Properties

Label 5376.2.c.r.2689.2
Level $5376$
Weight $2$
Character 5376.2689
Analytic conductor $42.928$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5376.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.9275761266\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5376.2689
Dual form 5376.2.c.r.2689.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} -2.00000 q^{15} -6.00000 q^{17} +4.00000i q^{19} +1.00000i q^{21} +1.00000 q^{25} -1.00000i q^{27} -2.00000i q^{29} +4.00000 q^{33} +2.00000i q^{35} -6.00000i q^{37} +2.00000 q^{39} -2.00000 q^{41} +4.00000i q^{43} -2.00000i q^{45} +1.00000 q^{49} -6.00000i q^{51} -6.00000i q^{53} +8.00000 q^{55} -4.00000 q^{57} -12.0000i q^{59} -2.00000i q^{61} -1.00000 q^{63} +4.00000 q^{65} +4.00000i q^{67} +6.00000 q^{73} +1.00000i q^{75} -4.00000i q^{77} -16.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -12.0000i q^{85} +2.00000 q^{87} +14.0000 q^{89} -2.00000i q^{91} -8.00000 q^{95} +18.0000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{7} - 2q^{9} - 4q^{15} - 12q^{17} + 2q^{25} + 8q^{33} + 4q^{39} - 4q^{41} + 2q^{49} + 16q^{55} - 8q^{57} - 2q^{63} + 8q^{65} + 12q^{73} - 32q^{79} + 2q^{81} + 4q^{87} + 28q^{89} - 16q^{95} + 36q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times\).

\(n\) \(1793\) \(2815\) \(4609\) \(5125\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 2.00000i − 0.371391i −0.982607 0.185695i \(-0.940546\pi\)
0.982607 0.185695i \(-0.0594537\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) − 2.00000i − 0.298142i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 6.00000i − 0.840168i
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) − 12.0000i − 1.56227i −0.624364 0.781133i \(-0.714642\pi\)
0.624364 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) − 2.00000i − 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408676\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) − 4.00000i − 0.455842i
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) − 12.0000i − 1.30158i
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) − 2.00000i − 0.209657i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
\(100\) 0 0
\(101\) − 14.0000i − 1.39305i −0.717532 0.696526i \(-0.754728\pi\)
0.717532 0.696526i \(-0.245272\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 0 0
\(109\) − 18.0000i − 1.72409i −0.506834 0.862044i \(-0.669184\pi\)
0.506834 0.862044i \(-0.330816\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) − 2.00000i − 0.180334i
\(124\) 0 0
\(125\) 12.0000i 1.07331i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 4.00000i 0.349482i 0.984614 + 0.174741i \(0.0559088\pi\)
−0.984614 + 0.174741i \(0.944091\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 0 0
\(135\) 2.00000 0.172133
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) − 12.0000i − 1.01783i −0.860818 0.508913i \(-0.830047\pi\)
0.860818 0.508913i \(-0.169953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) − 6.00000i − 0.491539i −0.969328 0.245770i \(-0.920959\pi\)
0.969328 0.245770i \(-0.0790407\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 0 0
\(165\) 8.00000i 0.622799i
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) − 10.0000i − 0.760286i −0.924928 0.380143i \(-0.875875\pi\)
0.924928 0.380143i \(-0.124125\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) − 4.00000i − 0.298974i −0.988764 0.149487i \(-0.952238\pi\)
0.988764 0.149487i \(-0.0477622\pi\)
\(180\) 0 0
\(181\) 26.0000i 1.93256i 0.257485 + 0.966282i \(0.417106\pi\)
−0.257485 + 0.966282i \(0.582894\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 24.0000i 1.75505i
\(188\) 0 0
\(189\) − 1.00000i − 0.0727393i
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 4.00000i 0.286446i
\(196\) 0 0
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) − 2.00000i − 0.140372i
\(204\) 0 0
\(205\) − 4.00000i − 0.279372i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 4.00000i 0.275371i 0.990476 + 0.137686i \(0.0439664\pi\)
−0.990476 + 0.137686i \(0.956034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 6.00000i 0.405442i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 16.0000i − 1.03931i
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 2.00000i 0.127775i
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 20.0000i 1.26239i 0.775625 + 0.631194i \(0.217435\pi\)
−0.775625 + 0.631194i \(0.782565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 12.0000 0.751469
\(256\) 0 0
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 0 0
\(259\) − 6.00000i − 0.372822i
\(260\) 0 0
\(261\) 2.00000i 0.123797i
\(262\) 0 0
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 14.0000i 0.856786i
\(268\) 0 0
\(269\) 6.00000i 0.365826i 0.983129 + 0.182913i \(0.0585527\pi\)
−0.983129 + 0.182913i \(0.941447\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) − 4.00000i − 0.241209i
\(276\) 0 0
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) 0 0
\(285\) − 8.00000i − 0.473879i
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 18.0000i 1.05518i
\(292\) 0 0
\(293\) − 14.0000i − 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 4.00000i 0.230556i
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) − 8.00000i − 0.455104i
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 0 0
\(315\) − 2.00000i − 0.112687i
\(316\) 0 0
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) 0 0
\(325\) − 2.00000i − 0.110940i
\(326\) 0 0
\(327\) 18.0000 0.995402
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000i 0.219860i 0.993939 + 0.109930i \(0.0350627\pi\)
−0.993939 + 0.109930i \(0.964937\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) − 14.0000i − 0.760376i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) 0 0
\(349\) − 2.00000i − 0.107058i −0.998566 0.0535288i \(-0.982953\pi\)
0.998566 0.0535288i \(-0.0170469\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 6.00000i − 0.317554i
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) 12.0000i 0.628109i
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) − 6.00000i − 0.311504i
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) − 12.0000i − 0.616399i −0.951322 0.308199i \(-0.900274\pi\)
0.951322 0.308199i \(-0.0997264\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) 0 0
\(387\) − 4.00000i − 0.203331i
\(388\) 0 0
\(389\) − 6.00000i − 0.304212i −0.988364 0.152106i \(-0.951394\pi\)
0.988364 0.152106i \(-0.0486055\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) − 32.0000i − 1.61009i
\(396\) 0 0
\(397\) − 18.0000i − 0.903394i −0.892171 0.451697i \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.00000i 0.0993808i
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 6.00000i 0.295958i
\(412\) 0 0
\(413\) − 12.0000i − 0.590481i
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) − 12.0000i − 0.586238i −0.956076 0.293119i \(-0.905307\pi\)
0.956076 0.293119i \(-0.0946933\pi\)
\(420\) 0 0
\(421\) − 38.0000i − 1.85201i −0.377515 0.926003i \(-0.623221\pi\)
0.377515 0.926003i \(-0.376779\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) − 2.00000i − 0.0967868i
\(428\) 0 0
\(429\) − 8.00000i − 0.386244i
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 4.00000i 0.191785i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) − 36.0000i − 1.71041i −0.518289 0.855206i \(-0.673431\pi\)
0.518289 0.855206i \(-0.326569\pi\)
\(444\) 0 0
\(445\) 28.0000i 1.32733i
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 8.00000i 0.376705i
\(452\) 0 0
\(453\) − 8.00000i − 0.375873i
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 6.00000i 0.280056i
\(460\) 0 0
\(461\) − 10.0000i − 0.465746i −0.972507 0.232873i \(-0.925187\pi\)
0.972507 0.232873i \(-0.0748127\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.0000i 1.66588i 0.553362 + 0.832941i \(0.313345\pi\)
−0.553362 + 0.832941i \(0.686655\pi\)
\(468\) 0 0
\(469\) 4.00000i 0.184703i
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) 4.00000i 0.183533i
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 36.0000i 1.63468i
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) − 20.0000i − 0.902587i −0.892375 0.451294i \(-0.850963\pi\)
0.892375 0.451294i \(-0.149037\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.00000i 0.179065i 0.995984 + 0.0895323i \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 0 0
\(501\) 8.00000i 0.357414i
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 28.0000 1.24598
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) − 10.0000i − 0.443242i −0.975133 0.221621i \(-0.928865\pi\)
0.975133 0.221621i \(-0.0711348\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) − 16.0000i − 0.705044i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) 1.00000i 0.0436436i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 12.0000i 0.520756i
\(532\) 0 0
\(533\) 4.00000i 0.173259i
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 0 0
\(537\) 4.00000 0.172613
\(538\) 0 0
\(539\) − 4.00000i − 0.172292i
\(540\) 0 0
\(541\) − 34.0000i − 1.46177i −0.682498 0.730887i \(-0.739107\pi\)
0.682498 0.730887i \(-0.260893\pi\)
\(542\) 0 0
\(543\) −26.0000 −1.11577
\(544\) 0 0
\(545\) 36.0000 1.54207
\(546\) 0 0
\(547\) 4.00000i 0.171028i 0.996337 + 0.0855138i \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) 0 0
\(549\) 2.00000i 0.0853579i
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) 12.0000i 0.509372i
\(556\) 0 0
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 0 0
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 0 0
\(565\) − 28.0000i − 1.17797i
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) 4.00000i 0.167395i 0.996491 + 0.0836974i \(0.0266729\pi\)
−0.996491 + 0.0836974i \(0.973327\pi\)
\(572\) 0 0
\(573\) − 8.00000i − 0.334205i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 0 0
\(579\) 2.00000i 0.0831172i
\(580\) 0 0
\(581\) − 12.0000i − 0.497844i
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) − 12.0000i − 0.491952i
\(596\) 0 0
\(597\) − 24.0000i − 0.982255i
\(598\) 0 0
\(599\) −48.0000 −1.96123 −0.980613 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) 6.00000 0.244745 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) 0 0
\(605\) − 10.0000i − 0.406558i
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) 0 0
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 20.0000i 0.803868i 0.915669 + 0.401934i \(0.131662\pi\)
−0.915669 + 0.401934i \(0.868338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 16.0000i 0.638978i
\(628\) 0 0
\(629\) 36.0000i 1.43541i
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 2.00000i − 0.0792429i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 20.0000i 0.788723i 0.918955 + 0.394362i \(0.129034\pi\)
−0.918955 + 0.394362i \(0.870966\pi\)
\(644\) 0 0
\(645\) − 8.00000i − 0.315000i
\(646\) 0 0
\(647\) 40.0000 1.57256 0.786281 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 12.0000i 0.467454i 0.972302 + 0.233727i \(0.0750921\pi\)
−0.972302 + 0.233727i \(0.924908\pi\)
\(660\) 0 0
\(661\) − 22.0000i − 0.855701i −0.903850 0.427850i \(-0.859271\pi\)
0.903850 0.427850i \(-0.140729\pi\)
\(662\) 0 0
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 16.0000i 0.618596i
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 0 0
\(675\) − 1.00000i − 0.0384900i
\(676\) 0 0
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 0 0
\(679\) 18.0000 0.690777
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 12.0000i 0.458496i
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 20.0000i 0.760836i 0.924815 + 0.380418i \(0.124220\pi\)
−0.924815 + 0.380418i \(0.875780\pi\)
\(692\) 0 0
\(693\) 4.00000i 0.151947i
\(694\) 0 0
\(695\) 24.0000 0.910372
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) 6.00000i 0.226941i
\(700\) 0 0
\(701\) 30.0000i 1.13308i 0.824033 + 0.566542i \(0.191719\pi\)
−0.824033 + 0.566542i \(0.808281\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 14.0000i − 0.526524i
\(708\) 0 0
\(709\) − 6.00000i − 0.225335i −0.993633 0.112667i \(-0.964061\pi\)
0.993633 0.112667i \(-0.0359394\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) − 16.0000i − 0.598366i
\(716\) 0 0
\(717\) 24.0000i 0.896296i
\(718\) 0 0
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 2.00000i 0.0743808i
\(724\) 0 0
\(725\) − 2.00000i − 0.0742781i
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 24.0000i − 0.887672i
\(732\) 0 0
\(733\) − 18.0000i − 0.664845i −0.943131 0.332423i \(-0.892134\pi\)
0.943131 0.332423i \(-0.107866\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 36.0000i 1.32428i 0.749380 + 0.662141i \(0.230352\pi\)
−0.749380 + 0.662141i \(0.769648\pi\)
\(740\) 0 0
\(741\) 8.00000i 0.293887i
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) − 4.00000i − 0.146157i
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) −20.0000 −0.728841
\(754\) 0 0
\(755\) − 16.0000i − 0.582300i
\(756\) 0 0
\(757\) 10.0000i 0.363456i 0.983349 + 0.181728i \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) − 18.0000i − 0.651644i
\(764\) 0 0
\(765\) 12.0000i 0.433861i
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 26.0000i 0.936367i
\(772\) 0 0
\(773\) − 14.0000i − 0.503545i −0.967786 0.251773i \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) − 8.00000i − 0.286630i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) − 44.0000i − 1.56843i −0.620489 0.784215i \(-0.713066\pi\)
0.620489 0.784215i \(-0.286934\pi\)
\(788\) 0 0
\(789\) − 16.0000i − 0.569615i
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 0 0
\(795\) 12.0000i 0.425596i
\(796\) 0 0
\(797\) − 26.0000i − 0.920967i −0.887668 0.460484i \(-0.847676\pi\)
0.887668 0.460484i \(-0.152324\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) − 24.0000i − 0.846942i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) − 44.0000i − 1.54505i −0.634985 0.772524i \(-0.718994\pi\)
0.634985 0.772524i \(-0.281006\pi\)
\(812\) 0 0
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 2.00000i 0.0698857i
\(820\) 0 0
\(821\) − 38.0000i − 1.32621i −0.748527 0.663105i \(-0.769238\pi\)
0.748527 0.663105i \(-0.230762\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) 14.0000i 0.486240i 0.969996 + 0.243120i \(0.0781709\pi\)
−0.969996 + 0.243120i \(0.921829\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 16.0000i 0.553703i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 22.0000i 0.757720i
\(844\) 0 0
\(845\) 18.0000i 0.619219i
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) 14.0000 0.478231 0.239115 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) 0 0
\(859\) − 44.0000i − 1.50126i −0.660722 0.750630i \(-0.729750\pi\)
0.660722 0.750630i \(-0.270250\pi\)
\(860\) 0 0
\(861\) − 2.00000i − 0.0681598i
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 20.0000 0.680020
\(866\) 0 0
\(867\) 19.0000i 0.645274i
\(868\) 0 0
\(869\) 64.0000i 2.17105i
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) −18.0000 −0.609208
\(874\) 0 0
\(875\) 12.0000i 0.405674i
\(876\) 0 0
\(877\) 46.0000i 1.55331i 0.629926 + 0.776655i \(0.283085\pi\)
−0.629926 + 0.776655i \(0.716915\pi\)
\(878\) 0 0
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) − 28.0000i − 0.942275i −0.882060 0.471138i \(-0.843844\pi\)
0.882060 0.471138i \(-0.156156\pi\)
\(884\) 0 0
\(885\) 24.0000i 0.806751i
\(886\) 0 0
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 4.00000i − 0.134005i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 36.0000i 1.19933i
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) 0 0
\(905\) −52.0000 −1.72854
\(906\) 0 0
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) 0 0
\(909\) 14.0000i 0.464351i
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) 0 0
\(915\) 4.00000i 0.132236i
\(916\) 0 0
\(917\) 4.00000i 0.132092i
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 6.00000i − 0.197279i
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) 4.00000i 0.131095i
\(932\) 0 0
\(933\) 24.0000i 0.785725i
\(934\) 0 0
\(935\) −48.0000 −1.56977
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) − 26.0000i − 0.848478i
\(940\) 0 0
\(941\) 38.0000i 1.23876i 0.785090 + 0.619382i \(0.212617\pi\)
−0.785090 + 0.619382i \(0.787383\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) 44.0000i 1.42981i 0.699223 + 0.714904i \(0.253530\pi\)
−0.699223 + 0.714904i \(0.746470\pi\)
\(948\) 0 0
\(949\) − 12.0000i − 0.389536i
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 0 0
\(955\) − 16.0000i − 0.517748i
\(956\) 0 0
\(957\) − 8.00000i − 0.258603i
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 4.00000i 0.128898i
\(964\) 0 0
\(965\) 4.00000i 0.128765i
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) − 12.0000i − 0.385098i −0.981287 0.192549i \(-0.938325\pi\)
0.981287 0.192549i \(-0.0616755\pi\)
\(972\) 0 0
\(973\) − 12.0000i − 0.384702i
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) 0 0
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) − 56.0000i − 1.78977i
\(980\) 0 0
\(981\) 18.0000i 0.574696i
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 44.0000 1.40196
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) − 48.0000i − 1.52170i
\(996\) 0 0
\(997\) 26.0000i 0.823428i 0.911313 + 0.411714i \(0.135070\pi\)
−0.911313 + 0.411714i \(0.864930\pi\)
\(998\) 0 0
\(999\) −6.00000 −0.189832
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 5376.2.c.r.2689.2 2
4.3 odd 2 5376.2.c.l.2689.1 2
8.3 odd 2 5376.2.c.l.2689.2 2
8.5 even 2 inner 5376.2.c.r.2689.1 2
16.3 odd 4 1344.2.a.s.1.1 1
16.5 even 4 21.2.a.a.1.1 1
16.11 odd 4 336.2.a.a.1.1 1
16.13 even 4 1344.2.a.g.1.1 1
48.5 odd 4 63.2.a.a.1.1 1
48.11 even 4 1008.2.a.l.1.1 1
48.29 odd 4 4032.2.a.h.1.1 1
48.35 even 4 4032.2.a.k.1.1 1
80.37 odd 4 525.2.d.a.274.1 2
80.53 odd 4 525.2.d.a.274.2 2
80.59 odd 4 8400.2.a.bn.1.1 1
80.69 even 4 525.2.a.d.1.1 1
112.5 odd 12 147.2.e.c.67.1 2
112.11 odd 12 2352.2.q.x.961.1 2
112.13 odd 4 9408.2.a.bv.1.1 1
112.27 even 4 2352.2.a.v.1.1 1
112.37 even 12 147.2.e.b.67.1 2
112.53 even 12 147.2.e.b.79.1 2
112.59 even 12 2352.2.q.e.961.1 2
112.69 odd 4 147.2.a.a.1.1 1
112.75 even 12 2352.2.q.e.1537.1 2
112.83 even 4 9408.2.a.m.1.1 1
112.101 odd 12 147.2.e.c.79.1 2
112.107 odd 12 2352.2.q.x.1537.1 2
144.5 odd 12 567.2.f.b.379.1 2
144.85 even 12 567.2.f.g.379.1 2
144.101 odd 12 567.2.f.b.190.1 2
144.133 even 12 567.2.f.g.190.1 2
176.21 odd 4 2541.2.a.j.1.1 1
208.181 even 4 3549.2.a.c.1.1 1
240.53 even 4 1575.2.d.a.1324.1 2
240.149 odd 4 1575.2.a.c.1.1 1
240.197 even 4 1575.2.d.a.1324.2 2
272.101 even 4 6069.2.a.b.1.1 1
304.37 odd 4 7581.2.a.d.1.1 1
336.5 even 12 441.2.e.b.361.1 2
336.53 odd 12 441.2.e.a.226.1 2
336.101 even 12 441.2.e.b.226.1 2
336.149 odd 12 441.2.e.a.361.1 2
336.251 odd 4 7056.2.a.p.1.1 1
336.293 even 4 441.2.a.f.1.1 1
528.197 even 4 7623.2.a.g.1.1 1
560.69 odd 4 3675.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.a.a.1.1 1 16.5 even 4
63.2.a.a.1.1 1 48.5 odd 4
147.2.a.a.1.1 1 112.69 odd 4
147.2.e.b.67.1 2 112.37 even 12
147.2.e.b.79.1 2 112.53 even 12
147.2.e.c.67.1 2 112.5 odd 12
147.2.e.c.79.1 2 112.101 odd 12
336.2.a.a.1.1 1 16.11 odd 4
441.2.a.f.1.1 1 336.293 even 4
441.2.e.a.226.1 2 336.53 odd 12
441.2.e.a.361.1 2 336.149 odd 12
441.2.e.b.226.1 2 336.101 even 12
441.2.e.b.361.1 2 336.5 even 12
525.2.a.d.1.1 1 80.69 even 4
525.2.d.a.274.1 2 80.37 odd 4
525.2.d.a.274.2 2 80.53 odd 4
567.2.f.b.190.1 2 144.101 odd 12
567.2.f.b.379.1 2 144.5 odd 12
567.2.f.g.190.1 2 144.133 even 12
567.2.f.g.379.1 2 144.85 even 12
1008.2.a.l.1.1 1 48.11 even 4
1344.2.a.g.1.1 1 16.13 even 4
1344.2.a.s.1.1 1 16.3 odd 4
1575.2.a.c.1.1 1 240.149 odd 4
1575.2.d.a.1324.1 2 240.53 even 4
1575.2.d.a.1324.2 2 240.197 even 4
2352.2.a.v.1.1 1 112.27 even 4
2352.2.q.e.961.1 2 112.59 even 12
2352.2.q.e.1537.1 2 112.75 even 12
2352.2.q.x.961.1 2 112.11 odd 12
2352.2.q.x.1537.1 2 112.107 odd 12
2541.2.a.j.1.1 1 176.21 odd 4
3549.2.a.c.1.1 1 208.181 even 4
3675.2.a.n.1.1 1 560.69 odd 4
4032.2.a.h.1.1 1 48.29 odd 4
4032.2.a.k.1.1 1 48.35 even 4
5376.2.c.l.2689.1 2 4.3 odd 2
5376.2.c.l.2689.2 2 8.3 odd 2
5376.2.c.r.2689.1 2 8.5 even 2 inner
5376.2.c.r.2689.2 2 1.1 even 1 trivial
6069.2.a.b.1.1 1 272.101 even 4
7056.2.a.p.1.1 1 336.251 odd 4
7581.2.a.d.1.1 1 304.37 odd 4
7623.2.a.g.1.1 1 528.197 even 4
8400.2.a.bn.1.1 1 80.59 odd 4
9408.2.a.m.1.1 1 112.83 even 4
9408.2.a.bv.1.1 1 112.13 odd 4