Properties

Label 3360.2.a.bb.1.1
Level $3360$
Weight $2$
Character 3360.1
Self dual yes
Analytic conductor $26.830$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.8297350792\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3360.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.46410 q^{11} +3.46410 q^{13} +1.00000 q^{15} +2.00000 q^{17} -5.46410 q^{19} -1.00000 q^{21} +6.92820 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} +5.46410 q^{31} +5.46410 q^{33} -1.00000 q^{35} +2.00000 q^{37} -3.46410 q^{39} -4.92820 q^{41} -4.00000 q^{43} -1.00000 q^{45} -10.9282 q^{47} +1.00000 q^{49} -2.00000 q^{51} -0.535898 q^{53} +5.46410 q^{55} +5.46410 q^{57} +13.8564 q^{59} -4.92820 q^{61} +1.00000 q^{63} -3.46410 q^{65} +6.92820 q^{67} -6.92820 q^{69} -16.3923 q^{71} +0.535898 q^{73} -1.00000 q^{75} -5.46410 q^{77} -4.00000 q^{79} +1.00000 q^{81} -14.9282 q^{83} -2.00000 q^{85} +2.00000 q^{87} -12.9282 q^{89} +3.46410 q^{91} -5.46410 q^{93} +5.46410 q^{95} +19.4641 q^{97} -5.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} - 4q^{11} + 2q^{15} + 4q^{17} - 4q^{19} - 2q^{21} + 2q^{25} - 2q^{27} - 4q^{29} + 4q^{31} + 4q^{33} - 2q^{35} + 4q^{37} + 4q^{41} - 8q^{43} - 2q^{45} - 8q^{47} + 2q^{49} - 4q^{51} - 8q^{53} + 4q^{55} + 4q^{57} + 4q^{61} + 2q^{63} - 12q^{71} + 8q^{73} - 2q^{75} - 4q^{77} - 8q^{79} + 2q^{81} - 16q^{83} - 4q^{85} + 4q^{87} - 12q^{89} - 4q^{93} + 4q^{95} + 32q^{97} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.46410 −1.64749 −0.823744 0.566961i \(-0.808119\pi\)
−0.823744 + 0.566961i \(0.808119\pi\)
\(12\) 0 0
\(13\) 3.46410 0.960769 0.480384 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −5.46410 −1.25355 −0.626775 0.779200i \(-0.715626\pi\)
−0.626775 + 0.779200i \(0.715626\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 5.46410 0.981382 0.490691 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 0 0
\(33\) 5.46410 0.951178
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −3.46410 −0.554700
\(40\) 0 0
\(41\) −4.92820 −0.769656 −0.384828 0.922988i \(-0.625739\pi\)
−0.384828 + 0.922988i \(0.625739\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −10.9282 −1.59404 −0.797021 0.603951i \(-0.793592\pi\)
−0.797021 + 0.603951i \(0.793592\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −0.535898 −0.0736113 −0.0368057 0.999322i \(-0.511718\pi\)
−0.0368057 + 0.999322i \(0.511718\pi\)
\(54\) 0 0
\(55\) 5.46410 0.736779
\(56\) 0 0
\(57\) 5.46410 0.723738
\(58\) 0 0
\(59\) 13.8564 1.80395 0.901975 0.431788i \(-0.142117\pi\)
0.901975 + 0.431788i \(0.142117\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) 6.92820 0.846415 0.423207 0.906033i \(-0.360904\pi\)
0.423207 + 0.906033i \(0.360904\pi\)
\(68\) 0 0
\(69\) −6.92820 −0.834058
\(70\) 0 0
\(71\) −16.3923 −1.94541 −0.972704 0.232048i \(-0.925457\pi\)
−0.972704 + 0.232048i \(0.925457\pi\)
\(72\) 0 0
\(73\) 0.535898 0.0627222 0.0313611 0.999508i \(-0.490016\pi\)
0.0313611 + 0.999508i \(0.490016\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −5.46410 −0.622692
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.9282 −1.63858 −0.819292 0.573377i \(-0.805633\pi\)
−0.819292 + 0.573377i \(0.805633\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −12.9282 −1.37039 −0.685193 0.728361i \(-0.740282\pi\)
−0.685193 + 0.728361i \(0.740282\pi\)
\(90\) 0 0
\(91\) 3.46410 0.363137
\(92\) 0 0
\(93\) −5.46410 −0.566601
\(94\) 0 0
\(95\) 5.46410 0.560605
\(96\) 0 0
\(97\) 19.4641 1.97628 0.988140 0.153555i \(-0.0490723\pi\)
0.988140 + 0.153555i \(0.0490723\pi\)
\(98\) 0 0
\(99\) −5.46410 −0.549163
\(100\) 0 0
\(101\) 15.8564 1.57777 0.788886 0.614540i \(-0.210658\pi\)
0.788886 + 0.614540i \(0.210658\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) −18.9282 −1.82986 −0.914929 0.403614i \(-0.867754\pi\)
−0.914929 + 0.403614i \(0.867754\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 10.3923 0.977626 0.488813 0.872389i \(-0.337430\pi\)
0.488813 + 0.872389i \(0.337430\pi\)
\(114\) 0 0
\(115\) −6.92820 −0.646058
\(116\) 0 0
\(117\) 3.46410 0.320256
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 18.8564 1.71422
\(122\) 0 0
\(123\) 4.92820 0.444361
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −5.07180 −0.450049 −0.225025 0.974353i \(-0.572246\pi\)
−0.225025 + 0.974353i \(0.572246\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 2.92820 0.255838 0.127919 0.991785i \(-0.459170\pi\)
0.127919 + 0.991785i \(0.459170\pi\)
\(132\) 0 0
\(133\) −5.46410 −0.473798
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −0.535898 −0.0457849 −0.0228924 0.999738i \(-0.507288\pi\)
−0.0228924 + 0.999738i \(0.507288\pi\)
\(138\) 0 0
\(139\) −5.46410 −0.463459 −0.231730 0.972780i \(-0.574438\pi\)
−0.231730 + 0.972780i \(0.574438\pi\)
\(140\) 0 0
\(141\) 10.9282 0.920321
\(142\) 0 0
\(143\) −18.9282 −1.58286
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −20.9282 −1.71451 −0.857253 0.514896i \(-0.827831\pi\)
−0.857253 + 0.514896i \(0.827831\pi\)
\(150\) 0 0
\(151\) −14.9282 −1.21484 −0.607420 0.794381i \(-0.707795\pi\)
−0.607420 + 0.794381i \(0.707795\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −5.46410 −0.438887
\(156\) 0 0
\(157\) 0.535898 0.0427693 0.0213847 0.999771i \(-0.493193\pi\)
0.0213847 + 0.999771i \(0.493193\pi\)
\(158\) 0 0
\(159\) 0.535898 0.0424995
\(160\) 0 0
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 0 0
\(165\) −5.46410 −0.425380
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −5.46410 −0.417850
\(172\) 0 0
\(173\) −24.9282 −1.89526 −0.947628 0.319376i \(-0.896527\pi\)
−0.947628 + 0.319376i \(0.896527\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −13.8564 −1.04151
\(178\) 0 0
\(179\) 2.53590 0.189542 0.0947710 0.995499i \(-0.469788\pi\)
0.0947710 + 0.995499i \(0.469788\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 4.92820 0.364303
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −10.9282 −0.799149
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 8.39230 0.607246 0.303623 0.952792i \(-0.401804\pi\)
0.303623 + 0.952792i \(0.401804\pi\)
\(192\) 0 0
\(193\) −11.8564 −0.853443 −0.426721 0.904383i \(-0.640332\pi\)
−0.426721 + 0.904383i \(0.640332\pi\)
\(194\) 0 0
\(195\) 3.46410 0.248069
\(196\) 0 0
\(197\) −19.4641 −1.38676 −0.693380 0.720572i \(-0.743879\pi\)
−0.693380 + 0.720572i \(0.743879\pi\)
\(198\) 0 0
\(199\) −0.392305 −0.0278098 −0.0139049 0.999903i \(-0.504426\pi\)
−0.0139049 + 0.999903i \(0.504426\pi\)
\(200\) 0 0
\(201\) −6.92820 −0.488678
\(202\) 0 0
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 4.92820 0.344201
\(206\) 0 0
\(207\) 6.92820 0.481543
\(208\) 0 0
\(209\) 29.8564 2.06521
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 16.3923 1.12318
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 5.46410 0.370927
\(218\) 0 0
\(219\) −0.535898 −0.0362127
\(220\) 0 0
\(221\) 6.92820 0.466041
\(222\) 0 0
\(223\) −26.9282 −1.80325 −0.901623 0.432523i \(-0.857623\pi\)
−0.901623 + 0.432523i \(0.857623\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 6.92820 0.459841 0.229920 0.973209i \(-0.426153\pi\)
0.229920 + 0.973209i \(0.426153\pi\)
\(228\) 0 0
\(229\) 22.7846 1.50565 0.752825 0.658221i \(-0.228691\pi\)
0.752825 + 0.658221i \(0.228691\pi\)
\(230\) 0 0
\(231\) 5.46410 0.359511
\(232\) 0 0
\(233\) −19.4641 −1.27514 −0.637568 0.770394i \(-0.720059\pi\)
−0.637568 + 0.770394i \(0.720059\pi\)
\(234\) 0 0
\(235\) 10.9282 0.712877
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 19.3205 1.24974 0.624870 0.780729i \(-0.285152\pi\)
0.624870 + 0.780729i \(0.285152\pi\)
\(240\) 0 0
\(241\) −14.7846 −0.952360 −0.476180 0.879348i \(-0.657979\pi\)
−0.476180 + 0.879348i \(0.657979\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −18.9282 −1.20437
\(248\) 0 0
\(249\) 14.9282 0.946036
\(250\) 0 0
\(251\) −2.14359 −0.135302 −0.0676512 0.997709i \(-0.521551\pi\)
−0.0676512 + 0.997709i \(0.521551\pi\)
\(252\) 0 0
\(253\) −37.8564 −2.38001
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 0 0
\(257\) −3.07180 −0.191613 −0.0958067 0.995400i \(-0.530543\pi\)
−0.0958067 + 0.995400i \(0.530543\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 17.8564 1.10107 0.550537 0.834811i \(-0.314423\pi\)
0.550537 + 0.834811i \(0.314423\pi\)
\(264\) 0 0
\(265\) 0.535898 0.0329200
\(266\) 0 0
\(267\) 12.9282 0.791193
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −13.4641 −0.817886 −0.408943 0.912560i \(-0.634103\pi\)
−0.408943 + 0.912560i \(0.634103\pi\)
\(272\) 0 0
\(273\) −3.46410 −0.209657
\(274\) 0 0
\(275\) −5.46410 −0.329498
\(276\) 0 0
\(277\) −11.8564 −0.712382 −0.356191 0.934413i \(-0.615925\pi\)
−0.356191 + 0.934413i \(0.615925\pi\)
\(278\) 0 0
\(279\) 5.46410 0.327127
\(280\) 0 0
\(281\) −30.7846 −1.83646 −0.918228 0.396052i \(-0.870380\pi\)
−0.918228 + 0.396052i \(0.870380\pi\)
\(282\) 0 0
\(283\) −17.8564 −1.06145 −0.530727 0.847543i \(-0.678081\pi\)
−0.530727 + 0.847543i \(0.678081\pi\)
\(284\) 0 0
\(285\) −5.46410 −0.323665
\(286\) 0 0
\(287\) −4.92820 −0.290903
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −19.4641 −1.14101
\(292\) 0 0
\(293\) −24.9282 −1.45632 −0.728161 0.685407i \(-0.759625\pi\)
−0.728161 + 0.685407i \(0.759625\pi\)
\(294\) 0 0
\(295\) −13.8564 −0.806751
\(296\) 0 0
\(297\) 5.46410 0.317059
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) −15.8564 −0.910927
\(304\) 0 0
\(305\) 4.92820 0.282188
\(306\) 0 0
\(307\) 22.9282 1.30858 0.654291 0.756243i \(-0.272967\pi\)
0.654291 + 0.756243i \(0.272967\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 17.3205 0.979013 0.489506 0.872000i \(-0.337177\pi\)
0.489506 + 0.872000i \(0.337177\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) 13.3205 0.748154 0.374077 0.927398i \(-0.377960\pi\)
0.374077 + 0.927398i \(0.377960\pi\)
\(318\) 0 0
\(319\) 10.9282 0.611862
\(320\) 0 0
\(321\) 18.9282 1.05647
\(322\) 0 0
\(323\) −10.9282 −0.608061
\(324\) 0 0
\(325\) 3.46410 0.192154
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) −10.9282 −0.602491
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −6.92820 −0.378528
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 0 0
\(339\) −10.3923 −0.564433
\(340\) 0 0
\(341\) −29.8564 −1.61682
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 6.92820 0.373002
\(346\) 0 0
\(347\) −13.8564 −0.743851 −0.371925 0.928263i \(-0.621302\pi\)
−0.371925 + 0.928263i \(0.621302\pi\)
\(348\) 0 0
\(349\) 16.9282 0.906146 0.453073 0.891473i \(-0.350328\pi\)
0.453073 + 0.891473i \(0.350328\pi\)
\(350\) 0 0
\(351\) −3.46410 −0.184900
\(352\) 0 0
\(353\) 4.14359 0.220541 0.110271 0.993902i \(-0.464828\pi\)
0.110271 + 0.993902i \(0.464828\pi\)
\(354\) 0 0
\(355\) 16.3923 0.870013
\(356\) 0 0
\(357\) −2.00000 −0.105851
\(358\) 0 0
\(359\) 10.5359 0.556063 0.278032 0.960572i \(-0.410318\pi\)
0.278032 + 0.960572i \(0.410318\pi\)
\(360\) 0 0
\(361\) 10.8564 0.571390
\(362\) 0 0
\(363\) −18.8564 −0.989705
\(364\) 0 0
\(365\) −0.535898 −0.0280502
\(366\) 0 0
\(367\) 8.78461 0.458553 0.229276 0.973361i \(-0.426364\pi\)
0.229276 + 0.973361i \(0.426364\pi\)
\(368\) 0 0
\(369\) −4.92820 −0.256552
\(370\) 0 0
\(371\) −0.535898 −0.0278225
\(372\) 0 0
\(373\) −19.0718 −0.987500 −0.493750 0.869604i \(-0.664374\pi\)
−0.493750 + 0.869604i \(0.664374\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −6.92820 −0.356821
\(378\) 0 0
\(379\) 30.9282 1.58868 0.794338 0.607477i \(-0.207818\pi\)
0.794338 + 0.607477i \(0.207818\pi\)
\(380\) 0 0
\(381\) 5.07180 0.259836
\(382\) 0 0
\(383\) 21.8564 1.11681 0.558405 0.829568i \(-0.311413\pi\)
0.558405 + 0.829568i \(0.311413\pi\)
\(384\) 0 0
\(385\) 5.46410 0.278476
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −18.7846 −0.952418 −0.476209 0.879332i \(-0.657989\pi\)
−0.476209 + 0.879332i \(0.657989\pi\)
\(390\) 0 0
\(391\) 13.8564 0.700749
\(392\) 0 0
\(393\) −2.92820 −0.147708
\(394\) 0 0
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) 11.4641 0.575367 0.287683 0.957726i \(-0.407115\pi\)
0.287683 + 0.957726i \(0.407115\pi\)
\(398\) 0 0
\(399\) 5.46410 0.273547
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 18.9282 0.942881
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −10.9282 −0.541691
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 0.535898 0.0264339
\(412\) 0 0
\(413\) 13.8564 0.681829
\(414\) 0 0
\(415\) 14.9282 0.732797
\(416\) 0 0
\(417\) 5.46410 0.267578
\(418\) 0 0
\(419\) −8.78461 −0.429156 −0.214578 0.976707i \(-0.568838\pi\)
−0.214578 + 0.976707i \(0.568838\pi\)
\(420\) 0 0
\(421\) −15.8564 −0.772794 −0.386397 0.922333i \(-0.626281\pi\)
−0.386397 + 0.922333i \(0.626281\pi\)
\(422\) 0 0
\(423\) −10.9282 −0.531347
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −4.92820 −0.238492
\(428\) 0 0
\(429\) 18.9282 0.913862
\(430\) 0 0
\(431\) −22.2487 −1.07168 −0.535841 0.844319i \(-0.680005\pi\)
−0.535841 + 0.844319i \(0.680005\pi\)
\(432\) 0 0
\(433\) 35.4641 1.70430 0.852148 0.523301i \(-0.175300\pi\)
0.852148 + 0.523301i \(0.175300\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 0 0
\(437\) −37.8564 −1.81092
\(438\) 0 0
\(439\) 27.3205 1.30394 0.651968 0.758246i \(-0.273943\pi\)
0.651968 + 0.758246i \(0.273943\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 12.9282 0.612856
\(446\) 0 0
\(447\) 20.9282 0.989870
\(448\) 0 0
\(449\) 23.8564 1.12585 0.562927 0.826507i \(-0.309675\pi\)
0.562927 + 0.826507i \(0.309675\pi\)
\(450\) 0 0
\(451\) 26.9282 1.26800
\(452\) 0 0
\(453\) 14.9282 0.701388
\(454\) 0 0
\(455\) −3.46410 −0.162400
\(456\) 0 0
\(457\) −14.7846 −0.691595 −0.345797 0.938309i \(-0.612392\pi\)
−0.345797 + 0.938309i \(0.612392\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 4.14359 0.192986 0.0964932 0.995334i \(-0.469237\pi\)
0.0964932 + 0.995334i \(0.469237\pi\)
\(462\) 0 0
\(463\) −34.9282 −1.62325 −0.811626 0.584178i \(-0.801417\pi\)
−0.811626 + 0.584178i \(0.801417\pi\)
\(464\) 0 0
\(465\) 5.46410 0.253392
\(466\) 0 0
\(467\) 30.9282 1.43119 0.715593 0.698517i \(-0.246156\pi\)
0.715593 + 0.698517i \(0.246156\pi\)
\(468\) 0 0
\(469\) 6.92820 0.319915
\(470\) 0 0
\(471\) −0.535898 −0.0246929
\(472\) 0 0
\(473\) 21.8564 1.00496
\(474\) 0 0
\(475\) −5.46410 −0.250710
\(476\) 0 0
\(477\) −0.535898 −0.0245371
\(478\) 0 0
\(479\) −13.8564 −0.633115 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(480\) 0 0
\(481\) 6.92820 0.315899
\(482\) 0 0
\(483\) −6.92820 −0.315244
\(484\) 0 0
\(485\) −19.4641 −0.883819
\(486\) 0 0
\(487\) −16.7846 −0.760583 −0.380292 0.924867i \(-0.624176\pi\)
−0.380292 + 0.924867i \(0.624176\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −13.4641 −0.607626 −0.303813 0.952732i \(-0.598260\pi\)
−0.303813 + 0.952732i \(0.598260\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) 5.46410 0.245593
\(496\) 0 0
\(497\) −16.3923 −0.735295
\(498\) 0 0
\(499\) 22.9282 1.02641 0.513204 0.858267i \(-0.328459\pi\)
0.513204 + 0.858267i \(0.328459\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −15.8564 −0.705601
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 21.7128 0.962404 0.481202 0.876610i \(-0.340200\pi\)
0.481202 + 0.876610i \(0.340200\pi\)
\(510\) 0 0
\(511\) 0.535898 0.0237067
\(512\) 0 0
\(513\) 5.46410 0.241246
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 59.7128 2.62617
\(518\) 0 0
\(519\) 24.9282 1.09423
\(520\) 0 0
\(521\) 28.6410 1.25479 0.627393 0.778703i \(-0.284122\pi\)
0.627393 + 0.778703i \(0.284122\pi\)
\(522\) 0 0
\(523\) 28.7846 1.25866 0.629332 0.777137i \(-0.283329\pi\)
0.629332 + 0.777137i \(0.283329\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 10.9282 0.476040
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 13.8564 0.601317
\(532\) 0 0
\(533\) −17.0718 −0.739462
\(534\) 0 0
\(535\) 18.9282 0.818338
\(536\) 0 0
\(537\) −2.53590 −0.109432
\(538\) 0 0
\(539\) −5.46410 −0.235356
\(540\) 0 0
\(541\) 0.143594 0.00617357 0.00308678 0.999995i \(-0.499017\pi\)
0.00308678 + 0.999995i \(0.499017\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 38.9282 1.66445 0.832225 0.554438i \(-0.187067\pi\)
0.832225 + 0.554438i \(0.187067\pi\)
\(548\) 0 0
\(549\) −4.92820 −0.210331
\(550\) 0 0
\(551\) 10.9282 0.465557
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) 2.00000 0.0848953
\(556\) 0 0
\(557\) 9.60770 0.407091 0.203546 0.979065i \(-0.434753\pi\)
0.203546 + 0.979065i \(0.434753\pi\)
\(558\) 0 0
\(559\) −13.8564 −0.586064
\(560\) 0 0
\(561\) 10.9282 0.461389
\(562\) 0 0
\(563\) 1.85641 0.0782382 0.0391191 0.999235i \(-0.487545\pi\)
0.0391191 + 0.999235i \(0.487545\pi\)
\(564\) 0 0
\(565\) −10.3923 −0.437208
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 7.85641 0.329358 0.164679 0.986347i \(-0.447341\pi\)
0.164679 + 0.986347i \(0.447341\pi\)
\(570\) 0 0
\(571\) −36.7846 −1.53939 −0.769694 0.638413i \(-0.779591\pi\)
−0.769694 + 0.638413i \(0.779591\pi\)
\(572\) 0 0
\(573\) −8.39230 −0.350594
\(574\) 0 0
\(575\) 6.92820 0.288926
\(576\) 0 0
\(577\) 33.3205 1.38715 0.693575 0.720384i \(-0.256034\pi\)
0.693575 + 0.720384i \(0.256034\pi\)
\(578\) 0 0
\(579\) 11.8564 0.492735
\(580\) 0 0
\(581\) −14.9282 −0.619326
\(582\) 0 0
\(583\) 2.92820 0.121274
\(584\) 0 0
\(585\) −3.46410 −0.143223
\(586\) 0 0
\(587\) 9.85641 0.406817 0.203409 0.979094i \(-0.434798\pi\)
0.203409 + 0.979094i \(0.434798\pi\)
\(588\) 0 0
\(589\) −29.8564 −1.23021
\(590\) 0 0
\(591\) 19.4641 0.800646
\(592\) 0 0
\(593\) −9.71281 −0.398857 −0.199429 0.979912i \(-0.563909\pi\)
−0.199429 + 0.979912i \(0.563909\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) 0 0
\(597\) 0.392305 0.0160560
\(598\) 0 0
\(599\) 26.5359 1.08423 0.542114 0.840305i \(-0.317624\pi\)
0.542114 + 0.840305i \(0.317624\pi\)
\(600\) 0 0
\(601\) 28.9282 1.18001 0.590003 0.807401i \(-0.299127\pi\)
0.590003 + 0.807401i \(0.299127\pi\)
\(602\) 0 0
\(603\) 6.92820 0.282138
\(604\) 0 0
\(605\) −18.8564 −0.766622
\(606\) 0 0
\(607\) −34.9282 −1.41769 −0.708846 0.705363i \(-0.750784\pi\)
−0.708846 + 0.705363i \(0.750784\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) −37.8564 −1.53151
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) −4.92820 −0.198724
\(616\) 0 0
\(617\) −13.6077 −0.547825 −0.273913 0.961755i \(-0.588318\pi\)
−0.273913 + 0.961755i \(0.588318\pi\)
\(618\) 0 0
\(619\) 32.3923 1.30196 0.650978 0.759096i \(-0.274359\pi\)
0.650978 + 0.759096i \(0.274359\pi\)
\(620\) 0 0
\(621\) −6.92820 −0.278019
\(622\) 0 0
\(623\) −12.9282 −0.517958
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −29.8564 −1.19235
\(628\) 0 0
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 15.7128 0.625517 0.312759 0.949833i \(-0.398747\pi\)
0.312759 + 0.949833i \(0.398747\pi\)
\(632\) 0 0
\(633\) 4.00000 0.158986
\(634\) 0 0
\(635\) 5.07180 0.201268
\(636\) 0 0
\(637\) 3.46410 0.137253
\(638\) 0 0
\(639\) −16.3923 −0.648470
\(640\) 0 0
\(641\) −38.7846 −1.53190 −0.765950 0.642900i \(-0.777731\pi\)
−0.765950 + 0.642900i \(0.777731\pi\)
\(642\) 0 0
\(643\) 44.7846 1.76613 0.883066 0.469248i \(-0.155475\pi\)
0.883066 + 0.469248i \(0.155475\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) 13.8564 0.544752 0.272376 0.962191i \(-0.412191\pi\)
0.272376 + 0.962191i \(0.412191\pi\)
\(648\) 0 0
\(649\) −75.7128 −2.97199
\(650\) 0 0
\(651\) −5.46410 −0.214155
\(652\) 0 0
\(653\) −27.4641 −1.07475 −0.537377 0.843342i \(-0.680585\pi\)
−0.537377 + 0.843342i \(0.680585\pi\)
\(654\) 0 0
\(655\) −2.92820 −0.114414
\(656\) 0 0
\(657\) 0.535898 0.0209074
\(658\) 0 0
\(659\) 37.4641 1.45939 0.729697 0.683771i \(-0.239661\pi\)
0.729697 + 0.683771i \(0.239661\pi\)
\(660\) 0 0
\(661\) −7.07180 −0.275061 −0.137531 0.990498i \(-0.543917\pi\)
−0.137531 + 0.990498i \(0.543917\pi\)
\(662\) 0 0
\(663\) −6.92820 −0.269069
\(664\) 0 0
\(665\) 5.46410 0.211889
\(666\) 0 0
\(667\) −13.8564 −0.536522
\(668\) 0 0
\(669\) 26.9282 1.04110
\(670\) 0 0
\(671\) 26.9282 1.03955
\(672\) 0 0
\(673\) 28.9282 1.11510 0.557550 0.830143i \(-0.311741\pi\)
0.557550 + 0.830143i \(0.311741\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 10.7846 0.414486 0.207243 0.978289i \(-0.433551\pi\)
0.207243 + 0.978289i \(0.433551\pi\)
\(678\) 0 0
\(679\) 19.4641 0.746964
\(680\) 0 0
\(681\) −6.92820 −0.265489
\(682\) 0 0
\(683\) 5.85641 0.224089 0.112045 0.993703i \(-0.464260\pi\)
0.112045 + 0.993703i \(0.464260\pi\)
\(684\) 0 0
\(685\) 0.535898 0.0204756
\(686\) 0 0
\(687\) −22.7846 −0.869287
\(688\) 0 0
\(689\) −1.85641 −0.0707235
\(690\) 0 0
\(691\) −6.24871 −0.237712 −0.118856 0.992911i \(-0.537923\pi\)
−0.118856 + 0.992911i \(0.537923\pi\)
\(692\) 0 0
\(693\) −5.46410 −0.207564
\(694\) 0 0
\(695\) 5.46410 0.207265
\(696\) 0 0
\(697\) −9.85641 −0.373338
\(698\) 0 0
\(699\) 19.4641 0.736200
\(700\) 0 0
\(701\) 19.8564 0.749966 0.374983 0.927032i \(-0.377649\pi\)
0.374983 + 0.927032i \(0.377649\pi\)
\(702\) 0 0
\(703\) −10.9282 −0.412165
\(704\) 0 0
\(705\) −10.9282 −0.411580
\(706\) 0 0
\(707\) 15.8564 0.596342
\(708\) 0 0
\(709\) −37.7128 −1.41633 −0.708167 0.706045i \(-0.750478\pi\)
−0.708167 + 0.706045i \(0.750478\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 37.8564 1.41773
\(714\) 0 0
\(715\) 18.9282 0.707875
\(716\) 0 0
\(717\) −19.3205 −0.721538
\(718\) 0 0
\(719\) −16.7846 −0.625960 −0.312980 0.949760i \(-0.601327\pi\)
−0.312980 + 0.949760i \(0.601327\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 14.7846 0.549846
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −10.9282 −0.405305 −0.202652 0.979251i \(-0.564956\pi\)
−0.202652 + 0.979251i \(0.564956\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −10.3923 −0.383849 −0.191924 0.981410i \(-0.561473\pi\)
−0.191924 + 0.981410i \(0.561473\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) −37.8564 −1.39446
\(738\) 0 0
\(739\) −12.7846 −0.470289 −0.235145 0.971960i \(-0.575556\pi\)
−0.235145 + 0.971960i \(0.575556\pi\)
\(740\) 0 0
\(741\) 18.9282 0.695345
\(742\) 0 0
\(743\) −25.8564 −0.948580 −0.474290 0.880369i \(-0.657295\pi\)
−0.474290 + 0.880369i \(0.657295\pi\)
\(744\) 0 0
\(745\) 20.9282 0.766750
\(746\) 0 0
\(747\) −14.9282 −0.546194
\(748\) 0 0
\(749\) −18.9282 −0.691621
\(750\) 0 0
\(751\) −45.5692 −1.66284 −0.831422 0.555641i \(-0.812473\pi\)
−0.831422 + 0.555641i \(0.812473\pi\)
\(752\) 0 0
\(753\) 2.14359 0.0781169
\(754\) 0 0
\(755\) 14.9282 0.543293
\(756\) 0 0
\(757\) −19.0718 −0.693176 −0.346588 0.938017i \(-0.612660\pi\)
−0.346588 + 0.938017i \(0.612660\pi\)
\(758\) 0 0
\(759\) 37.8564 1.37410
\(760\) 0 0
\(761\) −31.0718 −1.12635 −0.563176 0.826337i \(-0.690421\pi\)
−0.563176 + 0.826337i \(0.690421\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) 48.0000 1.73318
\(768\) 0 0
\(769\) −28.6410 −1.03282 −0.516411 0.856341i \(-0.672732\pi\)
−0.516411 + 0.856341i \(0.672732\pi\)
\(770\) 0 0
\(771\) 3.07180 0.110628
\(772\) 0 0
\(773\) −8.92820 −0.321125 −0.160563 0.987026i \(-0.551331\pi\)
−0.160563 + 0.987026i \(0.551331\pi\)
\(774\) 0 0
\(775\) 5.46410 0.196276
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) 0 0
\(779\) 26.9282 0.964803
\(780\) 0 0
\(781\) 89.5692 3.20504
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) −0.535898 −0.0191270
\(786\) 0 0
\(787\) −12.7846 −0.455722 −0.227861 0.973694i \(-0.573173\pi\)
−0.227861 + 0.973694i \(0.573173\pi\)
\(788\) 0 0
\(789\) −17.8564 −0.635705
\(790\) 0 0
\(791\) 10.3923 0.369508
\(792\) 0 0
\(793\) −17.0718 −0.606237
\(794\) 0 0
\(795\) −0.535898 −0.0190064
\(796\) 0 0
\(797\) 23.0718 0.817245 0.408622 0.912703i \(-0.366009\pi\)
0.408622 + 0.912703i \(0.366009\pi\)
\(798\) 0 0
\(799\) −21.8564 −0.773224
\(800\) 0 0
\(801\) −12.9282 −0.456796
\(802\) 0 0
\(803\) −2.92820 −0.103334
\(804\) 0 0
\(805\) −6.92820 −0.244187
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) −38.7846 −1.36359 −0.681797 0.731541i \(-0.738801\pi\)
−0.681797 + 0.731541i \(0.738801\pi\)
\(810\) 0 0
\(811\) −41.1769 −1.44592 −0.722959 0.690891i \(-0.757218\pi\)
−0.722959 + 0.690891i \(0.757218\pi\)
\(812\) 0 0
\(813\) 13.4641 0.472207
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) 21.8564 0.764659
\(818\) 0 0
\(819\) 3.46410 0.121046
\(820\) 0 0
\(821\) 25.7128 0.897383 0.448692 0.893687i \(-0.351890\pi\)
0.448692 + 0.893687i \(0.351890\pi\)
\(822\) 0 0
\(823\) 8.78461 0.306212 0.153106 0.988210i \(-0.451072\pi\)
0.153106 + 0.988210i \(0.451072\pi\)
\(824\) 0 0
\(825\) 5.46410 0.190236
\(826\) 0 0
\(827\) 10.9282 0.380011 0.190005 0.981783i \(-0.439149\pi\)
0.190005 + 0.981783i \(0.439149\pi\)
\(828\) 0 0
\(829\) 3.85641 0.133939 0.0669693 0.997755i \(-0.478667\pi\)
0.0669693 + 0.997755i \(0.478667\pi\)
\(830\) 0 0
\(831\) 11.8564 0.411294
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) −5.46410 −0.188867
\(838\) 0 0
\(839\) 32.7846 1.13185 0.565925 0.824457i \(-0.308519\pi\)
0.565925 + 0.824457i \(0.308519\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 30.7846 1.06028
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 18.8564 0.647914
\(848\) 0 0
\(849\) 17.8564 0.612830
\(850\) 0 0
\(851\) 13.8564 0.474991
\(852\) 0 0
\(853\) −18.3923 −0.629741 −0.314870 0.949135i \(-0.601961\pi\)
−0.314870 + 0.949135i \(0.601961\pi\)
\(854\) 0 0
\(855\) 5.46410 0.186868
\(856\) 0 0
\(857\) −11.0718 −0.378205 −0.189103 0.981957i \(-0.560558\pi\)
−0.189103 + 0.981957i \(0.560558\pi\)
\(858\) 0 0
\(859\) 23.6077 0.805484 0.402742 0.915314i \(-0.368057\pi\)
0.402742 + 0.915314i \(0.368057\pi\)
\(860\) 0 0
\(861\) 4.92820 0.167953
\(862\) 0 0
\(863\) 33.0718 1.12578 0.562889 0.826533i \(-0.309690\pi\)
0.562889 + 0.826533i \(0.309690\pi\)
\(864\) 0 0
\(865\) 24.9282 0.847584
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 21.8564 0.741428
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 0 0
\(873\) 19.4641 0.658760
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 0 0
\(879\) 24.9282 0.840807
\(880\) 0 0
\(881\) 46.7846 1.57621 0.788107 0.615539i \(-0.211061\pi\)
0.788107 + 0.615539i \(0.211061\pi\)
\(882\) 0 0
\(883\) −26.6410 −0.896542 −0.448271 0.893898i \(-0.647960\pi\)
−0.448271 + 0.893898i \(0.647960\pi\)
\(884\) 0 0
\(885\) 13.8564 0.465778
\(886\) 0 0
\(887\) 2.92820 0.0983194 0.0491597 0.998791i \(-0.484346\pi\)
0.0491597 + 0.998791i \(0.484346\pi\)
\(888\) 0 0
\(889\) −5.07180 −0.170103
\(890\) 0 0
\(891\) −5.46410 −0.183054
\(892\) 0 0
\(893\) 59.7128 1.99821
\(894\) 0 0
\(895\) −2.53590 −0.0847657
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 0 0
\(899\) −10.9282 −0.364476
\(900\) 0 0
\(901\) −1.07180 −0.0357067
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) 12.7846 0.424506 0.212253 0.977215i \(-0.431920\pi\)
0.212253 + 0.977215i \(0.431920\pi\)
\(908\) 0 0
\(909\) 15.8564 0.525924
\(910\) 0 0
\(911\) 22.2487 0.737133 0.368566 0.929601i \(-0.379849\pi\)
0.368566 + 0.929601i \(0.379849\pi\)
\(912\) 0 0
\(913\) 81.5692 2.69955
\(914\) 0 0
\(915\) −4.92820 −0.162921
\(916\) 0 0
\(917\) 2.92820 0.0966978
\(918\) 0 0
\(919\) 28.0000 0.923635 0.461817 0.886975i \(-0.347198\pi\)
0.461817 + 0.886975i \(0.347198\pi\)
\(920\) 0 0
\(921\) −22.9282 −0.755510
\(922\) 0 0
\(923\) −56.7846 −1.86909
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) 52.6410 1.72710 0.863548 0.504267i \(-0.168237\pi\)
0.863548 + 0.504267i \(0.168237\pi\)
\(930\) 0 0
\(931\) −5.46410 −0.179079
\(932\) 0 0
\(933\) −8.00000 −0.261908
\(934\) 0 0
\(935\) 10.9282 0.357390
\(936\) 0 0
\(937\) −27.1769 −0.887831 −0.443916 0.896069i \(-0.646411\pi\)
−0.443916 + 0.896069i \(0.646411\pi\)
\(938\) 0 0
\(939\) −17.3205 −0.565233
\(940\) 0 0
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 0 0
\(943\) −34.1436 −1.11187
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −16.0000 −0.519930 −0.259965 0.965618i \(-0.583711\pi\)
−0.259965 + 0.965618i \(0.583711\pi\)
\(948\) 0 0
\(949\) 1.85641 0.0602615
\(950\) 0 0
\(951\) −13.3205 −0.431947
\(952\) 0 0
\(953\) −49.3205 −1.59765 −0.798824 0.601565i \(-0.794544\pi\)
−0.798824 + 0.601565i \(0.794544\pi\)
\(954\) 0 0
\(955\) −8.39230 −0.271569
\(956\) 0 0
\(957\) −10.9282 −0.353259
\(958\) 0 0
\(959\) −0.535898 −0.0173051
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) −18.9282 −0.609953
\(964\) 0 0
\(965\) 11.8564 0.381671
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 0 0
\(969\) 10.9282 0.351064
\(970\) 0 0
\(971\) 10.1436 0.325523 0.162762 0.986665i \(-0.447960\pi\)
0.162762 + 0.986665i \(0.447960\pi\)
\(972\) 0 0
\(973\) −5.46410 −0.175171
\(974\) 0 0
\(975\) −3.46410 −0.110940
\(976\) 0 0
\(977\) −13.6077 −0.435349 −0.217674 0.976021i \(-0.569847\pi\)
−0.217674 + 0.976021i \(0.569847\pi\)
\(978\) 0 0
\(979\) 70.6410 2.25770
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) −40.7846 −1.30083 −0.650414 0.759580i \(-0.725404\pi\)
−0.650414 + 0.759580i \(0.725404\pi\)
\(984\) 0 0
\(985\) 19.4641 0.620178
\(986\) 0 0
\(987\) 10.9282 0.347849
\(988\) 0 0
\(989\) −27.7128 −0.881216
\(990\) 0 0
\(991\) 51.4256 1.63359 0.816794 0.576929i \(-0.195749\pi\)
0.816794 + 0.576929i \(0.195749\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 0.392305 0.0124369
\(996\) 0 0
\(997\) −0.248711 −0.00787677 −0.00393838 0.999992i \(-0.501254\pi\)
−0.00393838 + 0.999992i \(0.501254\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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 3360.2.a.bb.1.1 2
4.3 odd 2 3360.2.a.bf.1.2 yes 2
8.3 odd 2 6720.2.a.cq.1.1 2
8.5 even 2 6720.2.a.cz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.bb.1.1 2 1.1 even 1 trivial
3360.2.a.bf.1.2 yes 2 4.3 odd 2
6720.2.a.cq.1.1 2 8.3 odd 2
6720.2.a.cz.1.2 2 8.5 even 2