Properties

Label 3822.2.a.bt.1.2
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.70156 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.70156 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.70156 q^{10} +5.70156 q^{11} +1.00000 q^{12} -1.00000 q^{13} +3.70156 q^{15} +1.00000 q^{16} -3.70156 q^{17} +1.00000 q^{18} -5.70156 q^{19} +3.70156 q^{20} +5.70156 q^{22} -1.70156 q^{23} +1.00000 q^{24} +8.70156 q^{25} -1.00000 q^{26} +1.00000 q^{27} -3.70156 q^{29} +3.70156 q^{30} +1.00000 q^{32} +5.70156 q^{33} -3.70156 q^{34} +1.00000 q^{36} +4.29844 q^{37} -5.70156 q^{38} -1.00000 q^{39} +3.70156 q^{40} +9.40312 q^{41} +9.10469 q^{43} +5.70156 q^{44} +3.70156 q^{45} -1.70156 q^{46} +8.00000 q^{47} +1.00000 q^{48} +8.70156 q^{50} -3.70156 q^{51} -1.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +21.1047 q^{55} -5.70156 q^{57} -3.70156 q^{58} -10.8062 q^{59} +3.70156 q^{60} -7.70156 q^{61} +1.00000 q^{64} -3.70156 q^{65} +5.70156 q^{66} -7.40312 q^{67} -3.70156 q^{68} -1.70156 q^{69} -8.00000 q^{71} +1.00000 q^{72} +7.70156 q^{73} +4.29844 q^{74} +8.70156 q^{75} -5.70156 q^{76} -1.00000 q^{78} -3.40312 q^{79} +3.70156 q^{80} +1.00000 q^{81} +9.40312 q^{82} +0.596876 q^{83} -13.7016 q^{85} +9.10469 q^{86} -3.70156 q^{87} +5.70156 q^{88} -16.8062 q^{89} +3.70156 q^{90} -1.70156 q^{92} +8.00000 q^{94} -21.1047 q^{95} +1.00000 q^{96} -16.8062 q^{97} +5.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + q^{10} + 5 q^{11} + 2 q^{12} - 2 q^{13} + q^{15} + 2 q^{16} - q^{17} + 2 q^{18} - 5 q^{19} + q^{20} + 5 q^{22} + 3 q^{23} + 2 q^{24} + 11 q^{25} - 2 q^{26} + 2 q^{27} - q^{29} + q^{30} + 2 q^{32} + 5 q^{33} - q^{34} + 2 q^{36} + 15 q^{37} - 5 q^{38} - 2 q^{39} + q^{40} + 6 q^{41} - q^{43} + 5 q^{44} + q^{45} + 3 q^{46} + 16 q^{47} + 2 q^{48} + 11 q^{50} - q^{51} - 2 q^{52} - 4 q^{53} + 2 q^{54} + 23 q^{55} - 5 q^{57} - q^{58} + 4 q^{59} + q^{60} - 9 q^{61} + 2 q^{64} - q^{65} + 5 q^{66} - 2 q^{67} - q^{68} + 3 q^{69} - 16 q^{71} + 2 q^{72} + 9 q^{73} + 15 q^{74} + 11 q^{75} - 5 q^{76} - 2 q^{78} + 6 q^{79} + q^{80} + 2 q^{81} + 6 q^{82} + 14 q^{83} - 21 q^{85} - q^{86} - q^{87} + 5 q^{88} - 8 q^{89} + q^{90} + 3 q^{92} + 16 q^{94} - 23 q^{95} + 2 q^{96} - 8 q^{97} + 5 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 3.70156 1.65539 0.827694 0.561179i \(-0.189652\pi\)
0.827694 + 0.561179i \(0.189652\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.70156 1.17054
\(11\) 5.70156 1.71909 0.859543 0.511064i \(-0.170748\pi\)
0.859543 + 0.511064i \(0.170748\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.70156 0.955739
\(16\) 1.00000 0.250000
\(17\) −3.70156 −0.897761 −0.448880 0.893592i \(-0.648177\pi\)
−0.448880 + 0.893592i \(0.648177\pi\)
\(18\) 1.00000 0.235702
\(19\) −5.70156 −1.30803 −0.654014 0.756482i \(-0.726916\pi\)
−0.654014 + 0.756482i \(0.726916\pi\)
\(20\) 3.70156 0.827694
\(21\) 0 0
\(22\) 5.70156 1.21558
\(23\) −1.70156 −0.354800 −0.177400 0.984139i \(-0.556769\pi\)
−0.177400 + 0.984139i \(0.556769\pi\)
\(24\) 1.00000 0.204124
\(25\) 8.70156 1.74031
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.70156 −0.687363 −0.343681 0.939086i \(-0.611674\pi\)
−0.343681 + 0.939086i \(0.611674\pi\)
\(30\) 3.70156 0.675810
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.70156 0.992515
\(34\) −3.70156 −0.634813
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.29844 0.706659 0.353329 0.935499i \(-0.385049\pi\)
0.353329 + 0.935499i \(0.385049\pi\)
\(38\) −5.70156 −0.924916
\(39\) −1.00000 −0.160128
\(40\) 3.70156 0.585268
\(41\) 9.40312 1.46852 0.734261 0.678868i \(-0.237529\pi\)
0.734261 + 0.678868i \(0.237529\pi\)
\(42\) 0 0
\(43\) 9.10469 1.38845 0.694226 0.719757i \(-0.255747\pi\)
0.694226 + 0.719757i \(0.255747\pi\)
\(44\) 5.70156 0.859543
\(45\) 3.70156 0.551796
\(46\) −1.70156 −0.250882
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 8.70156 1.23059
\(51\) −3.70156 −0.518322
\(52\) −1.00000 −0.138675
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 21.1047 2.84576
\(56\) 0 0
\(57\) −5.70156 −0.755190
\(58\) −3.70156 −0.486039
\(59\) −10.8062 −1.40685 −0.703427 0.710768i \(-0.748348\pi\)
−0.703427 + 0.710768i \(0.748348\pi\)
\(60\) 3.70156 0.477870
\(61\) −7.70156 −0.986084 −0.493042 0.870006i \(-0.664115\pi\)
−0.493042 + 0.870006i \(0.664115\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.70156 −0.459122
\(66\) 5.70156 0.701814
\(67\) −7.40312 −0.904436 −0.452218 0.891908i \(-0.649367\pi\)
−0.452218 + 0.891908i \(0.649367\pi\)
\(68\) −3.70156 −0.448880
\(69\) −1.70156 −0.204844
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 7.70156 0.901400 0.450700 0.892676i \(-0.351174\pi\)
0.450700 + 0.892676i \(0.351174\pi\)
\(74\) 4.29844 0.499683
\(75\) 8.70156 1.00477
\(76\) −5.70156 −0.654014
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) −3.40312 −0.382881 −0.191441 0.981504i \(-0.561316\pi\)
−0.191441 + 0.981504i \(0.561316\pi\)
\(80\) 3.70156 0.413847
\(81\) 1.00000 0.111111
\(82\) 9.40312 1.03840
\(83\) 0.596876 0.0655156 0.0327578 0.999463i \(-0.489571\pi\)
0.0327578 + 0.999463i \(0.489571\pi\)
\(84\) 0 0
\(85\) −13.7016 −1.48614
\(86\) 9.10469 0.981784
\(87\) −3.70156 −0.396849
\(88\) 5.70156 0.607789
\(89\) −16.8062 −1.78146 −0.890729 0.454534i \(-0.849806\pi\)
−0.890729 + 0.454534i \(0.849806\pi\)
\(90\) 3.70156 0.390179
\(91\) 0 0
\(92\) −1.70156 −0.177400
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) −21.1047 −2.16530
\(96\) 1.00000 0.102062
\(97\) −16.8062 −1.70642 −0.853208 0.521571i \(-0.825346\pi\)
−0.853208 + 0.521571i \(0.825346\pi\)
\(98\) 0 0
\(99\) 5.70156 0.573029
\(100\) 8.70156 0.870156
\(101\) −9.40312 −0.935646 −0.467823 0.883822i \(-0.654961\pi\)
−0.467823 + 0.883822i \(0.654961\pi\)
\(102\) −3.70156 −0.366509
\(103\) 9.70156 0.955923 0.477962 0.878381i \(-0.341376\pi\)
0.477962 + 0.878381i \(0.341376\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 7.40312 0.715687 0.357844 0.933782i \(-0.383512\pi\)
0.357844 + 0.933782i \(0.383512\pi\)
\(108\) 1.00000 0.0962250
\(109\) 15.7016 1.50394 0.751968 0.659199i \(-0.229105\pi\)
0.751968 + 0.659199i \(0.229105\pi\)
\(110\) 21.1047 2.01225
\(111\) 4.29844 0.407990
\(112\) 0 0
\(113\) −20.8062 −1.95729 −0.978644 0.205564i \(-0.934097\pi\)
−0.978644 + 0.205564i \(0.934097\pi\)
\(114\) −5.70156 −0.534000
\(115\) −6.29844 −0.587332
\(116\) −3.70156 −0.343681
\(117\) −1.00000 −0.0924500
\(118\) −10.8062 −0.994796
\(119\) 0 0
\(120\) 3.70156 0.337905
\(121\) 21.5078 1.95526
\(122\) −7.70156 −0.697267
\(123\) 9.40312 0.847851
\(124\) 0 0
\(125\) 13.7016 1.22550
\(126\) 0 0
\(127\) −19.4031 −1.72175 −0.860874 0.508817i \(-0.830083\pi\)
−0.860874 + 0.508817i \(0.830083\pi\)
\(128\) 1.00000 0.0883883
\(129\) 9.10469 0.801623
\(130\) −3.70156 −0.324648
\(131\) 13.7016 1.19711 0.598556 0.801081i \(-0.295742\pi\)
0.598556 + 0.801081i \(0.295742\pi\)
\(132\) 5.70156 0.496257
\(133\) 0 0
\(134\) −7.40312 −0.639533
\(135\) 3.70156 0.318580
\(136\) −3.70156 −0.317406
\(137\) −12.2984 −1.05073 −0.525363 0.850878i \(-0.676071\pi\)
−0.525363 + 0.850878i \(0.676071\pi\)
\(138\) −1.70156 −0.144847
\(139\) 18.8062 1.59513 0.797563 0.603236i \(-0.206122\pi\)
0.797563 + 0.603236i \(0.206122\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) −8.00000 −0.671345
\(143\) −5.70156 −0.476789
\(144\) 1.00000 0.0833333
\(145\) −13.7016 −1.13785
\(146\) 7.70156 0.637386
\(147\) 0 0
\(148\) 4.29844 0.353329
\(149\) 2.59688 0.212744 0.106372 0.994326i \(-0.466077\pi\)
0.106372 + 0.994326i \(0.466077\pi\)
\(150\) 8.70156 0.710480
\(151\) 1.70156 0.138471 0.0692356 0.997600i \(-0.477944\pi\)
0.0692356 + 0.997600i \(0.477944\pi\)
\(152\) −5.70156 −0.462458
\(153\) −3.70156 −0.299254
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) −0.895314 −0.0714538 −0.0357269 0.999362i \(-0.511375\pi\)
−0.0357269 + 0.999362i \(0.511375\pi\)
\(158\) −3.40312 −0.270738
\(159\) −2.00000 −0.158610
\(160\) 3.70156 0.292634
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −7.40312 −0.579857 −0.289929 0.957048i \(-0.593632\pi\)
−0.289929 + 0.957048i \(0.593632\pi\)
\(164\) 9.40312 0.734261
\(165\) 21.1047 1.64300
\(166\) 0.596876 0.0463265
\(167\) −9.70156 −0.750729 −0.375365 0.926877i \(-0.622483\pi\)
−0.375365 + 0.926877i \(0.622483\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −13.7016 −1.05086
\(171\) −5.70156 −0.436009
\(172\) 9.10469 0.694226
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −3.70156 −0.280615
\(175\) 0 0
\(176\) 5.70156 0.429771
\(177\) −10.8062 −0.812247
\(178\) −16.8062 −1.25968
\(179\) −10.8062 −0.807697 −0.403848 0.914826i \(-0.632328\pi\)
−0.403848 + 0.914826i \(0.632328\pi\)
\(180\) 3.70156 0.275898
\(181\) 16.8062 1.24920 0.624599 0.780945i \(-0.285262\pi\)
0.624599 + 0.780945i \(0.285262\pi\)
\(182\) 0 0
\(183\) −7.70156 −0.569316
\(184\) −1.70156 −0.125441
\(185\) 15.9109 1.16980
\(186\) 0 0
\(187\) −21.1047 −1.54333
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) −21.1047 −1.53109
\(191\) 6.29844 0.455739 0.227869 0.973692i \(-0.426824\pi\)
0.227869 + 0.973692i \(0.426824\pi\)
\(192\) 1.00000 0.0721688
\(193\) −1.40312 −0.100999 −0.0504995 0.998724i \(-0.516081\pi\)
−0.0504995 + 0.998724i \(0.516081\pi\)
\(194\) −16.8062 −1.20662
\(195\) −3.70156 −0.265074
\(196\) 0 0
\(197\) 9.40312 0.669945 0.334972 0.942228i \(-0.391273\pi\)
0.334972 + 0.942228i \(0.391273\pi\)
\(198\) 5.70156 0.405192
\(199\) −2.89531 −0.205243 −0.102622 0.994720i \(-0.532723\pi\)
−0.102622 + 0.994720i \(0.532723\pi\)
\(200\) 8.70156 0.615293
\(201\) −7.40312 −0.522176
\(202\) −9.40312 −0.661602
\(203\) 0 0
\(204\) −3.70156 −0.259161
\(205\) 34.8062 2.43097
\(206\) 9.70156 0.675940
\(207\) −1.70156 −0.118267
\(208\) −1.00000 −0.0693375
\(209\) −32.5078 −2.24861
\(210\) 0 0
\(211\) −18.2984 −1.25972 −0.629858 0.776710i \(-0.716887\pi\)
−0.629858 + 0.776710i \(0.716887\pi\)
\(212\) −2.00000 −0.137361
\(213\) −8.00000 −0.548151
\(214\) 7.40312 0.506067
\(215\) 33.7016 2.29843
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 15.7016 1.06344
\(219\) 7.70156 0.520423
\(220\) 21.1047 1.42288
\(221\) 3.70156 0.248994
\(222\) 4.29844 0.288492
\(223\) 3.40312 0.227890 0.113945 0.993487i \(-0.463651\pi\)
0.113945 + 0.993487i \(0.463651\pi\)
\(224\) 0 0
\(225\) 8.70156 0.580104
\(226\) −20.8062 −1.38401
\(227\) −8.59688 −0.570595 −0.285297 0.958439i \(-0.592092\pi\)
−0.285297 + 0.958439i \(0.592092\pi\)
\(228\) −5.70156 −0.377595
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) −6.29844 −0.415307
\(231\) 0 0
\(232\) −3.70156 −0.243019
\(233\) 20.2094 1.32396 0.661980 0.749521i \(-0.269716\pi\)
0.661980 + 0.749521i \(0.269716\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 29.6125 1.93171
\(236\) −10.8062 −0.703427
\(237\) −3.40312 −0.221057
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 3.70156 0.238935
\(241\) −0.806248 −0.0519350 −0.0259675 0.999663i \(-0.508267\pi\)
−0.0259675 + 0.999663i \(0.508267\pi\)
\(242\) 21.5078 1.38257
\(243\) 1.00000 0.0641500
\(244\) −7.70156 −0.493042
\(245\) 0 0
\(246\) 9.40312 0.599521
\(247\) 5.70156 0.362782
\(248\) 0 0
\(249\) 0.596876 0.0378255
\(250\) 13.7016 0.866563
\(251\) 28.5078 1.79940 0.899699 0.436512i \(-0.143786\pi\)
0.899699 + 0.436512i \(0.143786\pi\)
\(252\) 0 0
\(253\) −9.70156 −0.609932
\(254\) −19.4031 −1.21746
\(255\) −13.7016 −0.858025
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 9.10469 0.566833
\(259\) 0 0
\(260\) −3.70156 −0.229561
\(261\) −3.70156 −0.229121
\(262\) 13.7016 0.846485
\(263\) −1.19375 −0.0736099 −0.0368049 0.999322i \(-0.511718\pi\)
−0.0368049 + 0.999322i \(0.511718\pi\)
\(264\) 5.70156 0.350907
\(265\) −7.40312 −0.454770
\(266\) 0 0
\(267\) −16.8062 −1.02853
\(268\) −7.40312 −0.452218
\(269\) −20.8062 −1.26858 −0.634290 0.773095i \(-0.718707\pi\)
−0.634290 + 0.773095i \(0.718707\pi\)
\(270\) 3.70156 0.225270
\(271\) 26.2094 1.59211 0.796053 0.605227i \(-0.206918\pi\)
0.796053 + 0.605227i \(0.206918\pi\)
\(272\) −3.70156 −0.224440
\(273\) 0 0
\(274\) −12.2984 −0.742976
\(275\) 49.6125 2.99175
\(276\) −1.70156 −0.102422
\(277\) −23.6125 −1.41874 −0.709369 0.704838i \(-0.751020\pi\)
−0.709369 + 0.704838i \(0.751020\pi\)
\(278\) 18.8062 1.12792
\(279\) 0 0
\(280\) 0 0
\(281\) −12.8062 −0.763957 −0.381978 0.924171i \(-0.624757\pi\)
−0.381978 + 0.924171i \(0.624757\pi\)
\(282\) 8.00000 0.476393
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −8.00000 −0.474713
\(285\) −21.1047 −1.25013
\(286\) −5.70156 −0.337140
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −3.29844 −0.194026
\(290\) −13.7016 −0.804583
\(291\) −16.8062 −0.985200
\(292\) 7.70156 0.450700
\(293\) −12.8062 −0.748149 −0.374075 0.927399i \(-0.622040\pi\)
−0.374075 + 0.927399i \(0.622040\pi\)
\(294\) 0 0
\(295\) −40.0000 −2.32889
\(296\) 4.29844 0.249842
\(297\) 5.70156 0.330838
\(298\) 2.59688 0.150433
\(299\) 1.70156 0.0984039
\(300\) 8.70156 0.502385
\(301\) 0 0
\(302\) 1.70156 0.0979139
\(303\) −9.40312 −0.540195
\(304\) −5.70156 −0.327007
\(305\) −28.5078 −1.63235
\(306\) −3.70156 −0.211604
\(307\) −18.8062 −1.07333 −0.536665 0.843796i \(-0.680316\pi\)
−0.536665 + 0.843796i \(0.680316\pi\)
\(308\) 0 0
\(309\) 9.70156 0.551903
\(310\) 0 0
\(311\) −27.4031 −1.55389 −0.776944 0.629569i \(-0.783231\pi\)
−0.776944 + 0.629569i \(0.783231\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 16.2094 0.916208 0.458104 0.888899i \(-0.348529\pi\)
0.458104 + 0.888899i \(0.348529\pi\)
\(314\) −0.895314 −0.0505255
\(315\) 0 0
\(316\) −3.40312 −0.191441
\(317\) −5.40312 −0.303470 −0.151735 0.988421i \(-0.548486\pi\)
−0.151735 + 0.988421i \(0.548486\pi\)
\(318\) −2.00000 −0.112154
\(319\) −21.1047 −1.18164
\(320\) 3.70156 0.206924
\(321\) 7.40312 0.413202
\(322\) 0 0
\(323\) 21.1047 1.17430
\(324\) 1.00000 0.0555556
\(325\) −8.70156 −0.482676
\(326\) −7.40312 −0.410021
\(327\) 15.7016 0.868298
\(328\) 9.40312 0.519201
\(329\) 0 0
\(330\) 21.1047 1.16177
\(331\) −6.20937 −0.341298 −0.170649 0.985332i \(-0.554586\pi\)
−0.170649 + 0.985332i \(0.554586\pi\)
\(332\) 0.596876 0.0327578
\(333\) 4.29844 0.235553
\(334\) −9.70156 −0.530846
\(335\) −27.4031 −1.49719
\(336\) 0 0
\(337\) 2.50781 0.136609 0.0683046 0.997665i \(-0.478241\pi\)
0.0683046 + 0.997665i \(0.478241\pi\)
\(338\) 1.00000 0.0543928
\(339\) −20.8062 −1.13004
\(340\) −13.7016 −0.743072
\(341\) 0 0
\(342\) −5.70156 −0.308305
\(343\) 0 0
\(344\) 9.10469 0.490892
\(345\) −6.29844 −0.339096
\(346\) 18.0000 0.967686
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −3.70156 −0.198425
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 5.70156 0.303894
\(353\) 4.80625 0.255811 0.127905 0.991786i \(-0.459175\pi\)
0.127905 + 0.991786i \(0.459175\pi\)
\(354\) −10.8062 −0.574346
\(355\) −29.6125 −1.57167
\(356\) −16.8062 −0.890729
\(357\) 0 0
\(358\) −10.8062 −0.571128
\(359\) 1.19375 0.0630038 0.0315019 0.999504i \(-0.489971\pi\)
0.0315019 + 0.999504i \(0.489971\pi\)
\(360\) 3.70156 0.195089
\(361\) 13.5078 0.710937
\(362\) 16.8062 0.883317
\(363\) 21.5078 1.12887
\(364\) 0 0
\(365\) 28.5078 1.49217
\(366\) −7.70156 −0.402567
\(367\) −21.6125 −1.12816 −0.564082 0.825719i \(-0.690770\pi\)
−0.564082 + 0.825719i \(0.690770\pi\)
\(368\) −1.70156 −0.0887001
\(369\) 9.40312 0.489507
\(370\) 15.9109 0.827170
\(371\) 0 0
\(372\) 0 0
\(373\) −6.59688 −0.341573 −0.170787 0.985308i \(-0.554631\pi\)
−0.170787 + 0.985308i \(0.554631\pi\)
\(374\) −21.1047 −1.09130
\(375\) 13.7016 0.707546
\(376\) 8.00000 0.412568
\(377\) 3.70156 0.190640
\(378\) 0 0
\(379\) 33.6125 1.72656 0.863279 0.504727i \(-0.168407\pi\)
0.863279 + 0.504727i \(0.168407\pi\)
\(380\) −21.1047 −1.08265
\(381\) −19.4031 −0.994052
\(382\) 6.29844 0.322256
\(383\) 27.9109 1.42618 0.713091 0.701071i \(-0.247295\pi\)
0.713091 + 0.701071i \(0.247295\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −1.40312 −0.0714171
\(387\) 9.10469 0.462817
\(388\) −16.8062 −0.853208
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) −3.70156 −0.187436
\(391\) 6.29844 0.318526
\(392\) 0 0
\(393\) 13.7016 0.691152
\(394\) 9.40312 0.473723
\(395\) −12.5969 −0.633818
\(396\) 5.70156 0.286514
\(397\) 24.8062 1.24499 0.622495 0.782624i \(-0.286119\pi\)
0.622495 + 0.782624i \(0.286119\pi\)
\(398\) −2.89531 −0.145129
\(399\) 0 0
\(400\) 8.70156 0.435078
\(401\) −20.8062 −1.03901 −0.519507 0.854466i \(-0.673884\pi\)
−0.519507 + 0.854466i \(0.673884\pi\)
\(402\) −7.40312 −0.369234
\(403\) 0 0
\(404\) −9.40312 −0.467823
\(405\) 3.70156 0.183932
\(406\) 0 0
\(407\) 24.5078 1.21481
\(408\) −3.70156 −0.183255
\(409\) −11.7016 −0.578605 −0.289303 0.957238i \(-0.593423\pi\)
−0.289303 + 0.957238i \(0.593423\pi\)
\(410\) 34.8062 1.71896
\(411\) −12.2984 −0.606637
\(412\) 9.70156 0.477962
\(413\) 0 0
\(414\) −1.70156 −0.0836272
\(415\) 2.20937 0.108454
\(416\) −1.00000 −0.0490290
\(417\) 18.8062 0.920946
\(418\) −32.5078 −1.59001
\(419\) −14.8953 −0.727684 −0.363842 0.931461i \(-0.618535\pi\)
−0.363842 + 0.931461i \(0.618535\pi\)
\(420\) 0 0
\(421\) −31.6125 −1.54070 −0.770349 0.637622i \(-0.779918\pi\)
−0.770349 + 0.637622i \(0.779918\pi\)
\(422\) −18.2984 −0.890754
\(423\) 8.00000 0.388973
\(424\) −2.00000 −0.0971286
\(425\) −32.2094 −1.56238
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 7.40312 0.357844
\(429\) −5.70156 −0.275274
\(430\) 33.7016 1.62523
\(431\) 26.2094 1.26246 0.631231 0.775595i \(-0.282550\pi\)
0.631231 + 0.775595i \(0.282550\pi\)
\(432\) 1.00000 0.0481125
\(433\) 24.2094 1.16343 0.581714 0.813393i \(-0.302382\pi\)
0.581714 + 0.813393i \(0.302382\pi\)
\(434\) 0 0
\(435\) −13.7016 −0.656940
\(436\) 15.7016 0.751968
\(437\) 9.70156 0.464089
\(438\) 7.70156 0.367995
\(439\) 32.5078 1.55151 0.775757 0.631032i \(-0.217368\pi\)
0.775757 + 0.631032i \(0.217368\pi\)
\(440\) 21.1047 1.00613
\(441\) 0 0
\(442\) 3.70156 0.176065
\(443\) −38.2094 −1.81538 −0.907691 0.419639i \(-0.862157\pi\)
−0.907691 + 0.419639i \(0.862157\pi\)
\(444\) 4.29844 0.203995
\(445\) −62.2094 −2.94901
\(446\) 3.40312 0.161143
\(447\) 2.59688 0.122828
\(448\) 0 0
\(449\) −27.1047 −1.27915 −0.639575 0.768729i \(-0.720889\pi\)
−0.639575 + 0.768729i \(0.720889\pi\)
\(450\) 8.70156 0.410196
\(451\) 53.6125 2.52451
\(452\) −20.8062 −0.978644
\(453\) 1.70156 0.0799464
\(454\) −8.59688 −0.403471
\(455\) 0 0
\(456\) −5.70156 −0.267000
\(457\) 13.4031 0.626972 0.313486 0.949593i \(-0.398503\pi\)
0.313486 + 0.949593i \(0.398503\pi\)
\(458\) −6.00000 −0.280362
\(459\) −3.70156 −0.172774
\(460\) −6.29844 −0.293666
\(461\) −37.3141 −1.73789 −0.868944 0.494910i \(-0.835201\pi\)
−0.868944 + 0.494910i \(0.835201\pi\)
\(462\) 0 0
\(463\) 23.3141 1.08350 0.541748 0.840541i \(-0.317763\pi\)
0.541748 + 0.840541i \(0.317763\pi\)
\(464\) −3.70156 −0.171841
\(465\) 0 0
\(466\) 20.2094 0.936181
\(467\) 36.5078 1.68938 0.844690 0.535256i \(-0.179785\pi\)
0.844690 + 0.535256i \(0.179785\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 29.6125 1.36592
\(471\) −0.895314 −0.0412539
\(472\) −10.8062 −0.497398
\(473\) 51.9109 2.38687
\(474\) −3.40312 −0.156311
\(475\) −49.6125 −2.27638
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 16.0000 0.731823
\(479\) 24.5078 1.11979 0.559895 0.828563i \(-0.310841\pi\)
0.559895 + 0.828563i \(0.310841\pi\)
\(480\) 3.70156 0.168952
\(481\) −4.29844 −0.195992
\(482\) −0.806248 −0.0367236
\(483\) 0 0
\(484\) 21.5078 0.977628
\(485\) −62.2094 −2.82478
\(486\) 1.00000 0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −7.70156 −0.348633
\(489\) −7.40312 −0.334781
\(490\) 0 0
\(491\) 39.4031 1.77824 0.889119 0.457676i \(-0.151318\pi\)
0.889119 + 0.457676i \(0.151318\pi\)
\(492\) 9.40312 0.423926
\(493\) 13.7016 0.617087
\(494\) 5.70156 0.256525
\(495\) 21.1047 0.948585
\(496\) 0 0
\(497\) 0 0
\(498\) 0.596876 0.0267466
\(499\) −26.8062 −1.20001 −0.600006 0.799995i \(-0.704835\pi\)
−0.600006 + 0.799995i \(0.704835\pi\)
\(500\) 13.7016 0.612752
\(501\) −9.70156 −0.433434
\(502\) 28.5078 1.27237
\(503\) 20.5969 0.918369 0.459185 0.888341i \(-0.348142\pi\)
0.459185 + 0.888341i \(0.348142\pi\)
\(504\) 0 0
\(505\) −34.8062 −1.54886
\(506\) −9.70156 −0.431287
\(507\) 1.00000 0.0444116
\(508\) −19.4031 −0.860874
\(509\) 5.91093 0.261998 0.130999 0.991383i \(-0.458182\pi\)
0.130999 + 0.991383i \(0.458182\pi\)
\(510\) −13.7016 −0.606715
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −5.70156 −0.251730
\(514\) 6.00000 0.264649
\(515\) 35.9109 1.58242
\(516\) 9.10469 0.400811
\(517\) 45.6125 2.00604
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) −3.70156 −0.162324
\(521\) 7.70156 0.337412 0.168706 0.985666i \(-0.446041\pi\)
0.168706 + 0.985666i \(0.446041\pi\)
\(522\) −3.70156 −0.162013
\(523\) −17.6125 −0.770141 −0.385070 0.922887i \(-0.625823\pi\)
−0.385070 + 0.922887i \(0.625823\pi\)
\(524\) 13.7016 0.598556
\(525\) 0 0
\(526\) −1.19375 −0.0520500
\(527\) 0 0
\(528\) 5.70156 0.248129
\(529\) −20.1047 −0.874117
\(530\) −7.40312 −0.321571
\(531\) −10.8062 −0.468951
\(532\) 0 0
\(533\) −9.40312 −0.407295
\(534\) −16.8062 −0.727277
\(535\) 27.4031 1.18474
\(536\) −7.40312 −0.319766
\(537\) −10.8062 −0.466324
\(538\) −20.8062 −0.897021
\(539\) 0 0
\(540\) 3.70156 0.159290
\(541\) −0.298438 −0.0128308 −0.00641542 0.999979i \(-0.502042\pi\)
−0.00641542 + 0.999979i \(0.502042\pi\)
\(542\) 26.2094 1.12579
\(543\) 16.8062 0.721225
\(544\) −3.70156 −0.158703
\(545\) 58.1203 2.48960
\(546\) 0 0
\(547\) −41.6125 −1.77922 −0.889611 0.456719i \(-0.849024\pi\)
−0.889611 + 0.456719i \(0.849024\pi\)
\(548\) −12.2984 −0.525363
\(549\) −7.70156 −0.328695
\(550\) 49.6125 2.11548
\(551\) 21.1047 0.899090
\(552\) −1.70156 −0.0724233
\(553\) 0 0
\(554\) −23.6125 −1.00320
\(555\) 15.9109 0.675382
\(556\) 18.8062 0.797563
\(557\) −37.4031 −1.58482 −0.792411 0.609988i \(-0.791174\pi\)
−0.792411 + 0.609988i \(0.791174\pi\)
\(558\) 0 0
\(559\) −9.10469 −0.385087
\(560\) 0 0
\(561\) −21.1047 −0.891041
\(562\) −12.8062 −0.540199
\(563\) −2.29844 −0.0968676 −0.0484338 0.998826i \(-0.515423\pi\)
−0.0484338 + 0.998826i \(0.515423\pi\)
\(564\) 8.00000 0.336861
\(565\) −77.0156 −3.24007
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) −21.1047 −0.883978
\(571\) −26.8062 −1.12181 −0.560903 0.827881i \(-0.689546\pi\)
−0.560903 + 0.827881i \(0.689546\pi\)
\(572\) −5.70156 −0.238394
\(573\) 6.29844 0.263121
\(574\) 0 0
\(575\) −14.8062 −0.617463
\(576\) 1.00000 0.0416667
\(577\) −16.8062 −0.699653 −0.349827 0.936814i \(-0.613760\pi\)
−0.349827 + 0.936814i \(0.613760\pi\)
\(578\) −3.29844 −0.137197
\(579\) −1.40312 −0.0583119
\(580\) −13.7016 −0.568926
\(581\) 0 0
\(582\) −16.8062 −0.696641
\(583\) −11.4031 −0.472269
\(584\) 7.70156 0.318693
\(585\) −3.70156 −0.153041
\(586\) −12.8062 −0.529021
\(587\) 29.0156 1.19760 0.598801 0.800898i \(-0.295644\pi\)
0.598801 + 0.800898i \(0.295644\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −40.0000 −1.64677
\(591\) 9.40312 0.386793
\(592\) 4.29844 0.176665
\(593\) −8.80625 −0.361629 −0.180815 0.983517i \(-0.557873\pi\)
−0.180815 + 0.983517i \(0.557873\pi\)
\(594\) 5.70156 0.233938
\(595\) 0 0
\(596\) 2.59688 0.106372
\(597\) −2.89531 −0.118497
\(598\) 1.70156 0.0695820
\(599\) 17.7016 0.723266 0.361633 0.932320i \(-0.382219\pi\)
0.361633 + 0.932320i \(0.382219\pi\)
\(600\) 8.70156 0.355240
\(601\) 2.59688 0.105929 0.0529644 0.998596i \(-0.483133\pi\)
0.0529644 + 0.998596i \(0.483133\pi\)
\(602\) 0 0
\(603\) −7.40312 −0.301479
\(604\) 1.70156 0.0692356
\(605\) 79.6125 3.23671
\(606\) −9.40312 −0.381976
\(607\) 24.5078 0.994741 0.497371 0.867538i \(-0.334299\pi\)
0.497371 + 0.867538i \(0.334299\pi\)
\(608\) −5.70156 −0.231229
\(609\) 0 0
\(610\) −28.5078 −1.15425
\(611\) −8.00000 −0.323645
\(612\) −3.70156 −0.149627
\(613\) −5.91093 −0.238740 −0.119370 0.992850i \(-0.538088\pi\)
−0.119370 + 0.992850i \(0.538088\pi\)
\(614\) −18.8062 −0.758958
\(615\) 34.8062 1.40352
\(616\) 0 0
\(617\) 13.9109 0.560033 0.280017 0.959995i \(-0.409660\pi\)
0.280017 + 0.959995i \(0.409660\pi\)
\(618\) 9.70156 0.390254
\(619\) −13.7016 −0.550712 −0.275356 0.961342i \(-0.588796\pi\)
−0.275356 + 0.961342i \(0.588796\pi\)
\(620\) 0 0
\(621\) −1.70156 −0.0682813
\(622\) −27.4031 −1.09877
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 7.20937 0.288375
\(626\) 16.2094 0.647857
\(627\) −32.5078 −1.29824
\(628\) −0.895314 −0.0357269
\(629\) −15.9109 −0.634411
\(630\) 0 0
\(631\) 24.5078 0.975641 0.487820 0.872944i \(-0.337792\pi\)
0.487820 + 0.872944i \(0.337792\pi\)
\(632\) −3.40312 −0.135369
\(633\) −18.2984 −0.727298
\(634\) −5.40312 −0.214585
\(635\) −71.8219 −2.85016
\(636\) −2.00000 −0.0793052
\(637\) 0 0
\(638\) −21.1047 −0.835543
\(639\) −8.00000 −0.316475
\(640\) 3.70156 0.146317
\(641\) −23.1938 −0.916098 −0.458049 0.888927i \(-0.651452\pi\)
−0.458049 + 0.888927i \(0.651452\pi\)
\(642\) 7.40312 0.292178
\(643\) 11.3141 0.446183 0.223091 0.974798i \(-0.428385\pi\)
0.223091 + 0.974798i \(0.428385\pi\)
\(644\) 0 0
\(645\) 33.7016 1.32700
\(646\) 21.1047 0.830353
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 1.00000 0.0392837
\(649\) −61.6125 −2.41850
\(650\) −8.70156 −0.341303
\(651\) 0 0
\(652\) −7.40312 −0.289929
\(653\) 22.5078 0.880799 0.440399 0.897802i \(-0.354837\pi\)
0.440399 + 0.897802i \(0.354837\pi\)
\(654\) 15.7016 0.613980
\(655\) 50.7172 1.98168
\(656\) 9.40312 0.367130
\(657\) 7.70156 0.300467
\(658\) 0 0
\(659\) 29.0156 1.13029 0.565144 0.824992i \(-0.308821\pi\)
0.565144 + 0.824992i \(0.308821\pi\)
\(660\) 21.1047 0.821499
\(661\) −26.4187 −1.02757 −0.513785 0.857919i \(-0.671757\pi\)
−0.513785 + 0.857919i \(0.671757\pi\)
\(662\) −6.20937 −0.241334
\(663\) 3.70156 0.143757
\(664\) 0.596876 0.0231633
\(665\) 0 0
\(666\) 4.29844 0.166561
\(667\) 6.29844 0.243876
\(668\) −9.70156 −0.375365
\(669\) 3.40312 0.131572
\(670\) −27.4031 −1.05868
\(671\) −43.9109 −1.69516
\(672\) 0 0
\(673\) −1.91093 −0.0736611 −0.0368306 0.999322i \(-0.511726\pi\)
−0.0368306 + 0.999322i \(0.511726\pi\)
\(674\) 2.50781 0.0965973
\(675\) 8.70156 0.334923
\(676\) 1.00000 0.0384615
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −20.8062 −0.799059
\(679\) 0 0
\(680\) −13.7016 −0.525431
\(681\) −8.59688 −0.329433
\(682\) 0 0
\(683\) −20.5078 −0.784710 −0.392355 0.919814i \(-0.628339\pi\)
−0.392355 + 0.919814i \(0.628339\pi\)
\(684\) −5.70156 −0.218005
\(685\) −45.5234 −1.73936
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) 9.10469 0.347113
\(689\) 2.00000 0.0761939
\(690\) −6.29844 −0.239777
\(691\) 1.61250 0.0613423 0.0306711 0.999530i \(-0.490236\pi\)
0.0306711 + 0.999530i \(0.490236\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 69.6125 2.64055
\(696\) −3.70156 −0.140307
\(697\) −34.8062 −1.31838
\(698\) −30.0000 −1.13552
\(699\) 20.2094 0.764389
\(700\) 0 0
\(701\) 28.8062 1.08800 0.543998 0.839086i \(-0.316910\pi\)
0.543998 + 0.839086i \(0.316910\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −24.5078 −0.924330
\(704\) 5.70156 0.214886
\(705\) 29.6125 1.11527
\(706\) 4.80625 0.180886
\(707\) 0 0
\(708\) −10.8062 −0.406124
\(709\) 0.387503 0.0145530 0.00727649 0.999974i \(-0.497684\pi\)
0.00727649 + 0.999974i \(0.497684\pi\)
\(710\) −29.6125 −1.11134
\(711\) −3.40312 −0.127627
\(712\) −16.8062 −0.629841
\(713\) 0 0
\(714\) 0 0
\(715\) −21.1047 −0.789271
\(716\) −10.8062 −0.403848
\(717\) 16.0000 0.597531
\(718\) 1.19375 0.0445504
\(719\) −33.0156 −1.23127 −0.615637 0.788030i \(-0.711101\pi\)
−0.615637 + 0.788030i \(0.711101\pi\)
\(720\) 3.70156 0.137949
\(721\) 0 0
\(722\) 13.5078 0.502709
\(723\) −0.806248 −0.0299847
\(724\) 16.8062 0.624599
\(725\) −32.2094 −1.19623
\(726\) 21.5078 0.798230
\(727\) 2.89531 0.107381 0.0536906 0.998558i \(-0.482902\pi\)
0.0536906 + 0.998558i \(0.482902\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 28.5078 1.05512
\(731\) −33.7016 −1.24650
\(732\) −7.70156 −0.284658
\(733\) −33.4031 −1.23377 −0.616886 0.787052i \(-0.711606\pi\)
−0.616886 + 0.787052i \(0.711606\pi\)
\(734\) −21.6125 −0.797732
\(735\) 0 0
\(736\) −1.70156 −0.0627204
\(737\) −42.2094 −1.55480
\(738\) 9.40312 0.346134
\(739\) 1.79063 0.0658693 0.0329346 0.999458i \(-0.489515\pi\)
0.0329346 + 0.999458i \(0.489515\pi\)
\(740\) 15.9109 0.584898
\(741\) 5.70156 0.209452
\(742\) 0 0
\(743\) −41.0156 −1.50472 −0.752359 0.658754i \(-0.771084\pi\)
−0.752359 + 0.658754i \(0.771084\pi\)
\(744\) 0 0
\(745\) 9.61250 0.352175
\(746\) −6.59688 −0.241529
\(747\) 0.596876 0.0218385
\(748\) −21.1047 −0.771664
\(749\) 0 0
\(750\) 13.7016 0.500310
\(751\) 9.19375 0.335485 0.167742 0.985831i \(-0.446352\pi\)
0.167742 + 0.985831i \(0.446352\pi\)
\(752\) 8.00000 0.291730
\(753\) 28.5078 1.03888
\(754\) 3.70156 0.134803
\(755\) 6.29844 0.229224
\(756\) 0 0
\(757\) −46.4187 −1.68712 −0.843559 0.537037i \(-0.819544\pi\)
−0.843559 + 0.537037i \(0.819544\pi\)
\(758\) 33.6125 1.22086
\(759\) −9.70156 −0.352144
\(760\) −21.1047 −0.765547
\(761\) −45.4031 −1.64586 −0.822931 0.568141i \(-0.807663\pi\)
−0.822931 + 0.568141i \(0.807663\pi\)
\(762\) −19.4031 −0.702901
\(763\) 0 0
\(764\) 6.29844 0.227869
\(765\) −13.7016 −0.495381
\(766\) 27.9109 1.00846
\(767\) 10.8062 0.390191
\(768\) 1.00000 0.0360844
\(769\) −6.08907 −0.219577 −0.109789 0.993955i \(-0.535017\pi\)
−0.109789 + 0.993955i \(0.535017\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −1.40312 −0.0504995
\(773\) −29.3141 −1.05435 −0.527177 0.849756i \(-0.676749\pi\)
−0.527177 + 0.849756i \(0.676749\pi\)
\(774\) 9.10469 0.327261
\(775\) 0 0
\(776\) −16.8062 −0.603309
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) −53.6125 −1.92087
\(780\) −3.70156 −0.132537
\(781\) −45.6125 −1.63214
\(782\) 6.29844 0.225232
\(783\) −3.70156 −0.132283
\(784\) 0 0
\(785\) −3.31406 −0.118284
\(786\) 13.7016 0.488719
\(787\) 23.9109 0.852333 0.426166 0.904645i \(-0.359864\pi\)
0.426166 + 0.904645i \(0.359864\pi\)
\(788\) 9.40312 0.334972
\(789\) −1.19375 −0.0424987
\(790\) −12.5969 −0.448177
\(791\) 0 0
\(792\) 5.70156 0.202596
\(793\) 7.70156 0.273490
\(794\) 24.8062 0.880341
\(795\) −7.40312 −0.262562
\(796\) −2.89531 −0.102622
\(797\) 53.4031 1.89164 0.945818 0.324698i \(-0.105263\pi\)
0.945818 + 0.324698i \(0.105263\pi\)
\(798\) 0 0
\(799\) −29.6125 −1.04761
\(800\) 8.70156 0.307647
\(801\) −16.8062 −0.593820
\(802\) −20.8062 −0.734694
\(803\) 43.9109 1.54958
\(804\) −7.40312 −0.261088
\(805\) 0 0
\(806\) 0 0
\(807\) −20.8062 −0.732415
\(808\) −9.40312 −0.330801
\(809\) 39.6125 1.39270 0.696351 0.717702i \(-0.254806\pi\)
0.696351 + 0.717702i \(0.254806\pi\)
\(810\) 3.70156 0.130060
\(811\) 41.1047 1.44338 0.721690 0.692216i \(-0.243365\pi\)
0.721690 + 0.692216i \(0.243365\pi\)
\(812\) 0 0
\(813\) 26.2094 0.919203
\(814\) 24.5078 0.858998
\(815\) −27.4031 −0.959890
\(816\) −3.70156 −0.129581
\(817\) −51.9109 −1.81613
\(818\) −11.7016 −0.409136
\(819\) 0 0
\(820\) 34.8062 1.21549
\(821\) 41.4031 1.44498 0.722489 0.691382i \(-0.242998\pi\)
0.722489 + 0.691382i \(0.242998\pi\)
\(822\) −12.2984 −0.428957
\(823\) 30.8062 1.07384 0.536919 0.843634i \(-0.319588\pi\)
0.536919 + 0.843634i \(0.319588\pi\)
\(824\) 9.70156 0.337970
\(825\) 49.6125 1.72729
\(826\) 0 0
\(827\) 8.08907 0.281284 0.140642 0.990060i \(-0.455083\pi\)
0.140642 + 0.990060i \(0.455083\pi\)
\(828\) −1.70156 −0.0591334
\(829\) −21.3141 −0.740268 −0.370134 0.928978i \(-0.620688\pi\)
−0.370134 + 0.928978i \(0.620688\pi\)
\(830\) 2.20937 0.0766884
\(831\) −23.6125 −0.819108
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) 18.8062 0.651207
\(835\) −35.9109 −1.24275
\(836\) −32.5078 −1.12431
\(837\) 0 0
\(838\) −14.8953 −0.514550
\(839\) −9.19375 −0.317404 −0.158702 0.987327i \(-0.550731\pi\)
−0.158702 + 0.987327i \(0.550731\pi\)
\(840\) 0 0
\(841\) −15.2984 −0.527532
\(842\) −31.6125 −1.08944
\(843\) −12.8062 −0.441071
\(844\) −18.2984 −0.629858
\(845\) 3.70156 0.127338
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) −4.00000 −0.137280
\(850\) −32.2094 −1.10477
\(851\) −7.31406 −0.250723
\(852\) −8.00000 −0.274075
\(853\) −16.2094 −0.554998 −0.277499 0.960726i \(-0.589506\pi\)
−0.277499 + 0.960726i \(0.589506\pi\)
\(854\) 0 0
\(855\) −21.1047 −0.721765
\(856\) 7.40312 0.253034
\(857\) −8.80625 −0.300816 −0.150408 0.988624i \(-0.548059\pi\)
−0.150408 + 0.988624i \(0.548059\pi\)
\(858\) −5.70156 −0.194648
\(859\) −13.1938 −0.450165 −0.225082 0.974340i \(-0.572265\pi\)
−0.225082 + 0.974340i \(0.572265\pi\)
\(860\) 33.7016 1.14921
\(861\) 0 0
\(862\) 26.2094 0.892695
\(863\) −35.4031 −1.20514 −0.602568 0.798067i \(-0.705856\pi\)
−0.602568 + 0.798067i \(0.705856\pi\)
\(864\) 1.00000 0.0340207
\(865\) 66.6281 2.26542
\(866\) 24.2094 0.822668
\(867\) −3.29844 −0.112021
\(868\) 0 0
\(869\) −19.4031 −0.658206
\(870\) −13.7016 −0.464526
\(871\) 7.40312 0.250845
\(872\) 15.7016 0.531722
\(873\) −16.8062 −0.568805
\(874\) 9.70156 0.328160
\(875\) 0 0
\(876\) 7.70156 0.260212
\(877\) 51.6125 1.74283 0.871415 0.490546i \(-0.163203\pi\)
0.871415 + 0.490546i \(0.163203\pi\)
\(878\) 32.5078 1.09709
\(879\) −12.8062 −0.431944
\(880\) 21.1047 0.711439
\(881\) −49.3141 −1.66143 −0.830716 0.556696i \(-0.812069\pi\)
−0.830716 + 0.556696i \(0.812069\pi\)
\(882\) 0 0
\(883\) 33.1047 1.11406 0.557031 0.830492i \(-0.311941\pi\)
0.557031 + 0.830492i \(0.311941\pi\)
\(884\) 3.70156 0.124497
\(885\) −40.0000 −1.34459
\(886\) −38.2094 −1.28367
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 4.29844 0.144246
\(889\) 0 0
\(890\) −62.2094 −2.08526
\(891\) 5.70156 0.191010
\(892\) 3.40312 0.113945
\(893\) −45.6125 −1.52636
\(894\) 2.59688 0.0868525
\(895\) −40.0000 −1.33705
\(896\) 0 0
\(897\) 1.70156 0.0568135
\(898\) −27.1047 −0.904495
\(899\) 0 0
\(900\) 8.70156 0.290052
\(901\) 7.40312 0.246634
\(902\) 53.6125 1.78510
\(903\) 0 0
\(904\) −20.8062 −0.692006
\(905\) 62.2094 2.06791
\(906\) 1.70156 0.0565306
\(907\) 25.6125 0.850449 0.425225 0.905088i \(-0.360195\pi\)
0.425225 + 0.905088i \(0.360195\pi\)
\(908\) −8.59688 −0.285297
\(909\) −9.40312 −0.311882
\(910\) 0 0
\(911\) −3.91093 −0.129575 −0.0647875 0.997899i \(-0.520637\pi\)
−0.0647875 + 0.997899i \(0.520637\pi\)
\(912\) −5.70156 −0.188798
\(913\) 3.40312 0.112627
\(914\) 13.4031 0.443336
\(915\) −28.5078 −0.942439
\(916\) −6.00000 −0.198246
\(917\) 0 0
\(918\) −3.70156 −0.122170
\(919\) −27.4031 −0.903946 −0.451973 0.892032i \(-0.649280\pi\)
−0.451973 + 0.892032i \(0.649280\pi\)
\(920\) −6.29844 −0.207653
\(921\) −18.8062 −0.619687
\(922\) −37.3141 −1.22887
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 37.4031 1.22981
\(926\) 23.3141 0.766148
\(927\) 9.70156 0.318641
\(928\) −3.70156 −0.121510
\(929\) 31.0156 1.01759 0.508795 0.860888i \(-0.330091\pi\)
0.508795 + 0.860888i \(0.330091\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 20.2094 0.661980
\(933\) −27.4031 −0.897138
\(934\) 36.5078 1.19457
\(935\) −78.1203 −2.55481
\(936\) −1.00000 −0.0326860
\(937\) −3.19375 −0.104335 −0.0521677 0.998638i \(-0.516613\pi\)
−0.0521677 + 0.998638i \(0.516613\pi\)
\(938\) 0 0
\(939\) 16.2094 0.528973
\(940\) 29.6125 0.965853
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) −0.895314 −0.0291709
\(943\) −16.0000 −0.521032
\(944\) −10.8062 −0.351713
\(945\) 0 0
\(946\) 51.9109 1.68777
\(947\) 36.5078 1.18634 0.593172 0.805076i \(-0.297875\pi\)
0.593172 + 0.805076i \(0.297875\pi\)
\(948\) −3.40312 −0.110528
\(949\) −7.70156 −0.250003
\(950\) −49.6125 −1.60964
\(951\) −5.40312 −0.175208
\(952\) 0 0
\(953\) 49.8219 1.61389 0.806944 0.590628i \(-0.201120\pi\)
0.806944 + 0.590628i \(0.201120\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 23.3141 0.754425
\(956\) 16.0000 0.517477
\(957\) −21.1047 −0.682218
\(958\) 24.5078 0.791811
\(959\) 0 0
\(960\) 3.70156 0.119467
\(961\) −31.0000 −1.00000
\(962\) −4.29844 −0.138587
\(963\) 7.40312 0.238562
\(964\) −0.806248 −0.0259675
\(965\) −5.19375 −0.167193
\(966\) 0 0
\(967\) 18.7172 0.601904 0.300952 0.953639i \(-0.402696\pi\)
0.300952 + 0.953639i \(0.402696\pi\)
\(968\) 21.5078 0.691287
\(969\) 21.1047 0.677980
\(970\) −62.2094 −1.99742
\(971\) −29.1938 −0.936872 −0.468436 0.883497i \(-0.655182\pi\)
−0.468436 + 0.883497i \(0.655182\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) −8.70156 −0.278673
\(976\) −7.70156 −0.246521
\(977\) 28.7172 0.918744 0.459372 0.888244i \(-0.348074\pi\)
0.459372 + 0.888244i \(0.348074\pi\)
\(978\) −7.40312 −0.236726
\(979\) −95.8219 −3.06248
\(980\) 0 0
\(981\) 15.7016 0.501312
\(982\) 39.4031 1.25740
\(983\) 18.8953 0.602667 0.301333 0.953519i \(-0.402568\pi\)
0.301333 + 0.953519i \(0.402568\pi\)
\(984\) 9.40312 0.299761
\(985\) 34.8062 1.10902
\(986\) 13.7016 0.436347
\(987\) 0 0
\(988\) 5.70156 0.181391
\(989\) −15.4922 −0.492623
\(990\) 21.1047 0.670751
\(991\) −51.4031 −1.63287 −0.816437 0.577434i \(-0.804054\pi\)
−0.816437 + 0.577434i \(0.804054\pi\)
\(992\) 0 0
\(993\) −6.20937 −0.197048
\(994\) 0 0
\(995\) −10.7172 −0.339758
\(996\) 0.596876 0.0189127
\(997\) −12.8062 −0.405578 −0.202789 0.979222i \(-0.565001\pi\)
−0.202789 + 0.979222i \(0.565001\pi\)
\(998\) −26.8062 −0.848537
\(999\) 4.29844 0.135997
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 3822.2.a.bt.1.2 2
7.6 odd 2 546.2.a.i.1.1 2
21.20 even 2 1638.2.a.w.1.2 2
28.27 even 2 4368.2.a.bg.1.1 2
91.90 odd 2 7098.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.i.1.1 2 7.6 odd 2
1638.2.a.w.1.2 2 21.20 even 2
3822.2.a.bt.1.2 2 1.1 even 1 trivial
4368.2.a.bg.1.1 2 28.27 even 2
7098.2.a.bh.1.2 2 91.90 odd 2