Properties

Label 6930.2.a.ca.1.2
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +5.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} +0.732051 q^{19} -1.00000 q^{20} -1.00000 q^{22} -4.73205 q^{23} +1.00000 q^{25} +5.46410 q^{26} +1.00000 q^{28} +1.26795 q^{29} -4.92820 q^{31} +1.00000 q^{32} +3.46410 q^{34} -1.00000 q^{35} +6.73205 q^{37} +0.732051 q^{38} -1.00000 q^{40} +1.26795 q^{41} +8.92820 q^{43} -1.00000 q^{44} -4.73205 q^{46} +1.00000 q^{49} +1.00000 q^{50} +5.46410 q^{52} +1.26795 q^{53} +1.00000 q^{55} +1.00000 q^{56} +1.26795 q^{58} -13.8564 q^{59} +2.00000 q^{61} -4.92820 q^{62} +1.00000 q^{64} -5.46410 q^{65} +2.92820 q^{67} +3.46410 q^{68} -1.00000 q^{70} -2.53590 q^{71} +4.53590 q^{73} +6.73205 q^{74} +0.732051 q^{76} -1.00000 q^{77} +3.26795 q^{79} -1.00000 q^{80} +1.26795 q^{82} -16.3923 q^{83} -3.46410 q^{85} +8.92820 q^{86} -1.00000 q^{88} +8.53590 q^{89} +5.46410 q^{91} -4.73205 q^{92} -0.732051 q^{95} +16.1962 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{20} - 2 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{26} + 2 q^{28} + 6 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{35} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 4 q^{43} - 2 q^{44} - 6 q^{46} + 2 q^{49} + 2 q^{50} + 4 q^{52} + 6 q^{53} + 2 q^{55} + 2 q^{56} + 6 q^{58} + 4 q^{61} + 4 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 2 q^{70} - 12 q^{71} + 16 q^{73} + 10 q^{74} - 2 q^{76} - 2 q^{77} + 10 q^{79} - 2 q^{80} + 6 q^{82} - 12 q^{83} + 4 q^{86} - 2 q^{88} + 24 q^{89} + 4 q^{91} - 6 q^{92} + 2 q^{95} + 22 q^{97} + 2 q^{98} + O(q^{100}) \)

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
\(13\) 5.46410 1.51547 0.757735 0.652563i \(-0.226306\pi\)
0.757735 + 0.652563i \(0.226306\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 0.732051 0.167944 0.0839720 0.996468i \(-0.473239\pi\)
0.0839720 + 0.996468i \(0.473239\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −4.73205 −0.986701 −0.493350 0.869831i \(-0.664228\pi\)
−0.493350 + 0.869831i \(0.664228\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 5.46410 1.07160
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 1.26795 0.235452 0.117726 0.993046i \(-0.462440\pi\)
0.117726 + 0.993046i \(0.462440\pi\)
\(30\) 0 0
\(31\) −4.92820 −0.885131 −0.442566 0.896736i \(-0.645932\pi\)
−0.442566 + 0.896736i \(0.645932\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.46410 0.594089
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 6.73205 1.10674 0.553371 0.832935i \(-0.313341\pi\)
0.553371 + 0.832935i \(0.313341\pi\)
\(38\) 0.732051 0.118754
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 1.26795 0.198020 0.0990102 0.995086i \(-0.468432\pi\)
0.0990102 + 0.995086i \(0.468432\pi\)
\(42\) 0 0
\(43\) 8.92820 1.36154 0.680769 0.732498i \(-0.261646\pi\)
0.680769 + 0.732498i \(0.261646\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −4.73205 −0.697703
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 5.46410 0.757735
\(53\) 1.26795 0.174166 0.0870831 0.996201i \(-0.472245\pi\)
0.0870831 + 0.996201i \(0.472245\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 1.26795 0.166490
\(59\) −13.8564 −1.80395 −0.901975 0.431788i \(-0.857883\pi\)
−0.901975 + 0.431788i \(0.857883\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −4.92820 −0.625882
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.46410 −0.677738
\(66\) 0 0
\(67\) 2.92820 0.357737 0.178868 0.983873i \(-0.442756\pi\)
0.178868 + 0.983873i \(0.442756\pi\)
\(68\) 3.46410 0.420084
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −2.53590 −0.300956 −0.150478 0.988613i \(-0.548081\pi\)
−0.150478 + 0.988613i \(0.548081\pi\)
\(72\) 0 0
\(73\) 4.53590 0.530887 0.265443 0.964126i \(-0.414482\pi\)
0.265443 + 0.964126i \(0.414482\pi\)
\(74\) 6.73205 0.782585
\(75\) 0 0
\(76\) 0.732051 0.0839720
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 3.26795 0.367673 0.183837 0.982957i \(-0.441148\pi\)
0.183837 + 0.982957i \(0.441148\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 1.26795 0.140022
\(83\) −16.3923 −1.79929 −0.899645 0.436623i \(-0.856174\pi\)
−0.899645 + 0.436623i \(0.856174\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) 8.92820 0.962753
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 8.53590 0.904803 0.452402 0.891814i \(-0.350567\pi\)
0.452402 + 0.891814i \(0.350567\pi\)
\(90\) 0 0
\(91\) 5.46410 0.572793
\(92\) −4.73205 −0.493350
\(93\) 0 0
\(94\) 0 0
\(95\) −0.732051 −0.0751068
\(96\) 0 0
\(97\) 16.1962 1.64447 0.822235 0.569148i \(-0.192727\pi\)
0.822235 + 0.569148i \(0.192727\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 17.4641 1.72079 0.860395 0.509629i \(-0.170217\pi\)
0.860395 + 0.509629i \(0.170217\pi\)
\(104\) 5.46410 0.535799
\(105\) 0 0
\(106\) 1.26795 0.123154
\(107\) 12.9282 1.24982 0.624908 0.780698i \(-0.285136\pi\)
0.624908 + 0.780698i \(0.285136\pi\)
\(108\) 0 0
\(109\) −12.1962 −1.16818 −0.584090 0.811689i \(-0.698548\pi\)
−0.584090 + 0.811689i \(0.698548\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 0.928203 0.0873180 0.0436590 0.999046i \(-0.486098\pi\)
0.0436590 + 0.999046i \(0.486098\pi\)
\(114\) 0 0
\(115\) 4.73205 0.441266
\(116\) 1.26795 0.117726
\(117\) 0 0
\(118\) −13.8564 −1.27559
\(119\) 3.46410 0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −4.92820 −0.442566
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −17.8564 −1.58450 −0.792250 0.610197i \(-0.791090\pi\)
−0.792250 + 0.610197i \(0.791090\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −5.46410 −0.479233
\(131\) 0.339746 0.0296837 0.0148419 0.999890i \(-0.495276\pi\)
0.0148419 + 0.999890i \(0.495276\pi\)
\(132\) 0 0
\(133\) 0.732051 0.0634769
\(134\) 2.92820 0.252958
\(135\) 0 0
\(136\) 3.46410 0.297044
\(137\) 19.8564 1.69645 0.848224 0.529638i \(-0.177672\pi\)
0.848224 + 0.529638i \(0.177672\pi\)
\(138\) 0 0
\(139\) −6.19615 −0.525551 −0.262775 0.964857i \(-0.584638\pi\)
−0.262775 + 0.964857i \(0.584638\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −2.53590 −0.212808
\(143\) −5.46410 −0.456931
\(144\) 0 0
\(145\) −1.26795 −0.105297
\(146\) 4.53590 0.375394
\(147\) 0 0
\(148\) 6.73205 0.553371
\(149\) 10.7321 0.879204 0.439602 0.898193i \(-0.355120\pi\)
0.439602 + 0.898193i \(0.355120\pi\)
\(150\) 0 0
\(151\) −18.1962 −1.48078 −0.740391 0.672177i \(-0.765360\pi\)
−0.740391 + 0.672177i \(0.765360\pi\)
\(152\) 0.732051 0.0593772
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 4.92820 0.395843
\(156\) 0 0
\(157\) −14.3923 −1.14863 −0.574315 0.818634i \(-0.694732\pi\)
−0.574315 + 0.818634i \(0.694732\pi\)
\(158\) 3.26795 0.259984
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −4.73205 −0.372938
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 1.26795 0.0990102
\(165\) 0 0
\(166\) −16.3923 −1.27229
\(167\) −13.8564 −1.07224 −0.536120 0.844141i \(-0.680111\pi\)
−0.536120 + 0.844141i \(0.680111\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) −3.46410 −0.265684
\(171\) 0 0
\(172\) 8.92820 0.680769
\(173\) 12.9282 0.982913 0.491457 0.870902i \(-0.336465\pi\)
0.491457 + 0.870902i \(0.336465\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 8.53590 0.639793
\(179\) −7.85641 −0.587215 −0.293608 0.955926i \(-0.594856\pi\)
−0.293608 + 0.955926i \(0.594856\pi\)
\(180\) 0 0
\(181\) 15.8564 1.17860 0.589299 0.807915i \(-0.299404\pi\)
0.589299 + 0.807915i \(0.299404\pi\)
\(182\) 5.46410 0.405026
\(183\) 0 0
\(184\) −4.73205 −0.348851
\(185\) −6.73205 −0.494950
\(186\) 0 0
\(187\) −3.46410 −0.253320
\(188\) 0 0
\(189\) 0 0
\(190\) −0.732051 −0.0531085
\(191\) 6.92820 0.501307 0.250654 0.968077i \(-0.419354\pi\)
0.250654 + 0.968077i \(0.419354\pi\)
\(192\) 0 0
\(193\) −18.5359 −1.33424 −0.667122 0.744949i \(-0.732474\pi\)
−0.667122 + 0.744949i \(0.732474\pi\)
\(194\) 16.1962 1.16282
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 1.60770 0.114544 0.0572718 0.998359i \(-0.481760\pi\)
0.0572718 + 0.998359i \(0.481760\pi\)
\(198\) 0 0
\(199\) 16.7846 1.18983 0.594915 0.803789i \(-0.297186\pi\)
0.594915 + 0.803789i \(0.297186\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 1.26795 0.0889926
\(204\) 0 0
\(205\) −1.26795 −0.0885574
\(206\) 17.4641 1.21678
\(207\) 0 0
\(208\) 5.46410 0.378867
\(209\) −0.732051 −0.0506370
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 1.26795 0.0870831
\(213\) 0 0
\(214\) 12.9282 0.883754
\(215\) −8.92820 −0.608898
\(216\) 0 0
\(217\) −4.92820 −0.334548
\(218\) −12.1962 −0.826028
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 18.9282 1.27325
\(222\) 0 0
\(223\) 12.3923 0.829850 0.414925 0.909856i \(-0.363808\pi\)
0.414925 + 0.909856i \(0.363808\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 0.928203 0.0617432
\(227\) 13.8564 0.919682 0.459841 0.888001i \(-0.347906\pi\)
0.459841 + 0.888001i \(0.347906\pi\)
\(228\) 0 0
\(229\) −6.53590 −0.431904 −0.215952 0.976404i \(-0.569286\pi\)
−0.215952 + 0.976404i \(0.569286\pi\)
\(230\) 4.73205 0.312022
\(231\) 0 0
\(232\) 1.26795 0.0832449
\(233\) 19.8564 1.30084 0.650418 0.759576i \(-0.274594\pi\)
0.650418 + 0.759576i \(0.274594\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −13.8564 −0.901975
\(237\) 0 0
\(238\) 3.46410 0.224544
\(239\) 14.1962 0.918273 0.459136 0.888366i \(-0.348159\pi\)
0.459136 + 0.888366i \(0.348159\pi\)
\(240\) 0 0
\(241\) −7.80385 −0.502690 −0.251345 0.967898i \(-0.580873\pi\)
−0.251345 + 0.967898i \(0.580873\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) −4.92820 −0.312941
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 4.73205 0.297501
\(254\) −17.8564 −1.12041
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 28.9808 1.80777 0.903885 0.427775i \(-0.140703\pi\)
0.903885 + 0.427775i \(0.140703\pi\)
\(258\) 0 0
\(259\) 6.73205 0.418309
\(260\) −5.46410 −0.338869
\(261\) 0 0
\(262\) 0.339746 0.0209896
\(263\) −5.07180 −0.312740 −0.156370 0.987699i \(-0.549979\pi\)
−0.156370 + 0.987699i \(0.549979\pi\)
\(264\) 0 0
\(265\) −1.26795 −0.0778895
\(266\) 0.732051 0.0448849
\(267\) 0 0
\(268\) 2.92820 0.178868
\(269\) 16.3923 0.999456 0.499728 0.866182i \(-0.333433\pi\)
0.499728 + 0.866182i \(0.333433\pi\)
\(270\) 0 0
\(271\) 3.60770 0.219152 0.109576 0.993978i \(-0.465051\pi\)
0.109576 + 0.993978i \(0.465051\pi\)
\(272\) 3.46410 0.210042
\(273\) 0 0
\(274\) 19.8564 1.19957
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 0.143594 0.00862770 0.00431385 0.999991i \(-0.498627\pi\)
0.00431385 + 0.999991i \(0.498627\pi\)
\(278\) −6.19615 −0.371621
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −10.3923 −0.619953 −0.309976 0.950744i \(-0.600321\pi\)
−0.309976 + 0.950744i \(0.600321\pi\)
\(282\) 0 0
\(283\) 2.92820 0.174064 0.0870318 0.996206i \(-0.472262\pi\)
0.0870318 + 0.996206i \(0.472262\pi\)
\(284\) −2.53590 −0.150478
\(285\) 0 0
\(286\) −5.46410 −0.323099
\(287\) 1.26795 0.0748447
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) −1.26795 −0.0744565
\(291\) 0 0
\(292\) 4.53590 0.265443
\(293\) 9.46410 0.552899 0.276449 0.961028i \(-0.410842\pi\)
0.276449 + 0.961028i \(0.410842\pi\)
\(294\) 0 0
\(295\) 13.8564 0.806751
\(296\) 6.73205 0.391293
\(297\) 0 0
\(298\) 10.7321 0.621691
\(299\) −25.8564 −1.49531
\(300\) 0 0
\(301\) 8.92820 0.514613
\(302\) −18.1962 −1.04707
\(303\) 0 0
\(304\) 0.732051 0.0419860
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −3.32051 −0.189511 −0.0947557 0.995501i \(-0.530207\pi\)
−0.0947557 + 0.995501i \(0.530207\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 4.92820 0.279903
\(311\) 19.8564 1.12595 0.562977 0.826473i \(-0.309656\pi\)
0.562977 + 0.826473i \(0.309656\pi\)
\(312\) 0 0
\(313\) 30.7321 1.73708 0.868539 0.495621i \(-0.165059\pi\)
0.868539 + 0.495621i \(0.165059\pi\)
\(314\) −14.3923 −0.812205
\(315\) 0 0
\(316\) 3.26795 0.183837
\(317\) −26.4449 −1.48529 −0.742646 0.669684i \(-0.766429\pi\)
−0.742646 + 0.669684i \(0.766429\pi\)
\(318\) 0 0
\(319\) −1.26795 −0.0709915
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −4.73205 −0.263707
\(323\) 2.53590 0.141101
\(324\) 0 0
\(325\) 5.46410 0.303094
\(326\) 8.00000 0.443079
\(327\) 0 0
\(328\) 1.26795 0.0700108
\(329\) 0 0
\(330\) 0 0
\(331\) 8.92820 0.490738 0.245369 0.969430i \(-0.421091\pi\)
0.245369 + 0.969430i \(0.421091\pi\)
\(332\) −16.3923 −0.899645
\(333\) 0 0
\(334\) −13.8564 −0.758189
\(335\) −2.92820 −0.159985
\(336\) 0 0
\(337\) 10.7846 0.587475 0.293738 0.955886i \(-0.405101\pi\)
0.293738 + 0.955886i \(0.405101\pi\)
\(338\) 16.8564 0.916868
\(339\) 0 0
\(340\) −3.46410 −0.187867
\(341\) 4.92820 0.266877
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 8.92820 0.481376
\(345\) 0 0
\(346\) 12.9282 0.695025
\(347\) 19.8564 1.06595 0.532974 0.846132i \(-0.321074\pi\)
0.532974 + 0.846132i \(0.321074\pi\)
\(348\) 0 0
\(349\) 9.60770 0.514288 0.257144 0.966373i \(-0.417219\pi\)
0.257144 + 0.966373i \(0.417219\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −26.4449 −1.40752 −0.703759 0.710439i \(-0.748497\pi\)
−0.703759 + 0.710439i \(0.748497\pi\)
\(354\) 0 0
\(355\) 2.53590 0.134592
\(356\) 8.53590 0.452402
\(357\) 0 0
\(358\) −7.85641 −0.415224
\(359\) 4.73205 0.249748 0.124874 0.992173i \(-0.460147\pi\)
0.124874 + 0.992173i \(0.460147\pi\)
\(360\) 0 0
\(361\) −18.4641 −0.971795
\(362\) 15.8564 0.833394
\(363\) 0 0
\(364\) 5.46410 0.286397
\(365\) −4.53590 −0.237420
\(366\) 0 0
\(367\) 22.5359 1.17636 0.588182 0.808728i \(-0.299844\pi\)
0.588182 + 0.808728i \(0.299844\pi\)
\(368\) −4.73205 −0.246675
\(369\) 0 0
\(370\) −6.73205 −0.349983
\(371\) 1.26795 0.0658286
\(372\) 0 0
\(373\) 9.60770 0.497468 0.248734 0.968572i \(-0.419986\pi\)
0.248734 + 0.968572i \(0.419986\pi\)
\(374\) −3.46410 −0.179124
\(375\) 0 0
\(376\) 0 0
\(377\) 6.92820 0.356821
\(378\) 0 0
\(379\) −31.7128 −1.62898 −0.814489 0.580179i \(-0.802983\pi\)
−0.814489 + 0.580179i \(0.802983\pi\)
\(380\) −0.732051 −0.0375534
\(381\) 0 0
\(382\) 6.92820 0.354478
\(383\) 4.39230 0.224436 0.112218 0.993684i \(-0.464204\pi\)
0.112218 + 0.993684i \(0.464204\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) −18.5359 −0.943452
\(387\) 0 0
\(388\) 16.1962 0.822235
\(389\) −24.2487 −1.22946 −0.614729 0.788738i \(-0.710735\pi\)
−0.614729 + 0.788738i \(0.710735\pi\)
\(390\) 0 0
\(391\) −16.3923 −0.828994
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 1.60770 0.0809945
\(395\) −3.26795 −0.164428
\(396\) 0 0
\(397\) −17.6077 −0.883705 −0.441852 0.897088i \(-0.645679\pi\)
−0.441852 + 0.897088i \(0.645679\pi\)
\(398\) 16.7846 0.841336
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −37.1769 −1.85653 −0.928263 0.371924i \(-0.878698\pi\)
−0.928263 + 0.371924i \(0.878698\pi\)
\(402\) 0 0
\(403\) −26.9282 −1.34139
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 1.26795 0.0629273
\(407\) −6.73205 −0.333695
\(408\) 0 0
\(409\) 11.8038 0.583663 0.291831 0.956470i \(-0.405735\pi\)
0.291831 + 0.956470i \(0.405735\pi\)
\(410\) −1.26795 −0.0626195
\(411\) 0 0
\(412\) 17.4641 0.860395
\(413\) −13.8564 −0.681829
\(414\) 0 0
\(415\) 16.3923 0.804667
\(416\) 5.46410 0.267900
\(417\) 0 0
\(418\) −0.732051 −0.0358058
\(419\) 8.78461 0.429156 0.214578 0.976707i \(-0.431162\pi\)
0.214578 + 0.976707i \(0.431162\pi\)
\(420\) 0 0
\(421\) −30.7846 −1.50035 −0.750175 0.661239i \(-0.770031\pi\)
−0.750175 + 0.661239i \(0.770031\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 1.26795 0.0615771
\(425\) 3.46410 0.168034
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 12.9282 0.624908
\(429\) 0 0
\(430\) −8.92820 −0.430556
\(431\) −35.6603 −1.71769 −0.858847 0.512232i \(-0.828819\pi\)
−0.858847 + 0.512232i \(0.828819\pi\)
\(432\) 0 0
\(433\) −2.73205 −0.131294 −0.0656470 0.997843i \(-0.520911\pi\)
−0.0656470 + 0.997843i \(0.520911\pi\)
\(434\) −4.92820 −0.236561
\(435\) 0 0
\(436\) −12.1962 −0.584090
\(437\) −3.46410 −0.165710
\(438\) 0 0
\(439\) 15.6077 0.744915 0.372457 0.928049i \(-0.378515\pi\)
0.372457 + 0.928049i \(0.378515\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 18.9282 0.900323
\(443\) 20.7846 0.987507 0.493753 0.869602i \(-0.335625\pi\)
0.493753 + 0.869602i \(0.335625\pi\)
\(444\) 0 0
\(445\) −8.53590 −0.404640
\(446\) 12.3923 0.586793
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 0.679492 0.0320672 0.0160336 0.999871i \(-0.494896\pi\)
0.0160336 + 0.999871i \(0.494896\pi\)
\(450\) 0 0
\(451\) −1.26795 −0.0597054
\(452\) 0.928203 0.0436590
\(453\) 0 0
\(454\) 13.8564 0.650313
\(455\) −5.46410 −0.256161
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) −6.53590 −0.305402
\(459\) 0 0
\(460\) 4.73205 0.220633
\(461\) 5.32051 0.247801 0.123900 0.992295i \(-0.460460\pi\)
0.123900 + 0.992295i \(0.460460\pi\)
\(462\) 0 0
\(463\) −38.9808 −1.81159 −0.905795 0.423717i \(-0.860725\pi\)
−0.905795 + 0.423717i \(0.860725\pi\)
\(464\) 1.26795 0.0588631
\(465\) 0 0
\(466\) 19.8564 0.919830
\(467\) −0.339746 −0.0157216 −0.00786078 0.999969i \(-0.502502\pi\)
−0.00786078 + 0.999969i \(0.502502\pi\)
\(468\) 0 0
\(469\) 2.92820 0.135212
\(470\) 0 0
\(471\) 0 0
\(472\) −13.8564 −0.637793
\(473\) −8.92820 −0.410519
\(474\) 0 0
\(475\) 0.732051 0.0335888
\(476\) 3.46410 0.158777
\(477\) 0 0
\(478\) 14.1962 0.649317
\(479\) 13.8564 0.633115 0.316558 0.948573i \(-0.397473\pi\)
0.316558 + 0.948573i \(0.397473\pi\)
\(480\) 0 0
\(481\) 36.7846 1.67723
\(482\) −7.80385 −0.355456
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −16.1962 −0.735429
\(486\) 0 0
\(487\) 29.1244 1.31975 0.659875 0.751375i \(-0.270609\pi\)
0.659875 + 0.751375i \(0.270609\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −29.0718 −1.31199 −0.655996 0.754764i \(-0.727751\pi\)
−0.655996 + 0.754764i \(0.727751\pi\)
\(492\) 0 0
\(493\) 4.39230 0.197819
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −4.92820 −0.221283
\(497\) −2.53590 −0.113751
\(498\) 0 0
\(499\) 2.00000 0.0895323 0.0447661 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) 5.07180 0.226140 0.113070 0.993587i \(-0.463932\pi\)
0.113070 + 0.993587i \(0.463932\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 4.73205 0.210365
\(507\) 0 0
\(508\) −17.8564 −0.792250
\(509\) 38.7846 1.71910 0.859549 0.511054i \(-0.170745\pi\)
0.859549 + 0.511054i \(0.170745\pi\)
\(510\) 0 0
\(511\) 4.53590 0.200656
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 28.9808 1.27829
\(515\) −17.4641 −0.769560
\(516\) 0 0
\(517\) 0 0
\(518\) 6.73205 0.295789
\(519\) 0 0
\(520\) −5.46410 −0.239617
\(521\) 10.3923 0.455295 0.227648 0.973744i \(-0.426897\pi\)
0.227648 + 0.973744i \(0.426897\pi\)
\(522\) 0 0
\(523\) −15.3205 −0.669919 −0.334960 0.942233i \(-0.608723\pi\)
−0.334960 + 0.942233i \(0.608723\pi\)
\(524\) 0.339746 0.0148419
\(525\) 0 0
\(526\) −5.07180 −0.221141
\(527\) −17.0718 −0.743659
\(528\) 0 0
\(529\) −0.607695 −0.0264215
\(530\) −1.26795 −0.0550762
\(531\) 0 0
\(532\) 0.732051 0.0317384
\(533\) 6.92820 0.300094
\(534\) 0 0
\(535\) −12.9282 −0.558935
\(536\) 2.92820 0.126479
\(537\) 0 0
\(538\) 16.3923 0.706722
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −7.12436 −0.306300 −0.153150 0.988203i \(-0.548942\pi\)
−0.153150 + 0.988203i \(0.548942\pi\)
\(542\) 3.60770 0.154964
\(543\) 0 0
\(544\) 3.46410 0.148522
\(545\) 12.1962 0.522426
\(546\) 0 0
\(547\) −36.7846 −1.57280 −0.786398 0.617720i \(-0.788057\pi\)
−0.786398 + 0.617720i \(0.788057\pi\)
\(548\) 19.8564 0.848224
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 0.928203 0.0395428
\(552\) 0 0
\(553\) 3.26795 0.138967
\(554\) 0.143594 0.00610070
\(555\) 0 0
\(556\) −6.19615 −0.262775
\(557\) −36.2487 −1.53591 −0.767954 0.640505i \(-0.778725\pi\)
−0.767954 + 0.640505i \(0.778725\pi\)
\(558\) 0 0
\(559\) 48.7846 2.06337
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −10.3923 −0.438373
\(563\) −20.7846 −0.875967 −0.437983 0.898983i \(-0.644307\pi\)
−0.437983 + 0.898983i \(0.644307\pi\)
\(564\) 0 0
\(565\) −0.928203 −0.0390498
\(566\) 2.92820 0.123082
\(567\) 0 0
\(568\) −2.53590 −0.106404
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 0.392305 0.0164174 0.00820872 0.999966i \(-0.497387\pi\)
0.00820872 + 0.999966i \(0.497387\pi\)
\(572\) −5.46410 −0.228466
\(573\) 0 0
\(574\) 1.26795 0.0529232
\(575\) −4.73205 −0.197340
\(576\) 0 0
\(577\) 25.6603 1.06825 0.534125 0.845405i \(-0.320641\pi\)
0.534125 + 0.845405i \(0.320641\pi\)
\(578\) −5.00000 −0.207973
\(579\) 0 0
\(580\) −1.26795 −0.0526487
\(581\) −16.3923 −0.680067
\(582\) 0 0
\(583\) −1.26795 −0.0525131
\(584\) 4.53590 0.187697
\(585\) 0 0
\(586\) 9.46410 0.390958
\(587\) −22.9808 −0.948518 −0.474259 0.880385i \(-0.657284\pi\)
−0.474259 + 0.880385i \(0.657284\pi\)
\(588\) 0 0
\(589\) −3.60770 −0.148652
\(590\) 13.8564 0.570459
\(591\) 0 0
\(592\) 6.73205 0.276686
\(593\) −39.4641 −1.62060 −0.810298 0.586018i \(-0.800695\pi\)
−0.810298 + 0.586018i \(0.800695\pi\)
\(594\) 0 0
\(595\) −3.46410 −0.142014
\(596\) 10.7321 0.439602
\(597\) 0 0
\(598\) −25.8564 −1.05735
\(599\) −44.1051 −1.80209 −0.901043 0.433730i \(-0.857197\pi\)
−0.901043 + 0.433730i \(0.857197\pi\)
\(600\) 0 0
\(601\) −30.4449 −1.24187 −0.620936 0.783861i \(-0.713247\pi\)
−0.620936 + 0.783861i \(0.713247\pi\)
\(602\) 8.92820 0.363886
\(603\) 0 0
\(604\) −18.1962 −0.740391
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −6.78461 −0.275379 −0.137689 0.990475i \(-0.543968\pi\)
−0.137689 + 0.990475i \(0.543968\pi\)
\(608\) 0.732051 0.0296886
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 0 0
\(613\) −35.1769 −1.42078 −0.710391 0.703807i \(-0.751482\pi\)
−0.710391 + 0.703807i \(0.751482\pi\)
\(614\) −3.32051 −0.134005
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −15.4641 −0.622561 −0.311281 0.950318i \(-0.600758\pi\)
−0.311281 + 0.950318i \(0.600758\pi\)
\(618\) 0 0
\(619\) 2.92820 0.117694 0.0588472 0.998267i \(-0.481258\pi\)
0.0588472 + 0.998267i \(0.481258\pi\)
\(620\) 4.92820 0.197921
\(621\) 0 0
\(622\) 19.8564 0.796169
\(623\) 8.53590 0.341984
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 30.7321 1.22830
\(627\) 0 0
\(628\) −14.3923 −0.574315
\(629\) 23.3205 0.929850
\(630\) 0 0
\(631\) 28.7846 1.14590 0.572949 0.819591i \(-0.305799\pi\)
0.572949 + 0.819591i \(0.305799\pi\)
\(632\) 3.26795 0.129992
\(633\) 0 0
\(634\) −26.4449 −1.05026
\(635\) 17.8564 0.708610
\(636\) 0 0
\(637\) 5.46410 0.216496
\(638\) −1.26795 −0.0501986
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 7.85641 0.310309 0.155155 0.987890i \(-0.450412\pi\)
0.155155 + 0.987890i \(0.450412\pi\)
\(642\) 0 0
\(643\) −11.2679 −0.444365 −0.222182 0.975005i \(-0.571318\pi\)
−0.222182 + 0.975005i \(0.571318\pi\)
\(644\) −4.73205 −0.186469
\(645\) 0 0
\(646\) 2.53590 0.0997736
\(647\) 25.1769 0.989807 0.494903 0.868948i \(-0.335203\pi\)
0.494903 + 0.868948i \(0.335203\pi\)
\(648\) 0 0
\(649\) 13.8564 0.543912
\(650\) 5.46410 0.214320
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 20.1962 0.790337 0.395168 0.918609i \(-0.370686\pi\)
0.395168 + 0.918609i \(0.370686\pi\)
\(654\) 0 0
\(655\) −0.339746 −0.0132750
\(656\) 1.26795 0.0495051
\(657\) 0 0
\(658\) 0 0
\(659\) 42.2487 1.64578 0.822888 0.568204i \(-0.192361\pi\)
0.822888 + 0.568204i \(0.192361\pi\)
\(660\) 0 0
\(661\) −23.8564 −0.927907 −0.463953 0.885860i \(-0.653569\pi\)
−0.463953 + 0.885860i \(0.653569\pi\)
\(662\) 8.92820 0.347004
\(663\) 0 0
\(664\) −16.3923 −0.636145
\(665\) −0.732051 −0.0283877
\(666\) 0 0
\(667\) −6.00000 −0.232321
\(668\) −13.8564 −0.536120
\(669\) 0 0
\(670\) −2.92820 −0.113126
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −47.8564 −1.84473 −0.922364 0.386321i \(-0.873746\pi\)
−0.922364 + 0.386321i \(0.873746\pi\)
\(674\) 10.7846 0.415408
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) 7.85641 0.301946 0.150973 0.988538i \(-0.451759\pi\)
0.150973 + 0.988538i \(0.451759\pi\)
\(678\) 0 0
\(679\) 16.1962 0.621551
\(680\) −3.46410 −0.132842
\(681\) 0 0
\(682\) 4.92820 0.188711
\(683\) 6.24871 0.239100 0.119550 0.992828i \(-0.461855\pi\)
0.119550 + 0.992828i \(0.461855\pi\)
\(684\) 0 0
\(685\) −19.8564 −0.758674
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 8.92820 0.340385
\(689\) 6.92820 0.263944
\(690\) 0 0
\(691\) −3.32051 −0.126318 −0.0631590 0.998003i \(-0.520118\pi\)
−0.0631590 + 0.998003i \(0.520118\pi\)
\(692\) 12.9282 0.491457
\(693\) 0 0
\(694\) 19.8564 0.753739
\(695\) 6.19615 0.235033
\(696\) 0 0
\(697\) 4.39230 0.166370
\(698\) 9.60770 0.363657
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −3.80385 −0.143669 −0.0718347 0.997417i \(-0.522885\pi\)
−0.0718347 + 0.997417i \(0.522885\pi\)
\(702\) 0 0
\(703\) 4.92820 0.185871
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −26.4449 −0.995266
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 18.3923 0.690738 0.345369 0.938467i \(-0.387754\pi\)
0.345369 + 0.938467i \(0.387754\pi\)
\(710\) 2.53590 0.0951706
\(711\) 0 0
\(712\) 8.53590 0.319896
\(713\) 23.3205 0.873360
\(714\) 0 0
\(715\) 5.46410 0.204346
\(716\) −7.85641 −0.293608
\(717\) 0 0
\(718\) 4.73205 0.176599
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 17.4641 0.650397
\(722\) −18.4641 −0.687163
\(723\) 0 0
\(724\) 15.8564 0.589299
\(725\) 1.26795 0.0470905
\(726\) 0 0
\(727\) 30.6410 1.13641 0.568206 0.822886i \(-0.307638\pi\)
0.568206 + 0.822886i \(0.307638\pi\)
\(728\) 5.46410 0.202513
\(729\) 0 0
\(730\) −4.53590 −0.167881
\(731\) 30.9282 1.14392
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 22.5359 0.831815
\(735\) 0 0
\(736\) −4.73205 −0.174426
\(737\) −2.92820 −0.107862
\(738\) 0 0
\(739\) 45.8564 1.68686 0.843428 0.537243i \(-0.180534\pi\)
0.843428 + 0.537243i \(0.180534\pi\)
\(740\) −6.73205 −0.247475
\(741\) 0 0
\(742\) 1.26795 0.0465479
\(743\) 51.7128 1.89716 0.948580 0.316539i \(-0.102521\pi\)
0.948580 + 0.316539i \(0.102521\pi\)
\(744\) 0 0
\(745\) −10.7321 −0.393192
\(746\) 9.60770 0.351763
\(747\) 0 0
\(748\) −3.46410 −0.126660
\(749\) 12.9282 0.472386
\(750\) 0 0
\(751\) −46.2487 −1.68764 −0.843820 0.536627i \(-0.819698\pi\)
−0.843820 + 0.536627i \(0.819698\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 6.92820 0.252310
\(755\) 18.1962 0.662226
\(756\) 0 0
\(757\) −49.3731 −1.79449 −0.897247 0.441528i \(-0.854436\pi\)
−0.897247 + 0.441528i \(0.854436\pi\)
\(758\) −31.7128 −1.15186
\(759\) 0 0
\(760\) −0.732051 −0.0265543
\(761\) −22.7321 −0.824036 −0.412018 0.911176i \(-0.635176\pi\)
−0.412018 + 0.911176i \(0.635176\pi\)
\(762\) 0 0
\(763\) −12.1962 −0.441530
\(764\) 6.92820 0.250654
\(765\) 0 0
\(766\) 4.39230 0.158700
\(767\) −75.7128 −2.73383
\(768\) 0 0
\(769\) 34.4449 1.24211 0.621057 0.783766i \(-0.286704\pi\)
0.621057 + 0.783766i \(0.286704\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) −18.5359 −0.667122
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) −4.92820 −0.177026
\(776\) 16.1962 0.581408
\(777\) 0 0
\(778\) −24.2487 −0.869358
\(779\) 0.928203 0.0332563
\(780\) 0 0
\(781\) 2.53590 0.0907416
\(782\) −16.3923 −0.586188
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 14.3923 0.513683
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 1.60770 0.0572718
\(789\) 0 0
\(790\) −3.26795 −0.116268
\(791\) 0.928203 0.0330031
\(792\) 0 0
\(793\) 10.9282 0.388072
\(794\) −17.6077 −0.624874
\(795\) 0 0
\(796\) 16.7846 0.594915
\(797\) −41.3205 −1.46365 −0.731824 0.681494i \(-0.761331\pi\)
−0.731824 + 0.681494i \(0.761331\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −37.1769 −1.31276
\(803\) −4.53590 −0.160068
\(804\) 0 0
\(805\) 4.73205 0.166783
\(806\) −26.9282 −0.948506
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 29.3205 1.03085 0.515427 0.856933i \(-0.327633\pi\)
0.515427 + 0.856933i \(0.327633\pi\)
\(810\) 0 0
\(811\) 26.5885 0.933647 0.466824 0.884351i \(-0.345398\pi\)
0.466824 + 0.884351i \(0.345398\pi\)
\(812\) 1.26795 0.0444963
\(813\) 0 0
\(814\) −6.73205 −0.235958
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 6.53590 0.228662
\(818\) 11.8038 0.412712
\(819\) 0 0
\(820\) −1.26795 −0.0442787
\(821\) −13.9474 −0.486769 −0.243385 0.969930i \(-0.578258\pi\)
−0.243385 + 0.969930i \(0.578258\pi\)
\(822\) 0 0
\(823\) −13.8038 −0.481172 −0.240586 0.970628i \(-0.577340\pi\)
−0.240586 + 0.970628i \(0.577340\pi\)
\(824\) 17.4641 0.608391
\(825\) 0 0
\(826\) −13.8564 −0.482126
\(827\) 29.5692 1.02822 0.514111 0.857724i \(-0.328122\pi\)
0.514111 + 0.857724i \(0.328122\pi\)
\(828\) 0 0
\(829\) −36.1051 −1.25398 −0.626991 0.779026i \(-0.715714\pi\)
−0.626991 + 0.779026i \(0.715714\pi\)
\(830\) 16.3923 0.568985
\(831\) 0 0
\(832\) 5.46410 0.189434
\(833\) 3.46410 0.120024
\(834\) 0 0
\(835\) 13.8564 0.479521
\(836\) −0.732051 −0.0253185
\(837\) 0 0
\(838\) 8.78461 0.303459
\(839\) −2.78461 −0.0961354 −0.0480677 0.998844i \(-0.515306\pi\)
−0.0480677 + 0.998844i \(0.515306\pi\)
\(840\) 0 0
\(841\) −27.3923 −0.944562
\(842\) −30.7846 −1.06091
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) −16.8564 −0.579878
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 1.26795 0.0435416
\(849\) 0 0
\(850\) 3.46410 0.118818
\(851\) −31.8564 −1.09202
\(852\) 0 0
\(853\) 7.07180 0.242134 0.121067 0.992644i \(-0.461368\pi\)
0.121067 + 0.992644i \(0.461368\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) 12.9282 0.441877
\(857\) −43.1769 −1.47490 −0.737448 0.675404i \(-0.763969\pi\)
−0.737448 + 0.675404i \(0.763969\pi\)
\(858\) 0 0
\(859\) −0.784610 −0.0267705 −0.0133853 0.999910i \(-0.504261\pi\)
−0.0133853 + 0.999910i \(0.504261\pi\)
\(860\) −8.92820 −0.304449
\(861\) 0 0
\(862\) −35.6603 −1.21459
\(863\) 6.58846 0.224274 0.112137 0.993693i \(-0.464230\pi\)
0.112137 + 0.993693i \(0.464230\pi\)
\(864\) 0 0
\(865\) −12.9282 −0.439572
\(866\) −2.73205 −0.0928389
\(867\) 0 0
\(868\) −4.92820 −0.167274
\(869\) −3.26795 −0.110858
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) −12.1962 −0.413014
\(873\) 0 0
\(874\) −3.46410 −0.117175
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 39.1769 1.32291 0.661455 0.749985i \(-0.269939\pi\)
0.661455 + 0.749985i \(0.269939\pi\)
\(878\) 15.6077 0.526734
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −11.0718 −0.373018 −0.186509 0.982453i \(-0.559717\pi\)
−0.186509 + 0.982453i \(0.559717\pi\)
\(882\) 0 0
\(883\) 21.1769 0.712660 0.356330 0.934360i \(-0.384028\pi\)
0.356330 + 0.934360i \(0.384028\pi\)
\(884\) 18.9282 0.636624
\(885\) 0 0
\(886\) 20.7846 0.698273
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) −17.8564 −0.598885
\(890\) −8.53590 −0.286124
\(891\) 0 0
\(892\) 12.3923 0.414925
\(893\) 0 0
\(894\) 0 0
\(895\) 7.85641 0.262611
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 0.679492 0.0226749
\(899\) −6.24871 −0.208406
\(900\) 0 0
\(901\) 4.39230 0.146329
\(902\) −1.26795 −0.0422181
\(903\) 0 0
\(904\) 0.928203 0.0308716
\(905\) −15.8564 −0.527085
\(906\) 0 0
\(907\) 19.3205 0.641527 0.320763 0.947159i \(-0.396061\pi\)
0.320763 + 0.947159i \(0.396061\pi\)
\(908\) 13.8564 0.459841
\(909\) 0 0
\(910\) −5.46410 −0.181133
\(911\) −21.4641 −0.711137 −0.355569 0.934650i \(-0.615713\pi\)
−0.355569 + 0.934650i \(0.615713\pi\)
\(912\) 0 0
\(913\) 16.3923 0.542506
\(914\) −34.0000 −1.12462
\(915\) 0 0
\(916\) −6.53590 −0.215952
\(917\) 0.339746 0.0112194
\(918\) 0 0
\(919\) −26.9808 −0.890013 −0.445007 0.895527i \(-0.646799\pi\)
−0.445007 + 0.895527i \(0.646799\pi\)
\(920\) 4.73205 0.156011
\(921\) 0 0
\(922\) 5.32051 0.175222
\(923\) −13.8564 −0.456089
\(924\) 0 0
\(925\) 6.73205 0.221348
\(926\) −38.9808 −1.28099
\(927\) 0 0
\(928\) 1.26795 0.0416225
\(929\) 50.1051 1.64390 0.821948 0.569563i \(-0.192887\pi\)
0.821948 + 0.569563i \(0.192887\pi\)
\(930\) 0 0
\(931\) 0.732051 0.0239920
\(932\) 19.8564 0.650418
\(933\) 0 0
\(934\) −0.339746 −0.0111168
\(935\) 3.46410 0.113288
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 2.92820 0.0956092
\(939\) 0 0
\(940\) 0 0
\(941\) −47.5692 −1.55071 −0.775356 0.631524i \(-0.782430\pi\)
−0.775356 + 0.631524i \(0.782430\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) −13.8564 −0.450988
\(945\) 0 0
\(946\) −8.92820 −0.290281
\(947\) −37.1769 −1.20809 −0.604044 0.796951i \(-0.706445\pi\)
−0.604044 + 0.796951i \(0.706445\pi\)
\(948\) 0 0
\(949\) 24.7846 0.804542
\(950\) 0.732051 0.0237509
\(951\) 0 0
\(952\) 3.46410 0.112272
\(953\) −23.3205 −0.755425 −0.377713 0.925923i \(-0.623289\pi\)
−0.377713 + 0.925923i \(0.623289\pi\)
\(954\) 0 0
\(955\) −6.92820 −0.224191
\(956\) 14.1962 0.459136
\(957\) 0 0
\(958\) 13.8564 0.447680
\(959\) 19.8564 0.641197
\(960\) 0 0
\(961\) −6.71281 −0.216542
\(962\) 36.7846 1.18598
\(963\) 0 0
\(964\) −7.80385 −0.251345
\(965\) 18.5359 0.596692
\(966\) 0 0
\(967\) −53.8564 −1.73191 −0.865953 0.500126i \(-0.833287\pi\)
−0.865953 + 0.500126i \(0.833287\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −16.1962 −0.520027
\(971\) −22.1436 −0.710622 −0.355311 0.934748i \(-0.615625\pi\)
−0.355311 + 0.934748i \(0.615625\pi\)
\(972\) 0 0
\(973\) −6.19615 −0.198640
\(974\) 29.1244 0.933205
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −12.9282 −0.413610 −0.206805 0.978382i \(-0.566307\pi\)
−0.206805 + 0.978382i \(0.566307\pi\)
\(978\) 0 0
\(979\) −8.53590 −0.272808
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −29.0718 −0.927718
\(983\) 28.3923 0.905574 0.452787 0.891619i \(-0.350430\pi\)
0.452787 + 0.891619i \(0.350430\pi\)
\(984\) 0 0
\(985\) −1.60770 −0.0512254
\(986\) 4.39230 0.139879
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −42.2487 −1.34343
\(990\) 0 0
\(991\) −9.07180 −0.288175 −0.144088 0.989565i \(-0.546025\pi\)
−0.144088 + 0.989565i \(0.546025\pi\)
\(992\) −4.92820 −0.156471
\(993\) 0 0
\(994\) −2.53590 −0.0804338
\(995\) −16.7846 −0.532108
\(996\) 0 0
\(997\) 3.85641 0.122134 0.0610668 0.998134i \(-0.480550\pi\)
0.0610668 + 0.998134i \(0.480550\pi\)
\(998\) 2.00000 0.0633089
\(999\) 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.ca.1.2 2
3.2 odd 2 770.2.a.h.1.1 2
12.11 even 2 6160.2.a.v.1.2 2
15.2 even 4 3850.2.c.s.1849.2 4
15.8 even 4 3850.2.c.s.1849.3 4
15.14 odd 2 3850.2.a.bm.1.2 2
21.20 even 2 5390.2.a.bk.1.2 2
33.32 even 2 8470.2.a.ce.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.h.1.1 2 3.2 odd 2
3850.2.a.bm.1.2 2 15.14 odd 2
3850.2.c.s.1849.2 4 15.2 even 4
3850.2.c.s.1849.3 4 15.8 even 4
5390.2.a.bk.1.2 2 21.20 even 2
6160.2.a.v.1.2 2 12.11 even 2
6930.2.a.ca.1.2 2 1.1 even 1 trivial
8470.2.a.ce.1.1 2 33.32 even 2