Properties

Label 3360.2.a.bb.1.2
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 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3360.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.46410 q^{11} -3.46410 q^{13} +1.00000 q^{15} +2.00000 q^{17} +1.46410 q^{19} -1.00000 q^{21} -6.92820 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -1.46410 q^{31} -1.46410 q^{33} -1.00000 q^{35} +2.00000 q^{37} +3.46410 q^{39} +8.92820 q^{41} -4.00000 q^{43} -1.00000 q^{45} +2.92820 q^{47} +1.00000 q^{49} -2.00000 q^{51} -7.46410 q^{53} -1.46410 q^{55} -1.46410 q^{57} -13.8564 q^{59} +8.92820 q^{61} +1.00000 q^{63} +3.46410 q^{65} -6.92820 q^{67} +6.92820 q^{69} +4.39230 q^{71} +7.46410 q^{73} -1.00000 q^{75} +1.46410 q^{77} -4.00000 q^{79} +1.00000 q^{81} -1.07180 q^{83} -2.00000 q^{85} +2.00000 q^{87} +0.928203 q^{89} -3.46410 q^{91} +1.46410 q^{93} -1.46410 q^{95} +12.5359 q^{97} +1.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} - 4q^{11} + 2q^{15} + 4q^{17} - 4q^{19} - 2q^{21} + 2q^{25} - 2q^{27} - 4q^{29} + 4q^{31} + 4q^{33} - 2q^{35} + 4q^{37} + 4q^{41} - 8q^{43} - 2q^{45} - 8q^{47} + 2q^{49} - 4q^{51} - 8q^{53} + 4q^{55} + 4q^{57} + 4q^{61} + 2q^{63} - 12q^{71} + 8q^{73} - 2q^{75} - 4q^{77} - 8q^{79} + 2q^{81} - 16q^{83} - 4q^{85} + 4q^{87} - 12q^{89} - 4q^{93} + 4q^{95} + 32q^{97} - 4q^{99} + O(q^{100}) \)

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
\(11\) 1.46410 0.441443 0.220722 0.975337i \(-0.429159\pi\)
0.220722 + 0.975337i \(0.429159\pi\)
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) 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
\(19\) 1.46410 0.335888 0.167944 0.985797i \(-0.446287\pi\)
0.167944 + 0.985797i \(0.446287\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 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
\(31\) −1.46410 −0.262960 −0.131480 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) 0 0
\(33\) −1.46410 −0.254867
\(34\) 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
\(41\) 8.92820 1.39435 0.697176 0.716900i \(-0.254440\pi\)
0.697176 + 0.716900i \(0.254440\pi\)
\(42\) 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
\(47\) 2.92820 0.427122 0.213561 0.976930i \(-0.431494\pi\)
0.213561 + 0.976930i \(0.431494\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −7.46410 −1.02527 −0.512637 0.858606i \(-0.671331\pi\)
−0.512637 + 0.858606i \(0.671331\pi\)
\(54\) 0 0
\(55\) −1.46410 −0.197419
\(56\) 0 0
\(57\) −1.46410 −0.193925
\(58\) 0 0
\(59\) −13.8564 −1.80395 −0.901975 0.431788i \(-0.857883\pi\)
−0.901975 + 0.431788i \(0.857883\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) 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 \(-0.639096\pi\)
−0.423207 + 0.906033i \(0.639096\pi\)
\(68\) 0 0
\(69\) 6.92820 0.834058
\(70\) 0 0
\(71\) 4.39230 0.521271 0.260635 0.965437i \(-0.416068\pi\)
0.260635 + 0.965437i \(0.416068\pi\)
\(72\) 0 0
\(73\) 7.46410 0.873607 0.436804 0.899557i \(-0.356111\pi\)
0.436804 + 0.899557i \(0.356111\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 1.46410 0.166850
\(78\) 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
\(83\) −1.07180 −0.117645 −0.0588225 0.998268i \(-0.518735\pi\)
−0.0588225 + 0.998268i \(0.518735\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 0.928203 0.0983893 0.0491947 0.998789i \(-0.484335\pi\)
0.0491947 + 0.998789i \(0.484335\pi\)
\(90\) 0 0
\(91\) −3.46410 −0.363137
\(92\) 0 0
\(93\) 1.46410 0.151820
\(94\) 0 0
\(95\) −1.46410 −0.150214
\(96\) 0 0
\(97\) 12.5359 1.27283 0.636414 0.771348i \(-0.280417\pi\)
0.636414 + 0.771348i \(0.280417\pi\)
\(98\) 0 0
\(99\) 1.46410 0.147148
\(100\) 0 0
\(101\) −11.8564 −1.17976 −0.589878 0.807492i \(-0.700824\pi\)
−0.589878 + 0.807492i \(0.700824\pi\)
\(102\) 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
\(107\) −5.07180 −0.490309 −0.245155 0.969484i \(-0.578839\pi\)
−0.245155 + 0.969484i \(0.578839\pi\)
\(108\) 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 \(-0.662570\pi\)
−0.488813 + 0.872389i \(0.662570\pi\)
\(114\) 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
\(121\) −8.85641 −0.805128
\(122\) 0 0
\(123\) −8.92820 −0.805029
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −18.9282 −1.67961 −0.839803 0.542891i \(-0.817330\pi\)
−0.839803 + 0.542891i \(0.817330\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −10.9282 −0.954802 −0.477401 0.878686i \(-0.658421\pi\)
−0.477401 + 0.878686i \(0.658421\pi\)
\(132\) 0 0
\(133\) 1.46410 0.126954
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −7.46410 −0.637701 −0.318851 0.947805i \(-0.603297\pi\)
−0.318851 + 0.947805i \(0.603297\pi\)
\(138\) 0 0
\(139\) 1.46410 0.124183 0.0620917 0.998070i \(-0.480223\pi\)
0.0620917 + 0.998070i \(0.480223\pi\)
\(140\) 0 0
\(141\) −2.92820 −0.246599
\(142\) 0 0
\(143\) −5.07180 −0.424125
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −7.07180 −0.579344 −0.289672 0.957126i \(-0.593546\pi\)
−0.289672 + 0.957126i \(0.593546\pi\)
\(150\) 0 0
\(151\) −1.07180 −0.0872216 −0.0436108 0.999049i \(-0.513886\pi\)
−0.0436108 + 0.999049i \(0.513886\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 1.46410 0.117599
\(156\) 0 0
\(157\) 7.46410 0.595700 0.297850 0.954613i \(-0.403730\pi\)
0.297850 + 0.954613i \(0.403730\pi\)
\(158\) 0 0
\(159\) 7.46410 0.591942
\(160\) 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
\(165\) 1.46410 0.113980
\(166\) 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
\(171\) 1.46410 0.111963
\(172\) 0 0
\(173\) −11.0718 −0.841773 −0.420887 0.907113i \(-0.638281\pi\)
−0.420887 + 0.907113i \(0.638281\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 13.8564 1.04151
\(178\) 0 0
\(179\) 9.46410 0.707380 0.353690 0.935363i \(-0.384927\pi\)
0.353690 + 0.935363i \(0.384927\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −8.92820 −0.659992
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 2.92820 0.214131
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −12.3923 −0.896676 −0.448338 0.893864i \(-0.647984\pi\)
−0.448338 + 0.893864i \(0.647984\pi\)
\(192\) 0 0
\(193\) 15.8564 1.14137 0.570685 0.821169i \(-0.306678\pi\)
0.570685 + 0.821169i \(0.306678\pi\)
\(194\) 0 0
\(195\) −3.46410 −0.248069
\(196\) 0 0
\(197\) −12.5359 −0.893146 −0.446573 0.894747i \(-0.647356\pi\)
−0.446573 + 0.894747i \(0.647356\pi\)
\(198\) 0 0
\(199\) 20.3923 1.44557 0.722786 0.691072i \(-0.242861\pi\)
0.722786 + 0.691072i \(0.242861\pi\)
\(200\) 0 0
\(201\) 6.92820 0.488678
\(202\) 0 0
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) −8.92820 −0.623573
\(206\) 0 0
\(207\) −6.92820 −0.481543
\(208\) 0 0
\(209\) 2.14359 0.148275
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) −4.39230 −0.300956
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −1.46410 −0.0993897
\(218\) 0 0
\(219\) −7.46410 −0.504377
\(220\) 0 0
\(221\) −6.92820 −0.466041
\(222\) 0 0
\(223\) −13.0718 −0.875352 −0.437676 0.899133i \(-0.644198\pi\)
−0.437676 + 0.899133i \(0.644198\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −6.92820 −0.459841 −0.229920 0.973209i \(-0.573847\pi\)
−0.229920 + 0.973209i \(0.573847\pi\)
\(228\) 0 0
\(229\) −18.7846 −1.24132 −0.620661 0.784079i \(-0.713136\pi\)
−0.620661 + 0.784079i \(0.713136\pi\)
\(230\) 0 0
\(231\) −1.46410 −0.0963308
\(232\) 0 0
\(233\) −12.5359 −0.821254 −0.410627 0.911803i \(-0.634690\pi\)
−0.410627 + 0.911803i \(0.634690\pi\)
\(234\) 0 0
\(235\) −2.92820 −0.191015
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) −15.3205 −0.991001 −0.495501 0.868608i \(-0.665015\pi\)
−0.495501 + 0.868608i \(0.665015\pi\)
\(240\) 0 0
\(241\) 26.7846 1.72535 0.862674 0.505760i \(-0.168788\pi\)
0.862674 + 0.505760i \(0.168788\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −5.07180 −0.322711
\(248\) 0 0
\(249\) 1.07180 0.0679224
\(250\) 0 0
\(251\) −29.8564 −1.88452 −0.942260 0.334883i \(-0.891303\pi\)
−0.942260 + 0.334883i \(0.891303\pi\)
\(252\) 0 0
\(253\) −10.1436 −0.637722
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 0 0
\(257\) −16.9282 −1.05595 −0.527976 0.849259i \(-0.677049\pi\)
−0.527976 + 0.849259i \(0.677049\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −9.85641 −0.607772 −0.303886 0.952708i \(-0.598284\pi\)
−0.303886 + 0.952708i \(0.598284\pi\)
\(264\) 0 0
\(265\) 7.46410 0.458516
\(266\) 0 0
\(267\) −0.928203 −0.0568051
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −6.53590 −0.397028 −0.198514 0.980098i \(-0.563611\pi\)
−0.198514 + 0.980098i \(0.563611\pi\)
\(272\) 0 0
\(273\) 3.46410 0.209657
\(274\) 0 0
\(275\) 1.46410 0.0882886
\(276\) 0 0
\(277\) 15.8564 0.952719 0.476360 0.879251i \(-0.341956\pi\)
0.476360 + 0.879251i \(0.341956\pi\)
\(278\) 0 0
\(279\) −1.46410 −0.0876535
\(280\) 0 0
\(281\) 10.7846 0.643356 0.321678 0.946849i \(-0.395753\pi\)
0.321678 + 0.946849i \(0.395753\pi\)
\(282\) 0 0
\(283\) 9.85641 0.585903 0.292951 0.956127i \(-0.405363\pi\)
0.292951 + 0.956127i \(0.405363\pi\)
\(284\) 0 0
\(285\) 1.46410 0.0867259
\(286\) 0 0
\(287\) 8.92820 0.527015
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −12.5359 −0.734867
\(292\) 0 0
\(293\) −11.0718 −0.646821 −0.323411 0.946259i \(-0.604830\pi\)
−0.323411 + 0.946259i \(0.604830\pi\)
\(294\) 0 0
\(295\) 13.8564 0.806751
\(296\) 0 0
\(297\) −1.46410 −0.0849558
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 11.8564 0.681133
\(304\) 0 0
\(305\) −8.92820 −0.511227
\(306\) 0 0
\(307\) 9.07180 0.517755 0.258877 0.965910i \(-0.416647\pi\)
0.258877 + 0.965910i \(0.416647\pi\)
\(308\) 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 \(-0.662823\pi\)
−0.489506 + 0.872000i \(0.662823\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) −21.3205 −1.19748 −0.598740 0.800944i \(-0.704332\pi\)
−0.598740 + 0.800944i \(0.704332\pi\)
\(318\) 0 0
\(319\) −2.92820 −0.163948
\(320\) 0 0
\(321\) 5.07180 0.283080
\(322\) 0 0
\(323\) 2.92820 0.162930
\(324\) 0 0
\(325\) −3.46410 −0.192154
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) 2.92820 0.161437
\(330\) 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
\(341\) −2.14359 −0.116082
\(342\) 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 \(-0.378698\pi\)
0.371925 + 0.928263i \(0.378698\pi\)
\(348\) 0 0
\(349\) 3.07180 0.164430 0.0822148 0.996615i \(-0.473801\pi\)
0.0822148 + 0.996615i \(0.473801\pi\)
\(350\) 0 0
\(351\) 3.46410 0.184900
\(352\) 0 0
\(353\) 31.8564 1.69555 0.847773 0.530360i \(-0.177943\pi\)
0.847773 + 0.530360i \(0.177943\pi\)
\(354\) 0 0
\(355\) −4.39230 −0.233119
\(356\) 0 0
\(357\) −2.00000 −0.105851
\(358\) 0 0
\(359\) 17.4641 0.921720 0.460860 0.887473i \(-0.347541\pi\)
0.460860 + 0.887473i \(0.347541\pi\)
\(360\) 0 0
\(361\) −16.8564 −0.887179
\(362\) 0 0
\(363\) 8.85641 0.464841
\(364\) 0 0
\(365\) −7.46410 −0.390689
\(366\) 0 0
\(367\) −32.7846 −1.71134 −0.855671 0.517520i \(-0.826855\pi\)
−0.855671 + 0.517520i \(0.826855\pi\)
\(368\) 0 0
\(369\) 8.92820 0.464784
\(370\) 0 0
\(371\) −7.46410 −0.387517
\(372\) 0 0
\(373\) −32.9282 −1.70496 −0.852479 0.522762i \(-0.824902\pi\)
−0.852479 + 0.522762i \(0.824902\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 6.92820 0.356821
\(378\) 0 0
\(379\) 17.0718 0.876919 0.438460 0.898751i \(-0.355524\pi\)
0.438460 + 0.898751i \(0.355524\pi\)
\(380\) 0 0
\(381\) 18.9282 0.969721
\(382\) 0 0
\(383\) −5.85641 −0.299248 −0.149624 0.988743i \(-0.547806\pi\)
−0.149624 + 0.988743i \(0.547806\pi\)
\(384\) 0 0
\(385\) −1.46410 −0.0746175
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 22.7846 1.15523 0.577613 0.816311i \(-0.303984\pi\)
0.577613 + 0.816311i \(0.303984\pi\)
\(390\) 0 0
\(391\) −13.8564 −0.700749
\(392\) 0 0
\(393\) 10.9282 0.551255
\(394\) 0 0
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) 4.53590 0.227650 0.113825 0.993501i \(-0.463690\pi\)
0.113825 + 0.993501i \(0.463690\pi\)
\(398\) 0 0
\(399\) −1.46410 −0.0732968
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 5.07180 0.252644
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 2.92820 0.145146
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 7.46410 0.368177
\(412\) 0 0
\(413\) −13.8564 −0.681829
\(414\) 0 0
\(415\) 1.07180 0.0526124
\(416\) 0 0
\(417\) −1.46410 −0.0716974
\(418\) 0 0
\(419\) 32.7846 1.60163 0.800816 0.598910i \(-0.204399\pi\)
0.800816 + 0.598910i \(0.204399\pi\)
\(420\) 0 0
\(421\) 11.8564 0.577846 0.288923 0.957352i \(-0.406703\pi\)
0.288923 + 0.957352i \(0.406703\pi\)
\(422\) 0 0
\(423\) 2.92820 0.142374
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 8.92820 0.432066
\(428\) 0 0
\(429\) 5.07180 0.244869
\(430\) 0 0
\(431\) 26.2487 1.26436 0.632178 0.774823i \(-0.282161\pi\)
0.632178 + 0.774823i \(0.282161\pi\)
\(432\) 0 0
\(433\) 28.5359 1.37135 0.685674 0.727909i \(-0.259508\pi\)
0.685674 + 0.727909i \(0.259508\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 0 0
\(437\) −10.1436 −0.485234
\(438\) 0 0
\(439\) −7.32051 −0.349389 −0.174694 0.984623i \(-0.555894\pi\)
−0.174694 + 0.984623i \(0.555894\pi\)
\(440\) 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
\(445\) −0.928203 −0.0440011
\(446\) 0 0
\(447\) 7.07180 0.334485
\(448\) 0 0
\(449\) −3.85641 −0.181995 −0.0909975 0.995851i \(-0.529006\pi\)
−0.0909975 + 0.995851i \(0.529006\pi\)
\(450\) 0 0
\(451\) 13.0718 0.615527
\(452\) 0 0
\(453\) 1.07180 0.0503574
\(454\) 0 0
\(455\) 3.46410 0.162400
\(456\) 0 0
\(457\) 26.7846 1.25293 0.626466 0.779449i \(-0.284501\pi\)
0.626466 + 0.779449i \(0.284501\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 31.8564 1.48370 0.741850 0.670565i \(-0.233948\pi\)
0.741850 + 0.670565i \(0.233948\pi\)
\(462\) 0 0
\(463\) −21.0718 −0.979289 −0.489645 0.871922i \(-0.662874\pi\)
−0.489645 + 0.871922i \(0.662874\pi\)
\(464\) 0 0
\(465\) −1.46410 −0.0678961
\(466\) 0 0
\(467\) 17.0718 0.789989 0.394994 0.918684i \(-0.370747\pi\)
0.394994 + 0.918684i \(0.370747\pi\)
\(468\) 0 0
\(469\) −6.92820 −0.319915
\(470\) 0 0
\(471\) −7.46410 −0.343928
\(472\) 0 0
\(473\) −5.85641 −0.269278
\(474\) 0 0
\(475\) 1.46410 0.0671776
\(476\) 0 0
\(477\) −7.46410 −0.341758
\(478\) 0 0
\(479\) 13.8564 0.633115 0.316558 0.948573i \(-0.397473\pi\)
0.316558 + 0.948573i \(0.397473\pi\)
\(480\) 0 0
\(481\) −6.92820 −0.315899
\(482\) 0 0
\(483\) 6.92820 0.315244
\(484\) 0 0
\(485\) −12.5359 −0.569226
\(486\) 0 0
\(487\) 24.7846 1.12310 0.561549 0.827444i \(-0.310206\pi\)
0.561549 + 0.827444i \(0.310206\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −6.53590 −0.294961 −0.147480 0.989065i \(-0.547116\pi\)
−0.147480 + 0.989065i \(0.547116\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) −1.46410 −0.0658065
\(496\) 0 0
\(497\) 4.39230 0.197022
\(498\) 0 0
\(499\) 9.07180 0.406109 0.203055 0.979167i \(-0.434913\pi\)
0.203055 + 0.979167i \(0.434913\pi\)
\(500\) 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
\(505\) 11.8564 0.527603
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −33.7128 −1.49429 −0.747147 0.664659i \(-0.768577\pi\)
−0.747147 + 0.664659i \(0.768577\pi\)
\(510\) 0 0
\(511\) 7.46410 0.330192
\(512\) 0 0
\(513\) −1.46410 −0.0646417
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 4.28719 0.188550
\(518\) 0 0
\(519\) 11.0718 0.485998
\(520\) 0 0
\(521\) −40.6410 −1.78052 −0.890258 0.455457i \(-0.849476\pi\)
−0.890258 + 0.455457i \(0.849476\pi\)
\(522\) 0 0
\(523\) −12.7846 −0.559032 −0.279516 0.960141i \(-0.590174\pi\)
−0.279516 + 0.960141i \(0.590174\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) −2.92820 −0.127555
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) −13.8564 −0.601317
\(532\) 0 0
\(533\) −30.9282 −1.33965
\(534\) 0 0
\(535\) 5.07180 0.219273
\(536\) 0 0
\(537\) −9.46410 −0.408406
\(538\) 0 0
\(539\) 1.46410 0.0630633
\(540\) 0 0
\(541\) 27.8564 1.19764 0.598820 0.800883i \(-0.295636\pi\)
0.598820 + 0.800883i \(0.295636\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 25.0718 1.07199 0.535996 0.844220i \(-0.319936\pi\)
0.535996 + 0.844220i \(0.319936\pi\)
\(548\) 0 0
\(549\) 8.92820 0.381046
\(550\) 0 0
\(551\) −2.92820 −0.124746
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) 2.00000 0.0848953
\(556\) 0 0
\(557\) 30.3923 1.28776 0.643882 0.765125i \(-0.277323\pi\)
0.643882 + 0.765125i \(0.277323\pi\)
\(558\) 0 0
\(559\) 13.8564 0.586064
\(560\) 0 0
\(561\) −2.92820 −0.123629
\(562\) 0 0
\(563\) −25.8564 −1.08972 −0.544859 0.838528i \(-0.683417\pi\)
−0.544859 + 0.838528i \(0.683417\pi\)
\(564\) 0 0
\(565\) 10.3923 0.437208
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −19.8564 −0.832424 −0.416212 0.909268i \(-0.636643\pi\)
−0.416212 + 0.909268i \(0.636643\pi\)
\(570\) 0 0
\(571\) 4.78461 0.200230 0.100115 0.994976i \(-0.468079\pi\)
0.100115 + 0.994976i \(0.468079\pi\)
\(572\) 0 0
\(573\) 12.3923 0.517696
\(574\) 0 0
\(575\) −6.92820 −0.288926
\(576\) 0 0
\(577\) −1.32051 −0.0549735 −0.0274867 0.999622i \(-0.508750\pi\)
−0.0274867 + 0.999622i \(0.508750\pi\)
\(578\) 0 0
\(579\) −15.8564 −0.658970
\(580\) 0 0
\(581\) −1.07180 −0.0444656
\(582\) 0 0
\(583\) −10.9282 −0.452600
\(584\) 0 0
\(585\) 3.46410 0.143223
\(586\) 0 0
\(587\) −17.8564 −0.737013 −0.368506 0.929625i \(-0.620131\pi\)
−0.368506 + 0.929625i \(0.620131\pi\)
\(588\) 0 0
\(589\) −2.14359 −0.0883252
\(590\) 0 0
\(591\) 12.5359 0.515658
\(592\) 0 0
\(593\) 45.7128 1.87720 0.938600 0.345007i \(-0.112123\pi\)
0.938600 + 0.345007i \(0.112123\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) 0 0
\(597\) −20.3923 −0.834601
\(598\) 0 0
\(599\) 33.4641 1.36731 0.683653 0.729807i \(-0.260390\pi\)
0.683653 + 0.729807i \(0.260390\pi\)
\(600\) 0 0
\(601\) 15.0718 0.614791 0.307396 0.951582i \(-0.400542\pi\)
0.307396 + 0.951582i \(0.400542\pi\)
\(602\) 0 0
\(603\) −6.92820 −0.282138
\(604\) 0 0
\(605\) 8.85641 0.360064
\(606\) 0 0
\(607\) −21.0718 −0.855278 −0.427639 0.903950i \(-0.640655\pi\)
−0.427639 + 0.903950i \(0.640655\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) −10.1436 −0.410366
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 8.92820 0.360020
\(616\) 0 0
\(617\) −34.3923 −1.38458 −0.692291 0.721618i \(-0.743399\pi\)
−0.692291 + 0.721618i \(0.743399\pi\)
\(618\) 0 0
\(619\) 11.6077 0.466553 0.233276 0.972410i \(-0.425055\pi\)
0.233276 + 0.972410i \(0.425055\pi\)
\(620\) 0 0
\(621\) 6.92820 0.278019
\(622\) 0 0
\(623\) 0.928203 0.0371877
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −2.14359 −0.0856069
\(628\) 0 0
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −39.7128 −1.58094 −0.790471 0.612499i \(-0.790164\pi\)
−0.790471 + 0.612499i \(0.790164\pi\)
\(632\) 0 0
\(633\) 4.00000 0.158986
\(634\) 0 0
\(635\) 18.9282 0.751143
\(636\) 0 0
\(637\) −3.46410 −0.137253
\(638\) 0 0
\(639\) 4.39230 0.173757
\(640\) 0 0
\(641\) 2.78461 0.109985 0.0549927 0.998487i \(-0.482486\pi\)
0.0549927 + 0.998487i \(0.482486\pi\)
\(642\) 0 0
\(643\) 3.21539 0.126803 0.0634013 0.997988i \(-0.479805\pi\)
0.0634013 + 0.997988i \(0.479805\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) −13.8564 −0.544752 −0.272376 0.962191i \(-0.587809\pi\)
−0.272376 + 0.962191i \(0.587809\pi\)
\(648\) 0 0
\(649\) −20.2872 −0.796342
\(650\) 0 0
\(651\) 1.46410 0.0573827
\(652\) 0 0
\(653\) −20.5359 −0.803632 −0.401816 0.915720i \(-0.631621\pi\)
−0.401816 + 0.915720i \(0.631621\pi\)
\(654\) 0 0
\(655\) 10.9282 0.427000
\(656\) 0 0
\(657\) 7.46410 0.291202
\(658\) 0 0
\(659\) 30.5359 1.18951 0.594755 0.803907i \(-0.297249\pi\)
0.594755 + 0.803907i \(0.297249\pi\)
\(660\) 0 0
\(661\) −20.9282 −0.814013 −0.407006 0.913425i \(-0.633427\pi\)
−0.407006 + 0.913425i \(0.633427\pi\)
\(662\) 0 0
\(663\) 6.92820 0.269069
\(664\) 0 0
\(665\) −1.46410 −0.0567754
\(666\) 0 0
\(667\) 13.8564 0.536522
\(668\) 0 0
\(669\) 13.0718 0.505385
\(670\) 0 0
\(671\) 13.0718 0.504631
\(672\) 0 0
\(673\) 15.0718 0.580975 0.290488 0.956879i \(-0.406183\pi\)
0.290488 + 0.956879i \(0.406183\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −30.7846 −1.18315 −0.591574 0.806250i \(-0.701493\pi\)
−0.591574 + 0.806250i \(0.701493\pi\)
\(678\) 0 0
\(679\) 12.5359 0.481084
\(680\) 0 0
\(681\) 6.92820 0.265489
\(682\) 0 0
\(683\) −21.8564 −0.836312 −0.418156 0.908375i \(-0.637323\pi\)
−0.418156 + 0.908375i \(0.637323\pi\)
\(684\) 0 0
\(685\) 7.46410 0.285189
\(686\) 0 0
\(687\) 18.7846 0.716678
\(688\) 0 0
\(689\) 25.8564 0.985051
\(690\) 0 0
\(691\) 42.2487 1.60722 0.803608 0.595158i \(-0.202911\pi\)
0.803608 + 0.595158i \(0.202911\pi\)
\(692\) 0 0
\(693\) 1.46410 0.0556166
\(694\) 0 0
\(695\) −1.46410 −0.0555365
\(696\) 0 0
\(697\) 17.8564 0.676360
\(698\) 0 0
\(699\) 12.5359 0.474151
\(700\) 0 0
\(701\) −7.85641 −0.296732 −0.148366 0.988932i \(-0.547401\pi\)
−0.148366 + 0.988932i \(0.547401\pi\)
\(702\) 0 0
\(703\) 2.92820 0.110439
\(704\) 0 0
\(705\) 2.92820 0.110283
\(706\) 0 0
\(707\) −11.8564 −0.445906
\(708\) 0 0
\(709\) 17.7128 0.665219 0.332609 0.943065i \(-0.392071\pi\)
0.332609 + 0.943065i \(0.392071\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 10.1436 0.379881
\(714\) 0 0
\(715\) 5.07180 0.189674
\(716\) 0 0
\(717\) 15.3205 0.572155
\(718\) 0 0
\(719\) 24.7846 0.924310 0.462155 0.886799i \(-0.347076\pi\)
0.462155 + 0.886799i \(0.347076\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) −26.7846 −0.996130
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 2.92820 0.108601 0.0543005 0.998525i \(-0.482707\pi\)
0.0543005 + 0.998525i \(0.482707\pi\)
\(728\) 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 \(-0.438527\pi\)
0.191924 + 0.981410i \(0.438527\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) −10.1436 −0.373644
\(738\) 0 0
\(739\) 28.7846 1.05886 0.529429 0.848354i \(-0.322406\pi\)
0.529429 + 0.848354i \(0.322406\pi\)
\(740\) 0 0
\(741\) 5.07180 0.186317
\(742\) 0 0
\(743\) 1.85641 0.0681049 0.0340525 0.999420i \(-0.489159\pi\)
0.0340525 + 0.999420i \(0.489159\pi\)
\(744\) 0 0
\(745\) 7.07180 0.259091
\(746\) 0 0
\(747\) −1.07180 −0.0392150
\(748\) 0 0
\(749\) −5.07180 −0.185319
\(750\) 0 0
\(751\) 37.5692 1.37092 0.685460 0.728110i \(-0.259601\pi\)
0.685460 + 0.728110i \(0.259601\pi\)
\(752\) 0 0
\(753\) 29.8564 1.08803
\(754\) 0 0
\(755\) 1.07180 0.0390067
\(756\) 0 0
\(757\) −32.9282 −1.19680 −0.598398 0.801199i \(-0.704196\pi\)
−0.598398 + 0.801199i \(0.704196\pi\)
\(758\) 0 0
\(759\) 10.1436 0.368189
\(760\) 0 0
\(761\) −44.9282 −1.62865 −0.814323 0.580412i \(-0.802892\pi\)
−0.814323 + 0.580412i \(0.802892\pi\)
\(762\) 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
\(769\) 40.6410 1.46555 0.732776 0.680470i \(-0.238224\pi\)
0.732776 + 0.680470i \(0.238224\pi\)
\(770\) 0 0
\(771\) 16.9282 0.609654
\(772\) 0 0
\(773\) 4.92820 0.177255 0.0886276 0.996065i \(-0.471752\pi\)
0.0886276 + 0.996065i \(0.471752\pi\)
\(774\) 0 0
\(775\) −1.46410 −0.0525921
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) 0 0
\(779\) 13.0718 0.468346
\(780\) 0 0
\(781\) 6.43078 0.230111
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) −7.46410 −0.266405
\(786\) 0 0
\(787\) 28.7846 1.02606 0.513030 0.858371i \(-0.328523\pi\)
0.513030 + 0.858371i \(0.328523\pi\)
\(788\) 0 0
\(789\) 9.85641 0.350897
\(790\) 0 0
\(791\) −10.3923 −0.369508
\(792\) 0 0
\(793\) −30.9282 −1.09829
\(794\) 0 0
\(795\) −7.46410 −0.264724
\(796\) 0 0
\(797\) 36.9282 1.30806 0.654032 0.756467i \(-0.273076\pi\)
0.654032 + 0.756467i \(0.273076\pi\)
\(798\) 0 0
\(799\) 5.85641 0.207185
\(800\) 0 0
\(801\) 0.928203 0.0327964
\(802\) 0 0
\(803\) 10.9282 0.385648
\(804\) 0 0
\(805\) 6.92820 0.244187
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) 2.78461 0.0979017 0.0489508 0.998801i \(-0.484412\pi\)
0.0489508 + 0.998801i \(0.484412\pi\)
\(810\) 0 0
\(811\) 21.1769 0.743622 0.371811 0.928308i \(-0.378737\pi\)
0.371811 + 0.928308i \(0.378737\pi\)
\(812\) 0 0
\(813\) 6.53590 0.229224
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) −5.85641 −0.204890
\(818\) 0 0
\(819\) −3.46410 −0.121046
\(820\) 0 0
\(821\) −29.7128 −1.03698 −0.518492 0.855082i \(-0.673506\pi\)
−0.518492 + 0.855082i \(0.673506\pi\)
\(822\) 0 0
\(823\) −32.7846 −1.14280 −0.571400 0.820672i \(-0.693599\pi\)
−0.571400 + 0.820672i \(0.693599\pi\)
\(824\) 0 0
\(825\) −1.46410 −0.0509735
\(826\) 0 0
\(827\) −2.92820 −0.101824 −0.0509118 0.998703i \(-0.516213\pi\)
−0.0509118 + 0.998703i \(0.516213\pi\)
\(828\) 0 0
\(829\) −23.8564 −0.828567 −0.414284 0.910148i \(-0.635968\pi\)
−0.414284 + 0.910148i \(0.635968\pi\)
\(830\) 0 0
\(831\) −15.8564 −0.550053
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 1.46410 0.0506068
\(838\) 0 0
\(839\) −8.78461 −0.303278 −0.151639 0.988436i \(-0.548455\pi\)
−0.151639 + 0.988436i \(0.548455\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −10.7846 −0.371442
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −8.85641 −0.304310
\(848\) 0 0
\(849\) −9.85641 −0.338271
\(850\) 0 0
\(851\) −13.8564 −0.474991
\(852\) 0 0
\(853\) 2.39230 0.0819110 0.0409555 0.999161i \(-0.486960\pi\)
0.0409555 + 0.999161i \(0.486960\pi\)
\(854\) 0 0
\(855\) −1.46410 −0.0500712
\(856\) 0 0
\(857\) −24.9282 −0.851531 −0.425766 0.904833i \(-0.639995\pi\)
−0.425766 + 0.904833i \(0.639995\pi\)
\(858\) 0 0
\(859\) 44.3923 1.51465 0.757323 0.653041i \(-0.226507\pi\)
0.757323 + 0.653041i \(0.226507\pi\)
\(860\) 0 0
\(861\) −8.92820 −0.304272
\(862\) 0 0
\(863\) 46.9282 1.59745 0.798727 0.601693i \(-0.205507\pi\)
0.798727 + 0.601693i \(0.205507\pi\)
\(864\) 0 0
\(865\) 11.0718 0.376452
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −5.85641 −0.198665
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 0 0
\(873\) 12.5359 0.424276
\(874\) 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
\(879\) 11.0718 0.373442
\(880\) 0 0
\(881\) 5.21539 0.175711 0.0878555 0.996133i \(-0.471999\pi\)
0.0878555 + 0.996133i \(0.471999\pi\)
\(882\) 0 0
\(883\) 42.6410 1.43498 0.717492 0.696567i \(-0.245290\pi\)
0.717492 + 0.696567i \(0.245290\pi\)
\(884\) 0 0
\(885\) −13.8564 −0.465778
\(886\) 0 0
\(887\) −10.9282 −0.366933 −0.183467 0.983026i \(-0.558732\pi\)
−0.183467 + 0.983026i \(0.558732\pi\)
\(888\) 0 0
\(889\) −18.9282 −0.634832
\(890\) 0 0
\(891\) 1.46410 0.0490492
\(892\) 0 0
\(893\) 4.28719 0.143465
\(894\) 0 0
\(895\) −9.46410 −0.316350
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 0 0
\(899\) 2.92820 0.0976610
\(900\) 0 0
\(901\) −14.9282 −0.497331
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) −28.7846 −0.955777 −0.477889 0.878420i \(-0.658598\pi\)
−0.477889 + 0.878420i \(0.658598\pi\)
\(908\) 0 0
\(909\) −11.8564 −0.393252
\(910\) 0 0
\(911\) −26.2487 −0.869659 −0.434829 0.900513i \(-0.643191\pi\)
−0.434829 + 0.900513i \(0.643191\pi\)
\(912\) 0 0
\(913\) −1.56922 −0.0519336
\(914\) 0 0
\(915\) 8.92820 0.295157
\(916\) 0 0
\(917\) −10.9282 −0.360881
\(918\) 0 0
\(919\) 28.0000 0.923635 0.461817 0.886975i \(-0.347198\pi\)
0.461817 + 0.886975i \(0.347198\pi\)
\(920\) 0 0
\(921\) −9.07180 −0.298926
\(922\) 0 0
\(923\) −15.2154 −0.500821
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −16.6410 −0.545974 −0.272987 0.962018i \(-0.588012\pi\)
−0.272987 + 0.962018i \(0.588012\pi\)
\(930\) 0 0
\(931\) 1.46410 0.0479840
\(932\) 0 0
\(933\) −8.00000 −0.261908
\(934\) 0 0
\(935\) −2.92820 −0.0957625
\(936\) 0 0
\(937\) 35.1769 1.14918 0.574590 0.818442i \(-0.305162\pi\)
0.574590 + 0.818442i \(0.305162\pi\)
\(938\) 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
\(943\) −61.8564 −2.01432
\(944\) 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
\(949\) −25.8564 −0.839334
\(950\) 0 0
\(951\) 21.3205 0.691365
\(952\) 0 0
\(953\) −14.6795 −0.475515 −0.237758 0.971324i \(-0.576412\pi\)
−0.237758 + 0.971324i \(0.576412\pi\)
\(954\) 0 0
\(955\) 12.3923 0.401006
\(956\) 0 0
\(957\) 2.92820 0.0946554
\(958\) 0 0
\(959\) −7.46410 −0.241028
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) 0 0
\(963\) −5.07180 −0.163436
\(964\) 0 0
\(965\) −15.8564 −0.510436
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 0 0
\(969\) −2.92820 −0.0940674
\(970\) 0 0
\(971\) 37.8564 1.21487 0.607435 0.794369i \(-0.292199\pi\)
0.607435 + 0.794369i \(0.292199\pi\)
\(972\) 0 0
\(973\) 1.46410 0.0469369
\(974\) 0 0
\(975\) 3.46410 0.110940
\(976\) 0 0
\(977\) −34.3923 −1.10031 −0.550154 0.835063i \(-0.685431\pi\)
−0.550154 + 0.835063i \(0.685431\pi\)
\(978\) 0 0
\(979\) 1.35898 0.0434333
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) 0.784610 0.0250252 0.0125126 0.999922i \(-0.496017\pi\)
0.0125126 + 0.999922i \(0.496017\pi\)
\(984\) 0 0
\(985\) 12.5359 0.399427
\(986\) 0 0
\(987\) −2.92820 −0.0932057
\(988\) 0 0
\(989\) 27.7128 0.881216
\(990\) 0 0
\(991\) −59.4256 −1.88772 −0.943859 0.330350i \(-0.892833\pi\)
−0.943859 + 0.330350i \(0.892833\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) −20.3923 −0.646480
\(996\) 0 0
\(997\) 48.2487 1.52805 0.764026 0.645185i \(-0.223220\pi\)
0.764026 + 0.645185i \(0.223220\pi\)
\(998\) 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.1.2 2
4.3 odd 2 3360.2.a.bf.1.1 yes 2
8.3 odd 2 6720.2.a.cq.1.2 2
8.5 even 2 6720.2.a.cz.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.bb.1.2 2 1.1 even 1 trivial
3360.2.a.bf.1.1 yes 2 4.3 odd 2
6720.2.a.cq.1.2 2 8.3 odd 2
6720.2.a.cz.1.1 2 8.5 even 2