Properties

Label 546.2.a.i.1.1
Level $546$
Weight $2$
Character 546.1
Self dual yes
Analytic conductor $4.360$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.70156 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.70156 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.70156 q^{10} +5.70156 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} +3.70156 q^{15} +1.00000 q^{16} +3.70156 q^{17} +1.00000 q^{18} +5.70156 q^{19} -3.70156 q^{20} -1.00000 q^{21} +5.70156 q^{22} -1.70156 q^{23} -1.00000 q^{24} +8.70156 q^{25} +1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -3.70156 q^{29} +3.70156 q^{30} +1.00000 q^{32} -5.70156 q^{33} +3.70156 q^{34} -3.70156 q^{35} +1.00000 q^{36} +4.29844 q^{37} +5.70156 q^{38} -1.00000 q^{39} -3.70156 q^{40} -9.40312 q^{41} -1.00000 q^{42} +9.10469 q^{43} +5.70156 q^{44} -3.70156 q^{45} -1.70156 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +8.70156 q^{50} -3.70156 q^{51} +1.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -21.1047 q^{55} +1.00000 q^{56} -5.70156 q^{57} -3.70156 q^{58} +10.8062 q^{59} +3.70156 q^{60} +7.70156 q^{61} +1.00000 q^{63} +1.00000 q^{64} -3.70156 q^{65} -5.70156 q^{66} -7.40312 q^{67} +3.70156 q^{68} +1.70156 q^{69} -3.70156 q^{70} -8.00000 q^{71} +1.00000 q^{72} -7.70156 q^{73} +4.29844 q^{74} -8.70156 q^{75} +5.70156 q^{76} +5.70156 q^{77} -1.00000 q^{78} -3.40312 q^{79} -3.70156 q^{80} +1.00000 q^{81} -9.40312 q^{82} -0.596876 q^{83} -1.00000 q^{84} -13.7016 q^{85} +9.10469 q^{86} +3.70156 q^{87} +5.70156 q^{88} +16.8062 q^{89} -3.70156 q^{90} +1.00000 q^{91} -1.70156 q^{92} -8.00000 q^{94} -21.1047 q^{95} -1.00000 q^{96} +16.8062 q^{97} +1.00000 q^{98} +5.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} - q^{10} + 5q^{11} - 2q^{12} + 2q^{13} + 2q^{14} + q^{15} + 2q^{16} + q^{17} + 2q^{18} + 5q^{19} - q^{20} - 2q^{21} + 5q^{22} + 3q^{23} - 2q^{24} + 11q^{25} + 2q^{26} - 2q^{27} + 2q^{28} - q^{29} + q^{30} + 2q^{32} - 5q^{33} + q^{34} - q^{35} + 2q^{36} + 15q^{37} + 5q^{38} - 2q^{39} - q^{40} - 6q^{41} - 2q^{42} - q^{43} + 5q^{44} - q^{45} + 3q^{46} - 16q^{47} - 2q^{48} + 2q^{49} + 11q^{50} - q^{51} + 2q^{52} - 4q^{53} - 2q^{54} - 23q^{55} + 2q^{56} - 5q^{57} - q^{58} - 4q^{59} + q^{60} + 9q^{61} + 2q^{63} + 2q^{64} - q^{65} - 5q^{66} - 2q^{67} + q^{68} - 3q^{69} - q^{70} - 16q^{71} + 2q^{72} - 9q^{73} + 15q^{74} - 11q^{75} + 5q^{76} + 5q^{77} - 2q^{78} + 6q^{79} - q^{80} + 2q^{81} - 6q^{82} - 14q^{83} - 2q^{84} - 21q^{85} - q^{86} + q^{87} + 5q^{88} + 8q^{89} - q^{90} + 2q^{91} + 3q^{92} - 16q^{94} - 23q^{95} - 2q^{96} + 8q^{97} + 2q^{98} + 5q^{99} + O(q^{100}) \)

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
\(5\) −3.70156 −1.65539 −0.827694 0.561179i \(-0.810348\pi\)
−0.827694 + 0.561179i \(0.810348\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 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
\(14\) 1.00000 0.267261
\(15\) 3.70156 0.955739
\(16\) 1.00000 0.250000
\(17\) 3.70156 0.897761 0.448880 0.893592i \(-0.351823\pi\)
0.448880 + 0.893592i \(0.351823\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.70156 1.30803 0.654014 0.756482i \(-0.273084\pi\)
0.654014 + 0.756482i \(0.273084\pi\)
\(20\) −3.70156 −0.827694
\(21\) −1.00000 −0.218218
\(22\) 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
\(28\) 1.00000 0.188982
\(29\) −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
\(35\) −3.70156 −0.625678
\(36\) 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 \(-0.762471\pi\)
−0.734261 + 0.678868i \(0.762471\pi\)
\(42\) −1.00000 −0.154303
\(43\) 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 \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 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
\(56\) 1.00000 0.133631
\(57\) −5.70156 −0.755190
\(58\) −3.70156 −0.486039
\(59\) 10.8062 1.40685 0.703427 0.710768i \(-0.251652\pi\)
0.703427 + 0.710768i \(0.251652\pi\)
\(60\) 3.70156 0.477870
\(61\) 7.70156 0.986084 0.493042 0.870006i \(-0.335885\pi\)
0.493042 + 0.870006i \(0.335885\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 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
\(70\) −3.70156 −0.442421
\(71\) −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 \(-0.648826\pi\)
−0.450700 + 0.892676i \(0.648826\pi\)
\(74\) 4.29844 0.499683
\(75\) −8.70156 −1.00477
\(76\) 5.70156 0.654014
\(77\) 5.70156 0.649753
\(78\) −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 \(-0.510429\pi\)
−0.0327578 + 0.999463i \(0.510429\pi\)
\(84\) −1.00000 −0.109109
\(85\) −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 \(-0.150194\pi\)
0.890729 + 0.454534i \(0.150194\pi\)
\(90\) −3.70156 −0.390179
\(91\) 1.00000 0.104828
\(92\) −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 \(-0.174654\pi\)
0.853208 + 0.521571i \(0.174654\pi\)
\(98\) 1.00000 0.101015
\(99\) 5.70156 0.573029
\(100\) 8.70156 0.870156
\(101\) 9.40312 0.935646 0.467823 0.883822i \(-0.345039\pi\)
0.467823 + 0.883822i \(0.345039\pi\)
\(102\) −3.70156 −0.366509
\(103\) −9.70156 −0.955923 −0.477962 0.878381i \(-0.658624\pi\)
−0.477962 + 0.878381i \(0.658624\pi\)
\(104\) 1.00000 0.0980581
\(105\) 3.70156 0.361235
\(106\) −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
\(112\) 1.00000 0.0944911
\(113\) −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
\(119\) 3.70156 0.339322
\(120\) 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
\(126\) 1.00000 0.0890871
\(127\) −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 \(-0.704258\pi\)
−0.598556 + 0.801081i \(0.704258\pi\)
\(132\) −5.70156 −0.496257
\(133\) 5.70156 0.494388
\(134\) −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 \(-0.793878\pi\)
−0.797563 + 0.603236i \(0.793878\pi\)
\(140\) −3.70156 −0.312839
\(141\) 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
\(147\) −1.00000 −0.0824786
\(148\) 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
\(154\) 5.70156 0.459445
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 0.895314 0.0714538 0.0357269 0.999362i \(-0.488625\pi\)
0.0357269 + 0.999362i \(0.488625\pi\)
\(158\) −3.40312 −0.270738
\(159\) 2.00000 0.158610
\(160\) −3.70156 −0.292634
\(161\) −1.70156 −0.134102
\(162\) 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 \(-0.377517\pi\)
0.375365 + 0.926877i \(0.377517\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 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 \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 3.70156 0.280615
\(175\) 8.70156 0.657776
\(176\) 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 \(-0.714738\pi\)
−0.624599 + 0.780945i \(0.714738\pi\)
\(182\) 1.00000 0.0741249
\(183\) −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
\(189\) −1.00000 −0.0727393
\(190\) −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
\(196\) 1.00000 0.0714286
\(197\) 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 \(-0.467277\pi\)
0.102622 + 0.994720i \(0.467277\pi\)
\(200\) 8.70156 0.615293
\(201\) 7.40312 0.522176
\(202\) 9.40312 0.661602
\(203\) −3.70156 −0.259799
\(204\) −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
\(210\) 3.70156 0.255432
\(211\) −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 \(-0.536349\pi\)
−0.113945 + 0.993487i \(0.536349\pi\)
\(224\) 1.00000 0.0668153
\(225\) 8.70156 0.580104
\(226\) −20.8062 −1.38401
\(227\) 8.59688 0.570595 0.285297 0.958439i \(-0.407908\pi\)
0.285297 + 0.958439i \(0.407908\pi\)
\(228\) −5.70156 −0.377595
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 6.29844 0.415307
\(231\) −5.70156 −0.375135
\(232\) −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
\(238\) 3.70156 0.239937
\(239\) 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 \(-0.491733\pi\)
0.0259675 + 0.999663i \(0.491733\pi\)
\(242\) 21.5078 1.38257
\(243\) −1.00000 −0.0641500
\(244\) 7.70156 0.493042
\(245\) −3.70156 −0.236484
\(246\) 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 \(-0.856214\pi\)
−0.899699 + 0.436512i \(0.856214\pi\)
\(252\) 1.00000 0.0629941
\(253\) −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 \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −9.10469 −0.566833
\(259\) 4.29844 0.267092
\(260\) −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
\(266\) 5.70156 0.349585
\(267\) −16.8062 −1.02853
\(268\) −7.40312 −0.452218
\(269\) 20.8062 1.26858 0.634290 0.773095i \(-0.281293\pi\)
0.634290 + 0.773095i \(0.281293\pi\)
\(270\) 3.70156 0.225270
\(271\) −26.2094 −1.59211 −0.796053 0.605227i \(-0.793082\pi\)
−0.796053 + 0.605227i \(0.793082\pi\)
\(272\) 3.70156 0.224440
\(273\) −1.00000 −0.0605228
\(274\) −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
\(280\) −3.70156 −0.221211
\(281\) −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 \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −8.00000 −0.474713
\(285\) 21.1047 1.25013
\(286\) 5.70156 0.337140
\(287\) −9.40312 −0.555049
\(288\) 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 \(-0.377960\pi\)
0.374075 + 0.927399i \(0.377960\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −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
\(301\) 9.10469 0.524785
\(302\) 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 \(-0.319684\pi\)
0.536665 + 0.843796i \(0.319684\pi\)
\(308\) 5.70156 0.324877
\(309\) 9.70156 0.551903
\(310\) 0 0
\(311\) 27.4031 1.55389 0.776944 0.629569i \(-0.216769\pi\)
0.776944 + 0.629569i \(0.216769\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −16.2094 −0.916208 −0.458104 0.888899i \(-0.651471\pi\)
−0.458104 + 0.888899i \(0.651471\pi\)
\(314\) 0.895314 0.0505255
\(315\) −3.70156 −0.208559
\(316\) −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
\(322\) −1.70156 −0.0948243
\(323\) 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
\(329\) −8.00000 −0.441054
\(330\) 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
\(336\) −1.00000 −0.0545545
\(337\) 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
\(343\) 1.00000 0.0539949
\(344\) 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 \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 8.70156 0.465118
\(351\) −1.00000 −0.0533761
\(352\) 5.70156 0.303894
\(353\) −4.80625 −0.255811 −0.127905 0.991786i \(-0.540825\pi\)
−0.127905 + 0.991786i \(0.540825\pi\)
\(354\) −10.8062 −0.574346
\(355\) 29.6125 1.57167
\(356\) 16.8062 0.890729
\(357\) −3.70156 −0.195907
\(358\) −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
\(364\) 1.00000 0.0524142
\(365\) 28.5078 1.49217
\(366\) −7.70156 −0.402567
\(367\) 21.6125 1.12816 0.564082 0.825719i \(-0.309230\pi\)
0.564082 + 0.825719i \(0.309230\pi\)
\(368\) −1.70156 −0.0887001
\(369\) −9.40312 −0.489507
\(370\) −15.9109 −0.827170
\(371\) −2.00000 −0.103835
\(372\) 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
\(378\) −1.00000 −0.0514344
\(379\) 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 \(-0.752705\pi\)
−0.713091 + 0.701071i \(0.752705\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −21.1047 −1.07559
\(386\) −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
\(392\) 1.00000 0.0505076
\(393\) 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 \(-0.713881\pi\)
−0.622495 + 0.782624i \(0.713881\pi\)
\(398\) 2.89531 0.145129
\(399\) −5.70156 −0.285435
\(400\) 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
\(406\) −3.70156 −0.183705
\(407\) 24.5078 1.21481
\(408\) −3.70156 −0.183255
\(409\) 11.7016 0.578605 0.289303 0.957238i \(-0.406577\pi\)
0.289303 + 0.957238i \(0.406577\pi\)
\(410\) 34.8062 1.71896
\(411\) 12.2984 0.606637
\(412\) −9.70156 −0.477962
\(413\) 10.8062 0.531741
\(414\) −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 \(-0.381465\pi\)
0.363842 + 0.931461i \(0.381465\pi\)
\(420\) 3.70156 0.180618
\(421\) −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
\(427\) 7.70156 0.372705
\(428\) 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 \(-0.697618\pi\)
−0.581714 + 0.813393i \(0.697618\pi\)
\(434\) 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 \(-0.782632\pi\)
−0.775757 + 0.631032i \(0.782632\pi\)
\(440\) −21.1047 −1.00613
\(441\) 1.00000 0.0476190
\(442\) 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
\(448\) 1.00000 0.0472456
\(449\) −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
\(455\) −3.70156 −0.173532
\(456\) −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 \(-0.164799\pi\)
0.868944 + 0.494910i \(0.164799\pi\)
\(462\) −5.70156 −0.265261
\(463\) 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 \(-0.820215\pi\)
−0.844690 + 0.535256i \(0.820215\pi\)
\(468\) 1.00000 0.0462250
\(469\) −7.40312 −0.341845
\(470\) 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
\(476\) 3.70156 0.169661
\(477\) −2.00000 −0.0915737
\(478\) 16.0000 0.731823
\(479\) −24.5078 −1.11979 −0.559895 0.828563i \(-0.689159\pi\)
−0.559895 + 0.828563i \(0.689159\pi\)
\(480\) 3.70156 0.168952
\(481\) 4.29844 0.195992
\(482\) 0.806248 0.0367236
\(483\) 1.70156 0.0774238
\(484\) 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
\(490\) −3.70156 −0.167220
\(491\) 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
\(497\) −8.00000 −0.358849
\(498\) 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 \(-0.651858\pi\)
−0.459185 + 0.888341i \(0.651858\pi\)
\(504\) 1.00000 0.0445435
\(505\) −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 \(-0.541818\pi\)
−0.130999 + 0.991383i \(0.541818\pi\)
\(510\) 13.7016 0.606715
\(511\) −7.70156 −0.340697
\(512\) 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
\(518\) 4.29844 0.188863
\(519\) 18.0000 0.790112
\(520\) −3.70156 −0.162324
\(521\) −7.70156 −0.337412 −0.168706 0.985666i \(-0.553959\pi\)
−0.168706 + 0.985666i \(0.553959\pi\)
\(522\) −3.70156 −0.162013
\(523\) 17.6125 0.770141 0.385070 0.922887i \(-0.374177\pi\)
0.385070 + 0.922887i \(0.374177\pi\)
\(524\) −13.7016 −0.598556
\(525\) −8.70156 −0.379767
\(526\) −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
\(532\) 5.70156 0.247194
\(533\) −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
\(539\) 5.70156 0.245584
\(540\) 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
\(546\) −1.00000 −0.0427960
\(547\) −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
\(553\) −3.40312 −0.144716
\(554\) −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
\(560\) −3.70156 −0.156420
\(561\) −21.1047 −0.891041
\(562\) −12.8062 −0.540199
\(563\) 2.29844 0.0968676 0.0484338 0.998826i \(-0.484577\pi\)
0.0484338 + 0.998826i \(0.484577\pi\)
\(564\) 8.00000 0.336861
\(565\) 77.0156 3.24007
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) −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
\(574\) −9.40312 −0.392479
\(575\) −14.8062 −0.617463
\(576\) 1.00000 0.0416667
\(577\) 16.8062 0.699653 0.349827 0.936814i \(-0.386240\pi\)
0.349827 + 0.936814i \(0.386240\pi\)
\(578\) −3.29844 −0.137197
\(579\) 1.40312 0.0583119
\(580\) 13.7016 0.568926
\(581\) −0.596876 −0.0247626
\(582\) −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 \(-0.704356\pi\)
−0.598801 + 0.800898i \(0.704356\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 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 \(-0.442127\pi\)
0.180815 + 0.983517i \(0.442127\pi\)
\(594\) −5.70156 −0.233938
\(595\) −13.7016 −0.561709
\(596\) 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 \(-0.516867\pi\)
−0.0529644 + 0.998596i \(0.516867\pi\)
\(602\) 9.10469 0.371079
\(603\) −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 \(-0.665701\pi\)
−0.497371 + 0.867538i \(0.665701\pi\)
\(608\) 5.70156 0.231229
\(609\) 3.70156 0.149995
\(610\) −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
\(616\) 5.70156 0.229722
\(617\) 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 \(-0.411204\pi\)
0.275356 + 0.961342i \(0.411204\pi\)
\(620\) 0 0
\(621\) 1.70156 0.0682813
\(622\) 27.4031 1.09877
\(623\) 16.8062 0.673328
\(624\) −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
\(630\) −3.70156 −0.147474
\(631\) 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
\(637\) 1.00000 0.0396214
\(638\) −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 \(-0.571615\pi\)
−0.223091 + 0.974798i \(0.571615\pi\)
\(644\) −1.70156 −0.0670509
\(645\) 33.7016 1.32700
\(646\) 21.1047 0.830353
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 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
\(658\) −8.00000 −0.311872
\(659\) 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 \(-0.328243\pi\)
0.513785 + 0.857919i \(0.328243\pi\)
\(662\) −6.20937 −0.241334
\(663\) −3.70156 −0.143757
\(664\) −0.596876 −0.0231633
\(665\) −21.1047 −0.818405
\(666\) 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
\(672\) −1.00000 −0.0385758
\(673\) −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 \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 20.8062 0.799059
\(679\) 16.8062 0.644965
\(680\) −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
\(686\) 1.00000 0.0381802
\(687\) −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 \(-0.509764\pi\)
−0.0306711 + 0.999530i \(0.509764\pi\)
\(692\) −18.0000 −0.684257
\(693\) 5.70156 0.216584
\(694\) 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
\(700\) 8.70156 0.328888
\(701\) 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
\(707\) 9.40312 0.353641
\(708\) −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
\(714\) −3.70156 −0.138527
\(715\) −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 \(-0.288899\pi\)
0.615637 + 0.788030i \(0.288899\pi\)
\(720\) −3.70156 −0.137949
\(721\) −9.70156 −0.361305
\(722\) 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 \(-0.517098\pi\)
−0.0536906 + 0.998558i \(0.517098\pi\)
\(728\) 1.00000 0.0370625
\(729\) 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 \(-0.288394\pi\)
0.616886 + 0.787052i \(0.288394\pi\)
\(734\) 21.6125 0.797732
\(735\) 3.70156 0.136534
\(736\) −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
\(742\) −2.00000 −0.0734223
\(743\) −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
\(749\) 7.40312 0.270504
\(750\) 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
\(756\) −1.00000 −0.0363696
\(757\) −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 \(-0.192337\pi\)
0.822931 + 0.568141i \(0.192337\pi\)
\(762\) 19.4031 0.702901
\(763\) 15.7016 0.568435
\(764\) 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 \(-0.464983\pi\)
0.109789 + 0.993955i \(0.464983\pi\)
\(770\) −21.1047 −0.760560
\(771\) 6.00000 0.216085
\(772\) −1.40312 −0.0504995
\(773\) 29.3141 1.05435 0.527177 0.849756i \(-0.323251\pi\)
0.527177 + 0.849756i \(0.323251\pi\)
\(774\) 9.10469 0.327261
\(775\) 0 0
\(776\) 16.8062 0.603309
\(777\) −4.29844 −0.154206
\(778\) 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
\(784\) 1.00000 0.0357143
\(785\) −3.31406 −0.118284
\(786\) 13.7016 0.488719
\(787\) −23.9109 −0.852333 −0.426166 0.904645i \(-0.640136\pi\)
−0.426166 + 0.904645i \(0.640136\pi\)
\(788\) 9.40312 0.334972
\(789\) 1.19375 0.0424987
\(790\) 12.5969 0.448177
\(791\) −20.8062 −0.739785
\(792\) 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 \(-0.894737\pi\)
−0.945818 + 0.324698i \(0.894737\pi\)
\(798\) −5.70156 −0.201833
\(799\) −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
\(805\) 6.29844 0.221991
\(806\) 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 \(-0.756635\pi\)
−0.721690 + 0.692216i \(0.756635\pi\)
\(812\) −3.70156 −0.129899
\(813\) 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
\(819\) 1.00000 0.0349428
\(820\) 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
\(826\) 10.8062 0.375997
\(827\) 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 \(-0.379312\pi\)
0.370134 + 0.928978i \(0.379312\pi\)
\(830\) 2.20937 0.0766884
\(831\) 23.6125 0.819108
\(832\) 1.00000 0.0346688
\(833\) 3.70156 0.128252
\(834\) 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 \(-0.449269\pi\)
0.158702 + 0.987327i \(0.449269\pi\)
\(840\) 3.70156 0.127716
\(841\) −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
\(847\) 21.5078 0.739017
\(848\) −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 \(-0.410494\pi\)
0.277499 + 0.960726i \(0.410494\pi\)
\(854\) 7.70156 0.263542
\(855\) −21.1047 −0.721765
\(856\) 7.40312 0.253034
\(857\) 8.80625 0.300816 0.150408 0.988624i \(-0.451941\pi\)
0.150408 + 0.988624i \(0.451941\pi\)
\(858\) −5.70156 −0.194648
\(859\) 13.1938 0.450165 0.225082 0.974340i \(-0.427735\pi\)
0.225082 + 0.974340i \(0.427735\pi\)
\(860\) −33.7016 −1.14921
\(861\) 9.40312 0.320458
\(862\) 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
\(875\) −13.7016 −0.463197
\(876\) 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 \(-0.187931\pi\)
0.830716 + 0.556696i \(0.187931\pi\)
\(882\) 1.00000 0.0336718
\(883\) 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 \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −4.29844 −0.144246
\(889\) −19.4031 −0.650760
\(890\) −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
\(896\) 1.00000 0.0334077
\(897\) 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
\(903\) −9.10469 −0.302985
\(904\) −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
\(910\) −3.70156 −0.122706
\(911\) −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
\(917\) −13.7016 −0.452465
\(918\) −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
\(924\) −5.70156 −0.187568
\(925\) 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 \(-0.669909\pi\)
−0.508795 + 0.860888i \(0.669909\pi\)
\(930\) 0 0
\(931\) 5.70156 0.186861
\(932\) 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 \(-0.483387\pi\)
0.0521677 + 0.998638i \(0.483387\pi\)
\(938\) −7.40312 −0.241721
\(939\) 16.2094 0.528973
\(940\) 29.6125 0.965853
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) −0.895314 −0.0291709
\(943\) 16.0000 0.521032
\(944\) 10.8062 0.351713
\(945\) 3.70156 0.120412
\(946\) 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
\(952\) 3.70156 0.119968
\(953\) 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
\(959\) −12.2984 −0.397137
\(960\) 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
\(966\) 1.70156 0.0547469
\(967\) 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 \(-0.344818\pi\)
0.468436 + 0.883497i \(0.344818\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −18.8062 −0.602901
\(974\) −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
\(980\) −3.70156 −0.118242
\(981\) 15.7016 0.501312
\(982\) 39.4031 1.25740
\(983\) −18.8953 −0.602667 −0.301333 0.953519i \(-0.597432\pi\)
−0.301333 + 0.953519i \(0.597432\pi\)
\(984\) 9.40312 0.299761
\(985\) −34.8062 −1.10902
\(986\) −13.7016 −0.436347
\(987\) 8.00000 0.254643
\(988\) 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
\(994\) −8.00000 −0.253745
\(995\) −10.7172 −0.339758
\(996\) 0.596876 0.0189127
\(997\) 12.8062 0.405578 0.202789 0.979222i \(-0.434999\pi\)
0.202789 + 0.979222i \(0.434999\pi\)
\(998\) −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 546.2.a.i.1.1 2
3.2 odd 2 1638.2.a.w.1.2 2
4.3 odd 2 4368.2.a.bg.1.1 2
7.6 odd 2 3822.2.a.bt.1.2 2
13.12 even 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 1.1 even 1 trivial
1638.2.a.w.1.2 2 3.2 odd 2
3822.2.a.bt.1.2 2 7.6 odd 2
4368.2.a.bg.1.1 2 4.3 odd 2
7098.2.a.bh.1.2 2 13.12 even 2