Properties

Label 5376.2.c.y.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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5376,2,Mod(2689,5376)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5376, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5376.2689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(42.9275761266\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
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.y.2689.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} -2.00000i q^{11} +2.00000i q^{13} +4.00000 q^{17} +4.00000i q^{19} +1.00000i q^{21} +6.00000 q^{23} +5.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +2.00000 q^{33} -6.00000i q^{37} -2.00000 q^{39} -8.00000 q^{41} -8.00000i q^{43} -4.00000 q^{47} +1.00000 q^{49} +4.00000i q^{51} -6.00000i q^{53} -4.00000 q^{57} +14.0000i q^{61} -1.00000 q^{63} -4.00000i q^{67} +6.00000i q^{69} +2.00000 q^{71} +2.00000 q^{73} +5.00000i q^{75} -2.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -2.00000 q^{87} +2.00000i q^{91} +6.00000 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} - 2 q^{9} + 8 q^{17} + 12 q^{23} + 10 q^{25} + 4 q^{33} - 4 q^{39} - 16 q^{41} - 8 q^{47} + 2 q^{49} - 8 q^{57} - 2 q^{63} + 4 q^{71} + 4 q^{73} + 8 q^{79} + 2 q^{81} - 4 q^{87} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 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
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 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
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000i 0.560112i
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 14.0000i 1.79252i 0.443533 + 0.896258i \(0.353725\pi\)
−0.443533 + 0.896258i \(0.646275\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 6.00000i 0.722315i
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 5.00000i 0.577350i
\(76\) 0 0
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 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
\(85\) 0 0
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 2.00000i 0.209657i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 2.00000i 0.201008i
\(100\) 0 0
\(101\) 12.0000i 1.19404i 0.802225 + 0.597022i \(0.203650\pi\)
−0.802225 + 0.597022i \(0.796350\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000i 0.580042i 0.957020 + 0.290021i \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) 0 0
\(109\) 18.0000i 1.72409i 0.506834 + 0.862044i \(0.330816\pi\)
−0.506834 + 0.862044i \(0.669184\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.00000i − 0.184900i
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) − 8.00000i − 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) − 20.0000i − 1.74741i −0.486458 0.873704i \(-0.661711\pi\)
0.486458 0.873704i \(-0.338289\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 12.0000i 1.01783i 0.860818 + 0.508913i \(0.169953\pi\)
−0.860818 + 0.508913i \(0.830047\pi\)
\(140\) 0 0
\(141\) − 4.00000i − 0.336861i
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) 10.0000i 0.819232i 0.912258 + 0.409616i \(0.134337\pi\)
−0.912258 + 0.409616i \(0.865663\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 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
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) 12.0000i 0.912343i 0.889892 + 0.456172i \(0.150780\pi\)
−0.889892 + 0.456172i \(0.849220\pi\)
\(174\) 0 0
\(175\) 5.00000 0.377964
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 10.0000i − 0.747435i −0.927543 0.373718i \(-0.878083\pi\)
0.927543 0.373718i \(-0.121917\pi\)
\(180\) 0 0
\(181\) 14.0000i 1.04061i 0.853980 + 0.520306i \(0.174182\pi\)
−0.853980 + 0.520306i \(0.825818\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) − 1.00000i − 0.0727393i
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 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
\(205\) 0 0
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 24.0000i 1.65223i 0.563503 + 0.826114i \(0.309453\pi\)
−0.563503 + 0.826114i \(0.690547\pi\)
\(212\) 0 0
\(213\) 2.00000i 0.137038i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.00000i 0.135147i
\(220\) 0 0
\(221\) 8.00000i 0.538138i
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) 8.00000i 0.530979i 0.964114 + 0.265489i \(0.0855335\pi\)
−0.964114 + 0.265489i \(0.914466\pi\)
\(228\) 0 0
\(229\) − 2.00000i − 0.132164i −0.997814 0.0660819i \(-0.978950\pi\)
0.997814 0.0660819i \(-0.0210498\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) − 12.0000i − 0.754434i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 0 0
\(259\) − 6.00000i − 0.372822i
\(260\) 0 0
\(261\) − 2.00000i − 0.123797i
\(262\) 0 0
\(263\) 14.0000 0.863277 0.431638 0.902047i \(-0.357936\pi\)
0.431638 + 0.902047i \(0.357936\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 16.0000i − 0.975537i −0.872973 0.487769i \(-0.837811\pi\)
0.872973 0.487769i \(-0.162189\pi\)
\(270\) 0 0
\(271\) −32.0000 −1.94386 −0.971931 0.235267i \(-0.924404\pi\)
−0.971931 + 0.235267i \(0.924404\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) − 10.0000i − 0.603023i
\(276\) 0 0
\(277\) 10.0000i 0.600842i 0.953807 + 0.300421i \(0.0971271\pi\)
−0.953807 + 0.300421i \(0.902873\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 6.00000i 0.351726i
\(292\) 0 0
\(293\) 28.0000i 1.63578i 0.575376 + 0.817889i \(0.304856\pi\)
−0.575376 + 0.817889i \(0.695144\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) 12.0000i 0.693978i
\(300\) 0 0
\(301\) − 8.00000i − 0.461112i
\(302\) 0 0
\(303\) −12.0000 −0.689382
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 28.0000i 1.59804i 0.601302 + 0.799022i \(0.294649\pi\)
−0.601302 + 0.799022i \(0.705351\pi\)
\(308\) 0 0
\(309\) 8.00000i 0.455104i
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.0000i 0.786318i 0.919470 + 0.393159i \(0.128618\pi\)
−0.919470 + 0.393159i \(0.871382\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) 0 0
\(325\) 10.0000i 0.554700i
\(326\) 0 0
\(327\) −18.0000 −0.995402
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 16.0000i 0.879440i 0.898135 + 0.439720i \(0.144922\pi\)
−0.898135 + 0.439720i \(0.855078\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 0 0
\(339\) 18.0000i 0.977626i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 30.0000i − 1.61048i −0.592946 0.805242i \(-0.702035\pi\)
0.592946 0.805242i \(-0.297965\pi\)
\(348\) 0 0
\(349\) − 10.0000i − 0.535288i −0.963518 0.267644i \(-0.913755\pi\)
0.963518 0.267644i \(-0.0862451\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −36.0000 −1.91609 −0.958043 0.286623i \(-0.907467\pi\)
−0.958043 + 0.286623i \(0.907467\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.00000i 0.211702i
\(358\) 0 0
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) − 6.00000i − 0.311504i
\(372\) 0 0
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) − 16.0000i − 0.821865i −0.911666 0.410932i \(-0.865203\pi\)
0.911666 0.410932i \(-0.134797\pi\)
\(380\) 0 0
\(381\) 12.0000i 0.614779i
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000i 0.406663i
\(388\) 0 0
\(389\) − 26.0000i − 1.31825i −0.752032 0.659126i \(-0.770926\pi\)
0.752032 0.659126i \(-0.229074\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 20.0000 1.00887
\(394\) 0 0
\(395\) 0 0
\(396\) 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
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 18.0000i 0.887875i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) − 8.00000i − 0.390826i −0.980721 0.195413i \(-0.937395\pi\)
0.980721 0.195413i \(-0.0626047\pi\)
\(420\) 0 0
\(421\) 10.0000i 0.487370i 0.969854 + 0.243685i \(0.0783563\pi\)
−0.969854 + 0.243685i \(0.921644\pi\)
\(422\) 0 0
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) 20.0000 0.970143
\(426\) 0 0
\(427\) 14.0000i 0.677507i
\(428\) 0 0
\(429\) 4.00000i 0.193122i
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 24.0000i 1.14808i
\(438\) 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
\(443\) − 22.0000i − 1.04525i −0.852562 0.522626i \(-0.824953\pi\)
0.852562 0.522626i \(-0.175047\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 16.0000i 0.753411i
\(452\) 0 0
\(453\) − 16.0000i − 0.751746i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) − 4.00000i − 0.186704i
\(460\) 0 0
\(461\) 36.0000i 1.67669i 0.545142 + 0.838344i \(0.316476\pi\)
−0.545142 + 0.838344i \(0.683524\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 32.0000i − 1.48078i −0.672176 0.740392i \(-0.734640\pi\)
0.672176 0.740392i \(-0.265360\pi\)
\(468\) 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
\(475\) 20.0000i 0.917663i
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) − 6.00000i − 0.270776i −0.990793 0.135388i \(-0.956772\pi\)
0.990793 0.135388i \(-0.0432281\pi\)
\(492\) 0 0
\(493\) 8.00000i 0.360302i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.00000 0.0897123
\(498\) 0 0
\(499\) − 40.0000i − 1.79065i −0.445418 0.895323i \(-0.646945\pi\)
0.445418 0.895323i \(-0.353055\pi\)
\(500\) 0 0
\(501\) 12.0000i 0.536120i
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) − 16.0000i − 0.709188i −0.935020 0.354594i \(-0.884619\pi\)
0.935020 0.354594i \(-0.115381\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.00000i 0.351840i
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) − 28.0000i − 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) 0 0
\(525\) 5.00000i 0.218218i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 16.0000i − 0.693037i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 10.0000 0.431532
\(538\) 0 0
\(539\) − 2.00000i − 0.0861461i
\(540\) 0 0
\(541\) 14.0000i 0.601907i 0.953639 + 0.300954i \(0.0973049\pi\)
−0.953639 + 0.300954i \(0.902695\pi\)
\(542\) 0 0
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) 0 0
\(549\) − 14.0000i − 0.597505i
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22.0000i 0.932170i 0.884740 + 0.466085i \(0.154336\pi\)
−0.884740 + 0.466085i \(0.845664\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 16.0000i 0.674320i 0.941447 + 0.337160i \(0.109466\pi\)
−0.941447 + 0.337160i \(0.890534\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i 0.908223 + 0.418487i \(0.137439\pi\)
−0.908223 + 0.418487i \(0.862561\pi\)
\(572\) 0 0
\(573\) − 10.0000i − 0.417756i
\(574\) 0 0
\(575\) 30.0000 1.25109
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 0 0
\(579\) − 26.0000i − 1.08052i
\(580\) 0 0
\(581\) − 12.0000i − 0.497844i
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.0000i 0.990586i 0.868726 + 0.495293i \(0.164939\pi\)
−0.868726 + 0.495293i \(0.835061\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 0 0
\(593\) −4.00000 −0.164260 −0.0821302 0.996622i \(-0.526172\pi\)
−0.0821302 + 0.996622i \(0.526172\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 24.0000i − 0.982255i
\(598\) 0 0
\(599\) −38.0000 −1.55264 −0.776319 0.630340i \(-0.782915\pi\)
−0.776319 + 0.630340i \(0.782915\pi\)
\(600\) 0 0
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) − 8.00000i − 0.323645i
\(612\) 0 0
\(613\) 22.0000i 0.888572i 0.895885 + 0.444286i \(0.146543\pi\)
−0.895885 + 0.444286i \(0.853457\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) 28.0000i 1.12542i 0.826656 + 0.562708i \(0.190240\pi\)
−0.826656 + 0.562708i \(0.809760\pi\)
\(620\) 0 0
\(621\) − 6.00000i − 0.240772i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 8.00000i 0.319489i
\(628\) 0 0
\(629\) − 24.0000i − 0.956943i
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) −24.0000 −0.953914
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 22.0000 0.868948 0.434474 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 0 0
\(643\) − 4.00000i − 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 46.0000i − 1.80012i −0.435767 0.900060i \(-0.643523\pi\)
0.435767 0.900060i \(-0.356477\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) − 6.00000i − 0.233727i −0.993148 0.116863i \(-0.962716\pi\)
0.993148 0.116863i \(-0.0372840\pi\)
\(660\) 0 0
\(661\) − 30.0000i − 1.16686i −0.812162 0.583432i \(-0.801709\pi\)
0.812162 0.583432i \(-0.198291\pi\)
\(662\) 0 0
\(663\) −8.00000 −0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.0000i 0.464642i
\(668\) 0 0
\(669\) − 16.0000i − 0.618596i
\(670\) 0 0
\(671\) 28.0000 1.08093
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) − 5.00000i − 0.192450i
\(676\) 0 0
\(677\) 12.0000i 0.461197i 0.973049 + 0.230599i \(0.0740685\pi\)
−0.973049 + 0.230599i \(0.925932\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) − 30.0000i − 1.14792i −0.818884 0.573959i \(-0.805407\pi\)
0.818884 0.573959i \(-0.194593\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.00000 0.0763048
\(688\) 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
\(693\) 2.00000i 0.0759737i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −32.0000 −1.21209
\(698\) 0 0
\(699\) 26.0000i 0.983410i
\(700\) 0 0
\(701\) 26.0000i 0.982006i 0.871158 + 0.491003i \(0.163370\pi\)
−0.871158 + 0.491003i \(0.836630\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.0000i 0.451306i
\(708\) 0 0
\(709\) − 26.0000i − 0.976450i −0.872718 0.488225i \(-0.837644\pi\)
0.872718 0.488225i \(-0.162356\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 26.0000i 0.970988i
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 22.0000i 0.818189i
\(724\) 0 0
\(725\) 10.0000i 0.371391i
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 32.0000i − 1.18356i
\(732\) 0 0
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) − 36.0000i − 1.32428i −0.749380 0.662141i \(-0.769648\pi\)
0.749380 0.662141i \(-0.230352\pi\)
\(740\) 0 0
\(741\) − 8.00000i − 0.293887i
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) 6.00000i 0.219235i
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 10.0000i − 0.363456i −0.983349 0.181728i \(-0.941831\pi\)
0.983349 0.181728i \(-0.0581691\pi\)
\(758\) 0 0
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −40.0000 −1.45000 −0.724999 0.688749i \(-0.758160\pi\)
−0.724999 + 0.688749i \(0.758160\pi\)
\(762\) 0 0
\(763\) 18.0000i 0.651644i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) − 8.00000i − 0.288113i
\(772\) 0 0
\(773\) − 12.0000i − 0.431610i −0.976436 0.215805i \(-0.930762\pi\)
0.976436 0.215805i \(-0.0692376\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) − 32.0000i − 1.14652i
\(780\) 0 0
\(781\) − 4.00000i − 0.143131i
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 0 0
\(789\) 14.0000i 0.498413i
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) −28.0000 −0.994309
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000i 0.425062i 0.977154 + 0.212531i \(0.0681706\pi\)
−0.977154 + 0.212531i \(0.931829\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 4.00000i − 0.141157i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 16.0000 0.563227
\(808\) 0 0
\(809\) −50.0000 −1.75791 −0.878953 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 0 0
\(811\) − 4.00000i − 0.140459i −0.997531 0.0702295i \(-0.977627\pi\)
0.997531 0.0702295i \(-0.0223732\pi\)
\(812\) 0 0
\(813\) − 32.0000i − 1.12229i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 0 0
\(819\) − 2.00000i − 0.0698857i
\(820\) 0 0
\(821\) − 30.0000i − 1.04701i −0.852023 0.523504i \(-0.824625\pi\)
0.852023 0.523504i \(-0.175375\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) 10.0000 0.348155
\(826\) 0 0
\(827\) − 42.0000i − 1.46048i −0.683189 0.730242i \(-0.739408\pi\)
0.683189 0.730242i \(-0.260592\pi\)
\(828\) 0 0
\(829\) 50.0000i 1.73657i 0.496064 + 0.868286i \(0.334778\pi\)
−0.496064 + 0.868286i \(0.665222\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 2.00000i 0.0688837i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) − 36.0000i − 1.23406i
\(852\) 0 0
\(853\) − 30.0000i − 1.02718i −0.858036 0.513590i \(-0.828315\pi\)
0.858036 0.513590i \(-0.171685\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 40.0000 1.36637 0.683187 0.730243i \(-0.260593\pi\)
0.683187 + 0.730243i \(0.260593\pi\)
\(858\) 0 0
\(859\) − 4.00000i − 0.136478i −0.997669 0.0682391i \(-0.978262\pi\)
0.997669 0.0682391i \(-0.0217381\pi\)
\(860\) 0 0
\(861\) − 8.00000i − 0.272639i
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 1.00000i − 0.0339618i
\(868\) 0 0
\(869\) − 8.00000i − 0.271381i
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.00000i 0.0675352i 0.999430 + 0.0337676i \(0.0107506\pi\)
−0.999430 + 0.0337676i \(0.989249\pi\)
\(878\) 0 0
\(879\) −28.0000 −0.944417
\(880\) 0 0
\(881\) −12.0000 −0.404290 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) 0 0
\(883\) 56.0000i 1.88455i 0.334840 + 0.942275i \(0.391318\pi\)
−0.334840 + 0.942275i \(0.608682\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) − 2.00000i − 0.0670025i
\(892\) 0 0
\(893\) − 16.0000i − 0.535420i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) − 24.0000i − 0.799556i
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 16.0000i − 0.531271i −0.964073 0.265636i \(-0.914418\pi\)
0.964073 0.265636i \(-0.0855818\pi\)
\(908\) 0 0
\(909\) − 12.0000i − 0.398015i
\(910\) 0 0
\(911\) 14.0000 0.463841 0.231920 0.972735i \(-0.425499\pi\)
0.231920 + 0.972735i \(0.425499\pi\)
\(912\) 0 0
\(913\) −24.0000 −0.794284
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 20.0000i − 0.660458i
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 0 0
\(923\) 4.00000i 0.131662i
\(924\) 0 0
\(925\) − 30.0000i − 0.986394i
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −40.0000 −1.31236 −0.656179 0.754606i \(-0.727828\pi\)
−0.656179 + 0.754606i \(0.727828\pi\)
\(930\) 0 0
\(931\) 4.00000i 0.131095i
\(932\) 0 0
\(933\) 28.0000i 0.916679i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 0 0
\(939\) 14.0000i 0.456873i
\(940\) 0 0
\(941\) − 56.0000i − 1.82555i −0.408465 0.912774i \(-0.633936\pi\)
0.408465 0.912774i \(-0.366064\pi\)
\(942\) 0 0
\(943\) −48.0000 −1.56310
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 14.0000i − 0.454939i −0.973785 0.227469i \(-0.926955\pi\)
0.973785 0.227469i \(-0.0730452\pi\)
\(948\) 0 0
\(949\) 4.00000i 0.129845i
\(950\) 0 0
\(951\) −14.0000 −0.453981
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.00000i 0.129302i
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 6.00000i − 0.193347i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 0 0
\(969\) −16.0000 −0.513994
\(970\) 0 0
\(971\) − 28.0000i − 0.898563i −0.893390 0.449281i \(-0.851680\pi\)
0.893390 0.449281i \(-0.148320\pi\)
\(972\) 0 0
\(973\) 12.0000i 0.384702i
\(974\) 0 0
\(975\) −10.0000 −0.320256
\(976\) 0 0
\(977\) 54.0000 1.72761 0.863807 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 18.0000i − 0.574696i
\(982\) 0 0
\(983\) 32.0000 1.02064 0.510321 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 4.00000i − 0.127321i
\(988\) 0 0
\(989\) − 48.0000i − 1.52631i
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −16.0000 −0.507745
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 14.0000i − 0.443384i −0.975117 0.221692i \(-0.928842\pi\)
0.975117 0.221692i \(-0.0711580\pi\)
\(998\) 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.y.2689.2 2
4.3 odd 2 5376.2.c.j.2689.1 2
8.3 odd 2 5376.2.c.j.2689.2 2
8.5 even 2 inner 5376.2.c.y.2689.1 2
16.3 odd 4 672.2.a.g.1.1 yes 1
16.5 even 4 1344.2.a.p.1.1 1
16.11 odd 4 1344.2.a.e.1.1 1
16.13 even 4 672.2.a.c.1.1 1
48.5 odd 4 4032.2.a.q.1.1 1
48.11 even 4 4032.2.a.x.1.1 1
48.29 odd 4 2016.2.a.d.1.1 1
48.35 even 4 2016.2.a.i.1.1 1
112.13 odd 4 4704.2.a.z.1.1 1
112.27 even 4 9408.2.a.ch.1.1 1
112.69 odd 4 9408.2.a.w.1.1 1
112.83 even 4 4704.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.a.c.1.1 1 16.13 even 4
672.2.a.g.1.1 yes 1 16.3 odd 4
1344.2.a.e.1.1 1 16.11 odd 4
1344.2.a.p.1.1 1 16.5 even 4
2016.2.a.d.1.1 1 48.29 odd 4
2016.2.a.i.1.1 1 48.35 even 4
4032.2.a.q.1.1 1 48.5 odd 4
4032.2.a.x.1.1 1 48.11 even 4
4704.2.a.k.1.1 1 112.83 even 4
4704.2.a.z.1.1 1 112.13 odd 4
5376.2.c.j.2689.1 2 4.3 odd 2
5376.2.c.j.2689.2 2 8.3 odd 2
5376.2.c.y.2689.1 2 8.5 even 2 inner
5376.2.c.y.2689.2 2 1.1 even 1 trivial
9408.2.a.w.1.1 1 112.69 odd 4
9408.2.a.ch.1.1 1 112.27 even 4