Properties

Label 6930.2.a.ci.1.2
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} -0.694593 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.69459 q^{17} +5.51754 q^{19} +1.00000 q^{20} -1.00000 q^{22} +7.43376 q^{23} +1.00000 q^{25} +0.694593 q^{26} +1.00000 q^{28} +0.610815 q^{29} -7.51754 q^{31} -1.00000 q^{32} -2.69459 q^{34} +1.00000 q^{35} -1.30541 q^{37} -5.51754 q^{38} -1.00000 q^{40} +9.51754 q^{41} +5.51754 q^{43} +1.00000 q^{44} -7.43376 q^{46} +6.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -0.694593 q^{52} -4.12836 q^{53} +1.00000 q^{55} -1.00000 q^{56} -0.610815 q^{58} -3.51754 q^{59} -4.00000 q^{61} +7.51754 q^{62} +1.00000 q^{64} -0.694593 q^{65} -8.95130 q^{67} +2.69459 q^{68} -1.00000 q^{70} +4.77837 q^{71} +5.51754 q^{73} +1.30541 q^{74} +5.51754 q^{76} +1.00000 q^{77} -2.77837 q^{79} +1.00000 q^{80} -9.51754 q^{82} -4.73917 q^{83} +2.69459 q^{85} -5.51754 q^{86} -1.00000 q^{88} +8.04458 q^{89} -0.694593 q^{91} +7.43376 q^{92} -6.00000 q^{94} +5.51754 q^{95} -6.90673 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} + O(q^{10}) \) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} + 3 q^{11} - 3 q^{14} + 3 q^{16} + 6 q^{17} - 6 q^{19} + 3 q^{20} - 3 q^{22} + 6 q^{23} + 3 q^{25} + 3 q^{28} + 6 q^{29} - 3 q^{32} - 6 q^{34} + 3 q^{35} - 6 q^{37} + 6 q^{38} - 3 q^{40} + 6 q^{41} - 6 q^{43} + 3 q^{44} - 6 q^{46} + 18 q^{47} + 3 q^{49} - 3 q^{50} + 6 q^{53} + 3 q^{55} - 3 q^{56} - 6 q^{58} + 12 q^{59} - 12 q^{61} + 3 q^{64} + 12 q^{67} + 6 q^{68} - 3 q^{70} + 6 q^{71} - 6 q^{73} + 6 q^{74} - 6 q^{76} + 3 q^{77} + 3 q^{80} - 6 q^{82} + 6 q^{85} + 6 q^{86} - 3 q^{88} + 12 q^{89} + 6 q^{92} - 18 q^{94} - 6 q^{95} + 6 q^{97} - 3 q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −0.694593 −0.192645 −0.0963227 0.995350i \(-0.530708\pi\)
−0.0963227 + 0.995350i \(0.530708\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.69459 0.653535 0.326767 0.945105i \(-0.394041\pi\)
0.326767 + 0.945105i \(0.394041\pi\)
\(18\) 0 0
\(19\) 5.51754 1.26581 0.632905 0.774229i \(-0.281862\pi\)
0.632905 + 0.774229i \(0.281862\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 7.43376 1.55005 0.775023 0.631933i \(-0.217738\pi\)
0.775023 + 0.631933i \(0.217738\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.694593 0.136221
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 0.610815 0.113425 0.0567127 0.998391i \(-0.481938\pi\)
0.0567127 + 0.998391i \(0.481938\pi\)
\(30\) 0 0
\(31\) −7.51754 −1.35019 −0.675095 0.737731i \(-0.735897\pi\)
−0.675095 + 0.737731i \(0.735897\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.69459 −0.462119
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −1.30541 −0.214608 −0.107304 0.994226i \(-0.534222\pi\)
−0.107304 + 0.994226i \(0.534222\pi\)
\(38\) −5.51754 −0.895063
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 9.51754 1.48639 0.743195 0.669075i \(-0.233309\pi\)
0.743195 + 0.669075i \(0.233309\pi\)
\(42\) 0 0
\(43\) 5.51754 0.841417 0.420709 0.907196i \(-0.361782\pi\)
0.420709 + 0.907196i \(0.361782\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −7.43376 −1.09605
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −0.694593 −0.0963227
\(53\) −4.12836 −0.567073 −0.283537 0.958961i \(-0.591508\pi\)
−0.283537 + 0.958961i \(0.591508\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −0.610815 −0.0802039
\(59\) −3.51754 −0.457945 −0.228972 0.973433i \(-0.573537\pi\)
−0.228972 + 0.973433i \(0.573537\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 7.51754 0.954729
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.694593 −0.0861536
\(66\) 0 0
\(67\) −8.95130 −1.09358 −0.546788 0.837271i \(-0.684150\pi\)
−0.546788 + 0.837271i \(0.684150\pi\)
\(68\) 2.69459 0.326767
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 4.77837 0.567088 0.283544 0.958959i \(-0.408490\pi\)
0.283544 + 0.958959i \(0.408490\pi\)
\(72\) 0 0
\(73\) 5.51754 0.645779 0.322890 0.946437i \(-0.395346\pi\)
0.322890 + 0.946437i \(0.395346\pi\)
\(74\) 1.30541 0.151751
\(75\) 0 0
\(76\) 5.51754 0.632905
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −2.77837 −0.312591 −0.156296 0.987710i \(-0.549955\pi\)
−0.156296 + 0.987710i \(0.549955\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −9.51754 −1.05104
\(83\) −4.73917 −0.520192 −0.260096 0.965583i \(-0.583754\pi\)
−0.260096 + 0.965583i \(0.583754\pi\)
\(84\) 0 0
\(85\) 2.69459 0.292270
\(86\) −5.51754 −0.594972
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 8.04458 0.852724 0.426362 0.904553i \(-0.359795\pi\)
0.426362 + 0.904553i \(0.359795\pi\)
\(90\) 0 0
\(91\) −0.694593 −0.0728131
\(92\) 7.43376 0.775023
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 5.51754 0.566088
\(96\) 0 0
\(97\) −6.90673 −0.701272 −0.350636 0.936512i \(-0.614035\pi\)
−0.350636 + 0.936512i \(0.614035\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −7.64590 −0.760795 −0.380398 0.924823i \(-0.624213\pi\)
−0.380398 + 0.924823i \(0.624213\pi\)
\(102\) 0 0
\(103\) −7.51754 −0.740725 −0.370363 0.928887i \(-0.620767\pi\)
−0.370363 + 0.928887i \(0.620767\pi\)
\(104\) 0.694593 0.0681104
\(105\) 0 0
\(106\) 4.12836 0.400981
\(107\) −17.1634 −1.65925 −0.829626 0.558319i \(-0.811446\pi\)
−0.829626 + 0.558319i \(0.811446\pi\)
\(108\) 0 0
\(109\) −1.91622 −0.183541 −0.0917704 0.995780i \(-0.529253\pi\)
−0.0917704 + 0.995780i \(0.529253\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 12.2121 1.14882 0.574410 0.818567i \(-0.305231\pi\)
0.574410 + 0.818567i \(0.305231\pi\)
\(114\) 0 0
\(115\) 7.43376 0.693202
\(116\) 0.610815 0.0567127
\(117\) 0 0
\(118\) 3.51754 0.323816
\(119\) 2.69459 0.247013
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) −7.51754 −0.675095
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 11.5175 1.02202 0.511008 0.859576i \(-0.329272\pi\)
0.511008 + 0.859576i \(0.329272\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0.694593 0.0609198
\(131\) 5.38919 0.470855 0.235428 0.971892i \(-0.424351\pi\)
0.235428 + 0.971892i \(0.424351\pi\)
\(132\) 0 0
\(133\) 5.51754 0.478431
\(134\) 8.95130 0.773275
\(135\) 0 0
\(136\) −2.69459 −0.231059
\(137\) 13.4338 1.14772 0.573862 0.818952i \(-0.305445\pi\)
0.573862 + 0.818952i \(0.305445\pi\)
\(138\) 0 0
\(139\) −18.9067 −1.60365 −0.801824 0.597561i \(-0.796137\pi\)
−0.801824 + 0.597561i \(0.796137\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −4.77837 −0.400992
\(143\) −0.694593 −0.0580848
\(144\) 0 0
\(145\) 0.610815 0.0507254
\(146\) −5.51754 −0.456635
\(147\) 0 0
\(148\) −1.30541 −0.107304
\(149\) −0.610815 −0.0500399 −0.0250199 0.999687i \(-0.507965\pi\)
−0.0250199 + 0.999687i \(0.507965\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −5.51754 −0.447532
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) −7.51754 −0.603823
\(156\) 0 0
\(157\) 24.5526 1.95951 0.979756 0.200194i \(-0.0641572\pi\)
0.979756 + 0.200194i \(0.0641572\pi\)
\(158\) 2.77837 0.221035
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 7.43376 0.585863
\(162\) 0 0
\(163\) −1.91622 −0.150090 −0.0750450 0.997180i \(-0.523910\pi\)
−0.0750450 + 0.997180i \(0.523910\pi\)
\(164\) 9.51754 0.743195
\(165\) 0 0
\(166\) 4.73917 0.367831
\(167\) 24.2121 1.87359 0.936796 0.349877i \(-0.113777\pi\)
0.936796 + 0.349877i \(0.113777\pi\)
\(168\) 0 0
\(169\) −12.5175 −0.962888
\(170\) −2.69459 −0.206666
\(171\) 0 0
\(172\) 5.51754 0.420709
\(173\) 7.22163 0.549050 0.274525 0.961580i \(-0.411479\pi\)
0.274525 + 0.961580i \(0.411479\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −8.04458 −0.602967
\(179\) 14.2959 1.06853 0.534263 0.845318i \(-0.320589\pi\)
0.534263 + 0.845318i \(0.320589\pi\)
\(180\) 0 0
\(181\) −2.52704 −0.187833 −0.0939166 0.995580i \(-0.529939\pi\)
−0.0939166 + 0.995580i \(0.529939\pi\)
\(182\) 0.694593 0.0514866
\(183\) 0 0
\(184\) −7.43376 −0.548024
\(185\) −1.30541 −0.0959755
\(186\) 0 0
\(187\) 2.69459 0.197048
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −5.51754 −0.400284
\(191\) −0.610815 −0.0441970 −0.0220985 0.999756i \(-0.507035\pi\)
−0.0220985 + 0.999756i \(0.507035\pi\)
\(192\) 0 0
\(193\) −1.51754 −0.109235 −0.0546175 0.998507i \(-0.517394\pi\)
−0.0546175 + 0.998507i \(0.517394\pi\)
\(194\) 6.90673 0.495874
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −10.7392 −0.765134 −0.382567 0.923928i \(-0.624960\pi\)
−0.382567 + 0.923928i \(0.624960\pi\)
\(198\) 0 0
\(199\) −0.906726 −0.0642761 −0.0321381 0.999483i \(-0.510232\pi\)
−0.0321381 + 0.999483i \(0.510232\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 7.64590 0.537963
\(203\) 0.610815 0.0428708
\(204\) 0 0
\(205\) 9.51754 0.664734
\(206\) 7.51754 0.523772
\(207\) 0 0
\(208\) −0.694593 −0.0481613
\(209\) 5.51754 0.381656
\(210\) 0 0
\(211\) −27.3756 −1.88461 −0.942306 0.334753i \(-0.891347\pi\)
−0.942306 + 0.334753i \(0.891347\pi\)
\(212\) −4.12836 −0.283537
\(213\) 0 0
\(214\) 17.1634 1.17327
\(215\) 5.51754 0.376293
\(216\) 0 0
\(217\) −7.51754 −0.510324
\(218\) 1.91622 0.129783
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) −1.87164 −0.125900
\(222\) 0 0
\(223\) 21.6459 1.44952 0.724758 0.689003i \(-0.241951\pi\)
0.724758 + 0.689003i \(0.241951\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −12.2121 −0.812339
\(227\) 11.3500 0.753325 0.376662 0.926351i \(-0.377072\pi\)
0.376662 + 0.926351i \(0.377072\pi\)
\(228\) 0 0
\(229\) −6.69459 −0.442391 −0.221196 0.975229i \(-0.570996\pi\)
−0.221196 + 0.975229i \(0.570996\pi\)
\(230\) −7.43376 −0.490168
\(231\) 0 0
\(232\) −0.610815 −0.0401019
\(233\) 14.2567 0.933988 0.466994 0.884260i \(-0.345337\pi\)
0.466994 + 0.884260i \(0.345337\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) −3.51754 −0.228972
\(237\) 0 0
\(238\) −2.69459 −0.174665
\(239\) −3.72967 −0.241253 −0.120626 0.992698i \(-0.538490\pi\)
−0.120626 + 0.992698i \(0.538490\pi\)
\(240\) 0 0
\(241\) −14.7392 −0.949433 −0.474717 0.880139i \(-0.657449\pi\)
−0.474717 + 0.880139i \(0.657449\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −3.83244 −0.243853
\(248\) 7.51754 0.477364
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 18.3851 1.16046 0.580228 0.814454i \(-0.302964\pi\)
0.580228 + 0.814454i \(0.302964\pi\)
\(252\) 0 0
\(253\) 7.43376 0.467357
\(254\) −11.5175 −0.722675
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 3.09327 0.192953 0.0964766 0.995335i \(-0.469243\pi\)
0.0964766 + 0.995335i \(0.469243\pi\)
\(258\) 0 0
\(259\) −1.30541 −0.0811141
\(260\) −0.694593 −0.0430768
\(261\) 0 0
\(262\) −5.38919 −0.332945
\(263\) −6.61081 −0.407640 −0.203820 0.979008i \(-0.565336\pi\)
−0.203820 + 0.979008i \(0.565336\pi\)
\(264\) 0 0
\(265\) −4.12836 −0.253603
\(266\) −5.51754 −0.338302
\(267\) 0 0
\(268\) −8.95130 −0.546788
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 13.8135 0.839107 0.419554 0.907730i \(-0.362187\pi\)
0.419554 + 0.907730i \(0.362187\pi\)
\(272\) 2.69459 0.163384
\(273\) 0 0
\(274\) −13.4338 −0.811563
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 25.1634 1.51192 0.755962 0.654615i \(-0.227169\pi\)
0.755962 + 0.654615i \(0.227169\pi\)
\(278\) 18.9067 1.13395
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −15.9162 −0.949482 −0.474741 0.880125i \(-0.657458\pi\)
−0.474741 + 0.880125i \(0.657458\pi\)
\(282\) 0 0
\(283\) −10.2121 −0.607048 −0.303524 0.952824i \(-0.598163\pi\)
−0.303524 + 0.952824i \(0.598163\pi\)
\(284\) 4.77837 0.283544
\(285\) 0 0
\(286\) 0.694593 0.0410721
\(287\) 9.51754 0.561803
\(288\) 0 0
\(289\) −9.73917 −0.572892
\(290\) −0.610815 −0.0358683
\(291\) 0 0
\(292\) 5.51754 0.322890
\(293\) −26.6810 −1.55872 −0.779360 0.626577i \(-0.784455\pi\)
−0.779360 + 0.626577i \(0.784455\pi\)
\(294\) 0 0
\(295\) −3.51754 −0.204799
\(296\) 1.30541 0.0758753
\(297\) 0 0
\(298\) 0.610815 0.0353835
\(299\) −5.16344 −0.298609
\(300\) 0 0
\(301\) 5.51754 0.318026
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 5.51754 0.316453
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −17.2472 −0.984351 −0.492175 0.870496i \(-0.663798\pi\)
−0.492175 + 0.870496i \(0.663798\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) 7.51754 0.426968
\(311\) 12.1729 0.690264 0.345132 0.938554i \(-0.387834\pi\)
0.345132 + 0.938554i \(0.387834\pi\)
\(312\) 0 0
\(313\) −18.6810 −1.05591 −0.527956 0.849272i \(-0.677041\pi\)
−0.527956 + 0.849272i \(0.677041\pi\)
\(314\) −24.5526 −1.38558
\(315\) 0 0
\(316\) −2.77837 −0.156296
\(317\) 29.7743 1.67229 0.836144 0.548510i \(-0.184805\pi\)
0.836144 + 0.548510i \(0.184805\pi\)
\(318\) 0 0
\(319\) 0.610815 0.0341991
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −7.43376 −0.414267
\(323\) 14.8675 0.827251
\(324\) 0 0
\(325\) −0.694593 −0.0385291
\(326\) 1.91622 0.106130
\(327\) 0 0
\(328\) −9.51754 −0.525518
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 4.25671 0.233970 0.116985 0.993134i \(-0.462677\pi\)
0.116985 + 0.993134i \(0.462677\pi\)
\(332\) −4.73917 −0.260096
\(333\) 0 0
\(334\) −24.2121 −1.32483
\(335\) −8.95130 −0.489062
\(336\) 0 0
\(337\) 3.64590 0.198605 0.0993023 0.995057i \(-0.468339\pi\)
0.0993023 + 0.995057i \(0.468339\pi\)
\(338\) 12.5175 0.680864
\(339\) 0 0
\(340\) 2.69459 0.146135
\(341\) −7.51754 −0.407098
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −5.51754 −0.297486
\(345\) 0 0
\(346\) −7.22163 −0.388237
\(347\) −27.9418 −1.49999 −0.749997 0.661441i \(-0.769945\pi\)
−0.749997 + 0.661441i \(0.769945\pi\)
\(348\) 0 0
\(349\) −25.4783 −1.36382 −0.681912 0.731434i \(-0.738851\pi\)
−0.681912 + 0.731434i \(0.738851\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −17.1634 −0.913518 −0.456759 0.889591i \(-0.650990\pi\)
−0.456759 + 0.889591i \(0.650990\pi\)
\(354\) 0 0
\(355\) 4.77837 0.253610
\(356\) 8.04458 0.426362
\(357\) 0 0
\(358\) −14.2959 −0.755562
\(359\) −31.8188 −1.67933 −0.839667 0.543102i \(-0.817250\pi\)
−0.839667 + 0.543102i \(0.817250\pi\)
\(360\) 0 0
\(361\) 11.4433 0.602277
\(362\) 2.52704 0.132818
\(363\) 0 0
\(364\) −0.694593 −0.0364066
\(365\) 5.51754 0.288801
\(366\) 0 0
\(367\) −27.7743 −1.44980 −0.724902 0.688852i \(-0.758115\pi\)
−0.724902 + 0.688852i \(0.758115\pi\)
\(368\) 7.43376 0.387512
\(369\) 0 0
\(370\) 1.30541 0.0678649
\(371\) −4.12836 −0.214334
\(372\) 0 0
\(373\) 0.739170 0.0382728 0.0191364 0.999817i \(-0.493908\pi\)
0.0191364 + 0.999817i \(0.493908\pi\)
\(374\) −2.69459 −0.139334
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −0.424267 −0.0218509
\(378\) 0 0
\(379\) 24.5134 1.25917 0.629585 0.776932i \(-0.283225\pi\)
0.629585 + 0.776932i \(0.283225\pi\)
\(380\) 5.51754 0.283044
\(381\) 0 0
\(382\) 0.610815 0.0312520
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 1.51754 0.0772408
\(387\) 0 0
\(388\) −6.90673 −0.350636
\(389\) 6.65002 0.337169 0.168585 0.985687i \(-0.446080\pi\)
0.168585 + 0.985687i \(0.446080\pi\)
\(390\) 0 0
\(391\) 20.0310 1.01301
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 10.7392 0.541032
\(395\) −2.77837 −0.139795
\(396\) 0 0
\(397\) 34.9067 1.75192 0.875959 0.482385i \(-0.160229\pi\)
0.875959 + 0.482385i \(0.160229\pi\)
\(398\) 0.906726 0.0454501
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 5.22163 0.260108
\(404\) −7.64590 −0.380398
\(405\) 0 0
\(406\) −0.610815 −0.0303142
\(407\) −1.30541 −0.0647066
\(408\) 0 0
\(409\) 3.64590 0.180278 0.0901390 0.995929i \(-0.471269\pi\)
0.0901390 + 0.995929i \(0.471269\pi\)
\(410\) −9.51754 −0.470038
\(411\) 0 0
\(412\) −7.51754 −0.370363
\(413\) −3.51754 −0.173087
\(414\) 0 0
\(415\) −4.73917 −0.232637
\(416\) 0.694593 0.0340552
\(417\) 0 0
\(418\) −5.51754 −0.269872
\(419\) 18.3851 0.898169 0.449085 0.893489i \(-0.351750\pi\)
0.449085 + 0.893489i \(0.351750\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 27.3756 1.33262
\(423\) 0 0
\(424\) 4.12836 0.200491
\(425\) 2.69459 0.130707
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) −17.1634 −0.829626
\(429\) 0 0
\(430\) −5.51754 −0.266079
\(431\) 15.3054 0.737236 0.368618 0.929581i \(-0.379831\pi\)
0.368618 + 0.929581i \(0.379831\pi\)
\(432\) 0 0
\(433\) −9.34998 −0.449332 −0.224666 0.974436i \(-0.572129\pi\)
−0.224666 + 0.974436i \(0.572129\pi\)
\(434\) 7.51754 0.360854
\(435\) 0 0
\(436\) −1.91622 −0.0917704
\(437\) 41.0161 1.96207
\(438\) 0 0
\(439\) 30.5526 1.45820 0.729099 0.684409i \(-0.239940\pi\)
0.729099 + 0.684409i \(0.239940\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 1.87164 0.0890250
\(443\) 36.1985 1.71984 0.859922 0.510426i \(-0.170512\pi\)
0.859922 + 0.510426i \(0.170512\pi\)
\(444\) 0 0
\(445\) 8.04458 0.381350
\(446\) −21.6459 −1.02496
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −1.22163 −0.0576522 −0.0288261 0.999584i \(-0.509177\pi\)
−0.0288261 + 0.999584i \(0.509177\pi\)
\(450\) 0 0
\(451\) 9.51754 0.448164
\(452\) 12.2121 0.574410
\(453\) 0 0
\(454\) −11.3500 −0.532681
\(455\) −0.694593 −0.0325630
\(456\) 0 0
\(457\) −11.0743 −0.518033 −0.259017 0.965873i \(-0.583398\pi\)
−0.259017 + 0.965873i \(0.583398\pi\)
\(458\) 6.69459 0.312818
\(459\) 0 0
\(460\) 7.43376 0.346601
\(461\) −7.87164 −0.366619 −0.183310 0.983055i \(-0.558681\pi\)
−0.183310 + 0.983055i \(0.558681\pi\)
\(462\) 0 0
\(463\) 24.1676 1.12316 0.561581 0.827422i \(-0.310193\pi\)
0.561581 + 0.827422i \(0.310193\pi\)
\(464\) 0.610815 0.0283564
\(465\) 0 0
\(466\) −14.2567 −0.660429
\(467\) 27.5175 1.27336 0.636680 0.771128i \(-0.280307\pi\)
0.636680 + 0.771128i \(0.280307\pi\)
\(468\) 0 0
\(469\) −8.95130 −0.413333
\(470\) −6.00000 −0.276759
\(471\) 0 0
\(472\) 3.51754 0.161908
\(473\) 5.51754 0.253697
\(474\) 0 0
\(475\) 5.51754 0.253162
\(476\) 2.69459 0.123506
\(477\) 0 0
\(478\) 3.72967 0.170591
\(479\) 26.0702 1.19118 0.595588 0.803290i \(-0.296919\pi\)
0.595588 + 0.803290i \(0.296919\pi\)
\(480\) 0 0
\(481\) 0.906726 0.0413432
\(482\) 14.7392 0.671351
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −6.90673 −0.313618
\(486\) 0 0
\(487\) 12.9649 0.587497 0.293748 0.955883i \(-0.405097\pi\)
0.293748 + 0.955883i \(0.405097\pi\)
\(488\) 4.00000 0.181071
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −36.4243 −1.64380 −0.821902 0.569629i \(-0.807087\pi\)
−0.821902 + 0.569629i \(0.807087\pi\)
\(492\) 0 0
\(493\) 1.64590 0.0741275
\(494\) 3.83244 0.172430
\(495\) 0 0
\(496\) −7.51754 −0.337548
\(497\) 4.77837 0.214339
\(498\) 0 0
\(499\) 19.1242 0.856118 0.428059 0.903751i \(-0.359197\pi\)
0.428059 + 0.903751i \(0.359197\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −18.3851 −0.820566
\(503\) 21.7689 0.970626 0.485313 0.874340i \(-0.338706\pi\)
0.485313 + 0.874340i \(0.338706\pi\)
\(504\) 0 0
\(505\) −7.64590 −0.340238
\(506\) −7.43376 −0.330471
\(507\) 0 0
\(508\) 11.5175 0.511008
\(509\) 22.1676 0.982560 0.491280 0.871002i \(-0.336529\pi\)
0.491280 + 0.871002i \(0.336529\pi\)
\(510\) 0 0
\(511\) 5.51754 0.244082
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −3.09327 −0.136438
\(515\) −7.51754 −0.331262
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 1.30541 0.0573563
\(519\) 0 0
\(520\) 0.694593 0.0304599
\(521\) −32.4688 −1.42249 −0.711243 0.702946i \(-0.751867\pi\)
−0.711243 + 0.702946i \(0.751867\pi\)
\(522\) 0 0
\(523\) −3.60132 −0.157475 −0.0787373 0.996895i \(-0.525089\pi\)
−0.0787373 + 0.996895i \(0.525089\pi\)
\(524\) 5.38919 0.235428
\(525\) 0 0
\(526\) 6.61081 0.288245
\(527\) −20.2567 −0.882396
\(528\) 0 0
\(529\) 32.2608 1.40264
\(530\) 4.12836 0.179324
\(531\) 0 0
\(532\) 5.51754 0.239216
\(533\) −6.61081 −0.286346
\(534\) 0 0
\(535\) −17.1634 −0.742040
\(536\) 8.95130 0.386637
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 0.951304 0.0408997 0.0204499 0.999791i \(-0.493490\pi\)
0.0204499 + 0.999791i \(0.493490\pi\)
\(542\) −13.8135 −0.593339
\(543\) 0 0
\(544\) −2.69459 −0.115530
\(545\) −1.91622 −0.0820819
\(546\) 0 0
\(547\) 38.1985 1.63325 0.816625 0.577168i \(-0.195842\pi\)
0.816625 + 0.577168i \(0.195842\pi\)
\(548\) 13.4338 0.573862
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 3.37019 0.143575
\(552\) 0 0
\(553\) −2.77837 −0.118148
\(554\) −25.1634 −1.06909
\(555\) 0 0
\(556\) −18.9067 −0.801824
\(557\) 8.29591 0.351509 0.175755 0.984434i \(-0.443763\pi\)
0.175755 + 0.984434i \(0.443763\pi\)
\(558\) 0 0
\(559\) −3.83244 −0.162095
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 15.9162 0.671385
\(563\) −15.0933 −0.636106 −0.318053 0.948073i \(-0.603029\pi\)
−0.318053 + 0.948073i \(0.603029\pi\)
\(564\) 0 0
\(565\) 12.2121 0.513768
\(566\) 10.2121 0.429248
\(567\) 0 0
\(568\) −4.77837 −0.200496
\(569\) 43.2080 1.81137 0.905687 0.423947i \(-0.139356\pi\)
0.905687 + 0.423947i \(0.139356\pi\)
\(570\) 0 0
\(571\) 13.5621 0.567557 0.283778 0.958890i \(-0.408412\pi\)
0.283778 + 0.958890i \(0.408412\pi\)
\(572\) −0.694593 −0.0290424
\(573\) 0 0
\(574\) −9.51754 −0.397254
\(575\) 7.43376 0.310009
\(576\) 0 0
\(577\) 19.3892 0.807182 0.403591 0.914939i \(-0.367762\pi\)
0.403591 + 0.914939i \(0.367762\pi\)
\(578\) 9.73917 0.405096
\(579\) 0 0
\(580\) 0.610815 0.0253627
\(581\) −4.73917 −0.196614
\(582\) 0 0
\(583\) −4.12836 −0.170979
\(584\) −5.51754 −0.228317
\(585\) 0 0
\(586\) 26.6810 1.10218
\(587\) 28.7392 1.18619 0.593096 0.805132i \(-0.297905\pi\)
0.593096 + 0.805132i \(0.297905\pi\)
\(588\) 0 0
\(589\) −41.4783 −1.70909
\(590\) 3.51754 0.144815
\(591\) 0 0
\(592\) −1.30541 −0.0536519
\(593\) 30.7837 1.26414 0.632068 0.774913i \(-0.282206\pi\)
0.632068 + 0.774913i \(0.282206\pi\)
\(594\) 0 0
\(595\) 2.69459 0.110468
\(596\) −0.610815 −0.0250199
\(597\) 0 0
\(598\) 5.16344 0.211149
\(599\) −5.20264 −0.212574 −0.106287 0.994335i \(-0.533896\pi\)
−0.106287 + 0.994335i \(0.533896\pi\)
\(600\) 0 0
\(601\) −6.68098 −0.272523 −0.136261 0.990673i \(-0.543509\pi\)
−0.136261 + 0.990673i \(0.543509\pi\)
\(602\) −5.51754 −0.224878
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 15.8324 0.642619 0.321310 0.946974i \(-0.395877\pi\)
0.321310 + 0.946974i \(0.395877\pi\)
\(608\) −5.51754 −0.223766
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) −4.16756 −0.168601
\(612\) 0 0
\(613\) −18.7202 −0.756101 −0.378050 0.925785i \(-0.623405\pi\)
−0.378050 + 0.925785i \(0.623405\pi\)
\(614\) 17.2472 0.696041
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −15.0797 −0.607084 −0.303542 0.952818i \(-0.598169\pi\)
−0.303542 + 0.952818i \(0.598169\pi\)
\(618\) 0 0
\(619\) −45.1498 −1.81472 −0.907362 0.420349i \(-0.861907\pi\)
−0.907362 + 0.420349i \(0.861907\pi\)
\(620\) −7.51754 −0.301912
\(621\) 0 0
\(622\) −12.1729 −0.488090
\(623\) 8.04458 0.322299
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 18.6810 0.746642
\(627\) 0 0
\(628\) 24.5526 0.979756
\(629\) −3.51754 −0.140254
\(630\) 0 0
\(631\) 7.42839 0.295719 0.147860 0.989008i \(-0.452762\pi\)
0.147860 + 0.989008i \(0.452762\pi\)
\(632\) 2.77837 0.110518
\(633\) 0 0
\(634\) −29.7743 −1.18249
\(635\) 11.5175 0.457060
\(636\) 0 0
\(637\) −0.694593 −0.0275208
\(638\) −0.610815 −0.0241824
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −36.8485 −1.45543 −0.727715 0.685880i \(-0.759418\pi\)
−0.727715 + 0.685880i \(0.759418\pi\)
\(642\) 0 0
\(643\) −32.5134 −1.28220 −0.641102 0.767456i \(-0.721522\pi\)
−0.641102 + 0.767456i \(0.721522\pi\)
\(644\) 7.43376 0.292931
\(645\) 0 0
\(646\) −14.8675 −0.584955
\(647\) −3.47834 −0.136748 −0.0683738 0.997660i \(-0.521781\pi\)
−0.0683738 + 0.997660i \(0.521781\pi\)
\(648\) 0 0
\(649\) −3.51754 −0.138076
\(650\) 0.694593 0.0272442
\(651\) 0 0
\(652\) −1.91622 −0.0750450
\(653\) −20.7202 −0.810843 −0.405422 0.914130i \(-0.632875\pi\)
−0.405422 + 0.914130i \(0.632875\pi\)
\(654\) 0 0
\(655\) 5.38919 0.210573
\(656\) 9.51754 0.371598
\(657\) 0 0
\(658\) −6.00000 −0.233904
\(659\) 8.25671 0.321636 0.160818 0.986984i \(-0.448587\pi\)
0.160818 + 0.986984i \(0.448587\pi\)
\(660\) 0 0
\(661\) −40.5972 −1.57905 −0.789524 0.613720i \(-0.789673\pi\)
−0.789524 + 0.613720i \(0.789673\pi\)
\(662\) −4.25671 −0.165442
\(663\) 0 0
\(664\) 4.73917 0.183915
\(665\) 5.51754 0.213961
\(666\) 0 0
\(667\) 4.54065 0.175815
\(668\) 24.2121 0.936796
\(669\) 0 0
\(670\) 8.95130 0.345819
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −22.3459 −0.861370 −0.430685 0.902502i \(-0.641728\pi\)
−0.430685 + 0.902502i \(0.641728\pi\)
\(674\) −3.64590 −0.140435
\(675\) 0 0
\(676\) −12.5175 −0.481444
\(677\) 15.9026 0.611187 0.305593 0.952162i \(-0.401145\pi\)
0.305593 + 0.952162i \(0.401145\pi\)
\(678\) 0 0
\(679\) −6.90673 −0.265056
\(680\) −2.69459 −0.103333
\(681\) 0 0
\(682\) 7.51754 0.287862
\(683\) −33.2526 −1.27238 −0.636188 0.771534i \(-0.719490\pi\)
−0.636188 + 0.771534i \(0.719490\pi\)
\(684\) 0 0
\(685\) 13.4338 0.513278
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 5.51754 0.210354
\(689\) 2.86753 0.109244
\(690\) 0 0
\(691\) 10.7338 0.408333 0.204166 0.978936i \(-0.434552\pi\)
0.204166 + 0.978936i \(0.434552\pi\)
\(692\) 7.22163 0.274525
\(693\) 0 0
\(694\) 27.9418 1.06066
\(695\) −18.9067 −0.717173
\(696\) 0 0
\(697\) 25.6459 0.971408
\(698\) 25.4783 0.964369
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −37.4593 −1.41482 −0.707410 0.706803i \(-0.750137\pi\)
−0.707410 + 0.706803i \(0.750137\pi\)
\(702\) 0 0
\(703\) −7.20264 −0.271653
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 17.1634 0.645954
\(707\) −7.64590 −0.287554
\(708\) 0 0
\(709\) 19.8135 0.744110 0.372055 0.928211i \(-0.378653\pi\)
0.372055 + 0.928211i \(0.378653\pi\)
\(710\) −4.77837 −0.179329
\(711\) 0 0
\(712\) −8.04458 −0.301483
\(713\) −55.8836 −2.09286
\(714\) 0 0
\(715\) −0.694593 −0.0259763
\(716\) 14.2959 0.534263
\(717\) 0 0
\(718\) 31.8188 1.18747
\(719\) 32.0838 1.19652 0.598262 0.801301i \(-0.295858\pi\)
0.598262 + 0.801301i \(0.295858\pi\)
\(720\) 0 0
\(721\) −7.51754 −0.279968
\(722\) −11.4433 −0.425874
\(723\) 0 0
\(724\) −2.52704 −0.0939166
\(725\) 0.610815 0.0226851
\(726\) 0 0
\(727\) 34.8675 1.29316 0.646582 0.762844i \(-0.276198\pi\)
0.646582 + 0.762844i \(0.276198\pi\)
\(728\) 0.694593 0.0257433
\(729\) 0 0
\(730\) −5.51754 −0.204213
\(731\) 14.8675 0.549895
\(732\) 0 0
\(733\) 29.0405 1.07263 0.536317 0.844017i \(-0.319815\pi\)
0.536317 + 0.844017i \(0.319815\pi\)
\(734\) 27.7743 1.02517
\(735\) 0 0
\(736\) −7.43376 −0.274012
\(737\) −8.95130 −0.329726
\(738\) 0 0
\(739\) 50.4107 1.85439 0.927193 0.374584i \(-0.122215\pi\)
0.927193 + 0.374584i \(0.122215\pi\)
\(740\) −1.30541 −0.0479877
\(741\) 0 0
\(742\) 4.12836 0.151557
\(743\) −10.7784 −0.395420 −0.197710 0.980261i \(-0.563350\pi\)
−0.197710 + 0.980261i \(0.563350\pi\)
\(744\) 0 0
\(745\) −0.610815 −0.0223785
\(746\) −0.739170 −0.0270629
\(747\) 0 0
\(748\) 2.69459 0.0985241
\(749\) −17.1634 −0.627138
\(750\) 0 0
\(751\) −7.94181 −0.289801 −0.144900 0.989446i \(-0.546286\pi\)
−0.144900 + 0.989446i \(0.546286\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 0.424267 0.0154509
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 21.3946 0.777599 0.388799 0.921322i \(-0.372890\pi\)
0.388799 + 0.921322i \(0.372890\pi\)
\(758\) −24.5134 −0.890368
\(759\) 0 0
\(760\) −5.51754 −0.200142
\(761\) −29.7743 −1.07932 −0.539658 0.841884i \(-0.681446\pi\)
−0.539658 + 0.841884i \(0.681446\pi\)
\(762\) 0 0
\(763\) −1.91622 −0.0693719
\(764\) −0.610815 −0.0220985
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) 2.44326 0.0882209
\(768\) 0 0
\(769\) −12.9459 −0.466842 −0.233421 0.972376i \(-0.574992\pi\)
−0.233421 + 0.972376i \(0.574992\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 0 0
\(772\) −1.51754 −0.0546175
\(773\) 2.90673 0.104548 0.0522738 0.998633i \(-0.483353\pi\)
0.0522738 + 0.998633i \(0.483353\pi\)
\(774\) 0 0
\(775\) −7.51754 −0.270038
\(776\) 6.90673 0.247937
\(777\) 0 0
\(778\) −6.65002 −0.238415
\(779\) 52.5134 1.88149
\(780\) 0 0
\(781\) 4.77837 0.170984
\(782\) −20.0310 −0.716306
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 24.5526 0.876321
\(786\) 0 0
\(787\) 29.1581 1.03937 0.519686 0.854357i \(-0.326049\pi\)
0.519686 + 0.854357i \(0.326049\pi\)
\(788\) −10.7392 −0.382567
\(789\) 0 0
\(790\) 2.77837 0.0988500
\(791\) 12.2121 0.434213
\(792\) 0 0
\(793\) 2.77837 0.0986628
\(794\) −34.9067 −1.23879
\(795\) 0 0
\(796\) −0.906726 −0.0321381
\(797\) 25.6851 0.909813 0.454906 0.890539i \(-0.349673\pi\)
0.454906 + 0.890539i \(0.349673\pi\)
\(798\) 0 0
\(799\) 16.1676 0.571967
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 0 0
\(803\) 5.51754 0.194710
\(804\) 0 0
\(805\) 7.43376 0.262006
\(806\) −5.22163 −0.183924
\(807\) 0 0
\(808\) 7.64590 0.268982
\(809\) 32.4296 1.14017 0.570083 0.821587i \(-0.306911\pi\)
0.570083 + 0.821587i \(0.306911\pi\)
\(810\) 0 0
\(811\) 16.2175 0.569474 0.284737 0.958606i \(-0.408094\pi\)
0.284737 + 0.958606i \(0.408094\pi\)
\(812\) 0.610815 0.0214354
\(813\) 0 0
\(814\) 1.30541 0.0457545
\(815\) −1.91622 −0.0671223
\(816\) 0 0
\(817\) 30.4433 1.06507
\(818\) −3.64590 −0.127476
\(819\) 0 0
\(820\) 9.51754 0.332367
\(821\) −16.7000 −0.582833 −0.291416 0.956596i \(-0.594127\pi\)
−0.291416 + 0.956596i \(0.594127\pi\)
\(822\) 0 0
\(823\) 17.9810 0.626779 0.313389 0.949625i \(-0.398536\pi\)
0.313389 + 0.949625i \(0.398536\pi\)
\(824\) 7.51754 0.261886
\(825\) 0 0
\(826\) 3.51754 0.122391
\(827\) −11.4284 −0.397404 −0.198702 0.980060i \(-0.563673\pi\)
−0.198702 + 0.980060i \(0.563673\pi\)
\(828\) 0 0
\(829\) 22.2431 0.772535 0.386267 0.922387i \(-0.373764\pi\)
0.386267 + 0.922387i \(0.373764\pi\)
\(830\) 4.73917 0.164499
\(831\) 0 0
\(832\) −0.694593 −0.0240807
\(833\) 2.69459 0.0933621
\(834\) 0 0
\(835\) 24.2121 0.837895
\(836\) 5.51754 0.190828
\(837\) 0 0
\(838\) −18.3851 −0.635102
\(839\) −13.8972 −0.479786 −0.239893 0.970799i \(-0.577112\pi\)
−0.239893 + 0.970799i \(0.577112\pi\)
\(840\) 0 0
\(841\) −28.6269 −0.987135
\(842\) −14.0000 −0.482472
\(843\) 0 0
\(844\) −27.3756 −0.942306
\(845\) −12.5175 −0.430617
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −4.12836 −0.141768
\(849\) 0 0
\(850\) −2.69459 −0.0924238
\(851\) −9.70409 −0.332652
\(852\) 0 0
\(853\) 26.5972 0.910671 0.455335 0.890320i \(-0.349519\pi\)
0.455335 + 0.890320i \(0.349519\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) 17.1634 0.586634
\(857\) −3.91622 −0.133776 −0.0668878 0.997761i \(-0.521307\pi\)
−0.0668878 + 0.997761i \(0.521307\pi\)
\(858\) 0 0
\(859\) −12.4688 −0.425431 −0.212716 0.977114i \(-0.568231\pi\)
−0.212716 + 0.977114i \(0.568231\pi\)
\(860\) 5.51754 0.188147
\(861\) 0 0
\(862\) −15.3054 −0.521304
\(863\) −37.5931 −1.27968 −0.639842 0.768507i \(-0.721000\pi\)
−0.639842 + 0.768507i \(0.721000\pi\)
\(864\) 0 0
\(865\) 7.22163 0.245543
\(866\) 9.34998 0.317725
\(867\) 0 0
\(868\) −7.51754 −0.255162
\(869\) −2.77837 −0.0942498
\(870\) 0 0
\(871\) 6.21751 0.210672
\(872\) 1.91622 0.0648915
\(873\) 0 0
\(874\) −41.0161 −1.38739
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −30.7202 −1.03735 −0.518673 0.854972i \(-0.673574\pi\)
−0.518673 + 0.854972i \(0.673574\pi\)
\(878\) −30.5526 −1.03110
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −3.95542 −0.133262 −0.0666308 0.997778i \(-0.521225\pi\)
−0.0666308 + 0.997778i \(0.521225\pi\)
\(882\) 0 0
\(883\) 49.3756 1.66162 0.830810 0.556556i \(-0.187878\pi\)
0.830810 + 0.556556i \(0.187878\pi\)
\(884\) −1.87164 −0.0629502
\(885\) 0 0
\(886\) −36.1985 −1.21611
\(887\) 4.30129 0.144423 0.0722116 0.997389i \(-0.476994\pi\)
0.0722116 + 0.997389i \(0.476994\pi\)
\(888\) 0 0
\(889\) 11.5175 0.386286
\(890\) −8.04458 −0.269655
\(891\) 0 0
\(892\) 21.6459 0.724758
\(893\) 33.1052 1.10782
\(894\) 0 0
\(895\) 14.2959 0.477860
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 1.22163 0.0407663
\(899\) −4.59182 −0.153146
\(900\) 0 0
\(901\) −11.1242 −0.370602
\(902\) −9.51754 −0.316899
\(903\) 0 0
\(904\) −12.2121 −0.406170
\(905\) −2.52704 −0.0840015
\(906\) 0 0
\(907\) 16.6946 0.554335 0.277167 0.960822i \(-0.410604\pi\)
0.277167 + 0.960822i \(0.410604\pi\)
\(908\) 11.3500 0.376662
\(909\) 0 0
\(910\) 0.694593 0.0230255
\(911\) −22.9649 −0.760862 −0.380431 0.924809i \(-0.624224\pi\)
−0.380431 + 0.924809i \(0.624224\pi\)
\(912\) 0 0
\(913\) −4.73917 −0.156844
\(914\) 11.0743 0.366305
\(915\) 0 0
\(916\) −6.69459 −0.221196
\(917\) 5.38919 0.177967
\(918\) 0 0
\(919\) −46.5836 −1.53665 −0.768325 0.640059i \(-0.778910\pi\)
−0.768325 + 0.640059i \(0.778910\pi\)
\(920\) −7.43376 −0.245084
\(921\) 0 0
\(922\) 7.87164 0.259239
\(923\) −3.31902 −0.109247
\(924\) 0 0
\(925\) −1.30541 −0.0429215
\(926\) −24.1676 −0.794195
\(927\) 0 0
\(928\) −0.610815 −0.0200510
\(929\) 39.5039 1.29608 0.648041 0.761606i \(-0.275589\pi\)
0.648041 + 0.761606i \(0.275589\pi\)
\(930\) 0 0
\(931\) 5.51754 0.180830
\(932\) 14.2567 0.466994
\(933\) 0 0
\(934\) −27.5175 −0.900401
\(935\) 2.69459 0.0881226
\(936\) 0 0
\(937\) 15.0743 0.492455 0.246228 0.969212i \(-0.420809\pi\)
0.246228 + 0.969212i \(0.420809\pi\)
\(938\) 8.95130 0.292270
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) −22.1676 −0.722642 −0.361321 0.932442i \(-0.617674\pi\)
−0.361321 + 0.932442i \(0.617674\pi\)
\(942\) 0 0
\(943\) 70.7511 2.30397
\(944\) −3.51754 −0.114486
\(945\) 0 0
\(946\) −5.51754 −0.179391
\(947\) −5.61493 −0.182461 −0.0912304 0.995830i \(-0.529080\pi\)
−0.0912304 + 0.995830i \(0.529080\pi\)
\(948\) 0 0
\(949\) −3.83244 −0.124406
\(950\) −5.51754 −0.179013
\(951\) 0 0
\(952\) −2.69459 −0.0873323
\(953\) −17.5757 −0.569334 −0.284667 0.958626i \(-0.591883\pi\)
−0.284667 + 0.958626i \(0.591883\pi\)
\(954\) 0 0
\(955\) −0.610815 −0.0197655
\(956\) −3.72967 −0.120626
\(957\) 0 0
\(958\) −26.0702 −0.842289
\(959\) 13.4338 0.433799
\(960\) 0 0
\(961\) 25.5134 0.823014
\(962\) −0.906726 −0.0292340
\(963\) 0 0
\(964\) −14.7392 −0.474717
\(965\) −1.51754 −0.0488514
\(966\) 0 0
\(967\) 11.5175 0.370379 0.185190 0.982703i \(-0.440710\pi\)
0.185190 + 0.982703i \(0.440710\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 6.90673 0.221762
\(971\) 24.2257 0.777441 0.388721 0.921356i \(-0.372917\pi\)
0.388721 + 0.921356i \(0.372917\pi\)
\(972\) 0 0
\(973\) −18.9067 −0.606122
\(974\) −12.9649 −0.415423
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) −14.7338 −0.471376 −0.235688 0.971829i \(-0.575734\pi\)
−0.235688 + 0.971829i \(0.575734\pi\)
\(978\) 0 0
\(979\) 8.04458 0.257106
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 36.4243 1.16235
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) −10.7392 −0.342178
\(986\) −1.64590 −0.0524160
\(987\) 0 0
\(988\) −3.83244 −0.121926
\(989\) 41.0161 1.30424
\(990\) 0 0
\(991\) 43.3500 1.37706 0.688529 0.725209i \(-0.258257\pi\)
0.688529 + 0.725209i \(0.258257\pi\)
\(992\) 7.51754 0.238682
\(993\) 0 0
\(994\) −4.77837 −0.151561
\(995\) −0.906726 −0.0287452
\(996\) 0 0
\(997\) −2.34049 −0.0741240 −0.0370620 0.999313i \(-0.511800\pi\)
−0.0370620 + 0.999313i \(0.511800\pi\)
\(998\) −19.1242 −0.605367
\(999\) 0 0
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 6930.2.a.ci.1.2 3
3.2 odd 2 6930.2.a.cj.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6930.2.a.ci.1.2 3 1.1 even 1 trivial
6930.2.a.cj.1.2 yes 3 3.2 odd 2