Properties

Label 361.2.a.h.1.2
Level $361$
Weight $2$
Character 361.1
Self dual yes
Analytic conductor $2.883$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 361.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.88259951297\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 361.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.34730 q^{2} +2.87939 q^{3} -0.184793 q^{4} +0.879385 q^{5} +3.87939 q^{6} +0.347296 q^{7} -2.94356 q^{8} +5.29086 q^{9} +O(q^{10})\) \(q+1.34730 q^{2} +2.87939 q^{3} -0.184793 q^{4} +0.879385 q^{5} +3.87939 q^{6} +0.347296 q^{7} -2.94356 q^{8} +5.29086 q^{9} +1.18479 q^{10} -2.22668 q^{11} -0.532089 q^{12} -2.57398 q^{13} +0.467911 q^{14} +2.53209 q^{15} -3.59627 q^{16} +0.467911 q^{17} +7.12836 q^{18} -0.162504 q^{20} +1.00000 q^{21} -3.00000 q^{22} -2.69459 q^{23} -8.47565 q^{24} -4.22668 q^{25} -3.46791 q^{26} +6.59627 q^{27} -0.0641778 q^{28} +6.87939 q^{29} +3.41147 q^{30} +7.10607 q^{31} +1.04189 q^{32} -6.41147 q^{33} +0.630415 q^{34} +0.305407 q^{35} -0.977711 q^{36} -4.94356 q^{37} -7.41147 q^{39} -2.58853 q^{40} -2.47565 q^{41} +1.34730 q^{42} +3.90167 q^{43} +0.411474 q^{44} +4.65270 q^{45} -3.63041 q^{46} -7.29086 q^{47} -10.3550 q^{48} -6.87939 q^{49} -5.69459 q^{50} +1.34730 q^{51} +0.475652 q^{52} +2.83750 q^{53} +8.88713 q^{54} -1.95811 q^{55} -1.02229 q^{56} +9.26857 q^{58} +6.30541 q^{59} -0.467911 q^{60} +9.12836 q^{61} +9.57398 q^{62} +1.83750 q^{63} +8.59627 q^{64} -2.26352 q^{65} -8.63816 q^{66} -7.67499 q^{67} -0.0864665 q^{68} -7.75877 q^{69} +0.411474 q^{70} +9.30541 q^{71} -15.5740 q^{72} +1.38919 q^{73} -6.66044 q^{74} -12.1702 q^{75} -0.773318 q^{77} -9.98545 q^{78} +11.8452 q^{79} -3.16250 q^{80} +3.12061 q^{81} -3.33544 q^{82} -14.8307 q^{83} -0.184793 q^{84} +0.411474 q^{85} +5.25671 q^{86} +19.8084 q^{87} +6.55438 q^{88} +10.2909 q^{89} +6.26857 q^{90} -0.893933 q^{91} +0.497941 q^{92} +20.4611 q^{93} -9.82295 q^{94} +3.00000 q^{96} -9.45336 q^{97} -9.26857 q^{98} -11.7811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 6 q^{6} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 6 q^{6} + 6 q^{8} + 3 q^{12} + 6 q^{14} + 3 q^{15} + 3 q^{16} + 6 q^{17} + 3 q^{18} - 3 q^{20} + 3 q^{21} - 9 q^{22} - 6 q^{23} - 6 q^{24} - 6 q^{25} - 15 q^{26} + 6 q^{27} + 9 q^{28} + 15 q^{29} + 9 q^{31} - 9 q^{33} + 9 q^{34} + 3 q^{35} - 9 q^{36} - 12 q^{39} - 18 q^{40} + 12 q^{41} + 3 q^{42} - 9 q^{44} + 15 q^{45} - 18 q^{46} - 6 q^{47} - 6 q^{48} - 15 q^{49} - 15 q^{50} + 3 q^{51} - 18 q^{52} + 6 q^{53} - 3 q^{54} - 9 q^{55} + 3 q^{56} + 18 q^{58} + 21 q^{59} - 6 q^{60} + 9 q^{61} + 21 q^{62} + 3 q^{63} + 12 q^{64} - 12 q^{65} - 9 q^{66} - 18 q^{67} + 15 q^{68} - 12 q^{69} - 9 q^{70} + 30 q^{71} - 39 q^{72} + 3 q^{74} - 15 q^{75} - 9 q^{77} - 12 q^{78} + 9 q^{79} - 12 q^{80} + 15 q^{81} + 18 q^{82} + 3 q^{84} - 9 q^{85} - 21 q^{86} + 21 q^{87} + 9 q^{88} + 15 q^{89} + 9 q^{90} - 15 q^{91} - 24 q^{92} + 24 q^{93} - 9 q^{94} + 9 q^{96} - 15 q^{97} - 18 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.34730 0.952682 0.476341 0.879261i \(-0.341963\pi\)
0.476341 + 0.879261i \(0.341963\pi\)
\(3\) 2.87939 1.66241 0.831207 0.555963i \(-0.187650\pi\)
0.831207 + 0.555963i \(0.187650\pi\)
\(4\) −0.184793 −0.0923963
\(5\) 0.879385 0.393273 0.196637 0.980476i \(-0.436998\pi\)
0.196637 + 0.980476i \(0.436998\pi\)
\(6\) 3.87939 1.58375
\(7\) 0.347296 0.131266 0.0656328 0.997844i \(-0.479093\pi\)
0.0656328 + 0.997844i \(0.479093\pi\)
\(8\) −2.94356 −1.04071
\(9\) 5.29086 1.76362
\(10\) 1.18479 0.374664
\(11\) −2.22668 −0.671370 −0.335685 0.941974i \(-0.608968\pi\)
−0.335685 + 0.941974i \(0.608968\pi\)
\(12\) −0.532089 −0.153601
\(13\) −2.57398 −0.713893 −0.356947 0.934125i \(-0.616182\pi\)
−0.356947 + 0.934125i \(0.616182\pi\)
\(14\) 0.467911 0.125055
\(15\) 2.53209 0.653783
\(16\) −3.59627 −0.899067
\(17\) 0.467911 0.113485 0.0567426 0.998389i \(-0.481929\pi\)
0.0567426 + 0.998389i \(0.481929\pi\)
\(18\) 7.12836 1.68017
\(19\) 0 0
\(20\) −0.162504 −0.0363370
\(21\) 1.00000 0.218218
\(22\) −3.00000 −0.639602
\(23\) −2.69459 −0.561861 −0.280931 0.959728i \(-0.590643\pi\)
−0.280931 + 0.959728i \(0.590643\pi\)
\(24\) −8.47565 −1.73009
\(25\) −4.22668 −0.845336
\(26\) −3.46791 −0.680113
\(27\) 6.59627 1.26945
\(28\) −0.0641778 −0.0121285
\(29\) 6.87939 1.27747 0.638735 0.769427i \(-0.279458\pi\)
0.638735 + 0.769427i \(0.279458\pi\)
\(30\) 3.41147 0.622847
\(31\) 7.10607 1.27629 0.638144 0.769917i \(-0.279703\pi\)
0.638144 + 0.769917i \(0.279703\pi\)
\(32\) 1.04189 0.184182
\(33\) −6.41147 −1.11609
\(34\) 0.630415 0.108115
\(35\) 0.305407 0.0516233
\(36\) −0.977711 −0.162952
\(37\) −4.94356 −0.812717 −0.406358 0.913714i \(-0.633202\pi\)
−0.406358 + 0.913714i \(0.633202\pi\)
\(38\) 0 0
\(39\) −7.41147 −1.18679
\(40\) −2.58853 −0.409282
\(41\) −2.47565 −0.386632 −0.193316 0.981137i \(-0.561924\pi\)
−0.193316 + 0.981137i \(0.561924\pi\)
\(42\) 1.34730 0.207892
\(43\) 3.90167 0.595000 0.297500 0.954722i \(-0.403847\pi\)
0.297500 + 0.954722i \(0.403847\pi\)
\(44\) 0.411474 0.0620321
\(45\) 4.65270 0.693584
\(46\) −3.63041 −0.535275
\(47\) −7.29086 −1.06348 −0.531741 0.846907i \(-0.678462\pi\)
−0.531741 + 0.846907i \(0.678462\pi\)
\(48\) −10.3550 −1.49462
\(49\) −6.87939 −0.982769
\(50\) −5.69459 −0.805337
\(51\) 1.34730 0.188659
\(52\) 0.475652 0.0659611
\(53\) 2.83750 0.389760 0.194880 0.980827i \(-0.437568\pi\)
0.194880 + 0.980827i \(0.437568\pi\)
\(54\) 8.88713 1.20938
\(55\) −1.95811 −0.264032
\(56\) −1.02229 −0.136609
\(57\) 0 0
\(58\) 9.26857 1.21702
\(59\) 6.30541 0.820894 0.410447 0.911884i \(-0.365373\pi\)
0.410447 + 0.911884i \(0.365373\pi\)
\(60\) −0.467911 −0.0604071
\(61\) 9.12836 1.16877 0.584383 0.811478i \(-0.301337\pi\)
0.584383 + 0.811478i \(0.301337\pi\)
\(62\) 9.57398 1.21590
\(63\) 1.83750 0.231503
\(64\) 8.59627 1.07453
\(65\) −2.26352 −0.280755
\(66\) −8.63816 −1.06328
\(67\) −7.67499 −0.937650 −0.468825 0.883291i \(-0.655322\pi\)
−0.468825 + 0.883291i \(0.655322\pi\)
\(68\) −0.0864665 −0.0104856
\(69\) −7.75877 −0.934046
\(70\) 0.411474 0.0491806
\(71\) 9.30541 1.10435 0.552174 0.833729i \(-0.313798\pi\)
0.552174 + 0.833729i \(0.313798\pi\)
\(72\) −15.5740 −1.83541
\(73\) 1.38919 0.162592 0.0812959 0.996690i \(-0.474094\pi\)
0.0812959 + 0.996690i \(0.474094\pi\)
\(74\) −6.66044 −0.774261
\(75\) −12.1702 −1.40530
\(76\) 0 0
\(77\) −0.773318 −0.0881278
\(78\) −9.98545 −1.13063
\(79\) 11.8452 1.33269 0.666347 0.745642i \(-0.267857\pi\)
0.666347 + 0.745642i \(0.267857\pi\)
\(80\) −3.16250 −0.353579
\(81\) 3.12061 0.346735
\(82\) −3.33544 −0.368337
\(83\) −14.8307 −1.62788 −0.813940 0.580949i \(-0.802681\pi\)
−0.813940 + 0.580949i \(0.802681\pi\)
\(84\) −0.184793 −0.0201625
\(85\) 0.411474 0.0446306
\(86\) 5.25671 0.566846
\(87\) 19.8084 2.12368
\(88\) 6.55438 0.698699
\(89\) 10.2909 1.09083 0.545414 0.838166i \(-0.316372\pi\)
0.545414 + 0.838166i \(0.316372\pi\)
\(90\) 6.26857 0.660765
\(91\) −0.893933 −0.0937097
\(92\) 0.497941 0.0519139
\(93\) 20.4611 2.12172
\(94\) −9.82295 −1.01316
\(95\) 0 0
\(96\) 3.00000 0.306186
\(97\) −9.45336 −0.959844 −0.479922 0.877311i \(-0.659335\pi\)
−0.479922 + 0.877311i \(0.659335\pi\)
\(98\) −9.26857 −0.936267
\(99\) −11.7811 −1.18404
\(100\) 0.781059 0.0781059
\(101\) −9.24897 −0.920307 −0.460153 0.887839i \(-0.652206\pi\)
−0.460153 + 0.887839i \(0.652206\pi\)
\(102\) 1.81521 0.179732
\(103\) 5.50980 0.542897 0.271448 0.962453i \(-0.412497\pi\)
0.271448 + 0.962453i \(0.412497\pi\)
\(104\) 7.57667 0.742953
\(105\) 0.879385 0.0858192
\(106\) 3.82295 0.371318
\(107\) −10.2344 −0.989399 −0.494699 0.869064i \(-0.664722\pi\)
−0.494699 + 0.869064i \(0.664722\pi\)
\(108\) −1.21894 −0.117293
\(109\) 1.82295 0.174607 0.0873034 0.996182i \(-0.472175\pi\)
0.0873034 + 0.996182i \(0.472175\pi\)
\(110\) −2.63816 −0.251538
\(111\) −14.2344 −1.35107
\(112\) −1.24897 −0.118017
\(113\) 17.6878 1.66393 0.831963 0.554830i \(-0.187217\pi\)
0.831963 + 0.554830i \(0.187217\pi\)
\(114\) 0 0
\(115\) −2.36959 −0.220965
\(116\) −1.27126 −0.118033
\(117\) −13.6186 −1.25904
\(118\) 8.49525 0.782051
\(119\) 0.162504 0.0148967
\(120\) −7.45336 −0.680396
\(121\) −6.04189 −0.549263
\(122\) 12.2986 1.11346
\(123\) −7.12836 −0.642742
\(124\) −1.31315 −0.117924
\(125\) −8.11381 −0.725721
\(126\) 2.47565 0.220549
\(127\) −11.6040 −1.02969 −0.514845 0.857284i \(-0.672150\pi\)
−0.514845 + 0.857284i \(0.672150\pi\)
\(128\) 9.49794 0.839507
\(129\) 11.2344 0.989136
\(130\) −3.04963 −0.267470
\(131\) 1.84524 0.161219 0.0806096 0.996746i \(-0.474313\pi\)
0.0806096 + 0.996746i \(0.474313\pi\)
\(132\) 1.18479 0.103123
\(133\) 0 0
\(134\) −10.3405 −0.893282
\(135\) 5.80066 0.499241
\(136\) −1.37733 −0.118105
\(137\) 0.255777 0.0218525 0.0109263 0.999940i \(-0.496522\pi\)
0.0109263 + 0.999940i \(0.496522\pi\)
\(138\) −10.4534 −0.889849
\(139\) 4.26352 0.361627 0.180813 0.983517i \(-0.442127\pi\)
0.180813 + 0.983517i \(0.442127\pi\)
\(140\) −0.0564370 −0.00476980
\(141\) −20.9932 −1.76795
\(142\) 12.5371 1.05209
\(143\) 5.73143 0.479286
\(144\) −19.0273 −1.58561
\(145\) 6.04963 0.502394
\(146\) 1.87164 0.154898
\(147\) −19.8084 −1.63377
\(148\) 0.913534 0.0750920
\(149\) 16.5594 1.35660 0.678301 0.734784i \(-0.262717\pi\)
0.678301 + 0.734784i \(0.262717\pi\)
\(150\) −16.3969 −1.33880
\(151\) 4.36184 0.354962 0.177481 0.984124i \(-0.443205\pi\)
0.177481 + 0.984124i \(0.443205\pi\)
\(152\) 0 0
\(153\) 2.47565 0.200145
\(154\) −1.04189 −0.0839578
\(155\) 6.24897 0.501929
\(156\) 1.36959 0.109655
\(157\) −9.61856 −0.767644 −0.383822 0.923407i \(-0.625392\pi\)
−0.383822 + 0.923407i \(0.625392\pi\)
\(158\) 15.9590 1.26963
\(159\) 8.17024 0.647943
\(160\) 0.916222 0.0724337
\(161\) −0.935822 −0.0737531
\(162\) 4.20439 0.330328
\(163\) 8.35504 0.654417 0.327209 0.944952i \(-0.393892\pi\)
0.327209 + 0.944952i \(0.393892\pi\)
\(164\) 0.457482 0.0357233
\(165\) −5.63816 −0.438930
\(166\) −19.9813 −1.55085
\(167\) −4.03508 −0.312244 −0.156122 0.987738i \(-0.549899\pi\)
−0.156122 + 0.987738i \(0.549899\pi\)
\(168\) −2.94356 −0.227101
\(169\) −6.37464 −0.490357
\(170\) 0.554378 0.0425188
\(171\) 0 0
\(172\) −0.721000 −0.0549758
\(173\) 20.1438 1.53151 0.765754 0.643134i \(-0.222366\pi\)
0.765754 + 0.643134i \(0.222366\pi\)
\(174\) 26.6878 2.02320
\(175\) −1.46791 −0.110964
\(176\) 8.00774 0.603606
\(177\) 18.1557 1.36467
\(178\) 13.8648 1.03921
\(179\) 11.5125 0.860484 0.430242 0.902714i \(-0.358428\pi\)
0.430242 + 0.902714i \(0.358428\pi\)
\(180\) −0.859785 −0.0640846
\(181\) −8.53714 −0.634561 −0.317280 0.948332i \(-0.602770\pi\)
−0.317280 + 0.948332i \(0.602770\pi\)
\(182\) −1.20439 −0.0892755
\(183\) 26.2841 1.94297
\(184\) 7.93170 0.584733
\(185\) −4.34730 −0.319620
\(186\) 27.5672 2.02132
\(187\) −1.04189 −0.0761905
\(188\) 1.34730 0.0982617
\(189\) 2.29086 0.166635
\(190\) 0 0
\(191\) 18.3354 1.32671 0.663353 0.748307i \(-0.269133\pi\)
0.663353 + 0.748307i \(0.269133\pi\)
\(192\) 24.7520 1.78632
\(193\) −0.297667 −0.0214265 −0.0107133 0.999943i \(-0.503410\pi\)
−0.0107133 + 0.999943i \(0.503410\pi\)
\(194\) −12.7365 −0.914426
\(195\) −6.51754 −0.466731
\(196\) 1.27126 0.0908042
\(197\) 13.1411 0.936268 0.468134 0.883657i \(-0.344926\pi\)
0.468134 + 0.883657i \(0.344926\pi\)
\(198\) −15.8726 −1.12802
\(199\) −0.256711 −0.0181978 −0.00909888 0.999959i \(-0.502896\pi\)
−0.00909888 + 0.999959i \(0.502896\pi\)
\(200\) 12.4415 0.879747
\(201\) −22.0993 −1.55876
\(202\) −12.4611 −0.876760
\(203\) 2.38919 0.167688
\(204\) −0.248970 −0.0174314
\(205\) −2.17705 −0.152052
\(206\) 7.42333 0.517208
\(207\) −14.2567 −0.990910
\(208\) 9.25671 0.641837
\(209\) 0 0
\(210\) 1.18479 0.0817585
\(211\) −2.44831 −0.168549 −0.0842743 0.996443i \(-0.526857\pi\)
−0.0842743 + 0.996443i \(0.526857\pi\)
\(212\) −0.524348 −0.0360124
\(213\) 26.7939 1.83588
\(214\) −13.7888 −0.942583
\(215\) 3.43107 0.233997
\(216\) −19.4165 −1.32113
\(217\) 2.46791 0.167533
\(218\) 2.45605 0.166345
\(219\) 4.00000 0.270295
\(220\) 0.361844 0.0243955
\(221\) −1.20439 −0.0810162
\(222\) −19.1780 −1.28714
\(223\) 8.50980 0.569858 0.284929 0.958549i \(-0.408030\pi\)
0.284929 + 0.958549i \(0.408030\pi\)
\(224\) 0.361844 0.0241767
\(225\) −22.3628 −1.49085
\(226\) 23.8307 1.58519
\(227\) 14.1506 0.939211 0.469606 0.882876i \(-0.344396\pi\)
0.469606 + 0.882876i \(0.344396\pi\)
\(228\) 0 0
\(229\) −20.5330 −1.35686 −0.678430 0.734665i \(-0.737339\pi\)
−0.678430 + 0.734665i \(0.737339\pi\)
\(230\) −3.19253 −0.210509
\(231\) −2.22668 −0.146505
\(232\) −20.2499 −1.32947
\(233\) 17.6509 1.15635 0.578176 0.815912i \(-0.303765\pi\)
0.578176 + 0.815912i \(0.303765\pi\)
\(234\) −18.3482 −1.19946
\(235\) −6.41147 −0.418238
\(236\) −1.16519 −0.0758475
\(237\) 34.1070 2.21549
\(238\) 0.218941 0.0141918
\(239\) 2.35235 0.152161 0.0760804 0.997102i \(-0.475759\pi\)
0.0760804 + 0.997102i \(0.475759\pi\)
\(240\) −9.10607 −0.587794
\(241\) −13.8007 −0.888979 −0.444489 0.895784i \(-0.646615\pi\)
−0.444489 + 0.895784i \(0.646615\pi\)
\(242\) −8.14022 −0.523273
\(243\) −10.8033 −0.693035
\(244\) −1.68685 −0.107990
\(245\) −6.04963 −0.386497
\(246\) −9.60401 −0.612329
\(247\) 0 0
\(248\) −20.9172 −1.32824
\(249\) −42.7033 −2.70621
\(250\) −10.9317 −0.691382
\(251\) −4.16519 −0.262905 −0.131452 0.991322i \(-0.541964\pi\)
−0.131452 + 0.991322i \(0.541964\pi\)
\(252\) −0.339556 −0.0213900
\(253\) 6.00000 0.377217
\(254\) −15.6340 −0.980967
\(255\) 1.18479 0.0741946
\(256\) −4.39599 −0.274750
\(257\) −0.667252 −0.0416220 −0.0208110 0.999783i \(-0.506625\pi\)
−0.0208110 + 0.999783i \(0.506625\pi\)
\(258\) 15.1361 0.942332
\(259\) −1.71688 −0.106682
\(260\) 0.418281 0.0259407
\(261\) 36.3979 2.25297
\(262\) 2.48608 0.153591
\(263\) 11.3996 0.702930 0.351465 0.936201i \(-0.385684\pi\)
0.351465 + 0.936201i \(0.385684\pi\)
\(264\) 18.8726 1.16153
\(265\) 2.49525 0.153282
\(266\) 0 0
\(267\) 29.6313 1.81341
\(268\) 1.41828 0.0866353
\(269\) −19.3901 −1.18224 −0.591118 0.806585i \(-0.701313\pi\)
−0.591118 + 0.806585i \(0.701313\pi\)
\(270\) 7.81521 0.475618
\(271\) −13.3942 −0.813642 −0.406821 0.913508i \(-0.633363\pi\)
−0.406821 + 0.913508i \(0.633363\pi\)
\(272\) −1.68273 −0.102031
\(273\) −2.57398 −0.155784
\(274\) 0.344608 0.0208185
\(275\) 9.41147 0.567533
\(276\) 1.43376 0.0863024
\(277\) 17.7469 1.06631 0.533154 0.846018i \(-0.321007\pi\)
0.533154 + 0.846018i \(0.321007\pi\)
\(278\) 5.74422 0.344516
\(279\) 37.5972 2.25089
\(280\) −0.898986 −0.0537247
\(281\) −18.2790 −1.09043 −0.545217 0.838295i \(-0.683553\pi\)
−0.545217 + 0.838295i \(0.683553\pi\)
\(282\) −28.2841 −1.68429
\(283\) 7.68779 0.456991 0.228496 0.973545i \(-0.426619\pi\)
0.228496 + 0.973545i \(0.426619\pi\)
\(284\) −1.71957 −0.102038
\(285\) 0 0
\(286\) 7.72193 0.456608
\(287\) −0.859785 −0.0507515
\(288\) 5.51249 0.324827
\(289\) −16.7811 −0.987121
\(290\) 8.15064 0.478622
\(291\) −27.2199 −1.59566
\(292\) −0.256711 −0.0150229
\(293\) 10.5030 0.613591 0.306796 0.951775i \(-0.400743\pi\)
0.306796 + 0.951775i \(0.400743\pi\)
\(294\) −26.6878 −1.55646
\(295\) 5.54488 0.322836
\(296\) 14.5517 0.845800
\(297\) −14.6878 −0.852272
\(298\) 22.3105 1.29241
\(299\) 6.93582 0.401109
\(300\) 2.24897 0.129844
\(301\) 1.35504 0.0781030
\(302\) 5.87670 0.338166
\(303\) −26.6313 −1.52993
\(304\) 0 0
\(305\) 8.02734 0.459644
\(306\) 3.33544 0.190674
\(307\) 11.6955 0.667499 0.333749 0.942662i \(-0.391686\pi\)
0.333749 + 0.942662i \(0.391686\pi\)
\(308\) 0.142903 0.00814268
\(309\) 15.8648 0.902519
\(310\) 8.41921 0.478179
\(311\) 15.9659 0.905340 0.452670 0.891678i \(-0.350471\pi\)
0.452670 + 0.891678i \(0.350471\pi\)
\(312\) 21.8161 1.23510
\(313\) −26.6287 −1.50514 −0.752570 0.658512i \(-0.771186\pi\)
−0.752570 + 0.658512i \(0.771186\pi\)
\(314\) −12.9590 −0.731321
\(315\) 1.61587 0.0910438
\(316\) −2.18891 −0.123136
\(317\) −29.5321 −1.65869 −0.829344 0.558739i \(-0.811285\pi\)
−0.829344 + 0.558739i \(0.811285\pi\)
\(318\) 11.0077 0.617283
\(319\) −15.3182 −0.857655
\(320\) 7.55943 0.422585
\(321\) −29.4688 −1.64479
\(322\) −1.26083 −0.0702633
\(323\) 0 0
\(324\) −0.576666 −0.0320370
\(325\) 10.8794 0.603480
\(326\) 11.2567 0.623452
\(327\) 5.24897 0.290269
\(328\) 7.28724 0.402370
\(329\) −2.53209 −0.139599
\(330\) −7.59627 −0.418161
\(331\) −27.6655 −1.52063 −0.760317 0.649553i \(-0.774956\pi\)
−0.760317 + 0.649553i \(0.774956\pi\)
\(332\) 2.74060 0.150410
\(333\) −26.1557 −1.43332
\(334\) −5.43645 −0.297469
\(335\) −6.74928 −0.368752
\(336\) −3.59627 −0.196192
\(337\) −17.8598 −0.972884 −0.486442 0.873713i \(-0.661706\pi\)
−0.486442 + 0.873713i \(0.661706\pi\)
\(338\) −8.58853 −0.467154
\(339\) 50.9299 2.76614
\(340\) −0.0760373 −0.00412370
\(341\) −15.8229 −0.856861
\(342\) 0 0
\(343\) −4.82026 −0.260270
\(344\) −11.4848 −0.619220
\(345\) −6.82295 −0.367335
\(346\) 27.1397 1.45904
\(347\) −5.80066 −0.311396 −0.155698 0.987805i \(-0.549763\pi\)
−0.155698 + 0.987805i \(0.549763\pi\)
\(348\) −3.66044 −0.196220
\(349\) 5.37227 0.287571 0.143786 0.989609i \(-0.454072\pi\)
0.143786 + 0.989609i \(0.454072\pi\)
\(350\) −1.97771 −0.105713
\(351\) −16.9786 −0.906253
\(352\) −2.31996 −0.123654
\(353\) −25.2344 −1.34309 −0.671546 0.740963i \(-0.734370\pi\)
−0.671546 + 0.740963i \(0.734370\pi\)
\(354\) 24.4611 1.30009
\(355\) 8.18304 0.434311
\(356\) −1.90167 −0.100789
\(357\) 0.467911 0.0247645
\(358\) 15.5107 0.819768
\(359\) 6.68685 0.352919 0.176459 0.984308i \(-0.443536\pi\)
0.176459 + 0.984308i \(0.443536\pi\)
\(360\) −13.6955 −0.721818
\(361\) 0 0
\(362\) −11.5021 −0.604535
\(363\) −17.3969 −0.913102
\(364\) 0.165192 0.00865842
\(365\) 1.22163 0.0639430
\(366\) 35.4124 1.85104
\(367\) 8.11886 0.423801 0.211901 0.977291i \(-0.432035\pi\)
0.211901 + 0.977291i \(0.432035\pi\)
\(368\) 9.69047 0.505151
\(369\) −13.0983 −0.681872
\(370\) −5.85710 −0.304496
\(371\) 0.985452 0.0511621
\(372\) −3.78106 −0.196039
\(373\) −34.8976 −1.80693 −0.903463 0.428665i \(-0.858984\pi\)
−0.903463 + 0.428665i \(0.858984\pi\)
\(374\) −1.40373 −0.0725853
\(375\) −23.3628 −1.20645
\(376\) 21.4611 1.10677
\(377\) −17.7074 −0.911977
\(378\) 3.08647 0.158751
\(379\) −1.70140 −0.0873950 −0.0436975 0.999045i \(-0.513914\pi\)
−0.0436975 + 0.999045i \(0.513914\pi\)
\(380\) 0 0
\(381\) −33.4124 −1.71177
\(382\) 24.7033 1.26393
\(383\) −2.93676 −0.150061 −0.0750306 0.997181i \(-0.523905\pi\)
−0.0750306 + 0.997181i \(0.523905\pi\)
\(384\) 27.3482 1.39561
\(385\) −0.680045 −0.0346583
\(386\) −0.401045 −0.0204127
\(387\) 20.6432 1.04935
\(388\) 1.74691 0.0886860
\(389\) −24.5672 −1.24561 −0.622803 0.782379i \(-0.714006\pi\)
−0.622803 + 0.782379i \(0.714006\pi\)
\(390\) −8.78106 −0.444646
\(391\) −1.26083 −0.0637629
\(392\) 20.2499 1.02277
\(393\) 5.31315 0.268013
\(394\) 17.7050 0.891966
\(395\) 10.4165 0.524112
\(396\) 2.17705 0.109401
\(397\) −31.8357 −1.59779 −0.798895 0.601470i \(-0.794582\pi\)
−0.798895 + 0.601470i \(0.794582\pi\)
\(398\) −0.345866 −0.0173367
\(399\) 0 0
\(400\) 15.2003 0.760014
\(401\) 0.0864665 0.00431793 0.00215896 0.999998i \(-0.499313\pi\)
0.00215896 + 0.999998i \(0.499313\pi\)
\(402\) −29.7743 −1.48500
\(403\) −18.2909 −0.911133
\(404\) 1.70914 0.0850329
\(405\) 2.74422 0.136362
\(406\) 3.21894 0.159753
\(407\) 11.0077 0.545633
\(408\) −3.96585 −0.196339
\(409\) −20.0060 −0.989232 −0.494616 0.869112i \(-0.664691\pi\)
−0.494616 + 0.869112i \(0.664691\pi\)
\(410\) −2.93313 −0.144857
\(411\) 0.736482 0.0363280
\(412\) −1.01817 −0.0501616
\(413\) 2.18984 0.107755
\(414\) −19.2080 −0.944022
\(415\) −13.0419 −0.640201
\(416\) −2.68180 −0.131486
\(417\) 12.2763 0.601174
\(418\) 0 0
\(419\) 25.4097 1.24135 0.620673 0.784070i \(-0.286859\pi\)
0.620673 + 0.784070i \(0.286859\pi\)
\(420\) −0.162504 −0.00792937
\(421\) −4.36959 −0.212961 −0.106480 0.994315i \(-0.533958\pi\)
−0.106480 + 0.994315i \(0.533958\pi\)
\(422\) −3.29860 −0.160573
\(423\) −38.5749 −1.87558
\(424\) −8.35235 −0.405626
\(425\) −1.97771 −0.0959331
\(426\) 36.0993 1.74901
\(427\) 3.17024 0.153419
\(428\) 1.89124 0.0914168
\(429\) 16.5030 0.796772
\(430\) 4.62267 0.222925
\(431\) 38.3063 1.84515 0.922576 0.385816i \(-0.126080\pi\)
0.922576 + 0.385816i \(0.126080\pi\)
\(432\) −23.7219 −1.14132
\(433\) 18.1310 0.871322 0.435661 0.900111i \(-0.356515\pi\)
0.435661 + 0.900111i \(0.356515\pi\)
\(434\) 3.32501 0.159605
\(435\) 17.4192 0.835187
\(436\) −0.336867 −0.0161330
\(437\) 0 0
\(438\) 5.38919 0.257505
\(439\) 6.09059 0.290688 0.145344 0.989381i \(-0.453571\pi\)
0.145344 + 0.989381i \(0.453571\pi\)
\(440\) 5.76382 0.274779
\(441\) −36.3979 −1.73323
\(442\) −1.62267 −0.0771827
\(443\) −29.8931 −1.42026 −0.710132 0.704068i \(-0.751365\pi\)
−0.710132 + 0.704068i \(0.751365\pi\)
\(444\) 2.63041 0.124834
\(445\) 9.04963 0.428994
\(446\) 11.4652 0.542894
\(447\) 47.6810 2.25523
\(448\) 2.98545 0.141049
\(449\) 11.2499 0.530916 0.265458 0.964122i \(-0.414477\pi\)
0.265458 + 0.964122i \(0.414477\pi\)
\(450\) −30.1293 −1.42031
\(451\) 5.51249 0.259573
\(452\) −3.26857 −0.153741
\(453\) 12.5594 0.590093
\(454\) 19.0651 0.894770
\(455\) −0.786112 −0.0368535
\(456\) 0 0
\(457\) −23.3901 −1.09414 −0.547072 0.837086i \(-0.684258\pi\)
−0.547072 + 0.837086i \(0.684258\pi\)
\(458\) −27.6641 −1.29266
\(459\) 3.08647 0.144064
\(460\) 0.437882 0.0204163
\(461\) −36.6236 −1.70573 −0.852866 0.522130i \(-0.825137\pi\)
−0.852866 + 0.522130i \(0.825137\pi\)
\(462\) −3.00000 −0.139573
\(463\) −42.9864 −1.99775 −0.998873 0.0474549i \(-0.984889\pi\)
−0.998873 + 0.0474549i \(0.984889\pi\)
\(464\) −24.7401 −1.14853
\(465\) 17.9932 0.834414
\(466\) 23.7811 1.10164
\(467\) 25.5963 1.18445 0.592227 0.805771i \(-0.298249\pi\)
0.592227 + 0.805771i \(0.298249\pi\)
\(468\) 2.51661 0.116330
\(469\) −2.66550 −0.123081
\(470\) −8.63816 −0.398448
\(471\) −27.6955 −1.27614
\(472\) −18.5604 −0.854310
\(473\) −8.68779 −0.399465
\(474\) 45.9522 2.11066
\(475\) 0 0
\(476\) −0.0300295 −0.00137640
\(477\) 15.0128 0.687389
\(478\) 3.16931 0.144961
\(479\) 38.1762 1.74432 0.872158 0.489224i \(-0.162720\pi\)
0.872158 + 0.489224i \(0.162720\pi\)
\(480\) 2.63816 0.120415
\(481\) 12.7246 0.580193
\(482\) −18.5936 −0.846914
\(483\) −2.69459 −0.122608
\(484\) 1.11650 0.0507498
\(485\) −8.31315 −0.377481
\(486\) −14.5553 −0.660242
\(487\) −7.76382 −0.351812 −0.175906 0.984407i \(-0.556286\pi\)
−0.175906 + 0.984407i \(0.556286\pi\)
\(488\) −26.8699 −1.21634
\(489\) 24.0574 1.08791
\(490\) −8.15064 −0.368209
\(491\) −36.7229 −1.65728 −0.828640 0.559782i \(-0.810885\pi\)
−0.828640 + 0.559782i \(0.810885\pi\)
\(492\) 1.31727 0.0593870
\(493\) 3.21894 0.144974
\(494\) 0 0
\(495\) −10.3601 −0.465651
\(496\) −25.5553 −1.14747
\(497\) 3.23173 0.144963
\(498\) −57.5340 −2.57816
\(499\) 4.92633 0.220533 0.110266 0.993902i \(-0.464830\pi\)
0.110266 + 0.993902i \(0.464830\pi\)
\(500\) 1.49937 0.0670539
\(501\) −11.6186 −0.519079
\(502\) −5.61175 −0.250465
\(503\) 32.9495 1.46915 0.734574 0.678529i \(-0.237382\pi\)
0.734574 + 0.678529i \(0.237382\pi\)
\(504\) −5.40879 −0.240926
\(505\) −8.13341 −0.361932
\(506\) 8.08378 0.359368
\(507\) −18.3550 −0.815176
\(508\) 2.14433 0.0951394
\(509\) 36.9350 1.63712 0.818558 0.574424i \(-0.194774\pi\)
0.818558 + 0.574424i \(0.194774\pi\)
\(510\) 1.59627 0.0706839
\(511\) 0.482459 0.0213427
\(512\) −24.9186 −1.10126
\(513\) 0 0
\(514\) −0.898986 −0.0396526
\(515\) 4.84524 0.213507
\(516\) −2.07604 −0.0913924
\(517\) 16.2344 0.713989
\(518\) −2.31315 −0.101634
\(519\) 58.0019 2.54600
\(520\) 6.66281 0.292183
\(521\) −9.29179 −0.407081 −0.203540 0.979067i \(-0.565245\pi\)
−0.203540 + 0.979067i \(0.565245\pi\)
\(522\) 49.0387 2.14637
\(523\) 28.4151 1.24251 0.621253 0.783610i \(-0.286624\pi\)
0.621253 + 0.783610i \(0.286624\pi\)
\(524\) −0.340986 −0.0148960
\(525\) −4.22668 −0.184468
\(526\) 15.3587 0.669669
\(527\) 3.32501 0.144840
\(528\) 23.0574 1.00344
\(529\) −15.7392 −0.684312
\(530\) 3.36184 0.146029
\(531\) 33.3610 1.44775
\(532\) 0 0
\(533\) 6.37227 0.276014
\(534\) 39.9222 1.72760
\(535\) −9.00000 −0.389104
\(536\) 22.5918 0.975818
\(537\) 33.1489 1.43048
\(538\) −26.1242 −1.12630
\(539\) 15.3182 0.659802
\(540\) −1.07192 −0.0461280
\(541\) 14.9855 0.644275 0.322137 0.946693i \(-0.395599\pi\)
0.322137 + 0.946693i \(0.395599\pi\)
\(542\) −18.0460 −0.775142
\(543\) −24.5817 −1.05490
\(544\) 0.487511 0.0209019
\(545\) 1.60307 0.0686681
\(546\) −3.46791 −0.148413
\(547\) 3.88713 0.166202 0.0831008 0.996541i \(-0.473518\pi\)
0.0831008 + 0.996541i \(0.473518\pi\)
\(548\) −0.0472658 −0.00201909
\(549\) 48.2968 2.06126
\(550\) 12.6800 0.540679
\(551\) 0 0
\(552\) 22.8384 0.972068
\(553\) 4.11381 0.174937
\(554\) 23.9103 1.01585
\(555\) −12.5175 −0.531340
\(556\) −0.787866 −0.0334130
\(557\) 13.2044 0.559488 0.279744 0.960075i \(-0.409750\pi\)
0.279744 + 0.960075i \(0.409750\pi\)
\(558\) 50.6546 2.14438
\(559\) −10.0428 −0.424766
\(560\) −1.09833 −0.0464127
\(561\) −3.00000 −0.126660
\(562\) −24.6272 −1.03884
\(563\) −10.7128 −0.451489 −0.225745 0.974187i \(-0.572481\pi\)
−0.225745 + 0.974187i \(0.572481\pi\)
\(564\) 3.87939 0.163352
\(565\) 15.5544 0.654378
\(566\) 10.3577 0.435368
\(567\) 1.08378 0.0455144
\(568\) −27.3911 −1.14930
\(569\) 13.4706 0.564717 0.282358 0.959309i \(-0.408883\pi\)
0.282358 + 0.959309i \(0.408883\pi\)
\(570\) 0 0
\(571\) 12.6655 0.530035 0.265017 0.964244i \(-0.414622\pi\)
0.265017 + 0.964244i \(0.414622\pi\)
\(572\) −1.05913 −0.0442843
\(573\) 52.7948 2.20553
\(574\) −1.15839 −0.0483501
\(575\) 11.3892 0.474962
\(576\) 45.4816 1.89507
\(577\) −10.5544 −0.439384 −0.219692 0.975569i \(-0.570505\pi\)
−0.219692 + 0.975569i \(0.570505\pi\)
\(578\) −22.6091 −0.940413
\(579\) −0.857097 −0.0356197
\(580\) −1.11793 −0.0464194
\(581\) −5.15064 −0.213685
\(582\) −36.6732 −1.52015
\(583\) −6.31820 −0.261673
\(584\) −4.08915 −0.169210
\(585\) −11.9760 −0.495145
\(586\) 14.1506 0.584558
\(587\) −19.1548 −0.790602 −0.395301 0.918552i \(-0.629360\pi\)
−0.395301 + 0.918552i \(0.629360\pi\)
\(588\) 3.66044 0.150954
\(589\) 0 0
\(590\) 7.47060 0.307560
\(591\) 37.8384 1.55647
\(592\) 17.7784 0.730687
\(593\) −8.69459 −0.357044 −0.178522 0.983936i \(-0.557132\pi\)
−0.178522 + 0.983936i \(0.557132\pi\)
\(594\) −19.7888 −0.811944
\(595\) 0.142903 0.00585847
\(596\) −3.06006 −0.125345
\(597\) −0.739170 −0.0302522
\(598\) 9.34461 0.382129
\(599\) −19.8316 −0.810298 −0.405149 0.914251i \(-0.632780\pi\)
−0.405149 + 0.914251i \(0.632780\pi\)
\(600\) 35.8239 1.46250
\(601\) 33.7615 1.37716 0.688579 0.725161i \(-0.258235\pi\)
0.688579 + 0.725161i \(0.258235\pi\)
\(602\) 1.82564 0.0744074
\(603\) −40.6073 −1.65366
\(604\) −0.806036 −0.0327971
\(605\) −5.31315 −0.216010
\(606\) −35.8803 −1.45754
\(607\) −35.2850 −1.43217 −0.716087 0.698011i \(-0.754068\pi\)
−0.716087 + 0.698011i \(0.754068\pi\)
\(608\) 0 0
\(609\) 6.87939 0.278767
\(610\) 10.8152 0.437895
\(611\) 18.7665 0.759212
\(612\) −0.457482 −0.0184926
\(613\) 18.4534 0.745324 0.372662 0.927967i \(-0.378445\pi\)
0.372662 + 0.927967i \(0.378445\pi\)
\(614\) 15.7573 0.635914
\(615\) −6.26857 −0.252773
\(616\) 2.27631 0.0917152
\(617\) −35.6854 −1.43664 −0.718320 0.695712i \(-0.755089\pi\)
−0.718320 + 0.695712i \(0.755089\pi\)
\(618\) 21.3746 0.859814
\(619\) 3.65951 0.147088 0.0735441 0.997292i \(-0.476569\pi\)
0.0735441 + 0.997292i \(0.476569\pi\)
\(620\) −1.15476 −0.0463764
\(621\) −17.7743 −0.713256
\(622\) 21.5107 0.862502
\(623\) 3.57398 0.143188
\(624\) 26.6536 1.06700
\(625\) 13.9982 0.559930
\(626\) −35.8767 −1.43392
\(627\) 0 0
\(628\) 1.77744 0.0709275
\(629\) −2.31315 −0.0922313
\(630\) 2.17705 0.0867358
\(631\) −0.793852 −0.0316028 −0.0158014 0.999875i \(-0.505030\pi\)
−0.0158014 + 0.999875i \(0.505030\pi\)
\(632\) −34.8672 −1.38694
\(633\) −7.04963 −0.280198
\(634\) −39.7885 −1.58020
\(635\) −10.2044 −0.404949
\(636\) −1.50980 −0.0598675
\(637\) 17.7074 0.701592
\(638\) −20.6382 −0.817072
\(639\) 49.2336 1.94765
\(640\) 8.35235 0.330156
\(641\) −29.3824 −1.16053 −0.580267 0.814426i \(-0.697052\pi\)
−0.580267 + 0.814426i \(0.697052\pi\)
\(642\) −39.7033 −1.56696
\(643\) 22.2139 0.876030 0.438015 0.898968i \(-0.355682\pi\)
0.438015 + 0.898968i \(0.355682\pi\)
\(644\) 0.172933 0.00681451
\(645\) 9.87939 0.389000
\(646\) 0 0
\(647\) 11.2591 0.442640 0.221320 0.975201i \(-0.428963\pi\)
0.221320 + 0.975201i \(0.428963\pi\)
\(648\) −9.18573 −0.360849
\(649\) −14.0401 −0.551123
\(650\) 14.6578 0.574924
\(651\) 7.10607 0.278509
\(652\) −1.54395 −0.0604657
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 7.07192 0.276534
\(655\) 1.62267 0.0634031
\(656\) 8.90310 0.347608
\(657\) 7.34998 0.286750
\(658\) −3.41147 −0.132993
\(659\) −28.0259 −1.09173 −0.545867 0.837872i \(-0.683800\pi\)
−0.545867 + 0.837872i \(0.683800\pi\)
\(660\) 1.04189 0.0405555
\(661\) −11.3678 −0.442157 −0.221079 0.975256i \(-0.570958\pi\)
−0.221079 + 0.975256i \(0.570958\pi\)
\(662\) −37.2736 −1.44868
\(663\) −3.46791 −0.134683
\(664\) 43.6551 1.69415
\(665\) 0 0
\(666\) −35.2395 −1.36550
\(667\) −18.5371 −0.717761
\(668\) 0.745653 0.0288502
\(669\) 24.5030 0.947340
\(670\) −9.09327 −0.351304
\(671\) −20.3259 −0.784674
\(672\) 1.04189 0.0401917
\(673\) 16.5672 0.638618 0.319309 0.947651i \(-0.396549\pi\)
0.319309 + 0.947651i \(0.396549\pi\)
\(674\) −24.0624 −0.926850
\(675\) −27.8803 −1.07311
\(676\) 1.17799 0.0453071
\(677\) 9.04963 0.347806 0.173903 0.984763i \(-0.444362\pi\)
0.173903 + 0.984763i \(0.444362\pi\)
\(678\) 68.6177 2.63525
\(679\) −3.28312 −0.125995
\(680\) −1.21120 −0.0464474
\(681\) 40.7452 1.56136
\(682\) −21.3182 −0.816316
\(683\) −8.73143 −0.334099 −0.167049 0.985949i \(-0.553424\pi\)
−0.167049 + 0.985949i \(0.553424\pi\)
\(684\) 0 0
\(685\) 0.224927 0.00859402
\(686\) −6.49432 −0.247954
\(687\) −59.1225 −2.25566
\(688\) −14.0315 −0.534944
\(689\) −7.30365 −0.278247
\(690\) −9.19253 −0.349954
\(691\) 34.7202 1.32082 0.660409 0.750906i \(-0.270383\pi\)
0.660409 + 0.750906i \(0.270383\pi\)
\(692\) −3.72243 −0.141506
\(693\) −4.09152 −0.155424
\(694\) −7.81521 −0.296661
\(695\) 3.74928 0.142218
\(696\) −58.3073 −2.21013
\(697\) −1.15839 −0.0438770
\(698\) 7.23804 0.273964
\(699\) 50.8239 1.92234
\(700\) 0.271259 0.0102526
\(701\) 39.4151 1.48869 0.744344 0.667797i \(-0.232762\pi\)
0.744344 + 0.667797i \(0.232762\pi\)
\(702\) −22.8753 −0.863371
\(703\) 0 0
\(704\) −19.1411 −0.721409
\(705\) −18.4611 −0.695285
\(706\) −33.9982 −1.27954
\(707\) −3.21213 −0.120805
\(708\) −3.35504 −0.126090
\(709\) −41.1215 −1.54435 −0.772176 0.635409i \(-0.780832\pi\)
−0.772176 + 0.635409i \(0.780832\pi\)
\(710\) 11.0250 0.413760
\(711\) 62.6715 2.35036
\(712\) −30.2918 −1.13523
\(713\) −19.1480 −0.717097
\(714\) 0.630415 0.0235927
\(715\) 5.04013 0.188490
\(716\) −2.12742 −0.0795055
\(717\) 6.77332 0.252954
\(718\) 9.00917 0.336219
\(719\) −42.3955 −1.58109 −0.790543 0.612407i \(-0.790201\pi\)
−0.790543 + 0.612407i \(0.790201\pi\)
\(720\) −16.7324 −0.623578
\(721\) 1.91353 0.0712637
\(722\) 0 0
\(723\) −39.7374 −1.47785
\(724\) 1.57760 0.0586310
\(725\) −29.0770 −1.07989
\(726\) −23.4388 −0.869896
\(727\) 51.6563 1.91583 0.957914 0.287057i \(-0.0926769\pi\)
0.957914 + 0.287057i \(0.0926769\pi\)
\(728\) 2.63135 0.0975243
\(729\) −40.4688 −1.49885
\(730\) 1.64590 0.0609174
\(731\) 1.82564 0.0675236
\(732\) −4.85710 −0.179523
\(733\) −22.9162 −0.846430 −0.423215 0.906029i \(-0.639098\pi\)
−0.423215 + 0.906029i \(0.639098\pi\)
\(734\) 10.9385 0.403748
\(735\) −17.4192 −0.642517
\(736\) −2.80747 −0.103485
\(737\) 17.0898 0.629510
\(738\) −17.6473 −0.649607
\(739\) 28.1266 1.03465 0.517327 0.855788i \(-0.326927\pi\)
0.517327 + 0.855788i \(0.326927\pi\)
\(740\) 0.803348 0.0295317
\(741\) 0 0
\(742\) 1.32770 0.0487413
\(743\) −6.13247 −0.224979 −0.112489 0.993653i \(-0.535882\pi\)
−0.112489 + 0.993653i \(0.535882\pi\)
\(744\) −60.2285 −2.20809
\(745\) 14.5621 0.533515
\(746\) −47.0173 −1.72143
\(747\) −78.4671 −2.87096
\(748\) 0.192533 0.00703972
\(749\) −3.55438 −0.129874
\(750\) −31.4766 −1.14936
\(751\) −5.64084 −0.205837 −0.102919 0.994690i \(-0.532818\pi\)
−0.102919 + 0.994690i \(0.532818\pi\)
\(752\) 26.2199 0.956140
\(753\) −11.9932 −0.437056
\(754\) −23.8571 −0.868824
\(755\) 3.83574 0.139597
\(756\) −0.423334 −0.0153965
\(757\) 15.6919 0.570332 0.285166 0.958478i \(-0.407951\pi\)
0.285166 + 0.958478i \(0.407951\pi\)
\(758\) −2.29229 −0.0832597
\(759\) 17.2763 0.627090
\(760\) 0 0
\(761\) 4.86484 0.176350 0.0881751 0.996105i \(-0.471896\pi\)
0.0881751 + 0.996105i \(0.471896\pi\)
\(762\) −45.0164 −1.63077
\(763\) 0.633103 0.0229199
\(764\) −3.38825 −0.122583
\(765\) 2.17705 0.0787115
\(766\) −3.95668 −0.142961
\(767\) −16.2300 −0.586031
\(768\) −12.6578 −0.456747
\(769\) 22.5321 0.812528 0.406264 0.913756i \(-0.366831\pi\)
0.406264 + 0.913756i \(0.366831\pi\)
\(770\) −0.916222 −0.0330183
\(771\) −1.92127 −0.0691930
\(772\) 0.0550065 0.00197973
\(773\) 26.4320 0.950693 0.475347 0.879799i \(-0.342323\pi\)
0.475347 + 0.879799i \(0.342323\pi\)
\(774\) 27.8125 0.999700
\(775\) −30.0351 −1.07889
\(776\) 27.8266 0.998916
\(777\) −4.94356 −0.177349
\(778\) −33.0993 −1.18667
\(779\) 0 0
\(780\) 1.20439 0.0431242
\(781\) −20.7202 −0.741426
\(782\) −1.69871 −0.0607458
\(783\) 45.3783 1.62169
\(784\) 24.7401 0.883575
\(785\) −8.45842 −0.301894
\(786\) 7.15839 0.255331
\(787\) 15.5577 0.554571 0.277286 0.960788i \(-0.410565\pi\)
0.277286 + 0.960788i \(0.410565\pi\)
\(788\) −2.42839 −0.0865077
\(789\) 32.8239 1.16856
\(790\) 14.0341 0.499313
\(791\) 6.14290 0.218417
\(792\) 34.6783 1.23224
\(793\) −23.4962 −0.834374
\(794\) −42.8922 −1.52219
\(795\) 7.18479 0.254818
\(796\) 0.0474383 0.00168141
\(797\) −33.4935 −1.18640 −0.593200 0.805055i \(-0.702136\pi\)
−0.593200 + 0.805055i \(0.702136\pi\)
\(798\) 0 0
\(799\) −3.41147 −0.120689
\(800\) −4.40373 −0.155695
\(801\) 54.4475 1.92381
\(802\) 0.116496 0.00411362
\(803\) −3.09327 −0.109159
\(804\) 4.08378 0.144024
\(805\) −0.822948 −0.0290051
\(806\) −24.6432 −0.868020
\(807\) −55.8316 −1.96537
\(808\) 27.2249 0.957770
\(809\) 41.1162 1.44557 0.722784 0.691074i \(-0.242862\pi\)
0.722784 + 0.691074i \(0.242862\pi\)
\(810\) 3.69728 0.129909
\(811\) −16.6878 −0.585987 −0.292994 0.956114i \(-0.594651\pi\)
−0.292994 + 0.956114i \(0.594651\pi\)
\(812\) −0.441504 −0.0154937
\(813\) −38.5672 −1.35261
\(814\) 14.8307 0.519815
\(815\) 7.34730 0.257365
\(816\) −4.84524 −0.169617
\(817\) 0 0
\(818\) −26.9540 −0.942424
\(819\) −4.72967 −0.165268
\(820\) 0.402303 0.0140490
\(821\) −31.3901 −1.09552 −0.547761 0.836635i \(-0.684520\pi\)
−0.547761 + 0.836635i \(0.684520\pi\)
\(822\) 0.992259 0.0346090
\(823\) 46.3259 1.61482 0.807410 0.589990i \(-0.200868\pi\)
0.807410 + 0.589990i \(0.200868\pi\)
\(824\) −16.2184 −0.564996
\(825\) 27.0993 0.943475
\(826\) 2.95037 0.102657
\(827\) −40.7588 −1.41732 −0.708661 0.705549i \(-0.750700\pi\)
−0.708661 + 0.705549i \(0.750700\pi\)
\(828\) 2.63453 0.0915564
\(829\) 35.4834 1.23239 0.616195 0.787594i \(-0.288673\pi\)
0.616195 + 0.787594i \(0.288673\pi\)
\(830\) −17.5713 −0.609908
\(831\) 51.1002 1.77265
\(832\) −22.1266 −0.767102
\(833\) −3.21894 −0.111530
\(834\) 16.5398 0.572727
\(835\) −3.54839 −0.122797
\(836\) 0 0
\(837\) 46.8735 1.62019
\(838\) 34.2344 1.18261
\(839\) 38.2026 1.31890 0.659451 0.751748i \(-0.270789\pi\)
0.659451 + 0.751748i \(0.270789\pi\)
\(840\) −2.58853 −0.0893126
\(841\) 18.3259 0.631929
\(842\) −5.88713 −0.202884
\(843\) −52.6323 −1.81275
\(844\) 0.452430 0.0155733
\(845\) −5.60576 −0.192844
\(846\) −51.9718 −1.78683
\(847\) −2.09833 −0.0720993
\(848\) −10.2044 −0.350420
\(849\) 22.1361 0.759709
\(850\) −2.66456 −0.0913938
\(851\) 13.3209 0.456634
\(852\) −4.95130 −0.169629
\(853\) −25.6016 −0.876584 −0.438292 0.898833i \(-0.644416\pi\)
−0.438292 + 0.898833i \(0.644416\pi\)
\(854\) 4.27126 0.146159
\(855\) 0 0
\(856\) 30.1257 1.02967
\(857\) 21.0865 0.720300 0.360150 0.932894i \(-0.382726\pi\)
0.360150 + 0.932894i \(0.382726\pi\)
\(858\) 22.2344 0.759071
\(859\) −19.5672 −0.667623 −0.333812 0.942640i \(-0.608335\pi\)
−0.333812 + 0.942640i \(0.608335\pi\)
\(860\) −0.634037 −0.0216205
\(861\) −2.47565 −0.0843700
\(862\) 51.6100 1.75784
\(863\) −4.94894 −0.168464 −0.0842319 0.996446i \(-0.526844\pi\)
−0.0842319 + 0.996446i \(0.526844\pi\)
\(864\) 6.87258 0.233810
\(865\) 17.7142 0.602301
\(866\) 24.4279 0.830093
\(867\) −48.3191 −1.64100
\(868\) −0.456052 −0.0154794
\(869\) −26.3756 −0.894730
\(870\) 23.4688 0.795668
\(871\) 19.7553 0.669381
\(872\) −5.36596 −0.181714
\(873\) −50.0164 −1.69280
\(874\) 0 0
\(875\) −2.81790 −0.0952623
\(876\) −0.739170 −0.0249742
\(877\) −1.21987 −0.0411922 −0.0205961 0.999788i \(-0.506556\pi\)
−0.0205961 + 0.999788i \(0.506556\pi\)
\(878\) 8.20582 0.276933
\(879\) 30.2422 1.02004
\(880\) 7.04189 0.237382
\(881\) 46.5030 1.56673 0.783363 0.621565i \(-0.213503\pi\)
0.783363 + 0.621565i \(0.213503\pi\)
\(882\) −49.0387 −1.65122
\(883\) 12.9249 0.434957 0.217479 0.976065i \(-0.430217\pi\)
0.217479 + 0.976065i \(0.430217\pi\)
\(884\) 0.222563 0.00748560
\(885\) 15.9659 0.536686
\(886\) −40.2749 −1.35306
\(887\) 23.2243 0.779796 0.389898 0.920858i \(-0.372510\pi\)
0.389898 + 0.920858i \(0.372510\pi\)
\(888\) 41.8999 1.40607
\(889\) −4.03003 −0.135163
\(890\) 12.1925 0.408695
\(891\) −6.94862 −0.232787
\(892\) −1.57255 −0.0526528
\(893\) 0 0
\(894\) 64.2404 2.14852
\(895\) 10.1239 0.338405
\(896\) 3.29860 0.110198
\(897\) 19.9709 0.666809
\(898\) 15.1570 0.505794
\(899\) 48.8854 1.63042
\(900\) 4.13247 0.137749
\(901\) 1.32770 0.0442320
\(902\) 7.42696 0.247291
\(903\) 3.90167 0.129840
\(904\) −52.0651 −1.73166
\(905\) −7.50744 −0.249556
\(906\) 16.9213 0.562172
\(907\) 39.9968 1.32807 0.664036 0.747700i \(-0.268842\pi\)
0.664036 + 0.747700i \(0.268842\pi\)
\(908\) −2.61493 −0.0867796
\(909\) −48.9350 −1.62307
\(910\) −1.05913 −0.0351097
\(911\) 18.7997 0.622863 0.311431 0.950269i \(-0.399192\pi\)
0.311431 + 0.950269i \(0.399192\pi\)
\(912\) 0 0
\(913\) 33.0232 1.09291
\(914\) −31.5134 −1.04237
\(915\) 23.1138 0.764119
\(916\) 3.79435 0.125369
\(917\) 0.640844 0.0211625
\(918\) 4.15839 0.137247
\(919\) 39.8316 1.31392 0.656962 0.753924i \(-0.271841\pi\)
0.656962 + 0.753924i \(0.271841\pi\)
\(920\) 6.97502 0.229960
\(921\) 33.6759 1.10966
\(922\) −49.3429 −1.62502
\(923\) −23.9519 −0.788387
\(924\) 0.411474 0.0135365
\(925\) 20.8949 0.687019
\(926\) −57.9154 −1.90322
\(927\) 29.1516 0.957463
\(928\) 7.16756 0.235287
\(929\) 26.9540 0.884332 0.442166 0.896933i \(-0.354210\pi\)
0.442166 + 0.896933i \(0.354210\pi\)
\(930\) 24.2422 0.794932
\(931\) 0 0
\(932\) −3.26176 −0.106843
\(933\) 45.9718 1.50505
\(934\) 34.4858 1.12841
\(935\) −0.916222 −0.0299637
\(936\) 40.0871 1.31029
\(937\) 2.62361 0.0857095 0.0428548 0.999081i \(-0.486355\pi\)
0.0428548 + 0.999081i \(0.486355\pi\)
\(938\) −3.59121 −0.117257
\(939\) −76.6742 −2.50217
\(940\) 1.18479 0.0386437
\(941\) 18.6696 0.608612 0.304306 0.952574i \(-0.401575\pi\)
0.304306 + 0.952574i \(0.401575\pi\)
\(942\) −37.3141 −1.21576
\(943\) 6.67087 0.217234
\(944\) −22.6759 −0.738039
\(945\) 2.01455 0.0655332
\(946\) −11.7050 −0.380563
\(947\) 8.39961 0.272951 0.136475 0.990643i \(-0.456423\pi\)
0.136475 + 0.990643i \(0.456423\pi\)
\(948\) −6.30272 −0.204703
\(949\) −3.57573 −0.116073
\(950\) 0 0
\(951\) −85.0343 −2.75742
\(952\) −0.478340 −0.0155031
\(953\) −33.6928 −1.09142 −0.545709 0.837975i \(-0.683740\pi\)
−0.545709 + 0.837975i \(0.683740\pi\)
\(954\) 20.2267 0.654863
\(955\) 16.1239 0.521758
\(956\) −0.434696 −0.0140591
\(957\) −44.1070 −1.42578
\(958\) 51.4347 1.66178
\(959\) 0.0888306 0.00286849
\(960\) 21.7665 0.702511
\(961\) 19.4962 0.628909
\(962\) 17.1438 0.552739
\(963\) −54.1489 −1.74492
\(964\) 2.55026 0.0821383
\(965\) −0.261764 −0.00842647
\(966\) −3.63041 −0.116807
\(967\) 11.7433 0.377639 0.188819 0.982012i \(-0.439534\pi\)
0.188819 + 0.982012i \(0.439534\pi\)
\(968\) 17.7847 0.571621
\(969\) 0 0
\(970\) −11.2003 −0.359619
\(971\) −12.8093 −0.411071 −0.205536 0.978650i \(-0.565894\pi\)
−0.205536 + 0.978650i \(0.565894\pi\)
\(972\) 1.99638 0.0640339
\(973\) 1.48070 0.0474692
\(974\) −10.4602 −0.335165
\(975\) 31.3259 1.00323
\(976\) −32.8280 −1.05080
\(977\) −14.5276 −0.464781 −0.232390 0.972623i \(-0.574655\pi\)
−0.232390 + 0.972623i \(0.574655\pi\)
\(978\) 32.4124 1.03643
\(979\) −22.9145 −0.732350
\(980\) 1.11793 0.0357108
\(981\) 9.64496 0.307940
\(982\) −49.4766 −1.57886
\(983\) 37.0502 1.18172 0.590860 0.806774i \(-0.298789\pi\)
0.590860 + 0.806774i \(0.298789\pi\)
\(984\) 20.9828 0.668906
\(985\) 11.5561 0.368209
\(986\) 4.33687 0.138114
\(987\) −7.29086 −0.232071
\(988\) 0 0
\(989\) −10.5134 −0.334307
\(990\) −13.9581 −0.443618
\(991\) −3.43140 −0.109002 −0.0545010 0.998514i \(-0.517357\pi\)
−0.0545010 + 0.998514i \(0.517357\pi\)
\(992\) 7.40373 0.235069
\(993\) −79.6596 −2.52792
\(994\) 4.35410 0.138104
\(995\) −0.225748 −0.00715669
\(996\) 7.89124 0.250044
\(997\) 12.7760 0.404620 0.202310 0.979322i \(-0.435155\pi\)
0.202310 + 0.979322i \(0.435155\pi\)
\(998\) 6.63722 0.210098
\(999\) −32.6091 −1.03170
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 361.2.a.h.1.2 3
3.2 odd 2 3249.2.a.s.1.2 3
4.3 odd 2 5776.2.a.bi.1.1 3
5.4 even 2 9025.2.a.x.1.2 3
19.2 odd 18 361.2.e.f.99.1 6
19.3 odd 18 19.2.e.a.9.1 6
19.4 even 9 361.2.e.a.54.1 6
19.5 even 9 361.2.e.a.234.1 6
19.6 even 9 361.2.e.h.245.1 6
19.7 even 3 361.2.c.h.68.2 6
19.8 odd 6 361.2.c.i.292.2 6
19.9 even 9 361.2.e.b.62.1 6
19.10 odd 18 361.2.e.f.62.1 6
19.11 even 3 361.2.c.h.292.2 6
19.12 odd 6 361.2.c.i.68.2 6
19.13 odd 18 19.2.e.a.17.1 yes 6
19.14 odd 18 361.2.e.g.234.1 6
19.15 odd 18 361.2.e.g.54.1 6
19.16 even 9 361.2.e.h.28.1 6
19.17 even 9 361.2.e.b.99.1 6
19.18 odd 2 361.2.a.g.1.2 3
57.32 even 18 171.2.u.c.55.1 6
57.41 even 18 171.2.u.c.28.1 6
57.56 even 2 3249.2.a.z.1.2 3
76.3 even 18 304.2.u.b.161.1 6
76.51 even 18 304.2.u.b.17.1 6
76.75 even 2 5776.2.a.br.1.3 3
95.3 even 36 475.2.u.a.199.1 12
95.13 even 36 475.2.u.a.74.2 12
95.22 even 36 475.2.u.a.199.2 12
95.32 even 36 475.2.u.a.74.1 12
95.79 odd 18 475.2.l.a.351.1 6
95.89 odd 18 475.2.l.a.226.1 6
95.94 odd 2 9025.2.a.bd.1.2 3
133.3 even 18 931.2.v.a.275.1 6
133.13 even 18 931.2.w.a.834.1 6
133.32 odd 18 931.2.x.a.226.1 6
133.41 even 18 931.2.w.a.883.1 6
133.51 odd 18 931.2.v.b.606.1 6
133.60 odd 18 931.2.v.b.275.1 6
133.79 odd 18 931.2.x.a.655.1 6
133.89 even 18 931.2.v.a.606.1 6
133.108 even 18 931.2.x.b.226.1 6
133.117 even 18 931.2.x.b.655.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
19.2.e.a.9.1 6 19.3 odd 18
19.2.e.a.17.1 yes 6 19.13 odd 18
171.2.u.c.28.1 6 57.41 even 18
171.2.u.c.55.1 6 57.32 even 18
304.2.u.b.17.1 6 76.51 even 18
304.2.u.b.161.1 6 76.3 even 18
361.2.a.g.1.2 3 19.18 odd 2
361.2.a.h.1.2 3 1.1 even 1 trivial
361.2.c.h.68.2 6 19.7 even 3
361.2.c.h.292.2 6 19.11 even 3
361.2.c.i.68.2 6 19.12 odd 6
361.2.c.i.292.2 6 19.8 odd 6
361.2.e.a.54.1 6 19.4 even 9
361.2.e.a.234.1 6 19.5 even 9
361.2.e.b.62.1 6 19.9 even 9
361.2.e.b.99.1 6 19.17 even 9
361.2.e.f.62.1 6 19.10 odd 18
361.2.e.f.99.1 6 19.2 odd 18
361.2.e.g.54.1 6 19.15 odd 18
361.2.e.g.234.1 6 19.14 odd 18
361.2.e.h.28.1 6 19.16 even 9
361.2.e.h.245.1 6 19.6 even 9
475.2.l.a.226.1 6 95.89 odd 18
475.2.l.a.351.1 6 95.79 odd 18
475.2.u.a.74.1 12 95.32 even 36
475.2.u.a.74.2 12 95.13 even 36
475.2.u.a.199.1 12 95.3 even 36
475.2.u.a.199.2 12 95.22 even 36
931.2.v.a.275.1 6 133.3 even 18
931.2.v.a.606.1 6 133.89 even 18
931.2.v.b.275.1 6 133.60 odd 18
931.2.v.b.606.1 6 133.51 odd 18
931.2.w.a.834.1 6 133.13 even 18
931.2.w.a.883.1 6 133.41 even 18
931.2.x.a.226.1 6 133.32 odd 18
931.2.x.a.655.1 6 133.79 odd 18
931.2.x.b.226.1 6 133.108 even 18
931.2.x.b.655.1 6 133.117 even 18
3249.2.a.s.1.2 3 3.2 odd 2
3249.2.a.z.1.2 3 57.56 even 2
5776.2.a.bi.1.1 3 4.3 odd 2
5776.2.a.br.1.3 3 76.75 even 2
9025.2.a.x.1.2 3 5.4 even 2
9025.2.a.bd.1.2 3 95.94 odd 2