Properties

Label 7942.2.a.x.1.1
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.4171892853\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 7942.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.56155 q^{6} -0.561553 q^{7} +1.00000 q^{8} +3.56155 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.56155 q^{6} -0.561553 q^{7} +1.00000 q^{8} +3.56155 q^{9} +2.00000 q^{10} +1.00000 q^{11} -2.56155 q^{12} -0.561553 q^{13} -0.561553 q^{14} -5.12311 q^{15} +1.00000 q^{16} -0.561553 q^{17} +3.56155 q^{18} +2.00000 q^{20} +1.43845 q^{21} +1.00000 q^{22} +1.43845 q^{23} -2.56155 q^{24} -1.00000 q^{25} -0.561553 q^{26} -1.43845 q^{27} -0.561553 q^{28} +5.68466 q^{29} -5.12311 q^{30} -2.00000 q^{31} +1.00000 q^{32} -2.56155 q^{33} -0.561553 q^{34} -1.12311 q^{35} +3.56155 q^{36} +5.12311 q^{37} +1.43845 q^{39} +2.00000 q^{40} -2.00000 q^{41} +1.43845 q^{42} +1.00000 q^{44} +7.12311 q^{45} +1.43845 q^{46} +8.00000 q^{47} -2.56155 q^{48} -6.68466 q^{49} -1.00000 q^{50} +1.43845 q^{51} -0.561553 q^{52} -12.8078 q^{53} -1.43845 q^{54} +2.00000 q^{55} -0.561553 q^{56} +5.68466 q^{58} +7.68466 q^{59} -5.12311 q^{60} +6.24621 q^{61} -2.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -1.12311 q^{65} -2.56155 q^{66} +7.68466 q^{67} -0.561553 q^{68} -3.68466 q^{69} -1.12311 q^{70} -6.00000 q^{71} +3.56155 q^{72} -9.68466 q^{73} +5.12311 q^{74} +2.56155 q^{75} -0.561553 q^{77} +1.43845 q^{78} +4.00000 q^{79} +2.00000 q^{80} -7.00000 q^{81} -2.00000 q^{82} +14.2462 q^{83} +1.43845 q^{84} -1.12311 q^{85} -14.5616 q^{87} +1.00000 q^{88} +0.876894 q^{89} +7.12311 q^{90} +0.315342 q^{91} +1.43845 q^{92} +5.12311 q^{93} +8.00000 q^{94} -2.56155 q^{96} +7.12311 q^{97} -6.68466 q^{98} +3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} + 2 q^{8} + 3 q^{9} + 4 q^{10} + 2 q^{11} - q^{12} + 3 q^{13} + 3 q^{14} - 2 q^{15} + 2 q^{16} + 3 q^{17} + 3 q^{18} + 4 q^{20} + 7 q^{21} + 2 q^{22} + 7 q^{23} - q^{24} - 2 q^{25} + 3 q^{26} - 7 q^{27} + 3 q^{28} - q^{29} - 2 q^{30} - 4 q^{31} + 2 q^{32} - q^{33} + 3 q^{34} + 6 q^{35} + 3 q^{36} + 2 q^{37} + 7 q^{39} + 4 q^{40} - 4 q^{41} + 7 q^{42} + 2 q^{44} + 6 q^{45} + 7 q^{46} + 16 q^{47} - q^{48} - q^{49} - 2 q^{50} + 7 q^{51} + 3 q^{52} - 5 q^{53} - 7 q^{54} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 2 q^{60} - 4 q^{61} - 4 q^{62} - 4 q^{63} + 2 q^{64} + 6 q^{65} - q^{66} + 3 q^{67} + 3 q^{68} + 5 q^{69} + 6 q^{70} - 12 q^{71} + 3 q^{72} - 7 q^{73} + 2 q^{74} + q^{75} + 3 q^{77} + 7 q^{78} + 8 q^{79} + 4 q^{80} - 14 q^{81} - 4 q^{82} + 12 q^{83} + 7 q^{84} + 6 q^{85} - 25 q^{87} + 2 q^{88} + 10 q^{89} + 6 q^{90} + 13 q^{91} + 7 q^{92} + 2 q^{93} + 16 q^{94} - q^{96} + 6 q^{97} - q^{98} + 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.56155 −1.47891 −0.739457 0.673204i \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −2.56155 −1.04575
\(7\) −0.561553 −0.212247 −0.106124 0.994353i \(-0.533844\pi\)
−0.106124 + 0.994353i \(0.533844\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.56155 1.18718
\(10\) 2.00000 0.632456
\(11\) 1.00000 0.301511
\(12\) −2.56155 −0.739457
\(13\) −0.561553 −0.155747 −0.0778734 0.996963i \(-0.524813\pi\)
−0.0778734 + 0.996963i \(0.524813\pi\)
\(14\) −0.561553 −0.150081
\(15\) −5.12311 −1.32278
\(16\) 1.00000 0.250000
\(17\) −0.561553 −0.136197 −0.0680983 0.997679i \(-0.521693\pi\)
−0.0680983 + 0.997679i \(0.521693\pi\)
\(18\) 3.56155 0.839466
\(19\) 0 0
\(20\) 2.00000 0.447214
\(21\) 1.43845 0.313895
\(22\) 1.00000 0.213201
\(23\) 1.43845 0.299937 0.149968 0.988691i \(-0.452083\pi\)
0.149968 + 0.988691i \(0.452083\pi\)
\(24\) −2.56155 −0.522875
\(25\) −1.00000 −0.200000
\(26\) −0.561553 −0.110130
\(27\) −1.43845 −0.276829
\(28\) −0.561553 −0.106124
\(29\) 5.68466 1.05561 0.527807 0.849364i \(-0.323014\pi\)
0.527807 + 0.849364i \(0.323014\pi\)
\(30\) −5.12311 −0.935347
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.56155 −0.445909
\(34\) −0.561553 −0.0963055
\(35\) −1.12311 −0.189839
\(36\) 3.56155 0.593592
\(37\) 5.12311 0.842233 0.421117 0.907006i \(-0.361638\pi\)
0.421117 + 0.907006i \(0.361638\pi\)
\(38\) 0 0
\(39\) 1.43845 0.230336
\(40\) 2.00000 0.316228
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 1.43845 0.221957
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 1.00000 0.150756
\(45\) 7.12311 1.06185
\(46\) 1.43845 0.212087
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −2.56155 −0.369728
\(49\) −6.68466 −0.954951
\(50\) −1.00000 −0.141421
\(51\) 1.43845 0.201423
\(52\) −0.561553 −0.0778734
\(53\) −12.8078 −1.75928 −0.879641 0.475638i \(-0.842217\pi\)
−0.879641 + 0.475638i \(0.842217\pi\)
\(54\) −1.43845 −0.195748
\(55\) 2.00000 0.269680
\(56\) −0.561553 −0.0750407
\(57\) 0 0
\(58\) 5.68466 0.746432
\(59\) 7.68466 1.00046 0.500229 0.865893i \(-0.333249\pi\)
0.500229 + 0.865893i \(0.333249\pi\)
\(60\) −5.12311 −0.661390
\(61\) 6.24621 0.799745 0.399873 0.916571i \(-0.369054\pi\)
0.399873 + 0.916571i \(0.369054\pi\)
\(62\) −2.00000 −0.254000
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) −1.12311 −0.139304
\(66\) −2.56155 −0.315305
\(67\) 7.68466 0.938830 0.469415 0.882978i \(-0.344465\pi\)
0.469415 + 0.882978i \(0.344465\pi\)
\(68\) −0.561553 −0.0680983
\(69\) −3.68466 −0.443581
\(70\) −1.12311 −0.134237
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 3.56155 0.419733
\(73\) −9.68466 −1.13350 −0.566752 0.823889i \(-0.691800\pi\)
−0.566752 + 0.823889i \(0.691800\pi\)
\(74\) 5.12311 0.595549
\(75\) 2.56155 0.295783
\(76\) 0 0
\(77\) −0.561553 −0.0639949
\(78\) 1.43845 0.162872
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 2.00000 0.223607
\(81\) −7.00000 −0.777778
\(82\) −2.00000 −0.220863
\(83\) 14.2462 1.56372 0.781862 0.623451i \(-0.214270\pi\)
0.781862 + 0.623451i \(0.214270\pi\)
\(84\) 1.43845 0.156947
\(85\) −1.12311 −0.121818
\(86\) 0 0
\(87\) −14.5616 −1.56116
\(88\) 1.00000 0.106600
\(89\) 0.876894 0.0929506 0.0464753 0.998919i \(-0.485201\pi\)
0.0464753 + 0.998919i \(0.485201\pi\)
\(90\) 7.12311 0.750841
\(91\) 0.315342 0.0330568
\(92\) 1.43845 0.149968
\(93\) 5.12311 0.531241
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −2.56155 −0.261437
\(97\) 7.12311 0.723242 0.361621 0.932325i \(-0.382223\pi\)
0.361621 + 0.932325i \(0.382223\pi\)
\(98\) −6.68466 −0.675252
\(99\) 3.56155 0.357950
\(100\) −1.00000 −0.100000
\(101\) −6.87689 −0.684277 −0.342138 0.939650i \(-0.611151\pi\)
−0.342138 + 0.939650i \(0.611151\pi\)
\(102\) 1.43845 0.142427
\(103\) 13.3693 1.31732 0.658659 0.752442i \(-0.271124\pi\)
0.658659 + 0.752442i \(0.271124\pi\)
\(104\) −0.561553 −0.0550648
\(105\) 2.87689 0.280756
\(106\) −12.8078 −1.24400
\(107\) 9.93087 0.960053 0.480027 0.877254i \(-0.340627\pi\)
0.480027 + 0.877254i \(0.340627\pi\)
\(108\) −1.43845 −0.138415
\(109\) 6.31534 0.604900 0.302450 0.953165i \(-0.402195\pi\)
0.302450 + 0.953165i \(0.402195\pi\)
\(110\) 2.00000 0.190693
\(111\) −13.1231 −1.24559
\(112\) −0.561553 −0.0530618
\(113\) 3.12311 0.293797 0.146899 0.989152i \(-0.453071\pi\)
0.146899 + 0.989152i \(0.453071\pi\)
\(114\) 0 0
\(115\) 2.87689 0.268272
\(116\) 5.68466 0.527807
\(117\) −2.00000 −0.184900
\(118\) 7.68466 0.707430
\(119\) 0.315342 0.0289073
\(120\) −5.12311 −0.467673
\(121\) 1.00000 0.0909091
\(122\) 6.24621 0.565505
\(123\) 5.12311 0.461935
\(124\) −2.00000 −0.179605
\(125\) −12.0000 −1.07331
\(126\) −2.00000 −0.178174
\(127\) −2.24621 −0.199319 −0.0996595 0.995022i \(-0.531775\pi\)
−0.0996595 + 0.995022i \(0.531775\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −1.12311 −0.0985029
\(131\) 1.12311 0.0981262 0.0490631 0.998796i \(-0.484376\pi\)
0.0490631 + 0.998796i \(0.484376\pi\)
\(132\) −2.56155 −0.222955
\(133\) 0 0
\(134\) 7.68466 0.663853
\(135\) −2.87689 −0.247604
\(136\) −0.561553 −0.0481528
\(137\) 12.5616 1.07321 0.536603 0.843835i \(-0.319707\pi\)
0.536603 + 0.843835i \(0.319707\pi\)
\(138\) −3.68466 −0.313659
\(139\) 7.36932 0.625057 0.312529 0.949908i \(-0.398824\pi\)
0.312529 + 0.949908i \(0.398824\pi\)
\(140\) −1.12311 −0.0949197
\(141\) −20.4924 −1.72577
\(142\) −6.00000 −0.503509
\(143\) −0.561553 −0.0469594
\(144\) 3.56155 0.296796
\(145\) 11.3693 0.944170
\(146\) −9.68466 −0.801508
\(147\) 17.1231 1.41229
\(148\) 5.12311 0.421117
\(149\) −6.87689 −0.563377 −0.281689 0.959506i \(-0.590894\pi\)
−0.281689 + 0.959506i \(0.590894\pi\)
\(150\) 2.56155 0.209150
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) −0.561553 −0.0452512
\(155\) −4.00000 −0.321288
\(156\) 1.43845 0.115168
\(157\) −14.4924 −1.15662 −0.578311 0.815817i \(-0.696288\pi\)
−0.578311 + 0.815817i \(0.696288\pi\)
\(158\) 4.00000 0.318223
\(159\) 32.8078 2.60182
\(160\) 2.00000 0.158114
\(161\) −0.807764 −0.0636607
\(162\) −7.00000 −0.549972
\(163\) 11.3693 0.890514 0.445257 0.895403i \(-0.353112\pi\)
0.445257 + 0.895403i \(0.353112\pi\)
\(164\) −2.00000 −0.156174
\(165\) −5.12311 −0.398833
\(166\) 14.2462 1.10572
\(167\) 0.630683 0.0488037 0.0244019 0.999702i \(-0.492232\pi\)
0.0244019 + 0.999702i \(0.492232\pi\)
\(168\) 1.43845 0.110979
\(169\) −12.6847 −0.975743
\(170\) −1.12311 −0.0861383
\(171\) 0 0
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −14.5616 −1.10391
\(175\) 0.561553 0.0424494
\(176\) 1.00000 0.0753778
\(177\) −19.6847 −1.47959
\(178\) 0.876894 0.0657260
\(179\) −5.75379 −0.430058 −0.215029 0.976608i \(-0.568985\pi\)
−0.215029 + 0.976608i \(0.568985\pi\)
\(180\) 7.12311 0.530925
\(181\) 10.8769 0.808473 0.404237 0.914654i \(-0.367537\pi\)
0.404237 + 0.914654i \(0.367537\pi\)
\(182\) 0.315342 0.0233747
\(183\) −16.0000 −1.18275
\(184\) 1.43845 0.106044
\(185\) 10.2462 0.753316
\(186\) 5.12311 0.375644
\(187\) −0.561553 −0.0410648
\(188\) 8.00000 0.583460
\(189\) 0.807764 0.0587562
\(190\) 0 0
\(191\) −8.31534 −0.601677 −0.300838 0.953675i \(-0.597267\pi\)
−0.300838 + 0.953675i \(0.597267\pi\)
\(192\) −2.56155 −0.184864
\(193\) −7.12311 −0.512732 −0.256366 0.966580i \(-0.582525\pi\)
−0.256366 + 0.966580i \(0.582525\pi\)
\(194\) 7.12311 0.511409
\(195\) 2.87689 0.206019
\(196\) −6.68466 −0.477476
\(197\) 18.2462 1.29999 0.649994 0.759939i \(-0.274771\pi\)
0.649994 + 0.759939i \(0.274771\pi\)
\(198\) 3.56155 0.253109
\(199\) 11.6847 0.828303 0.414152 0.910208i \(-0.364078\pi\)
0.414152 + 0.910208i \(0.364078\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −19.6847 −1.38845
\(202\) −6.87689 −0.483857
\(203\) −3.19224 −0.224051
\(204\) 1.43845 0.100711
\(205\) −4.00000 −0.279372
\(206\) 13.3693 0.931484
\(207\) 5.12311 0.356080
\(208\) −0.561553 −0.0389367
\(209\) 0 0
\(210\) 2.87689 0.198525
\(211\) 0.315342 0.0217090 0.0108545 0.999941i \(-0.496545\pi\)
0.0108545 + 0.999941i \(0.496545\pi\)
\(212\) −12.8078 −0.879641
\(213\) 15.3693 1.05309
\(214\) 9.93087 0.678860
\(215\) 0 0
\(216\) −1.43845 −0.0978739
\(217\) 1.12311 0.0762414
\(218\) 6.31534 0.427729
\(219\) 24.8078 1.67635
\(220\) 2.00000 0.134840
\(221\) 0.315342 0.0212122
\(222\) −13.1231 −0.880765
\(223\) −24.7386 −1.65662 −0.828311 0.560269i \(-0.810698\pi\)
−0.828311 + 0.560269i \(0.810698\pi\)
\(224\) −0.561553 −0.0375203
\(225\) −3.56155 −0.237437
\(226\) 3.12311 0.207746
\(227\) −28.1771 −1.87018 −0.935089 0.354412i \(-0.884681\pi\)
−0.935089 + 0.354412i \(0.884681\pi\)
\(228\) 0 0
\(229\) 12.8769 0.850929 0.425465 0.904975i \(-0.360111\pi\)
0.425465 + 0.904975i \(0.360111\pi\)
\(230\) 2.87689 0.189697
\(231\) 1.43845 0.0946429
\(232\) 5.68466 0.373216
\(233\) −28.7386 −1.88273 −0.941365 0.337389i \(-0.890456\pi\)
−0.941365 + 0.337389i \(0.890456\pi\)
\(234\) −2.00000 −0.130744
\(235\) 16.0000 1.04372
\(236\) 7.68466 0.500229
\(237\) −10.2462 −0.665563
\(238\) 0.315342 0.0204406
\(239\) 23.9309 1.54796 0.773980 0.633210i \(-0.218263\pi\)
0.773980 + 0.633210i \(0.218263\pi\)
\(240\) −5.12311 −0.330695
\(241\) 19.6155 1.26355 0.631774 0.775153i \(-0.282327\pi\)
0.631774 + 0.775153i \(0.282327\pi\)
\(242\) 1.00000 0.0642824
\(243\) 22.2462 1.42710
\(244\) 6.24621 0.399873
\(245\) −13.3693 −0.854134
\(246\) 5.12311 0.326637
\(247\) 0 0
\(248\) −2.00000 −0.127000
\(249\) −36.4924 −2.31261
\(250\) −12.0000 −0.758947
\(251\) 17.1231 1.08080 0.540400 0.841408i \(-0.318273\pi\)
0.540400 + 0.841408i \(0.318273\pi\)
\(252\) −2.00000 −0.125988
\(253\) 1.43845 0.0904344
\(254\) −2.24621 −0.140940
\(255\) 2.87689 0.180158
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) −2.87689 −0.178762
\(260\) −1.12311 −0.0696521
\(261\) 20.2462 1.25321
\(262\) 1.12311 0.0693857
\(263\) 8.87689 0.547373 0.273686 0.961819i \(-0.411757\pi\)
0.273686 + 0.961819i \(0.411757\pi\)
\(264\) −2.56155 −0.157653
\(265\) −25.6155 −1.57355
\(266\) 0 0
\(267\) −2.24621 −0.137466
\(268\) 7.68466 0.469415
\(269\) −2.87689 −0.175407 −0.0877037 0.996147i \(-0.527953\pi\)
−0.0877037 + 0.996147i \(0.527953\pi\)
\(270\) −2.87689 −0.175082
\(271\) 25.6847 1.56023 0.780116 0.625635i \(-0.215160\pi\)
0.780116 + 0.625635i \(0.215160\pi\)
\(272\) −0.561553 −0.0340491
\(273\) −0.807764 −0.0488881
\(274\) 12.5616 0.758871
\(275\) −1.00000 −0.0603023
\(276\) −3.68466 −0.221790
\(277\) 8.49242 0.510260 0.255130 0.966907i \(-0.417882\pi\)
0.255130 + 0.966907i \(0.417882\pi\)
\(278\) 7.36932 0.441982
\(279\) −7.12311 −0.426449
\(280\) −1.12311 −0.0671184
\(281\) 25.3693 1.51341 0.756703 0.653758i \(-0.226809\pi\)
0.756703 + 0.653758i \(0.226809\pi\)
\(282\) −20.4924 −1.22031
\(283\) 11.3693 0.675836 0.337918 0.941176i \(-0.390277\pi\)
0.337918 + 0.941176i \(0.390277\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −0.561553 −0.0332053
\(287\) 1.12311 0.0662948
\(288\) 3.56155 0.209867
\(289\) −16.6847 −0.981450
\(290\) 11.3693 0.667629
\(291\) −18.2462 −1.06961
\(292\) −9.68466 −0.566752
\(293\) −3.93087 −0.229644 −0.114822 0.993386i \(-0.536630\pi\)
−0.114822 + 0.993386i \(0.536630\pi\)
\(294\) 17.1231 0.998640
\(295\) 15.3693 0.894836
\(296\) 5.12311 0.297774
\(297\) −1.43845 −0.0834672
\(298\) −6.87689 −0.398368
\(299\) −0.807764 −0.0467142
\(300\) 2.56155 0.147891
\(301\) 0 0
\(302\) −4.00000 −0.230174
\(303\) 17.6155 1.01199
\(304\) 0 0
\(305\) 12.4924 0.715314
\(306\) −2.00000 −0.114332
\(307\) 22.2462 1.26966 0.634829 0.772653i \(-0.281070\pi\)
0.634829 + 0.772653i \(0.281070\pi\)
\(308\) −0.561553 −0.0319974
\(309\) −34.2462 −1.94820
\(310\) −4.00000 −0.227185
\(311\) −4.80776 −0.272623 −0.136312 0.990666i \(-0.543525\pi\)
−0.136312 + 0.990666i \(0.543525\pi\)
\(312\) 1.43845 0.0814360
\(313\) −0.561553 −0.0317408 −0.0158704 0.999874i \(-0.505052\pi\)
−0.0158704 + 0.999874i \(0.505052\pi\)
\(314\) −14.4924 −0.817855
\(315\) −4.00000 −0.225374
\(316\) 4.00000 0.225018
\(317\) 7.05398 0.396191 0.198095 0.980183i \(-0.436524\pi\)
0.198095 + 0.980183i \(0.436524\pi\)
\(318\) 32.8078 1.83977
\(319\) 5.68466 0.318280
\(320\) 2.00000 0.111803
\(321\) −25.4384 −1.41984
\(322\) −0.807764 −0.0450149
\(323\) 0 0
\(324\) −7.00000 −0.388889
\(325\) 0.561553 0.0311493
\(326\) 11.3693 0.629688
\(327\) −16.1771 −0.894595
\(328\) −2.00000 −0.110432
\(329\) −4.49242 −0.247675
\(330\) −5.12311 −0.282018
\(331\) 14.5616 0.800375 0.400188 0.916433i \(-0.368945\pi\)
0.400188 + 0.916433i \(0.368945\pi\)
\(332\) 14.2462 0.781862
\(333\) 18.2462 0.999886
\(334\) 0.630683 0.0345094
\(335\) 15.3693 0.839715
\(336\) 1.43845 0.0784737
\(337\) −11.1231 −0.605914 −0.302957 0.953004i \(-0.597974\pi\)
−0.302957 + 0.953004i \(0.597974\pi\)
\(338\) −12.6847 −0.689954
\(339\) −8.00000 −0.434500
\(340\) −1.12311 −0.0609090
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 7.68466 0.414933
\(344\) 0 0
\(345\) −7.36932 −0.396751
\(346\) −18.0000 −0.967686
\(347\) 9.61553 0.516189 0.258094 0.966120i \(-0.416905\pi\)
0.258094 + 0.966120i \(0.416905\pi\)
\(348\) −14.5616 −0.780581
\(349\) 21.6155 1.15705 0.578526 0.815664i \(-0.303628\pi\)
0.578526 + 0.815664i \(0.303628\pi\)
\(350\) 0.561553 0.0300163
\(351\) 0.807764 0.0431153
\(352\) 1.00000 0.0533002
\(353\) 22.1771 1.18037 0.590183 0.807269i \(-0.299055\pi\)
0.590183 + 0.807269i \(0.299055\pi\)
\(354\) −19.6847 −1.04623
\(355\) −12.0000 −0.636894
\(356\) 0.876894 0.0464753
\(357\) −0.807764 −0.0427514
\(358\) −5.75379 −0.304097
\(359\) −15.9309 −0.840799 −0.420400 0.907339i \(-0.638110\pi\)
−0.420400 + 0.907339i \(0.638110\pi\)
\(360\) 7.12311 0.375421
\(361\) 0 0
\(362\) 10.8769 0.571677
\(363\) −2.56155 −0.134447
\(364\) 0.315342 0.0165284
\(365\) −19.3693 −1.01384
\(366\) −16.0000 −0.836333
\(367\) 30.2462 1.57884 0.789420 0.613854i \(-0.210382\pi\)
0.789420 + 0.613854i \(0.210382\pi\)
\(368\) 1.43845 0.0749842
\(369\) −7.12311 −0.370814
\(370\) 10.2462 0.532675
\(371\) 7.19224 0.373402
\(372\) 5.12311 0.265621
\(373\) −3.43845 −0.178036 −0.0890180 0.996030i \(-0.528373\pi\)
−0.0890180 + 0.996030i \(0.528373\pi\)
\(374\) −0.561553 −0.0290372
\(375\) 30.7386 1.58734
\(376\) 8.00000 0.412568
\(377\) −3.19224 −0.164409
\(378\) 0.807764 0.0415469
\(379\) −6.56155 −0.337044 −0.168522 0.985698i \(-0.553899\pi\)
−0.168522 + 0.985698i \(0.553899\pi\)
\(380\) 0 0
\(381\) 5.75379 0.294776
\(382\) −8.31534 −0.425450
\(383\) 2.00000 0.102195 0.0510976 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) −2.56155 −0.130719
\(385\) −1.12311 −0.0572388
\(386\) −7.12311 −0.362557
\(387\) 0 0
\(388\) 7.12311 0.361621
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 2.87689 0.145677
\(391\) −0.807764 −0.0408504
\(392\) −6.68466 −0.337626
\(393\) −2.87689 −0.145120
\(394\) 18.2462 0.919231
\(395\) 8.00000 0.402524
\(396\) 3.56155 0.178975
\(397\) 38.9848 1.95659 0.978297 0.207209i \(-0.0664381\pi\)
0.978297 + 0.207209i \(0.0664381\pi\)
\(398\) 11.6847 0.585699
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 7.75379 0.387206 0.193603 0.981080i \(-0.437983\pi\)
0.193603 + 0.981080i \(0.437983\pi\)
\(402\) −19.6847 −0.981782
\(403\) 1.12311 0.0559459
\(404\) −6.87689 −0.342138
\(405\) −14.0000 −0.695666
\(406\) −3.19224 −0.158428
\(407\) 5.12311 0.253943
\(408\) 1.43845 0.0712137
\(409\) −16.2462 −0.803323 −0.401662 0.915788i \(-0.631567\pi\)
−0.401662 + 0.915788i \(0.631567\pi\)
\(410\) −4.00000 −0.197546
\(411\) −32.1771 −1.58718
\(412\) 13.3693 0.658659
\(413\) −4.31534 −0.212344
\(414\) 5.12311 0.251787
\(415\) 28.4924 1.39864
\(416\) −0.561553 −0.0275324
\(417\) −18.8769 −0.924405
\(418\) 0 0
\(419\) −14.8769 −0.726784 −0.363392 0.931636i \(-0.618381\pi\)
−0.363392 + 0.931636i \(0.618381\pi\)
\(420\) 2.87689 0.140378
\(421\) 29.9309 1.45874 0.729371 0.684119i \(-0.239813\pi\)
0.729371 + 0.684119i \(0.239813\pi\)
\(422\) 0.315342 0.0153506
\(423\) 28.4924 1.38535
\(424\) −12.8078 −0.622000
\(425\) 0.561553 0.0272393
\(426\) 15.3693 0.744646
\(427\) −3.50758 −0.169744
\(428\) 9.93087 0.480027
\(429\) 1.43845 0.0694489
\(430\) 0 0
\(431\) −31.8617 −1.53473 −0.767363 0.641213i \(-0.778432\pi\)
−0.767363 + 0.641213i \(0.778432\pi\)
\(432\) −1.43845 −0.0692073
\(433\) −6.63068 −0.318650 −0.159325 0.987226i \(-0.550932\pi\)
−0.159325 + 0.987226i \(0.550932\pi\)
\(434\) 1.12311 0.0539108
\(435\) −29.1231 −1.39635
\(436\) 6.31534 0.302450
\(437\) 0 0
\(438\) 24.8078 1.18536
\(439\) 9.61553 0.458924 0.229462 0.973318i \(-0.426303\pi\)
0.229462 + 0.973318i \(0.426303\pi\)
\(440\) 2.00000 0.0953463
\(441\) −23.8078 −1.13370
\(442\) 0.315342 0.0149993
\(443\) 39.8617 1.89389 0.946944 0.321398i \(-0.104153\pi\)
0.946944 + 0.321398i \(0.104153\pi\)
\(444\) −13.1231 −0.622795
\(445\) 1.75379 0.0831376
\(446\) −24.7386 −1.17141
\(447\) 17.6155 0.833186
\(448\) −0.561553 −0.0265309
\(449\) −23.6155 −1.11449 −0.557243 0.830350i \(-0.688141\pi\)
−0.557243 + 0.830350i \(0.688141\pi\)
\(450\) −3.56155 −0.167893
\(451\) −2.00000 −0.0941763
\(452\) 3.12311 0.146899
\(453\) 10.2462 0.481409
\(454\) −28.1771 −1.32242
\(455\) 0.630683 0.0295669
\(456\) 0 0
\(457\) 36.5616 1.71028 0.855139 0.518399i \(-0.173472\pi\)
0.855139 + 0.518399i \(0.173472\pi\)
\(458\) 12.8769 0.601698
\(459\) 0.807764 0.0377032
\(460\) 2.87689 0.134136
\(461\) 26.7386 1.24534 0.622671 0.782484i \(-0.286047\pi\)
0.622671 + 0.782484i \(0.286047\pi\)
\(462\) 1.43845 0.0669226
\(463\) 3.50758 0.163011 0.0815055 0.996673i \(-0.474027\pi\)
0.0815055 + 0.996673i \(0.474027\pi\)
\(464\) 5.68466 0.263904
\(465\) 10.2462 0.475157
\(466\) −28.7386 −1.33129
\(467\) 9.75379 0.451352 0.225676 0.974202i \(-0.427541\pi\)
0.225676 + 0.974202i \(0.427541\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −4.31534 −0.199264
\(470\) 16.0000 0.738025
\(471\) 37.1231 1.71054
\(472\) 7.68466 0.353715
\(473\) 0 0
\(474\) −10.2462 −0.470624
\(475\) 0 0
\(476\) 0.315342 0.0144537
\(477\) −45.6155 −2.08859
\(478\) 23.9309 1.09457
\(479\) −13.3693 −0.610860 −0.305430 0.952215i \(-0.598800\pi\)
−0.305430 + 0.952215i \(0.598800\pi\)
\(480\) −5.12311 −0.233837
\(481\) −2.87689 −0.131175
\(482\) 19.6155 0.893463
\(483\) 2.06913 0.0941487
\(484\) 1.00000 0.0454545
\(485\) 14.2462 0.646887
\(486\) 22.2462 1.00911
\(487\) −31.1231 −1.41032 −0.705161 0.709047i \(-0.749125\pi\)
−0.705161 + 0.709047i \(0.749125\pi\)
\(488\) 6.24621 0.282753
\(489\) −29.1231 −1.31699
\(490\) −13.3693 −0.603964
\(491\) −39.3693 −1.77671 −0.888356 0.459155i \(-0.848152\pi\)
−0.888356 + 0.459155i \(0.848152\pi\)
\(492\) 5.12311 0.230967
\(493\) −3.19224 −0.143771
\(494\) 0 0
\(495\) 7.12311 0.320160
\(496\) −2.00000 −0.0898027
\(497\) 3.36932 0.151135
\(498\) −36.4924 −1.63526
\(499\) 9.75379 0.436640 0.218320 0.975877i \(-0.429942\pi\)
0.218320 + 0.975877i \(0.429942\pi\)
\(500\) −12.0000 −0.536656
\(501\) −1.61553 −0.0721765
\(502\) 17.1231 0.764242
\(503\) 21.6847 0.966871 0.483436 0.875380i \(-0.339389\pi\)
0.483436 + 0.875380i \(0.339389\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −13.7538 −0.612036
\(506\) 1.43845 0.0639468
\(507\) 32.4924 1.44304
\(508\) −2.24621 −0.0996595
\(509\) 31.3693 1.39042 0.695210 0.718806i \(-0.255311\pi\)
0.695210 + 0.718806i \(0.255311\pi\)
\(510\) 2.87689 0.127391
\(511\) 5.43845 0.240583
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) 26.7386 1.17824
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) −2.87689 −0.126403
\(519\) 46.1080 2.02391
\(520\) −1.12311 −0.0492514
\(521\) 7.12311 0.312069 0.156034 0.987752i \(-0.450129\pi\)
0.156034 + 0.987752i \(0.450129\pi\)
\(522\) 20.2462 0.886153
\(523\) −31.0540 −1.35790 −0.678948 0.734187i \(-0.737564\pi\)
−0.678948 + 0.734187i \(0.737564\pi\)
\(524\) 1.12311 0.0490631
\(525\) −1.43845 −0.0627790
\(526\) 8.87689 0.387051
\(527\) 1.12311 0.0489232
\(528\) −2.56155 −0.111477
\(529\) −20.9309 −0.910038
\(530\) −25.6155 −1.11267
\(531\) 27.3693 1.18773
\(532\) 0 0
\(533\) 1.12311 0.0486471
\(534\) −2.24621 −0.0972031
\(535\) 19.8617 0.858698
\(536\) 7.68466 0.331927
\(537\) 14.7386 0.636019
\(538\) −2.87689 −0.124032
\(539\) −6.68466 −0.287929
\(540\) −2.87689 −0.123802
\(541\) −39.2311 −1.68667 −0.843337 0.537384i \(-0.819412\pi\)
−0.843337 + 0.537384i \(0.819412\pi\)
\(542\) 25.6847 1.10325
\(543\) −27.8617 −1.19566
\(544\) −0.561553 −0.0240764
\(545\) 12.6307 0.541039
\(546\) −0.807764 −0.0345691
\(547\) 42.7386 1.82737 0.913686 0.406421i \(-0.133223\pi\)
0.913686 + 0.406421i \(0.133223\pi\)
\(548\) 12.5616 0.536603
\(549\) 22.2462 0.949445
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) −3.68466 −0.156829
\(553\) −2.24621 −0.0955186
\(554\) 8.49242 0.360808
\(555\) −26.2462 −1.11409
\(556\) 7.36932 0.312529
\(557\) 17.1231 0.725529 0.362765 0.931881i \(-0.381833\pi\)
0.362765 + 0.931881i \(0.381833\pi\)
\(558\) −7.12311 −0.301545
\(559\) 0 0
\(560\) −1.12311 −0.0474599
\(561\) 1.43845 0.0607313
\(562\) 25.3693 1.07014
\(563\) 42.7386 1.80122 0.900609 0.434630i \(-0.143121\pi\)
0.900609 + 0.434630i \(0.143121\pi\)
\(564\) −20.4924 −0.862887
\(565\) 6.24621 0.262780
\(566\) 11.3693 0.477888
\(567\) 3.93087 0.165081
\(568\) −6.00000 −0.251754
\(569\) 21.3693 0.895848 0.447924 0.894072i \(-0.352163\pi\)
0.447924 + 0.894072i \(0.352163\pi\)
\(570\) 0 0
\(571\) −5.61553 −0.235003 −0.117501 0.993073i \(-0.537488\pi\)
−0.117501 + 0.993073i \(0.537488\pi\)
\(572\) −0.561553 −0.0234797
\(573\) 21.3002 0.889828
\(574\) 1.12311 0.0468775
\(575\) −1.43845 −0.0599874
\(576\) 3.56155 0.148398
\(577\) 41.0540 1.70910 0.854550 0.519370i \(-0.173833\pi\)
0.854550 + 0.519370i \(0.173833\pi\)
\(578\) −16.6847 −0.693990
\(579\) 18.2462 0.758287
\(580\) 11.3693 0.472085
\(581\) −8.00000 −0.331896
\(582\) −18.2462 −0.756330
\(583\) −12.8078 −0.530443
\(584\) −9.68466 −0.400754
\(585\) −4.00000 −0.165380
\(586\) −3.93087 −0.162383
\(587\) −40.4924 −1.67130 −0.835651 0.549261i \(-0.814909\pi\)
−0.835651 + 0.549261i \(0.814909\pi\)
\(588\) 17.1231 0.706145
\(589\) 0 0
\(590\) 15.3693 0.632745
\(591\) −46.7386 −1.92257
\(592\) 5.12311 0.210558
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) −1.43845 −0.0590202
\(595\) 0.630683 0.0258555
\(596\) −6.87689 −0.281689
\(597\) −29.9309 −1.22499
\(598\) −0.807764 −0.0330319
\(599\) 17.8617 0.729811 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(600\) 2.56155 0.104575
\(601\) −21.3693 −0.871673 −0.435836 0.900026i \(-0.643547\pi\)
−0.435836 + 0.900026i \(0.643547\pi\)
\(602\) 0 0
\(603\) 27.3693 1.11456
\(604\) −4.00000 −0.162758
\(605\) 2.00000 0.0813116
\(606\) 17.6155 0.715582
\(607\) 13.6155 0.552637 0.276319 0.961066i \(-0.410885\pi\)
0.276319 + 0.961066i \(0.410885\pi\)
\(608\) 0 0
\(609\) 8.17708 0.331352
\(610\) 12.4924 0.505803
\(611\) −4.49242 −0.181744
\(612\) −2.00000 −0.0808452
\(613\) 18.2462 0.736958 0.368479 0.929636i \(-0.379879\pi\)
0.368479 + 0.929636i \(0.379879\pi\)
\(614\) 22.2462 0.897784
\(615\) 10.2462 0.413167
\(616\) −0.561553 −0.0226256
\(617\) 44.7386 1.80111 0.900555 0.434743i \(-0.143161\pi\)
0.900555 + 0.434743i \(0.143161\pi\)
\(618\) −34.2462 −1.37758
\(619\) −9.75379 −0.392038 −0.196019 0.980600i \(-0.562801\pi\)
−0.196019 + 0.980600i \(0.562801\pi\)
\(620\) −4.00000 −0.160644
\(621\) −2.06913 −0.0830313
\(622\) −4.80776 −0.192774
\(623\) −0.492423 −0.0197285
\(624\) 1.43845 0.0575840
\(625\) −19.0000 −0.760000
\(626\) −0.561553 −0.0224442
\(627\) 0 0
\(628\) −14.4924 −0.578311
\(629\) −2.87689 −0.114709
\(630\) −4.00000 −0.159364
\(631\) 3.50758 0.139634 0.0698172 0.997560i \(-0.477758\pi\)
0.0698172 + 0.997560i \(0.477758\pi\)
\(632\) 4.00000 0.159111
\(633\) −0.807764 −0.0321057
\(634\) 7.05398 0.280149
\(635\) −4.49242 −0.178276
\(636\) 32.8078 1.30091
\(637\) 3.75379 0.148731
\(638\) 5.68466 0.225058
\(639\) −21.3693 −0.845357
\(640\) 2.00000 0.0790569
\(641\) −31.6155 −1.24874 −0.624369 0.781129i \(-0.714644\pi\)
−0.624369 + 0.781129i \(0.714644\pi\)
\(642\) −25.4384 −1.00398
\(643\) −12.6307 −0.498106 −0.249053 0.968490i \(-0.580119\pi\)
−0.249053 + 0.968490i \(0.580119\pi\)
\(644\) −0.807764 −0.0318304
\(645\) 0 0
\(646\) 0 0
\(647\) −47.5464 −1.86924 −0.934621 0.355646i \(-0.884261\pi\)
−0.934621 + 0.355646i \(0.884261\pi\)
\(648\) −7.00000 −0.274986
\(649\) 7.68466 0.301649
\(650\) 0.561553 0.0220259
\(651\) −2.87689 −0.112754
\(652\) 11.3693 0.445257
\(653\) −15.6155 −0.611083 −0.305541 0.952179i \(-0.598837\pi\)
−0.305541 + 0.952179i \(0.598837\pi\)
\(654\) −16.1771 −0.632574
\(655\) 2.24621 0.0877667
\(656\) −2.00000 −0.0780869
\(657\) −34.4924 −1.34568
\(658\) −4.49242 −0.175133
\(659\) 14.4233 0.561852 0.280926 0.959729i \(-0.409359\pi\)
0.280926 + 0.959729i \(0.409359\pi\)
\(660\) −5.12311 −0.199417
\(661\) 26.5616 1.03312 0.516562 0.856250i \(-0.327211\pi\)
0.516562 + 0.856250i \(0.327211\pi\)
\(662\) 14.5616 0.565951
\(663\) −0.807764 −0.0313710
\(664\) 14.2462 0.552860
\(665\) 0 0
\(666\) 18.2462 0.707026
\(667\) 8.17708 0.316618
\(668\) 0.630683 0.0244019
\(669\) 63.3693 2.45000
\(670\) 15.3693 0.593769
\(671\) 6.24621 0.241132
\(672\) 1.43845 0.0554893
\(673\) 27.1231 1.04552 0.522759 0.852480i \(-0.324903\pi\)
0.522759 + 0.852480i \(0.324903\pi\)
\(674\) −11.1231 −0.428446
\(675\) 1.43845 0.0553659
\(676\) −12.6847 −0.487871
\(677\) −6.17708 −0.237405 −0.118702 0.992930i \(-0.537873\pi\)
−0.118702 + 0.992930i \(0.537873\pi\)
\(678\) −8.00000 −0.307238
\(679\) −4.00000 −0.153506
\(680\) −1.12311 −0.0430691
\(681\) 72.1771 2.76583
\(682\) −2.00000 −0.0765840
\(683\) −28.4924 −1.09023 −0.545116 0.838361i \(-0.683514\pi\)
−0.545116 + 0.838361i \(0.683514\pi\)
\(684\) 0 0
\(685\) 25.1231 0.959905
\(686\) 7.68466 0.293402
\(687\) −32.9848 −1.25845
\(688\) 0 0
\(689\) 7.19224 0.274002
\(690\) −7.36932 −0.280545
\(691\) −2.38447 −0.0907096 −0.0453548 0.998971i \(-0.514442\pi\)
−0.0453548 + 0.998971i \(0.514442\pi\)
\(692\) −18.0000 −0.684257
\(693\) −2.00000 −0.0759737
\(694\) 9.61553 0.365000
\(695\) 14.7386 0.559068
\(696\) −14.5616 −0.551954
\(697\) 1.12311 0.0425407
\(698\) 21.6155 0.818160
\(699\) 73.6155 2.78439
\(700\) 0.561553 0.0212247
\(701\) −40.9848 −1.54798 −0.773988 0.633200i \(-0.781741\pi\)
−0.773988 + 0.633200i \(0.781741\pi\)
\(702\) 0.807764 0.0304871
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) −40.9848 −1.54358
\(706\) 22.1771 0.834645
\(707\) 3.86174 0.145236
\(708\) −19.6847 −0.739795
\(709\) −42.4924 −1.59584 −0.797918 0.602766i \(-0.794065\pi\)
−0.797918 + 0.602766i \(0.794065\pi\)
\(710\) −12.0000 −0.450352
\(711\) 14.2462 0.534275
\(712\) 0.876894 0.0328630
\(713\) −2.87689 −0.107741
\(714\) −0.807764 −0.0302298
\(715\) −1.12311 −0.0420018
\(716\) −5.75379 −0.215029
\(717\) −61.3002 −2.28930
\(718\) −15.9309 −0.594535
\(719\) −10.5616 −0.393879 −0.196940 0.980416i \(-0.563100\pi\)
−0.196940 + 0.980416i \(0.563100\pi\)
\(720\) 7.12311 0.265462
\(721\) −7.50758 −0.279597
\(722\) 0 0
\(723\) −50.2462 −1.86868
\(724\) 10.8769 0.404237
\(725\) −5.68466 −0.211123
\(726\) −2.56155 −0.0950681
\(727\) 22.4233 0.831634 0.415817 0.909448i \(-0.363496\pi\)
0.415817 + 0.909448i \(0.363496\pi\)
\(728\) 0.315342 0.0116873
\(729\) −35.9848 −1.33277
\(730\) −19.3693 −0.716891
\(731\) 0 0
\(732\) −16.0000 −0.591377
\(733\) 26.1080 0.964319 0.482160 0.876083i \(-0.339853\pi\)
0.482160 + 0.876083i \(0.339853\pi\)
\(734\) 30.2462 1.11641
\(735\) 34.2462 1.26319
\(736\) 1.43845 0.0530219
\(737\) 7.68466 0.283068
\(738\) −7.12311 −0.262205
\(739\) −44.9848 −1.65479 −0.827397 0.561617i \(-0.810179\pi\)
−0.827397 + 0.561617i \(0.810179\pi\)
\(740\) 10.2462 0.376658
\(741\) 0 0
\(742\) 7.19224 0.264035
\(743\) 42.2462 1.54986 0.774932 0.632045i \(-0.217784\pi\)
0.774932 + 0.632045i \(0.217784\pi\)
\(744\) 5.12311 0.187822
\(745\) −13.7538 −0.503900
\(746\) −3.43845 −0.125890
\(747\) 50.7386 1.85643
\(748\) −0.561553 −0.0205324
\(749\) −5.57671 −0.203768
\(750\) 30.7386 1.12242
\(751\) −41.2311 −1.50454 −0.752271 0.658853i \(-0.771042\pi\)
−0.752271 + 0.658853i \(0.771042\pi\)
\(752\) 8.00000 0.291730
\(753\) −43.8617 −1.59841
\(754\) −3.19224 −0.116254
\(755\) −8.00000 −0.291150
\(756\) 0.807764 0.0293781
\(757\) −42.4924 −1.54441 −0.772207 0.635371i \(-0.780847\pi\)
−0.772207 + 0.635371i \(0.780847\pi\)
\(758\) −6.56155 −0.238326
\(759\) −3.68466 −0.133745
\(760\) 0 0
\(761\) −42.8078 −1.55178 −0.775890 0.630868i \(-0.782699\pi\)
−0.775890 + 0.630868i \(0.782699\pi\)
\(762\) 5.75379 0.208438
\(763\) −3.54640 −0.128388
\(764\) −8.31534 −0.300838
\(765\) −4.00000 −0.144620
\(766\) 2.00000 0.0722629
\(767\) −4.31534 −0.155818
\(768\) −2.56155 −0.0924321
\(769\) −13.6847 −0.493481 −0.246741 0.969082i \(-0.579360\pi\)
−0.246741 + 0.969082i \(0.579360\pi\)
\(770\) −1.12311 −0.0404739
\(771\) −56.3542 −2.02955
\(772\) −7.12311 −0.256366
\(773\) −37.9309 −1.36428 −0.682139 0.731222i \(-0.738950\pi\)
−0.682139 + 0.731222i \(0.738950\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 7.12311 0.255705
\(777\) 7.36932 0.264373
\(778\) 18.0000 0.645331
\(779\) 0 0
\(780\) 2.87689 0.103009
\(781\) −6.00000 −0.214697
\(782\) −0.807764 −0.0288856
\(783\) −8.17708 −0.292225
\(784\) −6.68466 −0.238738
\(785\) −28.9848 −1.03451
\(786\) −2.87689 −0.102615
\(787\) −15.6847 −0.559098 −0.279549 0.960131i \(-0.590185\pi\)
−0.279549 + 0.960131i \(0.590185\pi\)
\(788\) 18.2462 0.649994
\(789\) −22.7386 −0.809517
\(790\) 8.00000 0.284627
\(791\) −1.75379 −0.0623575
\(792\) 3.56155 0.126554
\(793\) −3.50758 −0.124558
\(794\) 38.9848 1.38352
\(795\) 65.6155 2.32714
\(796\) 11.6847 0.414152
\(797\) 15.1922 0.538137 0.269068 0.963121i \(-0.413284\pi\)
0.269068 + 0.963121i \(0.413284\pi\)
\(798\) 0 0
\(799\) −4.49242 −0.158930
\(800\) −1.00000 −0.0353553
\(801\) 3.12311 0.110350
\(802\) 7.75379 0.273796
\(803\) −9.68466 −0.341764
\(804\) −19.6847 −0.694224
\(805\) −1.61553 −0.0569399
\(806\) 1.12311 0.0395597
\(807\) 7.36932 0.259412
\(808\) −6.87689 −0.241928
\(809\) 8.06913 0.283696 0.141848 0.989888i \(-0.454696\pi\)
0.141848 + 0.989888i \(0.454696\pi\)
\(810\) −14.0000 −0.491910
\(811\) −8.31534 −0.291991 −0.145996 0.989285i \(-0.546639\pi\)
−0.145996 + 0.989285i \(0.546639\pi\)
\(812\) −3.19224 −0.112026
\(813\) −65.7926 −2.30745
\(814\) 5.12311 0.179565
\(815\) 22.7386 0.796500
\(816\) 1.43845 0.0503557
\(817\) 0 0
\(818\) −16.2462 −0.568035
\(819\) 1.12311 0.0392445
\(820\) −4.00000 −0.139686
\(821\) −20.9848 −0.732376 −0.366188 0.930541i \(-0.619337\pi\)
−0.366188 + 0.930541i \(0.619337\pi\)
\(822\) −32.1771 −1.12230
\(823\) −13.9309 −0.485600 −0.242800 0.970076i \(-0.578066\pi\)
−0.242800 + 0.970076i \(0.578066\pi\)
\(824\) 13.3693 0.465742
\(825\) 2.56155 0.0891818
\(826\) −4.31534 −0.150150
\(827\) −49.9309 −1.73627 −0.868133 0.496331i \(-0.834680\pi\)
−0.868133 + 0.496331i \(0.834680\pi\)
\(828\) 5.12311 0.178040
\(829\) −12.1771 −0.422928 −0.211464 0.977386i \(-0.567823\pi\)
−0.211464 + 0.977386i \(0.567823\pi\)
\(830\) 28.4924 0.988986
\(831\) −21.7538 −0.754631
\(832\) −0.561553 −0.0194683
\(833\) 3.75379 0.130061
\(834\) −18.8769 −0.653653
\(835\) 1.26137 0.0436514
\(836\) 0 0
\(837\) 2.87689 0.0994400
\(838\) −14.8769 −0.513914
\(839\) 7.12311 0.245917 0.122958 0.992412i \(-0.460762\pi\)
0.122958 + 0.992412i \(0.460762\pi\)
\(840\) 2.87689 0.0992623
\(841\) 3.31534 0.114322
\(842\) 29.9309 1.03149
\(843\) −64.9848 −2.23820
\(844\) 0.315342 0.0108545
\(845\) −25.3693 −0.872731
\(846\) 28.4924 0.979590
\(847\) −0.561553 −0.0192952
\(848\) −12.8078 −0.439820
\(849\) −29.1231 −0.999502
\(850\) 0.561553 0.0192611
\(851\) 7.36932 0.252617
\(852\) 15.3693 0.526544
\(853\) −52.3542 −1.79257 −0.896286 0.443476i \(-0.853745\pi\)
−0.896286 + 0.443476i \(0.853745\pi\)
\(854\) −3.50758 −0.120027
\(855\) 0 0
\(856\) 9.93087 0.339430
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 1.43845 0.0491078
\(859\) 15.8617 0.541196 0.270598 0.962692i \(-0.412779\pi\)
0.270598 + 0.962692i \(0.412779\pi\)
\(860\) 0 0
\(861\) −2.87689 −0.0980443
\(862\) −31.8617 −1.08522
\(863\) −0.246211 −0.00838113 −0.00419056 0.999991i \(-0.501334\pi\)
−0.00419056 + 0.999991i \(0.501334\pi\)
\(864\) −1.43845 −0.0489370
\(865\) −36.0000 −1.22404
\(866\) −6.63068 −0.225320
\(867\) 42.7386 1.45148
\(868\) 1.12311 0.0381207
\(869\) 4.00000 0.135691
\(870\) −29.1231 −0.987366
\(871\) −4.31534 −0.146220
\(872\) 6.31534 0.213864
\(873\) 25.3693 0.858621
\(874\) 0 0
\(875\) 6.73863 0.227807
\(876\) 24.8078 0.838177
\(877\) 44.5616 1.50474 0.752368 0.658743i \(-0.228911\pi\)
0.752368 + 0.658743i \(0.228911\pi\)
\(878\) 9.61553 0.324508
\(879\) 10.0691 0.339623
\(880\) 2.00000 0.0674200
\(881\) 20.7386 0.698702 0.349351 0.936992i \(-0.386402\pi\)
0.349351 + 0.936992i \(0.386402\pi\)
\(882\) −23.8078 −0.801649
\(883\) −5.26137 −0.177059 −0.0885295 0.996074i \(-0.528217\pi\)
−0.0885295 + 0.996074i \(0.528217\pi\)
\(884\) 0.315342 0.0106061
\(885\) −39.3693 −1.32339
\(886\) 39.8617 1.33918
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) −13.1231 −0.440383
\(889\) 1.26137 0.0423049
\(890\) 1.75379 0.0587871
\(891\) −7.00000 −0.234509
\(892\) −24.7386 −0.828311
\(893\) 0 0
\(894\) 17.6155 0.589151
\(895\) −11.5076 −0.384656
\(896\) −0.561553 −0.0187602
\(897\) 2.06913 0.0690863
\(898\) −23.6155 −0.788060
\(899\) −11.3693 −0.379188
\(900\) −3.56155 −0.118718
\(901\) 7.19224 0.239608
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) 3.12311 0.103873
\(905\) 21.7538 0.723120
\(906\) 10.2462 0.340408
\(907\) 29.7926 0.989247 0.494624 0.869107i \(-0.335306\pi\)
0.494624 + 0.869107i \(0.335306\pi\)
\(908\) −28.1771 −0.935089
\(909\) −24.4924 −0.812362
\(910\) 0.630683 0.0209069
\(911\) −18.9848 −0.628996 −0.314498 0.949258i \(-0.601836\pi\)
−0.314498 + 0.949258i \(0.601836\pi\)
\(912\) 0 0
\(913\) 14.2462 0.471481
\(914\) 36.5616 1.20935
\(915\) −32.0000 −1.05789
\(916\) 12.8769 0.425465
\(917\) −0.630683 −0.0208270
\(918\) 0.807764 0.0266602
\(919\) 32.5616 1.07411 0.537053 0.843548i \(-0.319537\pi\)
0.537053 + 0.843548i \(0.319537\pi\)
\(920\) 2.87689 0.0948484
\(921\) −56.9848 −1.87771
\(922\) 26.7386 0.880590
\(923\) 3.36932 0.110902
\(924\) 1.43845 0.0473214
\(925\) −5.12311 −0.168447
\(926\) 3.50758 0.115266
\(927\) 47.6155 1.56390
\(928\) 5.68466 0.186608
\(929\) 11.9309 0.391439 0.195720 0.980660i \(-0.437296\pi\)
0.195720 + 0.980660i \(0.437296\pi\)
\(930\) 10.2462 0.335987
\(931\) 0 0
\(932\) −28.7386 −0.941365
\(933\) 12.3153 0.403186
\(934\) 9.75379 0.319154
\(935\) −1.12311 −0.0367295
\(936\) −2.00000 −0.0653720
\(937\) −0.699813 −0.0228619 −0.0114310 0.999935i \(-0.503639\pi\)
−0.0114310 + 0.999935i \(0.503639\pi\)
\(938\) −4.31534 −0.140901
\(939\) 1.43845 0.0469419
\(940\) 16.0000 0.521862
\(941\) −24.5616 −0.800684 −0.400342 0.916366i \(-0.631109\pi\)
−0.400342 + 0.916366i \(0.631109\pi\)
\(942\) 37.1231 1.20954
\(943\) −2.87689 −0.0936846
\(944\) 7.68466 0.250114
\(945\) 1.61553 0.0525531
\(946\) 0 0
\(947\) −36.9848 −1.20185 −0.600923 0.799307i \(-0.705200\pi\)
−0.600923 + 0.799307i \(0.705200\pi\)
\(948\) −10.2462 −0.332781
\(949\) 5.43845 0.176539
\(950\) 0 0
\(951\) −18.0691 −0.585932
\(952\) 0.315342 0.0102203
\(953\) 0.876894 0.0284054 0.0142027 0.999899i \(-0.495479\pi\)
0.0142027 + 0.999899i \(0.495479\pi\)
\(954\) −45.6155 −1.47686
\(955\) −16.6307 −0.538156
\(956\) 23.9309 0.773980
\(957\) −14.5616 −0.470708
\(958\) −13.3693 −0.431943
\(959\) −7.05398 −0.227785
\(960\) −5.12311 −0.165348
\(961\) −27.0000 −0.870968
\(962\) −2.87689 −0.0927548
\(963\) 35.3693 1.13976
\(964\) 19.6155 0.631774
\(965\) −14.2462 −0.458602
\(966\) 2.06913 0.0665732
\(967\) 53.8617 1.73208 0.866038 0.499978i \(-0.166658\pi\)
0.866038 + 0.499978i \(0.166658\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 14.2462 0.457418
\(971\) −2.24621 −0.0720843 −0.0360422 0.999350i \(-0.511475\pi\)
−0.0360422 + 0.999350i \(0.511475\pi\)
\(972\) 22.2462 0.713548
\(973\) −4.13826 −0.132667
\(974\) −31.1231 −0.997249
\(975\) −1.43845 −0.0460672
\(976\) 6.24621 0.199936
\(977\) −30.6307 −0.979962 −0.489981 0.871733i \(-0.662996\pi\)
−0.489981 + 0.871733i \(0.662996\pi\)
\(978\) −29.1231 −0.931254
\(979\) 0.876894 0.0280257
\(980\) −13.3693 −0.427067
\(981\) 22.4924 0.718128
\(982\) −39.3693 −1.25633
\(983\) 31.7538 1.01279 0.506394 0.862302i \(-0.330978\pi\)
0.506394 + 0.862302i \(0.330978\pi\)
\(984\) 5.12311 0.163319
\(985\) 36.4924 1.16275
\(986\) −3.19224 −0.101662
\(987\) 11.5076 0.366290
\(988\) 0 0
\(989\) 0 0
\(990\) 7.12311 0.226387
\(991\) −12.8769 −0.409048 −0.204524 0.978862i \(-0.565565\pi\)
−0.204524 + 0.978862i \(0.565565\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −37.3002 −1.18369
\(994\) 3.36932 0.106868
\(995\) 23.3693 0.740857
\(996\) −36.4924 −1.15631
\(997\) −28.6307 −0.906743 −0.453371 0.891322i \(-0.649779\pi\)
−0.453371 + 0.891322i \(0.649779\pi\)
\(998\) 9.75379 0.308751
\(999\) −7.36932 −0.233155
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 7942.2.a.x.1.1 2
19.18 odd 2 418.2.a.e.1.2 2
57.56 even 2 3762.2.a.y.1.1 2
76.75 even 2 3344.2.a.k.1.1 2
209.208 even 2 4598.2.a.bj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.e.1.2 2 19.18 odd 2
3344.2.a.k.1.1 2 76.75 even 2
3762.2.a.y.1.1 2 57.56 even 2
4598.2.a.bj.1.2 2 209.208 even 2
7942.2.a.x.1.1 2 1.1 even 1 trivial