Properties

Label 6762.2.a.bw.1.2
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.56155 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.56155 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.56155 q^{10} +5.12311 q^{11} +1.00000 q^{12} +3.56155 q^{13} +1.56155 q^{15} +1.00000 q^{16} +1.12311 q^{17} -1.00000 q^{18} +5.12311 q^{19} +1.56155 q^{20} -5.12311 q^{22} +1.00000 q^{23} -1.00000 q^{24} -2.56155 q^{25} -3.56155 q^{26} +1.00000 q^{27} +7.56155 q^{29} -1.56155 q^{30} -3.12311 q^{31} -1.00000 q^{32} +5.12311 q^{33} -1.12311 q^{34} +1.00000 q^{36} -1.56155 q^{37} -5.12311 q^{38} +3.56155 q^{39} -1.56155 q^{40} -3.56155 q^{41} +6.68466 q^{43} +5.12311 q^{44} +1.56155 q^{45} -1.00000 q^{46} -2.43845 q^{47} +1.00000 q^{48} +2.56155 q^{50} +1.12311 q^{51} +3.56155 q^{52} +14.2462 q^{53} -1.00000 q^{54} +8.00000 q^{55} +5.12311 q^{57} -7.56155 q^{58} -4.87689 q^{59} +1.56155 q^{60} -0.876894 q^{61} +3.12311 q^{62} +1.00000 q^{64} +5.56155 q^{65} -5.12311 q^{66} -1.12311 q^{67} +1.12311 q^{68} +1.00000 q^{69} -9.36932 q^{71} -1.00000 q^{72} -9.12311 q^{73} +1.56155 q^{74} -2.56155 q^{75} +5.12311 q^{76} -3.56155 q^{78} +14.2462 q^{79} +1.56155 q^{80} +1.00000 q^{81} +3.56155 q^{82} +9.12311 q^{83} +1.75379 q^{85} -6.68466 q^{86} +7.56155 q^{87} -5.12311 q^{88} -14.0000 q^{89} -1.56155 q^{90} +1.00000 q^{92} -3.12311 q^{93} +2.43845 q^{94} +8.00000 q^{95} -1.00000 q^{96} +12.4384 q^{97} +5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{8} + 2q^{9} + q^{10} + 2q^{11} + 2q^{12} + 3q^{13} - q^{15} + 2q^{16} - 6q^{17} - 2q^{18} + 2q^{19} - q^{20} - 2q^{22} + 2q^{23} - 2q^{24} - q^{25} - 3q^{26} + 2q^{27} + 11q^{29} + q^{30} + 2q^{31} - 2q^{32} + 2q^{33} + 6q^{34} + 2q^{36} + q^{37} - 2q^{38} + 3q^{39} + q^{40} - 3q^{41} + q^{43} + 2q^{44} - q^{45} - 2q^{46} - 9q^{47} + 2q^{48} + q^{50} - 6q^{51} + 3q^{52} + 12q^{53} - 2q^{54} + 16q^{55} + 2q^{57} - 11q^{58} - 18q^{59} - q^{60} - 10q^{61} - 2q^{62} + 2q^{64} + 7q^{65} - 2q^{66} + 6q^{67} - 6q^{68} + 2q^{69} + 6q^{71} - 2q^{72} - 10q^{73} - q^{74} - q^{75} + 2q^{76} - 3q^{78} + 12q^{79} - q^{80} + 2q^{81} + 3q^{82} + 10q^{83} + 20q^{85} - q^{86} + 11q^{87} - 2q^{88} - 28q^{89} + q^{90} + 2q^{92} + 2q^{93} + 9q^{94} + 16q^{95} - 2q^{96} + 29q^{97} + 2q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.56155 0.698348 0.349174 0.937058i \(-0.386462\pi\)
0.349174 + 0.937058i \(0.386462\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.56155 −0.493806
\(11\) 5.12311 1.54467 0.772337 0.635213i \(-0.219088\pi\)
0.772337 + 0.635213i \(0.219088\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.56155 0.987797 0.493899 0.869520i \(-0.335571\pi\)
0.493899 + 0.869520i \(0.335571\pi\)
\(14\) 0 0
\(15\) 1.56155 0.403191
\(16\) 1.00000 0.250000
\(17\) 1.12311 0.272393 0.136197 0.990682i \(-0.456512\pi\)
0.136197 + 0.990682i \(0.456512\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.12311 1.17532 0.587661 0.809108i \(-0.300049\pi\)
0.587661 + 0.809108i \(0.300049\pi\)
\(20\) 1.56155 0.349174
\(21\) 0 0
\(22\) −5.12311 −1.09225
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −2.56155 −0.512311
\(26\) −3.56155 −0.698478
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 7.56155 1.40415 0.702073 0.712105i \(-0.252258\pi\)
0.702073 + 0.712105i \(0.252258\pi\)
\(30\) −1.56155 −0.285099
\(31\) −3.12311 −0.560926 −0.280463 0.959865i \(-0.590488\pi\)
−0.280463 + 0.959865i \(0.590488\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.12311 0.891818
\(34\) −1.12311 −0.192611
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.56155 −0.256718 −0.128359 0.991728i \(-0.540971\pi\)
−0.128359 + 0.991728i \(0.540971\pi\)
\(38\) −5.12311 −0.831077
\(39\) 3.56155 0.570305
\(40\) −1.56155 −0.246903
\(41\) −3.56155 −0.556221 −0.278111 0.960549i \(-0.589708\pi\)
−0.278111 + 0.960549i \(0.589708\pi\)
\(42\) 0 0
\(43\) 6.68466 1.01940 0.509700 0.860352i \(-0.329756\pi\)
0.509700 + 0.860352i \(0.329756\pi\)
\(44\) 5.12311 0.772337
\(45\) 1.56155 0.232783
\(46\) −1.00000 −0.147442
\(47\) −2.43845 −0.355684 −0.177842 0.984059i \(-0.556912\pi\)
−0.177842 + 0.984059i \(0.556912\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 2.56155 0.362258
\(51\) 1.12311 0.157266
\(52\) 3.56155 0.493899
\(53\) 14.2462 1.95687 0.978434 0.206561i \(-0.0662271\pi\)
0.978434 + 0.206561i \(0.0662271\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 5.12311 0.678572
\(58\) −7.56155 −0.992881
\(59\) −4.87689 −0.634918 −0.317459 0.948272i \(-0.602830\pi\)
−0.317459 + 0.948272i \(0.602830\pi\)
\(60\) 1.56155 0.201596
\(61\) −0.876894 −0.112275 −0.0561374 0.998423i \(-0.517878\pi\)
−0.0561374 + 0.998423i \(0.517878\pi\)
\(62\) 3.12311 0.396635
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 5.56155 0.689826
\(66\) −5.12311 −0.630611
\(67\) −1.12311 −0.137209 −0.0686046 0.997644i \(-0.521855\pi\)
−0.0686046 + 0.997644i \(0.521855\pi\)
\(68\) 1.12311 0.136197
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −9.36932 −1.11193 −0.555967 0.831205i \(-0.687652\pi\)
−0.555967 + 0.831205i \(0.687652\pi\)
\(72\) −1.00000 −0.117851
\(73\) −9.12311 −1.06778 −0.533889 0.845554i \(-0.679270\pi\)
−0.533889 + 0.845554i \(0.679270\pi\)
\(74\) 1.56155 0.181527
\(75\) −2.56155 −0.295783
\(76\) 5.12311 0.587661
\(77\) 0 0
\(78\) −3.56155 −0.403266
\(79\) 14.2462 1.60282 0.801412 0.598113i \(-0.204082\pi\)
0.801412 + 0.598113i \(0.204082\pi\)
\(80\) 1.56155 0.174587
\(81\) 1.00000 0.111111
\(82\) 3.56155 0.393308
\(83\) 9.12311 1.00139 0.500695 0.865624i \(-0.333078\pi\)
0.500695 + 0.865624i \(0.333078\pi\)
\(84\) 0 0
\(85\) 1.75379 0.190225
\(86\) −6.68466 −0.720825
\(87\) 7.56155 0.810684
\(88\) −5.12311 −0.546125
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −1.56155 −0.164602
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −3.12311 −0.323851
\(94\) 2.43845 0.251507
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) 12.4384 1.26293 0.631466 0.775403i \(-0.282453\pi\)
0.631466 + 0.775403i \(0.282453\pi\)
\(98\) 0 0
\(99\) 5.12311 0.514891
\(100\) −2.56155 −0.256155
\(101\) −16.2462 −1.61656 −0.808279 0.588799i \(-0.799601\pi\)
−0.808279 + 0.588799i \(0.799601\pi\)
\(102\) −1.12311 −0.111204
\(103\) −18.9309 −1.86531 −0.932657 0.360764i \(-0.882516\pi\)
−0.932657 + 0.360764i \(0.882516\pi\)
\(104\) −3.56155 −0.349239
\(105\) 0 0
\(106\) −14.2462 −1.38371
\(107\) −5.12311 −0.495269 −0.247635 0.968853i \(-0.579653\pi\)
−0.247635 + 0.968853i \(0.579653\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.4384 0.999822 0.499911 0.866077i \(-0.333366\pi\)
0.499911 + 0.866077i \(0.333366\pi\)
\(110\) −8.00000 −0.762770
\(111\) −1.56155 −0.148216
\(112\) 0 0
\(113\) −2.68466 −0.252551 −0.126276 0.991995i \(-0.540302\pi\)
−0.126276 + 0.991995i \(0.540302\pi\)
\(114\) −5.12311 −0.479823
\(115\) 1.56155 0.145616
\(116\) 7.56155 0.702073
\(117\) 3.56155 0.329266
\(118\) 4.87689 0.448955
\(119\) 0 0
\(120\) −1.56155 −0.142550
\(121\) 15.2462 1.38602
\(122\) 0.876894 0.0793903
\(123\) −3.56155 −0.321134
\(124\) −3.12311 −0.280463
\(125\) −11.8078 −1.05612
\(126\) 0 0
\(127\) −16.6847 −1.48052 −0.740262 0.672318i \(-0.765299\pi\)
−0.740262 + 0.672318i \(0.765299\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.68466 0.588551
\(130\) −5.56155 −0.487780
\(131\) −4.87689 −0.426096 −0.213048 0.977042i \(-0.568339\pi\)
−0.213048 + 0.977042i \(0.568339\pi\)
\(132\) 5.12311 0.445909
\(133\) 0 0
\(134\) 1.12311 0.0970215
\(135\) 1.56155 0.134397
\(136\) −1.12311 −0.0963055
\(137\) −0.438447 −0.0374591 −0.0187295 0.999825i \(-0.505962\pi\)
−0.0187295 + 0.999825i \(0.505962\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −3.80776 −0.322970 −0.161485 0.986875i \(-0.551628\pi\)
−0.161485 + 0.986875i \(0.551628\pi\)
\(140\) 0 0
\(141\) −2.43845 −0.205354
\(142\) 9.36932 0.786256
\(143\) 18.2462 1.52582
\(144\) 1.00000 0.0833333
\(145\) 11.8078 0.980581
\(146\) 9.12311 0.755034
\(147\) 0 0
\(148\) −1.56155 −0.128359
\(149\) −6.24621 −0.511710 −0.255855 0.966715i \(-0.582357\pi\)
−0.255855 + 0.966715i \(0.582357\pi\)
\(150\) 2.56155 0.209150
\(151\) 7.31534 0.595314 0.297657 0.954673i \(-0.403795\pi\)
0.297657 + 0.954673i \(0.403795\pi\)
\(152\) −5.12311 −0.415539
\(153\) 1.12311 0.0907977
\(154\) 0 0
\(155\) −4.87689 −0.391722
\(156\) 3.56155 0.285152
\(157\) −14.2462 −1.13697 −0.568486 0.822693i \(-0.692471\pi\)
−0.568486 + 0.822693i \(0.692471\pi\)
\(158\) −14.2462 −1.13337
\(159\) 14.2462 1.12980
\(160\) −1.56155 −0.123452
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −3.56155 −0.278111
\(165\) 8.00000 0.622799
\(166\) −9.12311 −0.708090
\(167\) 14.2462 1.10240 0.551202 0.834372i \(-0.314169\pi\)
0.551202 + 0.834372i \(0.314169\pi\)
\(168\) 0 0
\(169\) −0.315342 −0.0242570
\(170\) −1.75379 −0.134509
\(171\) 5.12311 0.391774
\(172\) 6.68466 0.509700
\(173\) −18.4924 −1.40595 −0.702976 0.711213i \(-0.748146\pi\)
−0.702976 + 0.711213i \(0.748146\pi\)
\(174\) −7.56155 −0.573240
\(175\) 0 0
\(176\) 5.12311 0.386169
\(177\) −4.87689 −0.366570
\(178\) 14.0000 1.04934
\(179\) −3.80776 −0.284606 −0.142303 0.989823i \(-0.545451\pi\)
−0.142303 + 0.989823i \(0.545451\pi\)
\(180\) 1.56155 0.116391
\(181\) 15.1231 1.12409 0.562046 0.827106i \(-0.310014\pi\)
0.562046 + 0.827106i \(0.310014\pi\)
\(182\) 0 0
\(183\) −0.876894 −0.0648219
\(184\) −1.00000 −0.0737210
\(185\) −2.43845 −0.179278
\(186\) 3.12311 0.228997
\(187\) 5.75379 0.420759
\(188\) −2.43845 −0.177842
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.68466 0.193246 0.0966230 0.995321i \(-0.469196\pi\)
0.0966230 + 0.995321i \(0.469196\pi\)
\(194\) −12.4384 −0.893028
\(195\) 5.56155 0.398271
\(196\) 0 0
\(197\) 24.9309 1.77625 0.888125 0.459601i \(-0.152008\pi\)
0.888125 + 0.459601i \(0.152008\pi\)
\(198\) −5.12311 −0.364083
\(199\) 21.1771 1.50120 0.750602 0.660755i \(-0.229764\pi\)
0.750602 + 0.660755i \(0.229764\pi\)
\(200\) 2.56155 0.181129
\(201\) −1.12311 −0.0792178
\(202\) 16.2462 1.14308
\(203\) 0 0
\(204\) 1.12311 0.0786331
\(205\) −5.56155 −0.388436
\(206\) 18.9309 1.31898
\(207\) 1.00000 0.0695048
\(208\) 3.56155 0.246949
\(209\) 26.2462 1.81549
\(210\) 0 0
\(211\) −16.4924 −1.13539 −0.567693 0.823241i \(-0.692164\pi\)
−0.567693 + 0.823241i \(0.692164\pi\)
\(212\) 14.2462 0.978434
\(213\) −9.36932 −0.641975
\(214\) 5.12311 0.350208
\(215\) 10.4384 0.711896
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −10.4384 −0.706981
\(219\) −9.12311 −0.616482
\(220\) 8.00000 0.539360
\(221\) 4.00000 0.269069
\(222\) 1.56155 0.104805
\(223\) −3.12311 −0.209139 −0.104569 0.994518i \(-0.533346\pi\)
−0.104569 + 0.994518i \(0.533346\pi\)
\(224\) 0 0
\(225\) −2.56155 −0.170770
\(226\) 2.68466 0.178581
\(227\) 6.19224 0.410993 0.205497 0.978658i \(-0.434119\pi\)
0.205497 + 0.978658i \(0.434119\pi\)
\(228\) 5.12311 0.339286
\(229\) −2.24621 −0.148434 −0.0742169 0.997242i \(-0.523646\pi\)
−0.0742169 + 0.997242i \(0.523646\pi\)
\(230\) −1.56155 −0.102966
\(231\) 0 0
\(232\) −7.56155 −0.496440
\(233\) −18.4924 −1.21148 −0.605739 0.795663i \(-0.707123\pi\)
−0.605739 + 0.795663i \(0.707123\pi\)
\(234\) −3.56155 −0.232826
\(235\) −3.80776 −0.248391
\(236\) −4.87689 −0.317459
\(237\) 14.2462 0.925391
\(238\) 0 0
\(239\) 4.49242 0.290591 0.145295 0.989388i \(-0.453587\pi\)
0.145295 + 0.989388i \(0.453587\pi\)
\(240\) 1.56155 0.100798
\(241\) 20.0540 1.29179 0.645895 0.763426i \(-0.276484\pi\)
0.645895 + 0.763426i \(0.276484\pi\)
\(242\) −15.2462 −0.980064
\(243\) 1.00000 0.0641500
\(244\) −0.876894 −0.0561374
\(245\) 0 0
\(246\) 3.56155 0.227076
\(247\) 18.2462 1.16098
\(248\) 3.12311 0.198317
\(249\) 9.12311 0.578153
\(250\) 11.8078 0.746789
\(251\) 0.438447 0.0276745 0.0138373 0.999904i \(-0.495595\pi\)
0.0138373 + 0.999904i \(0.495595\pi\)
\(252\) 0 0
\(253\) 5.12311 0.322087
\(254\) 16.6847 1.04689
\(255\) 1.75379 0.109827
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) −6.68466 −0.416169
\(259\) 0 0
\(260\) 5.56155 0.344913
\(261\) 7.56155 0.468048
\(262\) 4.87689 0.301296
\(263\) 22.9309 1.41398 0.706989 0.707225i \(-0.250053\pi\)
0.706989 + 0.707225i \(0.250053\pi\)
\(264\) −5.12311 −0.315305
\(265\) 22.2462 1.36657
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) −1.12311 −0.0686046
\(269\) −12.2462 −0.746665 −0.373332 0.927698i \(-0.621785\pi\)
−0.373332 + 0.927698i \(0.621785\pi\)
\(270\) −1.56155 −0.0950331
\(271\) −14.2462 −0.865396 −0.432698 0.901539i \(-0.642438\pi\)
−0.432698 + 0.901539i \(0.642438\pi\)
\(272\) 1.12311 0.0680983
\(273\) 0 0
\(274\) 0.438447 0.0264876
\(275\) −13.1231 −0.791353
\(276\) 1.00000 0.0601929
\(277\) 13.1231 0.788491 0.394245 0.919005i \(-0.371006\pi\)
0.394245 + 0.919005i \(0.371006\pi\)
\(278\) 3.80776 0.228375
\(279\) −3.12311 −0.186975
\(280\) 0 0
\(281\) 4.93087 0.294151 0.147076 0.989125i \(-0.453014\pi\)
0.147076 + 0.989125i \(0.453014\pi\)
\(282\) 2.43845 0.145207
\(283\) 10.4924 0.623710 0.311855 0.950130i \(-0.399050\pi\)
0.311855 + 0.950130i \(0.399050\pi\)
\(284\) −9.36932 −0.555967
\(285\) 8.00000 0.473879
\(286\) −18.2462 −1.07892
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −15.7386 −0.925802
\(290\) −11.8078 −0.693376
\(291\) 12.4384 0.729155
\(292\) −9.12311 −0.533889
\(293\) −27.6155 −1.61332 −0.806658 0.591018i \(-0.798726\pi\)
−0.806658 + 0.591018i \(0.798726\pi\)
\(294\) 0 0
\(295\) −7.61553 −0.443393
\(296\) 1.56155 0.0907634
\(297\) 5.12311 0.297273
\(298\) 6.24621 0.361833
\(299\) 3.56155 0.205970
\(300\) −2.56155 −0.147891
\(301\) 0 0
\(302\) −7.31534 −0.420951
\(303\) −16.2462 −0.933320
\(304\) 5.12311 0.293830
\(305\) −1.36932 −0.0784069
\(306\) −1.12311 −0.0642037
\(307\) −2.93087 −0.167274 −0.0836368 0.996496i \(-0.526654\pi\)
−0.0836368 + 0.996496i \(0.526654\pi\)
\(308\) 0 0
\(309\) −18.9309 −1.07694
\(310\) 4.87689 0.276989
\(311\) −18.7386 −1.06257 −0.531285 0.847193i \(-0.678291\pi\)
−0.531285 + 0.847193i \(0.678291\pi\)
\(312\) −3.56155 −0.201633
\(313\) −4.24621 −0.240010 −0.120005 0.992773i \(-0.538291\pi\)
−0.120005 + 0.992773i \(0.538291\pi\)
\(314\) 14.2462 0.803960
\(315\) 0 0
\(316\) 14.2462 0.801412
\(317\) −17.8078 −1.00018 −0.500092 0.865972i \(-0.666700\pi\)
−0.500092 + 0.865972i \(0.666700\pi\)
\(318\) −14.2462 −0.798888
\(319\) 38.7386 2.16895
\(320\) 1.56155 0.0872935
\(321\) −5.12311 −0.285944
\(322\) 0 0
\(323\) 5.75379 0.320149
\(324\) 1.00000 0.0555556
\(325\) −9.12311 −0.506059
\(326\) 8.00000 0.443079
\(327\) 10.4384 0.577247
\(328\) 3.56155 0.196654
\(329\) 0 0
\(330\) −8.00000 −0.440386
\(331\) −24.4924 −1.34623 −0.673113 0.739540i \(-0.735043\pi\)
−0.673113 + 0.739540i \(0.735043\pi\)
\(332\) 9.12311 0.500695
\(333\) −1.56155 −0.0855726
\(334\) −14.2462 −0.779518
\(335\) −1.75379 −0.0958197
\(336\) 0 0
\(337\) 27.3693 1.49090 0.745451 0.666561i \(-0.232234\pi\)
0.745451 + 0.666561i \(0.232234\pi\)
\(338\) 0.315342 0.0171523
\(339\) −2.68466 −0.145811
\(340\) 1.75379 0.0951125
\(341\) −16.0000 −0.866449
\(342\) −5.12311 −0.277026
\(343\) 0 0
\(344\) −6.68466 −0.360413
\(345\) 1.56155 0.0840712
\(346\) 18.4924 0.994159
\(347\) 14.9309 0.801531 0.400766 0.916181i \(-0.368744\pi\)
0.400766 + 0.916181i \(0.368744\pi\)
\(348\) 7.56155 0.405342
\(349\) 16.2462 0.869640 0.434820 0.900517i \(-0.356812\pi\)
0.434820 + 0.900517i \(0.356812\pi\)
\(350\) 0 0
\(351\) 3.56155 0.190102
\(352\) −5.12311 −0.273062
\(353\) 33.8078 1.79941 0.899703 0.436503i \(-0.143783\pi\)
0.899703 + 0.436503i \(0.143783\pi\)
\(354\) 4.87689 0.259204
\(355\) −14.6307 −0.776516
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 3.80776 0.201247
\(359\) −6.43845 −0.339808 −0.169904 0.985461i \(-0.554346\pi\)
−0.169904 + 0.985461i \(0.554346\pi\)
\(360\) −1.56155 −0.0823011
\(361\) 7.24621 0.381380
\(362\) −15.1231 −0.794853
\(363\) 15.2462 0.800219
\(364\) 0 0
\(365\) −14.2462 −0.745681
\(366\) 0.876894 0.0458360
\(367\) 18.4384 0.962479 0.481240 0.876589i \(-0.340187\pi\)
0.481240 + 0.876589i \(0.340187\pi\)
\(368\) 1.00000 0.0521286
\(369\) −3.56155 −0.185407
\(370\) 2.43845 0.126769
\(371\) 0 0
\(372\) −3.12311 −0.161925
\(373\) 15.1231 0.783045 0.391522 0.920169i \(-0.371949\pi\)
0.391522 + 0.920169i \(0.371949\pi\)
\(374\) −5.75379 −0.297521
\(375\) −11.8078 −0.609750
\(376\) 2.43845 0.125753
\(377\) 26.9309 1.38701
\(378\) 0 0
\(379\) 20.0540 1.03010 0.515052 0.857159i \(-0.327773\pi\)
0.515052 + 0.857159i \(0.327773\pi\)
\(380\) 8.00000 0.410391
\(381\) −16.6847 −0.854781
\(382\) 20.0000 1.02329
\(383\) 6.24621 0.319166 0.159583 0.987184i \(-0.448985\pi\)
0.159583 + 0.987184i \(0.448985\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −2.68466 −0.136646
\(387\) 6.68466 0.339800
\(388\) 12.4384 0.631466
\(389\) −29.3693 −1.48908 −0.744542 0.667576i \(-0.767332\pi\)
−0.744542 + 0.667576i \(0.767332\pi\)
\(390\) −5.56155 −0.281620
\(391\) 1.12311 0.0567979
\(392\) 0 0
\(393\) −4.87689 −0.246007
\(394\) −24.9309 −1.25600
\(395\) 22.2462 1.11933
\(396\) 5.12311 0.257446
\(397\) 7.75379 0.389152 0.194576 0.980887i \(-0.437667\pi\)
0.194576 + 0.980887i \(0.437667\pi\)
\(398\) −21.1771 −1.06151
\(399\) 0 0
\(400\) −2.56155 −0.128078
\(401\) 16.2462 0.811297 0.405649 0.914029i \(-0.367046\pi\)
0.405649 + 0.914029i \(0.367046\pi\)
\(402\) 1.12311 0.0560154
\(403\) −11.1231 −0.554081
\(404\) −16.2462 −0.808279
\(405\) 1.56155 0.0775942
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) −1.12311 −0.0556020
\(409\) −33.6155 −1.66218 −0.831090 0.556137i \(-0.812283\pi\)
−0.831090 + 0.556137i \(0.812283\pi\)
\(410\) 5.56155 0.274666
\(411\) −0.438447 −0.0216270
\(412\) −18.9309 −0.932657
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 14.2462 0.699319
\(416\) −3.56155 −0.174619
\(417\) −3.80776 −0.186467
\(418\) −26.2462 −1.28374
\(419\) −23.8617 −1.16572 −0.582861 0.812572i \(-0.698067\pi\)
−0.582861 + 0.812572i \(0.698067\pi\)
\(420\) 0 0
\(421\) 33.1771 1.61695 0.808476 0.588529i \(-0.200293\pi\)
0.808476 + 0.588529i \(0.200293\pi\)
\(422\) 16.4924 0.802839
\(423\) −2.43845 −0.118561
\(424\) −14.2462 −0.691857
\(425\) −2.87689 −0.139550
\(426\) 9.36932 0.453945
\(427\) 0 0
\(428\) −5.12311 −0.247635
\(429\) 18.2462 0.880935
\(430\) −10.4384 −0.503387
\(431\) −3.31534 −0.159694 −0.0798472 0.996807i \(-0.525443\pi\)
−0.0798472 + 0.996807i \(0.525443\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.6847 −0.705700 −0.352850 0.935680i \(-0.614787\pi\)
−0.352850 + 0.935680i \(0.614787\pi\)
\(434\) 0 0
\(435\) 11.8078 0.566139
\(436\) 10.4384 0.499911
\(437\) 5.12311 0.245071
\(438\) 9.12311 0.435919
\(439\) 23.6155 1.12711 0.563554 0.826079i \(-0.309434\pi\)
0.563554 + 0.826079i \(0.309434\pi\)
\(440\) −8.00000 −0.381385
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 9.06913 0.430887 0.215444 0.976516i \(-0.430880\pi\)
0.215444 + 0.976516i \(0.430880\pi\)
\(444\) −1.56155 −0.0741080
\(445\) −21.8617 −1.03635
\(446\) 3.12311 0.147883
\(447\) −6.24621 −0.295436
\(448\) 0 0
\(449\) −0.246211 −0.0116194 −0.00580971 0.999983i \(-0.501849\pi\)
−0.00580971 + 0.999983i \(0.501849\pi\)
\(450\) 2.56155 0.120753
\(451\) −18.2462 −0.859181
\(452\) −2.68466 −0.126276
\(453\) 7.31534 0.343705
\(454\) −6.19224 −0.290616
\(455\) 0 0
\(456\) −5.12311 −0.239911
\(457\) 7.36932 0.344722 0.172361 0.985034i \(-0.444860\pi\)
0.172361 + 0.985034i \(0.444860\pi\)
\(458\) 2.24621 0.104959
\(459\) 1.12311 0.0524221
\(460\) 1.56155 0.0728078
\(461\) −40.7386 −1.89739 −0.948694 0.316197i \(-0.897594\pi\)
−0.948694 + 0.316197i \(0.897594\pi\)
\(462\) 0 0
\(463\) −13.5616 −0.630259 −0.315129 0.949049i \(-0.602048\pi\)
−0.315129 + 0.949049i \(0.602048\pi\)
\(464\) 7.56155 0.351036
\(465\) −4.87689 −0.226161
\(466\) 18.4924 0.856645
\(467\) −2.68466 −0.124231 −0.0621156 0.998069i \(-0.519785\pi\)
−0.0621156 + 0.998069i \(0.519785\pi\)
\(468\) 3.56155 0.164633
\(469\) 0 0
\(470\) 3.80776 0.175639
\(471\) −14.2462 −0.656431
\(472\) 4.87689 0.224477
\(473\) 34.2462 1.57464
\(474\) −14.2462 −0.654350
\(475\) −13.1231 −0.602129
\(476\) 0 0
\(477\) 14.2462 0.652289
\(478\) −4.49242 −0.205479
\(479\) −31.6155 −1.44455 −0.722275 0.691606i \(-0.756904\pi\)
−0.722275 + 0.691606i \(0.756904\pi\)
\(480\) −1.56155 −0.0712748
\(481\) −5.56155 −0.253585
\(482\) −20.0540 −0.913434
\(483\) 0 0
\(484\) 15.2462 0.693010
\(485\) 19.4233 0.881966
\(486\) −1.00000 −0.0453609
\(487\) −5.56155 −0.252018 −0.126009 0.992029i \(-0.540217\pi\)
−0.126009 + 0.992029i \(0.540217\pi\)
\(488\) 0.876894 0.0396951
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −26.7386 −1.20670 −0.603349 0.797477i \(-0.706167\pi\)
−0.603349 + 0.797477i \(0.706167\pi\)
\(492\) −3.56155 −0.160567
\(493\) 8.49242 0.382479
\(494\) −18.2462 −0.820936
\(495\) 8.00000 0.359573
\(496\) −3.12311 −0.140232
\(497\) 0 0
\(498\) −9.12311 −0.408816
\(499\) 41.3693 1.85194 0.925972 0.377591i \(-0.123247\pi\)
0.925972 + 0.377591i \(0.123247\pi\)
\(500\) −11.8078 −0.528059
\(501\) 14.2462 0.636474
\(502\) −0.438447 −0.0195689
\(503\) 32.1080 1.43162 0.715811 0.698294i \(-0.246057\pi\)
0.715811 + 0.698294i \(0.246057\pi\)
\(504\) 0 0
\(505\) −25.3693 −1.12892
\(506\) −5.12311 −0.227750
\(507\) −0.315342 −0.0140048
\(508\) −16.6847 −0.740262
\(509\) 3.36932 0.149342 0.0746712 0.997208i \(-0.476209\pi\)
0.0746712 + 0.997208i \(0.476209\pi\)
\(510\) −1.75379 −0.0776591
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 5.12311 0.226191
\(514\) −2.00000 −0.0882162
\(515\) −29.5616 −1.30264
\(516\) 6.68466 0.294276
\(517\) −12.4924 −0.549416
\(518\) 0 0
\(519\) −18.4924 −0.811727
\(520\) −5.56155 −0.243890
\(521\) −0.246211 −0.0107867 −0.00539336 0.999985i \(-0.501717\pi\)
−0.00539336 + 0.999985i \(0.501717\pi\)
\(522\) −7.56155 −0.330960
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) −4.87689 −0.213048
\(525\) 0 0
\(526\) −22.9309 −0.999833
\(527\) −3.50758 −0.152792
\(528\) 5.12311 0.222955
\(529\) 1.00000 0.0434783
\(530\) −22.2462 −0.966314
\(531\) −4.87689 −0.211639
\(532\) 0 0
\(533\) −12.6847 −0.549434
\(534\) 14.0000 0.605839
\(535\) −8.00000 −0.345870
\(536\) 1.12311 0.0485108
\(537\) −3.80776 −0.164317
\(538\) 12.2462 0.527972
\(539\) 0 0
\(540\) 1.56155 0.0671985
\(541\) −15.3693 −0.660779 −0.330389 0.943845i \(-0.607180\pi\)
−0.330389 + 0.943845i \(0.607180\pi\)
\(542\) 14.2462 0.611927
\(543\) 15.1231 0.648995
\(544\) −1.12311 −0.0481528
\(545\) 16.3002 0.698223
\(546\) 0 0
\(547\) −3.61553 −0.154589 −0.0772944 0.997008i \(-0.524628\pi\)
−0.0772944 + 0.997008i \(0.524628\pi\)
\(548\) −0.438447 −0.0187295
\(549\) −0.876894 −0.0374249
\(550\) 13.1231 0.559571
\(551\) 38.7386 1.65032
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −13.1231 −0.557547
\(555\) −2.43845 −0.103506
\(556\) −3.80776 −0.161485
\(557\) −28.9848 −1.22813 −0.614064 0.789257i \(-0.710466\pi\)
−0.614064 + 0.789257i \(0.710466\pi\)
\(558\) 3.12311 0.132212
\(559\) 23.8078 1.00696
\(560\) 0 0
\(561\) 5.75379 0.242925
\(562\) −4.93087 −0.207996
\(563\) −19.5616 −0.824421 −0.412211 0.911089i \(-0.635243\pi\)
−0.412211 + 0.911089i \(0.635243\pi\)
\(564\) −2.43845 −0.102677
\(565\) −4.19224 −0.176369
\(566\) −10.4924 −0.441029
\(567\) 0 0
\(568\) 9.36932 0.393128
\(569\) −44.5464 −1.86748 −0.933741 0.357949i \(-0.883476\pi\)
−0.933741 + 0.357949i \(0.883476\pi\)
\(570\) −8.00000 −0.335083
\(571\) 43.3693 1.81495 0.907475 0.420107i \(-0.138007\pi\)
0.907475 + 0.420107i \(0.138007\pi\)
\(572\) 18.2462 0.762912
\(573\) −20.0000 −0.835512
\(574\) 0 0
\(575\) −2.56155 −0.106824
\(576\) 1.00000 0.0416667
\(577\) 19.7538 0.822361 0.411180 0.911554i \(-0.365117\pi\)
0.411180 + 0.911554i \(0.365117\pi\)
\(578\) 15.7386 0.654641
\(579\) 2.68466 0.111571
\(580\) 11.8078 0.490291
\(581\) 0 0
\(582\) −12.4384 −0.515590
\(583\) 72.9848 3.02272
\(584\) 9.12311 0.377517
\(585\) 5.56155 0.229942
\(586\) 27.6155 1.14079
\(587\) 24.9848 1.03123 0.515617 0.856819i \(-0.327563\pi\)
0.515617 + 0.856819i \(0.327563\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 7.61553 0.313526
\(591\) 24.9309 1.02552
\(592\) −1.56155 −0.0641794
\(593\) −30.3002 −1.24428 −0.622140 0.782906i \(-0.713736\pi\)
−0.622140 + 0.782906i \(0.713736\pi\)
\(594\) −5.12311 −0.210204
\(595\) 0 0
\(596\) −6.24621 −0.255855
\(597\) 21.1771 0.866720
\(598\) −3.56155 −0.145643
\(599\) −10.7386 −0.438769 −0.219384 0.975639i \(-0.570405\pi\)
−0.219384 + 0.975639i \(0.570405\pi\)
\(600\) 2.56155 0.104575
\(601\) 3.36932 0.137437 0.0687187 0.997636i \(-0.478109\pi\)
0.0687187 + 0.997636i \(0.478109\pi\)
\(602\) 0 0
\(603\) −1.12311 −0.0457364
\(604\) 7.31534 0.297657
\(605\) 23.8078 0.967923
\(606\) 16.2462 0.659957
\(607\) 47.6155 1.93265 0.966327 0.257316i \(-0.0828381\pi\)
0.966327 + 0.257316i \(0.0828381\pi\)
\(608\) −5.12311 −0.207769
\(609\) 0 0
\(610\) 1.36932 0.0554420
\(611\) −8.68466 −0.351344
\(612\) 1.12311 0.0453989
\(613\) 3.80776 0.153794 0.0768971 0.997039i \(-0.475499\pi\)
0.0768971 + 0.997039i \(0.475499\pi\)
\(614\) 2.93087 0.118280
\(615\) −5.56155 −0.224263
\(616\) 0 0
\(617\) 31.7538 1.27836 0.639180 0.769057i \(-0.279274\pi\)
0.639180 + 0.769057i \(0.279274\pi\)
\(618\) 18.9309 0.761511
\(619\) −8.24621 −0.331443 −0.165722 0.986173i \(-0.552995\pi\)
−0.165722 + 0.986173i \(0.552995\pi\)
\(620\) −4.87689 −0.195861
\(621\) 1.00000 0.0401286
\(622\) 18.7386 0.751351
\(623\) 0 0
\(624\) 3.56155 0.142576
\(625\) −5.63068 −0.225227
\(626\) 4.24621 0.169713
\(627\) 26.2462 1.04817
\(628\) −14.2462 −0.568486
\(629\) −1.75379 −0.0699281
\(630\) 0 0
\(631\) 8.49242 0.338078 0.169039 0.985609i \(-0.445934\pi\)
0.169039 + 0.985609i \(0.445934\pi\)
\(632\) −14.2462 −0.566684
\(633\) −16.4924 −0.655515
\(634\) 17.8078 0.707237
\(635\) −26.0540 −1.03392
\(636\) 14.2462 0.564899
\(637\) 0 0
\(638\) −38.7386 −1.53368
\(639\) −9.36932 −0.370644
\(640\) −1.56155 −0.0617258
\(641\) −13.3153 −0.525924 −0.262962 0.964806i \(-0.584699\pi\)
−0.262962 + 0.964806i \(0.584699\pi\)
\(642\) 5.12311 0.202193
\(643\) 38.0000 1.49857 0.749287 0.662246i \(-0.230396\pi\)
0.749287 + 0.662246i \(0.230396\pi\)
\(644\) 0 0
\(645\) 10.4384 0.411013
\(646\) −5.75379 −0.226380
\(647\) 3.50758 0.137897 0.0689486 0.997620i \(-0.478036\pi\)
0.0689486 + 0.997620i \(0.478036\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −24.9848 −0.980741
\(650\) 9.12311 0.357838
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) 36.5464 1.43017 0.715086 0.699037i \(-0.246388\pi\)
0.715086 + 0.699037i \(0.246388\pi\)
\(654\) −10.4384 −0.408176
\(655\) −7.61553 −0.297563
\(656\) −3.56155 −0.139055
\(657\) −9.12311 −0.355926
\(658\) 0 0
\(659\) 5.61553 0.218750 0.109375 0.994001i \(-0.465115\pi\)
0.109375 + 0.994001i \(0.465115\pi\)
\(660\) 8.00000 0.311400
\(661\) −32.8769 −1.27876 −0.639381 0.768890i \(-0.720810\pi\)
−0.639381 + 0.768890i \(0.720810\pi\)
\(662\) 24.4924 0.951925
\(663\) 4.00000 0.155347
\(664\) −9.12311 −0.354045
\(665\) 0 0
\(666\) 1.56155 0.0605089
\(667\) 7.56155 0.292784
\(668\) 14.2462 0.551202
\(669\) −3.12311 −0.120746
\(670\) 1.75379 0.0677548
\(671\) −4.49242 −0.173428
\(672\) 0 0
\(673\) 22.6847 0.874429 0.437215 0.899357i \(-0.355965\pi\)
0.437215 + 0.899357i \(0.355965\pi\)
\(674\) −27.3693 −1.05423
\(675\) −2.56155 −0.0985942
\(676\) −0.315342 −0.0121285
\(677\) −2.63068 −0.101105 −0.0505527 0.998721i \(-0.516098\pi\)
−0.0505527 + 0.998721i \(0.516098\pi\)
\(678\) 2.68466 0.103104
\(679\) 0 0
\(680\) −1.75379 −0.0672547
\(681\) 6.19224 0.237287
\(682\) 16.0000 0.612672
\(683\) −7.50758 −0.287269 −0.143635 0.989631i \(-0.545879\pi\)
−0.143635 + 0.989631i \(0.545879\pi\)
\(684\) 5.12311 0.195887
\(685\) −0.684658 −0.0261595
\(686\) 0 0
\(687\) −2.24621 −0.0856983
\(688\) 6.68466 0.254850
\(689\) 50.7386 1.93299
\(690\) −1.56155 −0.0594473
\(691\) 50.0540 1.90414 0.952071 0.305876i \(-0.0989492\pi\)
0.952071 + 0.305876i \(0.0989492\pi\)
\(692\) −18.4924 −0.702976
\(693\) 0 0
\(694\) −14.9309 −0.566768
\(695\) −5.94602 −0.225546
\(696\) −7.56155 −0.286620
\(697\) −4.00000 −0.151511
\(698\) −16.2462 −0.614928
\(699\) −18.4924 −0.699448
\(700\) 0 0
\(701\) 34.2462 1.29346 0.646731 0.762718i \(-0.276136\pi\)
0.646731 + 0.762718i \(0.276136\pi\)
\(702\) −3.56155 −0.134422
\(703\) −8.00000 −0.301726
\(704\) 5.12311 0.193084
\(705\) −3.80776 −0.143409
\(706\) −33.8078 −1.27237
\(707\) 0 0
\(708\) −4.87689 −0.183285
\(709\) 10.6307 0.399244 0.199622 0.979873i \(-0.436029\pi\)
0.199622 + 0.979873i \(0.436029\pi\)
\(710\) 14.6307 0.549080
\(711\) 14.2462 0.534275
\(712\) 14.0000 0.524672
\(713\) −3.12311 −0.116961
\(714\) 0 0
\(715\) 28.4924 1.06556
\(716\) −3.80776 −0.142303
\(717\) 4.49242 0.167773
\(718\) 6.43845 0.240281
\(719\) −10.4384 −0.389288 −0.194644 0.980874i \(-0.562355\pi\)
−0.194644 + 0.980874i \(0.562355\pi\)
\(720\) 1.56155 0.0581956
\(721\) 0 0
\(722\) −7.24621 −0.269676
\(723\) 20.0540 0.745815
\(724\) 15.1231 0.562046
\(725\) −19.3693 −0.719358
\(726\) −15.2462 −0.565840
\(727\) 1.75379 0.0650444 0.0325222 0.999471i \(-0.489646\pi\)
0.0325222 + 0.999471i \(0.489646\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 14.2462 0.527276
\(731\) 7.50758 0.277678
\(732\) −0.876894 −0.0324109
\(733\) −25.7538 −0.951238 −0.475619 0.879651i \(-0.657776\pi\)
−0.475619 + 0.879651i \(0.657776\pi\)
\(734\) −18.4384 −0.680576
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −5.75379 −0.211944
\(738\) 3.56155 0.131103
\(739\) 23.6155 0.868711 0.434356 0.900741i \(-0.356976\pi\)
0.434356 + 0.900741i \(0.356976\pi\)
\(740\) −2.43845 −0.0896391
\(741\) 18.2462 0.670291
\(742\) 0 0
\(743\) −24.9848 −0.916605 −0.458303 0.888796i \(-0.651542\pi\)
−0.458303 + 0.888796i \(0.651542\pi\)
\(744\) 3.12311 0.114499
\(745\) −9.75379 −0.357351
\(746\) −15.1231 −0.553696
\(747\) 9.12311 0.333797
\(748\) 5.75379 0.210379
\(749\) 0 0
\(750\) 11.8078 0.431159
\(751\) 24.4924 0.893741 0.446871 0.894599i \(-0.352538\pi\)
0.446871 + 0.894599i \(0.352538\pi\)
\(752\) −2.43845 −0.0889210
\(753\) 0.438447 0.0159779
\(754\) −26.9309 −0.980764
\(755\) 11.4233 0.415736
\(756\) 0 0
\(757\) −25.3693 −0.922064 −0.461032 0.887384i \(-0.652521\pi\)
−0.461032 + 0.887384i \(0.652521\pi\)
\(758\) −20.0540 −0.728393
\(759\) 5.12311 0.185957
\(760\) −8.00000 −0.290191
\(761\) 3.26137 0.118224 0.0591122 0.998251i \(-0.481173\pi\)
0.0591122 + 0.998251i \(0.481173\pi\)
\(762\) 16.6847 0.604421
\(763\) 0 0
\(764\) −20.0000 −0.723575
\(765\) 1.75379 0.0634084
\(766\) −6.24621 −0.225685
\(767\) −17.3693 −0.627170
\(768\) 1.00000 0.0360844
\(769\) 47.1771 1.70125 0.850625 0.525774i \(-0.176224\pi\)
0.850625 + 0.525774i \(0.176224\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 2.68466 0.0966230
\(773\) 32.3002 1.16176 0.580878 0.813990i \(-0.302709\pi\)
0.580878 + 0.813990i \(0.302709\pi\)
\(774\) −6.68466 −0.240275
\(775\) 8.00000 0.287368
\(776\) −12.4384 −0.446514
\(777\) 0 0
\(778\) 29.3693 1.05294
\(779\) −18.2462 −0.653738
\(780\) 5.56155 0.199136
\(781\) −48.0000 −1.71758
\(782\) −1.12311 −0.0401622
\(783\) 7.56155 0.270228
\(784\) 0 0
\(785\) −22.2462 −0.794001
\(786\) 4.87689 0.173953
\(787\) 10.8769 0.387719 0.193860 0.981029i \(-0.437899\pi\)
0.193860 + 0.981029i \(0.437899\pi\)
\(788\) 24.9309 0.888125
\(789\) 22.9309 0.816361
\(790\) −22.2462 −0.791485
\(791\) 0 0
\(792\) −5.12311 −0.182042
\(793\) −3.12311 −0.110905
\(794\) −7.75379 −0.275172
\(795\) 22.2462 0.788992
\(796\) 21.1771 0.750602
\(797\) −27.4233 −0.971383 −0.485691 0.874130i \(-0.661432\pi\)
−0.485691 + 0.874130i \(0.661432\pi\)
\(798\) 0 0
\(799\) −2.73863 −0.0968859
\(800\) 2.56155 0.0905646
\(801\) −14.0000 −0.494666
\(802\) −16.2462 −0.573674
\(803\) −46.7386 −1.64937
\(804\) −1.12311 −0.0396089
\(805\) 0 0
\(806\) 11.1231 0.391795
\(807\) −12.2462 −0.431087
\(808\) 16.2462 0.571540
\(809\) 23.8617 0.838934 0.419467 0.907771i \(-0.362217\pi\)
0.419467 + 0.907771i \(0.362217\pi\)
\(810\) −1.56155 −0.0548674
\(811\) 24.6847 0.866796 0.433398 0.901203i \(-0.357315\pi\)
0.433398 + 0.901203i \(0.357315\pi\)
\(812\) 0 0
\(813\) −14.2462 −0.499636
\(814\) 8.00000 0.280400
\(815\) −12.4924 −0.437590
\(816\) 1.12311 0.0393166
\(817\) 34.2462 1.19812
\(818\) 33.6155 1.17534
\(819\) 0 0
\(820\) −5.56155 −0.194218
\(821\) −55.4773 −1.93617 −0.968085 0.250622i \(-0.919365\pi\)
−0.968085 + 0.250622i \(0.919365\pi\)
\(822\) 0.438447 0.0152926
\(823\) 23.3153 0.812722 0.406361 0.913713i \(-0.366798\pi\)
0.406361 + 0.913713i \(0.366798\pi\)
\(824\) 18.9309 0.659488
\(825\) −13.1231 −0.456888
\(826\) 0 0
\(827\) 48.7386 1.69481 0.847404 0.530948i \(-0.178164\pi\)
0.847404 + 0.530948i \(0.178164\pi\)
\(828\) 1.00000 0.0347524
\(829\) −24.7386 −0.859208 −0.429604 0.903017i \(-0.641347\pi\)
−0.429604 + 0.903017i \(0.641347\pi\)
\(830\) −14.2462 −0.494493
\(831\) 13.1231 0.455235
\(832\) 3.56155 0.123475
\(833\) 0 0
\(834\) 3.80776 0.131852
\(835\) 22.2462 0.769862
\(836\) 26.2462 0.907744
\(837\) −3.12311 −0.107950
\(838\) 23.8617 0.824290
\(839\) −7.61553 −0.262917 −0.131459 0.991322i \(-0.541966\pi\)
−0.131459 + 0.991322i \(0.541966\pi\)
\(840\) 0 0
\(841\) 28.1771 0.971623
\(842\) −33.1771 −1.14336
\(843\) 4.93087 0.169828
\(844\) −16.4924 −0.567693
\(845\) −0.492423 −0.0169398
\(846\) 2.43845 0.0838355
\(847\) 0 0
\(848\) 14.2462 0.489217
\(849\) 10.4924 0.360099
\(850\) 2.87689 0.0986767
\(851\) −1.56155 −0.0535293
\(852\) −9.36932 −0.320988
\(853\) 0.438447 0.0150121 0.00750607 0.999972i \(-0.497611\pi\)
0.00750607 + 0.999972i \(0.497611\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 5.12311 0.175104
\(857\) −1.31534 −0.0449312 −0.0224656 0.999748i \(-0.507152\pi\)
−0.0224656 + 0.999748i \(0.507152\pi\)
\(858\) −18.2462 −0.622915
\(859\) 33.1771 1.13199 0.565994 0.824410i \(-0.308493\pi\)
0.565994 + 0.824410i \(0.308493\pi\)
\(860\) 10.4384 0.355948
\(861\) 0 0
\(862\) 3.31534 0.112921
\(863\) 10.7386 0.365547 0.182774 0.983155i \(-0.441492\pi\)
0.182774 + 0.983155i \(0.441492\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −28.8769 −0.981844
\(866\) 14.6847 0.499005
\(867\) −15.7386 −0.534512
\(868\) 0 0
\(869\) 72.9848 2.47584
\(870\) −11.8078 −0.400321
\(871\) −4.00000 −0.135535
\(872\) −10.4384 −0.353490
\(873\) 12.4384 0.420978
\(874\) −5.12311 −0.173292
\(875\) 0 0
\(876\) −9.12311 −0.308241
\(877\) −35.8617 −1.21096 −0.605482 0.795859i \(-0.707020\pi\)
−0.605482 + 0.795859i \(0.707020\pi\)
\(878\) −23.6155 −0.796985
\(879\) −27.6155 −0.931449
\(880\) 8.00000 0.269680
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) −8.87689 −0.298731 −0.149366 0.988782i \(-0.547723\pi\)
−0.149366 + 0.988782i \(0.547723\pi\)
\(884\) 4.00000 0.134535
\(885\) −7.61553 −0.255993
\(886\) −9.06913 −0.304683
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 1.56155 0.0524023
\(889\) 0 0
\(890\) 21.8617 0.732807
\(891\) 5.12311 0.171630
\(892\) −3.12311 −0.104569
\(893\) −12.4924 −0.418043
\(894\) 6.24621 0.208905
\(895\) −5.94602 −0.198754
\(896\) 0 0
\(897\) 3.56155 0.118917
\(898\) 0.246211 0.00821618
\(899\) −23.6155 −0.787622
\(900\) −2.56155 −0.0853851
\(901\) 16.0000 0.533037
\(902\) 18.2462 0.607532
\(903\) 0 0
\(904\) 2.68466 0.0892904
\(905\) 23.6155 0.785007
\(906\) −7.31534 −0.243036
\(907\) 52.5464 1.74477 0.872387 0.488815i \(-0.162571\pi\)
0.872387 + 0.488815i \(0.162571\pi\)
\(908\) 6.19224 0.205497
\(909\) −16.2462 −0.538853
\(910\) 0 0
\(911\) −1.94602 −0.0644747 −0.0322373 0.999480i \(-0.510263\pi\)
−0.0322373 + 0.999480i \(0.510263\pi\)
\(912\) 5.12311 0.169643
\(913\) 46.7386 1.54682
\(914\) −7.36932 −0.243755
\(915\) −1.36932 −0.0452682
\(916\) −2.24621 −0.0742169
\(917\) 0 0
\(918\) −1.12311 −0.0370680
\(919\) −50.7386 −1.67371 −0.836857 0.547422i \(-0.815609\pi\)
−0.836857 + 0.547422i \(0.815609\pi\)
\(920\) −1.56155 −0.0514829
\(921\) −2.93087 −0.0965754
\(922\) 40.7386 1.34166
\(923\) −33.3693 −1.09836
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) 13.5616 0.445660
\(927\) −18.9309 −0.621771
\(928\) −7.56155 −0.248220
\(929\) 36.4384 1.19551 0.597753 0.801680i \(-0.296060\pi\)
0.597753 + 0.801680i \(0.296060\pi\)
\(930\) 4.87689 0.159920
\(931\) 0 0
\(932\) −18.4924 −0.605739
\(933\) −18.7386 −0.613475
\(934\) 2.68466 0.0878447
\(935\) 8.98485 0.293836
\(936\) −3.56155 −0.116413
\(937\) 27.1771 0.887837 0.443918 0.896067i \(-0.353588\pi\)
0.443918 + 0.896067i \(0.353588\pi\)
\(938\) 0 0
\(939\) −4.24621 −0.138570
\(940\) −3.80776 −0.124196
\(941\) −1.94602 −0.0634386 −0.0317193 0.999497i \(-0.510098\pi\)
−0.0317193 + 0.999497i \(0.510098\pi\)
\(942\) 14.2462 0.464167
\(943\) −3.56155 −0.115980
\(944\) −4.87689 −0.158729
\(945\) 0 0
\(946\) −34.2462 −1.11344
\(947\) −54.9309 −1.78501 −0.892507 0.451034i \(-0.851055\pi\)
−0.892507 + 0.451034i \(0.851055\pi\)
\(948\) 14.2462 0.462695
\(949\) −32.4924 −1.05475
\(950\) 13.1231 0.425770
\(951\) −17.8078 −0.577456
\(952\) 0 0
\(953\) 50.0000 1.61966 0.809829 0.586665i \(-0.199560\pi\)
0.809829 + 0.586665i \(0.199560\pi\)
\(954\) −14.2462 −0.461238
\(955\) −31.2311 −1.01061
\(956\) 4.49242 0.145295
\(957\) 38.7386 1.25224
\(958\) 31.6155 1.02145
\(959\) 0 0
\(960\) 1.56155 0.0503989
\(961\) −21.2462 −0.685362
\(962\) 5.56155 0.179312
\(963\) −5.12311 −0.165090
\(964\) 20.0540 0.645895
\(965\) 4.19224 0.134953
\(966\) 0 0
\(967\) −9.75379 −0.313661 −0.156830 0.987626i \(-0.550128\pi\)
−0.156830 + 0.987626i \(0.550128\pi\)
\(968\) −15.2462 −0.490032
\(969\) 5.75379 0.184838
\(970\) −19.4233 −0.623644
\(971\) 1.61553 0.0518448 0.0259224 0.999664i \(-0.491748\pi\)
0.0259224 + 0.999664i \(0.491748\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 5.56155 0.178204
\(975\) −9.12311 −0.292173
\(976\) −0.876894 −0.0280687
\(977\) −26.1922 −0.837964 −0.418982 0.907995i \(-0.637613\pi\)
−0.418982 + 0.907995i \(0.637613\pi\)
\(978\) 8.00000 0.255812
\(979\) −71.7235 −2.29229
\(980\) 0 0
\(981\) 10.4384 0.333274
\(982\) 26.7386 0.853264
\(983\) −33.8617 −1.08002 −0.540011 0.841658i \(-0.681580\pi\)
−0.540011 + 0.841658i \(0.681580\pi\)
\(984\) 3.56155 0.113538
\(985\) 38.9309 1.24044
\(986\) −8.49242 −0.270454
\(987\) 0 0
\(988\) 18.2462 0.580489
\(989\) 6.68466 0.212560
\(990\) −8.00000 −0.254257
\(991\) −6.24621 −0.198417 −0.0992087 0.995067i \(-0.531631\pi\)
−0.0992087 + 0.995067i \(0.531631\pi\)
\(992\) 3.12311 0.0991587
\(993\) −24.4924 −0.777244
\(994\) 0 0
\(995\) 33.0691 1.04836
\(996\) 9.12311 0.289077
\(997\) −51.4773 −1.63030 −0.815151 0.579249i \(-0.803346\pi\)
−0.815151 + 0.579249i \(0.803346\pi\)
\(998\) −41.3693 −1.30952
\(999\) −1.56155 −0.0494053
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 6762.2.a.bw.1.2 2
7.6 odd 2 966.2.a.l.1.1 2
21.20 even 2 2898.2.a.ba.1.2 2
28.27 even 2 7728.2.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.l.1.1 2 7.6 odd 2
2898.2.a.ba.1.2 2 21.20 even 2
6762.2.a.bw.1.2 2 1.1 even 1 trivial
7728.2.a.bm.1.1 2 28.27 even 2