0.543737
\(540\) 1.18479 0.0509854
\(541\) −28.3610 −1.21934 −0.609668 0.792657i \(-0.708697\pi\)
−0.609668 + 0.792657i \(0.708697\pi\)
\(542\) 1.87670 0.0806110
\(543\) 17.3063 0.742686
\(544\) 1.16250 0.0498419
\(545\) 19.4442 0.832898
\(546\) 2.06418 0.0883387
\(547\) 6.88032 0.294181 0.147091 0.989123i \(-0.453009\pi\)
0.147091 + 0.989123i \(0.453009\pi\)
\(548\) −11.2567 −0.480863
\(549\) −11.3473 −0.484291
\(550\) −6.75877 −0.288195
\(551\) 0 0
\(552\) −6.70233 −0.285270
\(553\) 8.58441 0.365046
\(554\) 4.87258 0.207016
\(555\) −8.15982 −0.346365
\(556\) −12.1702 −0.516133
\(557\) −14.1625 −0.600085 −0.300042 0.953926i \(-0.597001\pi\)
−0.300042 + 0.953926i \(0.597001\pi\)
\(558\) 1.95811 0.0828934
\(559\) 9.41147 0.398063
\(560\) 0.630415 0.0266399
\(561\) 2.18479 0.0922420
\(562\) 0.403733 0.0170305
\(563\) 36.2550 1.52796 0.763982 0.645237i \(-0.223242\pi\)
0.763982 + 0.645237i \(0.223242\pi\)
\(564\) 2.04189 0.0859790
\(565\) 6.79654 0.285933
\(566\) 17.5895 0.739340
\(567\) 0.532089 0.0223456
\(568\) −6.07192 −0.254772
\(569\) 30.2918 1.26990 0.634949 0.772554i \(-0.281021\pi\)
0.634949 + 0.772554i \(0.281021\pi\)
\(570\) 0 0
\(571\) 3.39094 0.141906 0.0709532 0.997480i \(-0.477396\pi\)
0.0709532 + 0.997480i \(0.477396\pi\)
\(572\) 7.29086 0.304846
\(573\) −15.1780 −0.634069
\(574\) 4.78106 0.199558
\(575\) 24.1034 1.00518
\(576\) 1.00000 0.0416667
\(577\) −22.3027 −0.928474 −0.464237 0.885711i \(-0.653671\pi\)
−0.464237 + 0.885711i \(0.653671\pi\)
\(578\) −15.6486 −0.650895
\(579\) −13.0077 −0.540583
\(580\) 4.76558 0.197880
\(581\) 6.15839 0.255493
\(582\) −5.87939 −0.243708
\(583\) −24.3773 −1.00961
\(584\) −0.327696 −0.0135602
\(585\) 4.59627 0.190032
\(586\) 13.1753 0.544267
\(587\) −28.7502 −1.18665 −0.593324 0.804964i \(-0.702185\pi\)
−0.593324 + 0.804964i \(0.702185\pi\)
\(588\) −6.71688 −0.277000
\(589\) 0 0
\(590\) 3.17530 0.130725
\(591\) −8.94862 −0.368097
\(592\) −6.88713 −0.283059
\(593\) 11.7760 0.483583 0.241791 0.970328i \(-0.422265\pi\)
0.241791 + 0.970328i \(0.422265\pi\)
\(594\) 1.87939 0.0771121
\(595\) 0.732860 0.0300443
\(596\) −16.3550 −0.669928
\(597\) −4.50475 −0.184367
\(598\) −26.0009 −1.06326
\(599\) 21.6313 0.883833 0.441916 0.897056i \(-0.354299\pi\)
0.441916 + 0.897056i \(0.354299\pi\)
\(600\) −3.59627 −0.146817
\(601\) 26.1935 1.06845 0.534227 0.845341i \(-0.320603\pi\)
0.534227 + 0.845341i \(0.320603\pi\)
\(602\) 1.29086 0.0526115
\(603\) 11.1702 0.454888
\(604\) −20.5226 −0.835052
\(605\) −8.84793 −0.359719
\(606\) 12.6459 0.513704
\(607\) 13.2249 0.536783 0.268392 0.963310i \(-0.413508\pi\)
0.268392 + 0.963310i \(0.413508\pi\)
\(608\) 0 0
\(609\) 2.14022 0.0867259
\(610\) −13.4442 −0.544339
\(611\) 7.92127 0.320460
\(612\) 1.16250 0.0469914
\(613\) −5.62267 −0.227098 −0.113549 0.993532i \(-0.536222\pi\)
−0.113549 + 0.993532i \(0.536222\pi\)
\(614\) −16.9017 −0.682096
\(615\) 10.6459 0.429284
\(616\) 1.00000 0.0402911
\(617\) 9.47834 0.381584 0.190792 0.981631i \(-0.438894\pi\)
0.190792 + 0.981631i \(0.438894\pi\)
\(618\) −2.96316 −0.119196
\(619\) −14.7493 −0.592823 −0.296412 0.955060i \(-0.595790\pi\)
−0.296412 + 0.955060i \(0.595790\pi\)
\(620\) 2.31996 0.0931716
\(621\) −6.70233 −0.268955
\(622\) −2.38144 −0.0954872
\(623\) 1.89393 0.0758788
\(624\) 3.87939 0.155300
\(625\) 5.91447 0.236579
\(626\) −29.8631 −1.19357
\(627\) 0 0
\(628\) −3.19934 −0.127668
\(629\) −8.00631 −0.319232
\(630\) 0.630415 0.0251163
\(631\) 25.0009 0.995271 0.497636 0.867386i \(-0.334202\pi\)
0.497636 + 0.867386i \(0.334202\pi\)
\(632\) 16.1334 0.641753
\(633\) 6.40373 0.254526
\(634\) 7.02229 0.278891
\(635\) 2.23585 0.0887271
\(636\) −12.9709 −0.514330
\(637\) −26.0574 −1.03243
\(638\) 7.55943 0.299281
\(639\) −6.07192 −0.240201
\(640\) 1.18479 0.0468330
\(641\) 43.7306 1.72726 0.863628 0.504130i \(-0.168187\pi\)
0.863628 + 0.504130i \(0.168187\pi\)
\(642\) −19.1138 −0.754362
\(643\) −33.4534 −1.31927 −0.659636 0.751585i \(-0.729290\pi\)
−0.659636 + 0.751585i \(0.729290\pi\)
\(644\) −3.56624 −0.140529
\(645\) 2.87433 0.113177
\(646\) 0 0
\(647\) −32.1266 −1.26303 −0.631514 0.775365i \(-0.717566\pi\)
−0.631514 + 0.775365i \(0.717566\pi\)
\(648\) 1.00000 0.0392837
\(649\) 5.03684 0.197713
\(650\) −13.9513 −0.547215
\(651\) 1.04189 0.0408349
\(652\) −24.0077 −0.940216
\(653\) 2.89630 0.113341 0.0566704 0.998393i \(-0.481952\pi\)
0.0566704 + 0.998393i \(0.481952\pi\)
\(654\) 16.4115 0.641739
\(655\) 12.6800 0.495450
\(656\) 8.98545 0.350823
\(657\) −0.327696 −0.0127846
\(658\) 1.08647 0.0423549
\(659\) −18.1935 −0.708717 −0.354358 0.935110i \(-0.615301\pi\)
−0.354358 + 0.935110i \(0.615301\pi\)
\(660\) 2.22668 0.0866735
\(661\) −21.4192 −0.833111 −0.416555 0.909110i \(-0.636763\pi\)
−0.416555 + 0.909110i \(0.636763\pi\)
\(662\) −27.1985 −1.05710
\(663\) 4.50980 0.175146
\(664\) 11.5740 0.449157
\(665\) 0 0
\(666\) −6.88713 −0.266871
\(667\) −26.9587 −1.04385
\(668\) −3.66044 −0.141627
\(669\) −3.24897 −0.125612
\(670\) 13.2344 0.511290
\(671\) −21.3259 −0.823279
\(672\) 0.532089 0.0205258
\(673\) −40.6914 −1.56854 −0.784269 0.620421i \(-0.786962\pi\)
−0.784269 + 0.620421i \(0.786962\pi\)
\(674\) 11.2172 0.432070
\(675\) −3.59627 −0.138420
\(676\) 2.04963 0.0788319
\(677\) 0.136096 0.00523061 0.00261530 0.999997i \(-0.499168\pi\)
0.00261530 + 0.999997i \(0.499168\pi\)
\(678\) 5.73648 0.220308
\(679\) −3.12836 −0.120055
\(680\) 1.37733 0.0528180
\(681\) 11.7023 0.448434
\(682\) 3.68004 0.140916
\(683\) 43.6459 1.67006 0.835032 0.550202i \(-0.185449\pi\)
0.835032 + 0.550202i \(0.185449\pi\)
\(684\) 0 0
\(685\) −13.3369 −0.509576
\(686\) −7.29860 −0.278662
\(687\) −22.9067 −0.873946
\(688\) 2.42602 0.0924912
\(689\) −50.3191 −1.91701
\(690\) −7.94087 −0.302304
\(691\) 15.8675 0.603629 0.301815 0.953367i \(-0.402408\pi\)
0.301815 + 0.953367i \(0.402408\pi\)
\(692\) 5.33275 0.202721
\(693\) 1.00000 0.0379869
\(694\) 1.93582 0.0734828
\(695\) −14.4192 −0.546952
\(696\) 4.02229 0.152464
\(697\) 10.4456 0.395656
\(698\) 14.3969 0.544932
\(699\) −3.19934 −0.121010
\(700\) −1.91353 −0.0723248
\(701\) 19.2098 0.725543 0.362771 0.931878i \(-0.381831\pi\)
0.362771 + 0.931878i \(0.381831\pi\)
\(702\) 3.87939 0.146418
\(703\) 0 0
\(704\) 1.87939 0.0708320
\(705\) 2.41921 0.0911129
\(706\) 15.7237 0.591769
\(707\) 6.72874 0.253060
\(708\) 2.68004 0.100722
\(709\) 33.0259 1.24031 0.620157 0.784478i \(-0.287069\pi\)
0.620157 + 0.784478i \(0.287069\pi\)
\(710\) −7.19396 −0.269985
\(711\) 16.1334 0.605050
\(712\) 3.55943 0.133395
\(713\) −13.1239 −0.491494
\(714\) 0.618555 0.0231489
\(715\) 8.63816 0.323049
\(716\) 25.2567 0.943888
\(717\) 6.17293 0.230532
\(718\) 15.5175 0.579109
\(719\) 39.4448 1.47104 0.735521 0.677501i \(-0.236937\pi\)
0.735521 + 0.677501i \(0.236937\pi\)
\(720\) 1.18479 0.0441546
\(721\) −1.57667 −0.0587181
\(722\) 0 0
\(723\) −1.53714 −0.0571669
\(724\) 17.3063 0.643185
\(725\) −14.4652 −0.537225
\(726\) −7.46791 −0.277160
\(727\) 18.2517 0.676917 0.338458 0.940981i \(-0.390095\pi\)
0.338458 + 0.940981i \(0.390095\pi\)
\(728\) 2.06418 0.0765035
\(729\) 1.00000 0.0370370
\(730\) −0.388252 −0.0143698
\(731\) 2.82026 0.104311
\(732\) −11.3473 −0.419408
\(733\) 15.3928 0.568546 0.284273 0.958743i \(-0.408248\pi\)
0.284273 + 0.958743i \(0.408248\pi\)
\(734\) −15.7442 −0.581130
\(735\) −7.95811 −0.293539
\(736\) −6.70233 −0.247051
\(737\) 20.9932 0.773294
\(738\) 8.98545 0.330759
\(739\) 15.1771 0.558297 0.279148 0.960248i \(-0.409948\pi\)
0.279148 + 0.960248i \(0.409948\pi\)
\(740\) −8.15982 −0.299961
\(741\) 0 0
\(742\) −6.90167 −0.253368
\(743\) 9.86659 0.361970 0.180985 0.983486i \(-0.442071\pi\)
0.180985 + 0.983486i \(0.442071\pi\)
\(744\) 1.95811 0.0717878
\(745\) −19.3773 −0.709930
\(746\) 29.6287 1.08478
\(747\) 11.5740 0.423470
\(748\) 2.18479 0.0798839
\(749\) −10.1702 −0.371613
\(750\) −10.1848 −0.371896
\(751\) 9.75877 0.356103 0.178051 0.984021i \(-0.443021\pi\)
0.178051 + 0.984021i \(0.443021\pi\)
\(752\) 2.04189 0.0744600
\(753\) 17.9436 0.653900
\(754\) 15.6040 0.568264
\(755\) −24.3150 −0.884914
\(756\) 0.532089 0.0193519
\(757\) −43.5039 −1.58118 −0.790589 0.612348i \(-0.790225\pi\)
−0.790589 + 0.612348i \(0.790225\pi\)
\(758\) −4.72462 −0.171606
\(759\) −12.5963 −0.457216
\(760\) 0 0
\(761\) 44.7137 1.62087 0.810435 0.585828i \(-0.199231\pi\)
0.810435 + 0.585828i \(0.199231\pi\)
\(762\) 1.88713 0.0683634
\(763\) 8.73236 0.316133
\(764\) −15.1780 −0.549120
\(765\) 1.37733 0.0497973
\(766\) 5.24123 0.189373
\(767\) 10.3969 0.375411
\(768\) 1.00000 0.0360844
\(769\) 22.1019 0.797017 0.398508 0.917165i \(-0.369528\pi\)
0.398508 + 0.917165i \(0.369528\pi\)
\(770\) 1.18479 0.0426970
\(771\) −29.4124 −1.05926
\(772\) −13.0077 −0.468159
\(773\) −42.1789 −1.51707 −0.758535 0.651632i \(-0.774085\pi\)
−0.758535 + 0.651632i \(0.774085\pi\)
\(774\) 2.42602 0.0872016
\(775\) −7.04189 −0.252952
\(776\) −5.87939 −0.211058
\(777\) −3.66456 −0.131465
\(778\) −15.0770 −0.540536
\(779\) 0 0
\(780\) 4.59627 0.164573
\(781\) −11.4115 −0.408335
\(782\) −7.79149 −0.278623
\(783\) 4.02229 0.143745
\(784\) −6.71688 −0.239889
\(785\) −3.79055 −0.135291
\(786\) 10.7023 0.381740
\(787\) 28.4793 1.01518 0.507588 0.861600i \(-0.330537\pi\)
0.507588 + 0.861600i \(0.330537\pi\)
\(788\) −8.94862 −0.318781
\(789\) 10.6827 0.380315
\(790\) 19.1147 0.680072
\(791\) 3.05232 0.108528
\(792\) 1.87939 0.0667810
\(793\) −44.0205 −1.56321
\(794\) −14.5449 −0.516179
\(795\) −15.3678 −0.545041
\(796\) −4.50475 −0.159667
\(797\) −16.6081 −0.588290 −0.294145 0.955761i \(-0.595035\pi\)
−0.294145 + 0.955761i \(0.595035\pi\)
\(798\) 0 0
\(799\) 2.37370 0.0839756
\(800\) −3.59627 −0.127147
\(801\) 3.55943 0.125766
\(802\) −2.85473 −0.100804
\(803\) −0.615867 −0.0217335
\(804\) 11.1702 0.393944
\(805\) −4.22525 −0.148921
\(806\) 7.59627 0.267567
\(807\) 20.5767 0.724333
\(808\) 12.6459 0.444881
\(809\) −13.6783 −0.480903 −0.240452 0.970661i \(-0.577296\pi\)
−0.240452 + 0.970661i \(0.577296\pi\)
\(810\) 1.18479 0.0416294
\(811\) 0.618231 0.0217090 0.0108545 0.999941i \(-0.496545\pi\)
0.0108545 + 0.999941i \(0.496545\pi\)
\(812\) 2.14022 0.0751068
\(813\) 1.87670 0.0658186
\(814\) −12.9436 −0.453672
\(815\) −28.4442 −0.996357
\(816\) 1.16250 0.0406958
\(817\) 0 0
\(818\) −34.2276 −1.19674
\(819\) 2.06418 0.0721282
\(820\) 10.6459 0.371771
\(821\) 6.11205 0.213312 0.106656 0.994296i \(-0.465986\pi\)
0.106656 + 0.994296i \(0.465986\pi\)
\(822\) −11.2567 −0.392623
\(823\) 30.6546 1.06855 0.534276 0.845310i \(-0.320584\pi\)
0.534276 + 0.845310i \(0.320584\pi\)
\(824\) −2.96316 −0.103227
\(825\) −6.75877 −0.235310
\(826\) 1.42602 0.0496177
\(827\) 50.7134 1.76348 0.881738 0.471739i \(-0.156373\pi\)
0.881738 + 0.471739i \(0.156373\pi\)
\(828\) −6.70233 −0.232922
\(829\) 13.0000 0.451509 0.225754 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(830\) 13.7128 0.475977
\(831\) 4.87258 0.169028
\(832\) 3.87939 0.134493
\(833\) −7.80840 −0.270545
\(834\) −12.1702 −0.421421
\(835\) −4.33687 −0.150083
\(836\) 0 0
\(837\) 1.95811 0.0676822
\(838\) −7.91891 −0.273554
\(839\) 21.4979 0.742191 0.371096 0.928595i \(-0.378982\pi\)
0.371096 + 0.928595i \(0.378982\pi\)
\(840\) 0.630415 0.0217514
\(841\) −12.8212 −0.442110
\(842\) −16.2098 −0.558626
\(843\) 0.403733 0.0139053
\(844\) 6.40373 0.220426
\(845\) 2.42839 0.0835390
\(846\) 2.04189 0.0702016
\(847\) −3.97359 −0.136534
\(848\) −12.9709 −0.445423
\(849\) 17.5895 0.603669
\(850\) −4.18067 −0.143396
\(851\) 46.1598 1.58234
\(852\) −6.07192 −0.208021
\(853\) −46.2276 −1.58280 −0.791402 0.611296i \(-0.790648\pi\)
−0.791402 + 0.611296i \(0.790648\pi\)
\(854\) −6.03777 −0.206608
\(855\) 0 0
\(856\) −19.1138 −0.653296
\(857\) 43.6623 1.49148 0.745738 0.666239i \(-0.232097\pi\)
0.745738 + 0.666239i \(0.232097\pi\)
\(858\) 7.29086 0.248906
\(859\) 20.1652 0.688027 0.344014 0.938965i \(-0.388213\pi\)
0.344014 + 0.938965i \(0.388213\pi\)
\(860\) 2.87433 0.0980139
\(861\) 4.78106 0.162938
\(862\) −25.4320 −0.866218
\(863\) 32.6932 1.11289 0.556444 0.830885i \(-0.312165\pi\)
0.556444 + 0.830885i \(0.312165\pi\)
\(864\) 1.00000 0.0340207
\(865\) 6.31820 0.214825
\(866\) −11.8479 −0.402609
\(867\) −15.6486 −0.531454
\(868\) 1.04189 0.0353640
\(869\) 30.3209 1.02857
\(870\) 4.76558 0.161568
\(871\) 43.3337 1.46831
\(872\) 16.4115 0.555763
\(873\) −5.87939 −0.198987
\(874\) 0 0
\(875\) −5.41921 −0.183203
\(876\) −0.327696 −0.0110718
\(877\) −30.2968 −1.02305 −0.511526 0.859268i \(-0.670920\pi\)
−0.511526 + 0.859268i \(0.670920\pi\)
\(878\) −25.8598 −0.872725
\(879\) 13.1753 0.444392
\(880\) 2.22668 0.0750614
\(881\) 23.5577 0.793678 0.396839 0.917888i \(-0.370107\pi\)
0.396839 + 0.917888i \(0.370107\pi\)
\(882\) −6.71688 −0.226169
\(883\) 36.8289 1.23939 0.619696 0.784842i \(-0.287256\pi\)
0.619696 + 0.784842i \(0.287256\pi\)
\(884\) 4.50980 0.151681
\(885\) 3.17530 0.106736
\(886\) 0.923029 0.0310098
\(887\) −4.30810 −0.144652 −0.0723258 0.997381i \(-0.523042\pi\)
−0.0723258 + 0.997381i \(0.523042\pi\)
\(888\) −6.88713 −0.231117
\(889\) 1.00412 0.0336771
\(890\) 4.21719 0.141360
\(891\) 1.87939 0.0629618
\(892\) −3.24897 −0.108784
\(893\) 0 0
\(894\) −16.3550 −0.546994
\(895\) 29.9240 1.00025
\(896\) 0.532089 0.0177758
\(897\) −26.0009 −0.868146
\(898\) −9.34461 −0.311834
\(899\) 7.87609 0.262682
\(900\) −3.59627 −0.119876
\(901\) −15.0787 −0.502345
\(902\) 16.8871 0.562280
\(903\) 1.29086 0.0429571
\(904\) 5.73648 0.190793
\(905\) 20.5044 0.681590
\(906\) −20.5226 −0.681817
\(907\) −23.3405 −0.775008 −0.387504 0.921868i \(-0.626663\pi\)
−0.387504 + 0.921868i \(0.626663\pi\)
\(908\) 11.7023 0.388356
\(909\) 12.6459 0.419438
\(910\) 2.44562 0.0810716
\(911\) −22.5631 −0.747547 −0.373774 0.927520i \(-0.621936\pi\)
−0.373774 + 0.927520i \(0.621936\pi\)
\(912\) 0 0
\(913\) 21.7520 0.719885
\(914\) 30.9009 1.02211
\(915\) −13.4442 −0.444451
\(916\) −22.9067 −0.756860
\(917\) 5.69459 0.188052
\(918\) 1.16250 0.0383683
\(919\) 4.25578 0.140385 0.0701926 0.997533i \(-0.477639\pi\)
0.0701926 + 0.997533i \(0.477639\pi\)
\(920\) −7.94087 −0.261803
\(921\) −16.9017 −0.556929
\(922\) −29.6878 −0.977715
\(923\) −23.5553 −0.775333
\(924\) 1.00000 0.0328976
\(925\) 24.7679 0.814365
\(926\) −11.8307 −0.388781
\(927\) −2.96316 −0.0973231
\(928\) 4.02229 0.132038
\(929\) 59.5245 1.95293 0.976467 0.215666i \(-0.0691924\pi\)
0.976467 + 0.215666i \(0.0691924\pi\)
\(930\) 2.31996 0.0760743
\(931\) 0 0
\(932\) −3.19934 −0.104798
\(933\) −2.38144 −0.0779650
\(934\) 38.3182 1.25381
\(935\) 2.58853 0.0846538
\(936\) 3.87939 0.126802
\(937\) 33.5090 1.09469 0.547345 0.836907i \(-0.315638\pi\)
0.547345 + 0.836907i \(0.315638\pi\)
\(938\) 5.94356 0.194064
\(939\) −29.8631 −0.974545
\(940\) 2.41921 0.0789061
\(941\) 5.52435 0.180089 0.0900443 0.995938i \(-0.471299\pi\)
0.0900443 + 0.995938i \(0.471299\pi\)
\(942\) −3.19934 −0.104240
\(943\) −60.2235 −1.96115
\(944\) 2.68004 0.0872280
\(945\) 0.630415 0.0205074
\(946\) 4.55943 0.148240
\(947\) −14.9050 −0.484347 −0.242173 0.970233i \(-0.577860\pi\)
−0.242173 + 0.970233i \(0.577860\pi\)
\(948\) 16.1334 0.523989
\(949\) −1.27126 −0.0412668
\(950\) 0 0
\(951\) 7.02229 0.227713
\(952\) 0.618555 0.0200475
\(953\) 46.3661 1.50194 0.750972 0.660334i \(-0.229585\pi\)
0.750972 + 0.660334i \(0.229585\pi\)
\(954\) −12.9709 −0.419949
\(955\) −17.9828 −0.581909
\(956\) 6.17293 0.199647
\(957\) 7.55943 0.244362
\(958\) 37.2841 1.20459
\(959\) −5.98957 −0.193413
\(960\) 1.18479 0.0382390
\(961\) −27.1658 −0.876316
\(962\) −26.7178 −0.861417
\(963\) −19.1138 −0.615934
\(964\) −1.53714 −0.0495080
\(965\) −15.4115 −0.496113
\(966\) −3.56624 −0.114742
\(967\) −23.2431 −0.747448 −0.373724 0.927540i \(-0.621919\pi\)
−0.373724 + 0.927540i \(0.621919\pi\)
\(968\) −7.46791 −0.240028
\(969\) 0 0
\(970\) −6.96585 −0.223660
\(971\) −4.93045 −0.158226 −0.0791128 0.996866i \(-0.525209\pi\)
−0.0791128 + 0.996866i \(0.525209\pi\)
\(972\) 1.00000 0.0320750
\(973\) −6.47565 −0.207600
\(974\) −32.9786 −1.05670
\(975\) −13.9513 −0.446799
\(976\) −11.3473 −0.363218
\(977\) −32.7442 −1.04758 −0.523790 0.851847i \(-0.675482\pi\)
−0.523790 + 0.851847i \(0.675482\pi\)
\(978\) −24.0077 −0.767683
\(979\) 6.68954 0.213799
\(980\) −7.95811 −0.254213
\(981\) 16.4115 0.523978
\(982\) 17.6810 0.564223
\(983\) 25.4406 0.811428 0.405714 0.914000i \(-0.367023\pi\)
0.405714 + 0.914000i \(0.367023\pi\)
\(984\) 8.98545 0.286446
\(985\) −10.6023 −0.337816
\(986\) 4.67593 0.148912
\(987\) 1.08647 0.0345826
\(988\) 0 0
\(989\) −16.2600 −0.517038
\(990\) 2.22668 0.0707686
\(991\) −10.1533 −0.322531 −0.161266 0.986911i \(-0.551558\pi\)
−0.161266 + 0.986911i \(0.551558\pi\)
\(992\) 1.95811 0.0621701
\(993\) −27.1985 −0.863119
\(994\) −3.23080 −0.102475
\(995\) −5.33719 −0.169200
\(996\) 11.5740 0.366736
\(997\) 55.7698 1.76625 0.883124 0.469140i \(-0.155436\pi\)
0.883124 + 0.469140i \(0.155436\pi\)
\(998\) 27.1239 0.858592
\(999\) −6.88713 −0.217899
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 2166.2.a.u.1.2 3
3.2 odd 2 6498.2.a.bn.1.2 3
19.2 odd 18 114.2.i.d.61.1 yes 6
19.10 odd 18 114.2.i.d.43.1 6
19.18 odd 2 2166.2.a.o.1.2 3
57.2 even 18 342.2.u.a.289.1 6
57.29 even 18 342.2.u.a.271.1 6
57.56 even 2 6498.2.a.bs.1.2 3
76.59 even 18 912.2.bo.f.289.1 6
76.67 even 18 912.2.bo.f.385.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.43.1 6 19.10 odd 18
114.2.i.d.61.1 yes 6 19.2 odd 18
342.2.u.a.271.1 6 57.29 even 18
342.2.u.a.289.1 6 57.2 even 18
912.2.bo.f.289.1 6 76.59 even 18
912.2.bo.f.385.1 6 76.67 even 18
2166.2.a.o.1.2 3 19.18 odd 2
2166.2.a.u.1.2 3 1.1 even 1 trivial
6498.2.a.bn.1.2 3 3.2 odd 2
6498.2.a.bs.1.2 3 57.56 even 2