Properties

Label 269.2.a.a.1.1
Level $269$
Weight $2$
Character 269.1
Self dual yes
Analytic conductor $2.148$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [269,2,Mod(1,269)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(269, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("269.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 269.a (trivial)

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: yes
Analytic conductor: \(2.14797581437\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 269.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} +1.00000 q^{5} -4.00000 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{4} +1.00000 q^{5} -4.00000 q^{7} -3.00000 q^{9} -3.00000 q^{11} +2.00000 q^{13} +4.00000 q^{16} -4.00000 q^{17} +2.00000 q^{19} -2.00000 q^{20} -1.00000 q^{23} -4.00000 q^{25} +8.00000 q^{28} -2.00000 q^{29} -8.00000 q^{31} -4.00000 q^{35} +6.00000 q^{36} +7.00000 q^{37} +11.0000 q^{41} +3.00000 q^{43} +6.00000 q^{44} -3.00000 q^{45} -9.00000 q^{47} +9.00000 q^{49} -4.00000 q^{52} +9.00000 q^{53} -3.00000 q^{55} +4.00000 q^{59} -1.00000 q^{61} +12.0000 q^{63} -8.00000 q^{64} +2.00000 q^{65} -5.00000 q^{67} +8.00000 q^{68} -6.00000 q^{71} -14.0000 q^{73} -4.00000 q^{76} +12.0000 q^{77} -8.00000 q^{79} +4.00000 q^{80} +9.00000 q^{81} +10.0000 q^{83} -4.00000 q^{85} -5.00000 q^{89} -8.00000 q^{91} +2.00000 q^{92} +2.00000 q^{95} -9.00000 q^{97} +9.00000 q^{99} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −2.00000 −1.00000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 8.00000 1.51186
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 6.00000 1.00000
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.0000 1.71791 0.858956 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) 0 0
\(43\) 3.00000 0.457496 0.228748 0.973486i \(-0.426537\pi\)
0.228748 + 0.973486i \(0.426537\pi\)
\(44\) 6.00000 0.904534
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 12.0000 1.51186
\(64\) −8.00000 −1.00000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −5.00000 −0.610847 −0.305424 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 4.00000 0.447214
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.00000 −0.529999 −0.264999 0.964249i \(-0.585372\pi\)
−0.264999 + 0.964249i \(0.585372\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −9.00000 −0.913812 −0.456906 0.889515i \(-0.651042\pi\)
−0.456906 + 0.889515i \(0.651042\pi\)
\(98\) 0 0
\(99\) 9.00000 0.904534
\(100\) 8.00000 0.800000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −5.00000 −0.492665 −0.246332 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.0000 1.35343 0.676716 0.736245i \(-0.263403\pi\)
0.676716 + 0.736245i \(0.263403\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −16.0000 −1.51186
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 4.00000 0.371391
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 0 0
\(124\) 16.0000 1.43684
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −17.0000 −1.50851 −0.754253 0.656584i \(-0.772001\pi\)
−0.754253 + 0.656584i \(0.772001\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 8.00000 0.676123
\(141\) 0 0
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) −12.0000 −1.00000
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) −14.0000 −1.15079
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) −11.0000 −0.895167 −0.447584 0.894242i \(-0.647715\pi\)
−0.447584 + 0.894242i \(0.647715\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −22.0000 −1.71791
\(165\) 0 0
\(166\) 0 0
\(167\) −4.00000 −0.309529 −0.154765 0.987951i \(-0.549462\pi\)
−0.154765 + 0.987951i \(0.549462\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) −6.00000 −0.457496
\(173\) 7.00000 0.532200 0.266100 0.963945i \(-0.414265\pi\)
0.266100 + 0.963945i \(0.414265\pi\)
\(174\) 0 0
\(175\) 16.0000 1.20949
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 6.00000 0.447214
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.00000 0.514650
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 18.0000 1.31278
\(189\) 0 0
\(190\) 0 0
\(191\) 25.0000 1.80894 0.904468 0.426541i \(-0.140268\pi\)
0.904468 + 0.426541i \(0.140268\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −18.0000 −1.28571
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 5.00000 0.354441 0.177220 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) 11.0000 0.768273
\(206\) 0 0
\(207\) 3.00000 0.208514
\(208\) 8.00000 0.554700
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −27.0000 −1.85876 −0.929378 0.369129i \(-0.879656\pi\)
−0.929378 + 0.369129i \(0.879656\pi\)
\(212\) −18.0000 −1.23625
\(213\) 0 0
\(214\) 0 0
\(215\) 3.00000 0.204598
\(216\) 0 0
\(217\) 32.0000 2.17230
\(218\) 0 0
\(219\) 0 0
\(220\) 6.00000 0.404520
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) 12.0000 0.800000
\(226\) 0 0
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) 0 0
\(239\) 27.0000 1.74648 0.873242 0.487286i \(-0.162013\pi\)
0.873242 + 0.487286i \(0.162013\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 9.00000 0.574989
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −24.0000 −1.51186
\(253\) 3.00000 0.188608
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) −28.0000 −1.73984
\(260\) −4.00000 −0.248069
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 1.00000 0.0616626 0.0308313 0.999525i \(-0.490185\pi\)
0.0308313 + 0.999525i \(0.490185\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) −1.00000 −0.0609711
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −16.0000 −0.970143
\(273\) 0 0
\(274\) 0 0
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) 30.0000 1.80253 0.901263 0.433273i \(-0.142641\pi\)
0.901263 + 0.433273i \(0.142641\pi\)
\(278\) 0 0
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) −44.0000 −2.59724
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 28.0000 1.63858
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −24.0000 −1.36753
\(309\) 0 0
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) −21.0000 −1.18699 −0.593495 0.804838i \(-0.702252\pi\)
−0.593495 + 0.804838i \(0.702252\pi\)
\(314\) 0 0
\(315\) 12.0000 0.676123
\(316\) 16.0000 0.900070
\(317\) −16.0000 −0.898650 −0.449325 0.893368i \(-0.648335\pi\)
−0.449325 + 0.893368i \(0.648335\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) −8.00000 −0.447214
\(321\) 0 0
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) −18.0000 −1.00000
\(325\) −8.00000 −0.443760
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 36.0000 1.98474
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −20.0000 −1.09764
\(333\) −21.0000 −1.15079
\(334\) 0 0
\(335\) −5.00000 −0.273179
\(336\) 0 0
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 8.00000 0.433861
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 3.00000 0.160586 0.0802932 0.996771i \(-0.474414\pi\)
0.0802932 + 0.996771i \(0.474414\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 31.0000 1.64996 0.824982 0.565159i \(-0.191185\pi\)
0.824982 + 0.565159i \(0.191185\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 16.0000 0.838628
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) −4.00000 −0.208514
\(369\) −33.0000 −1.71791
\(370\) 0 0
\(371\) −36.0000 −1.86903
\(372\) 0 0
\(373\) −30.0000 −1.55334 −0.776671 0.629907i \(-0.783093\pi\)
−0.776671 + 0.629907i \(0.783093\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 0 0
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) −9.00000 −0.457496
\(388\) 18.0000 0.913812
\(389\) −29.0000 −1.47036 −0.735179 0.677873i \(-0.762902\pi\)
−0.735179 + 0.677873i \(0.762902\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) −18.0000 −0.904534
\(397\) 30.0000 1.50566 0.752828 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −16.0000 −0.800000
\(401\) 32.0000 1.59800 0.799002 0.601329i \(-0.205362\pi\)
0.799002 + 0.601329i \(0.205362\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) 24.0000 1.19404
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) −21.0000 −1.04093
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 10.0000 0.492665
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) 10.0000 0.490881
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 21.0000 1.02348 0.511739 0.859141i \(-0.329002\pi\)
0.511739 + 0.859141i \(0.329002\pi\)
\(422\) 0 0
\(423\) 27.0000 1.31278
\(424\) 0 0
\(425\) 16.0000 0.776114
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) −28.0000 −1.35343
\(429\) 0 0
\(430\) 0 0
\(431\) −10.0000 −0.481683 −0.240842 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) 0 0
\(433\) −1.00000 −0.0480569 −0.0240285 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) −2.00000 −0.0956730
\(438\) 0 0
\(439\) −29.0000 −1.38409 −0.692047 0.721852i \(-0.743291\pi\)
−0.692047 + 0.721852i \(0.743291\pi\)
\(440\) 0 0
\(441\) −27.0000 −1.28571
\(442\) 0 0
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 0 0
\(445\) −5.00000 −0.237023
\(446\) 0 0
\(447\) 0 0
\(448\) 32.0000 1.51186
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) −33.0000 −1.55391
\(452\) −12.0000 −0.564433
\(453\) 0 0
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) −40.0000 −1.86299 −0.931493 0.363760i \(-0.881493\pi\)
−0.931493 + 0.363760i \(0.881493\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) 0 0
\(467\) −2.00000 −0.0925490 −0.0462745 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) 12.0000 0.554700
\(469\) 20.0000 0.923514
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −9.00000 −0.413820
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) −32.0000 −1.46672
\(477\) −27.0000 −1.23625
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 14.0000 0.638345
\(482\) 0 0
\(483\) 0 0
\(484\) 4.00000 0.181818
\(485\) −9.00000 −0.408669
\(486\) 0 0
\(487\) −5.00000 −0.226572 −0.113286 0.993562i \(-0.536138\pi\)
−0.113286 + 0.993562i \(0.536138\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 0 0
\(493\) 8.00000 0.360302
\(494\) 0 0
\(495\) 9.00000 0.404520
\(496\) −32.0000 −1.43684
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 18.0000 0.804984
\(501\) 0 0
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 34.0000 1.50851
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.00000 −0.220326
\(516\) 0 0
\(517\) 27.0000 1.18746
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 40.0000 1.74741
\(525\) 0 0
\(526\) 0 0
\(527\) 32.0000 1.39394
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 16.0000 0.693688
\(533\) 22.0000 0.952926
\(534\) 0 0
\(535\) 14.0000 0.605273
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −27.0000 −1.16297
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.00000 0.342682
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 24.0000 1.02523
\(549\) 3.00000 0.128037
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) 0 0
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −28.0000 −1.18640 −0.593199 0.805056i \(-0.702135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) −16.0000 −0.676123
\(561\) 0 0
\(562\) 0 0
\(563\) −39.0000 −1.64365 −0.821827 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) −36.0000 −1.51186
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 24.0000 1.00000
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) −40.0000 −1.65948
\(582\) 0 0
\(583\) −27.0000 −1.11823
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) −7.00000 −0.288921 −0.144460 0.989511i \(-0.546145\pi\)
−0.144460 + 0.989511i \(0.546145\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) 28.0000 1.15079
\(593\) 39.0000 1.60154 0.800769 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) 0 0
\(599\) 21.0000 0.858037 0.429018 0.903296i \(-0.358860\pi\)
0.429018 + 0.903296i \(0.358860\pi\)
\(600\) 0 0
\(601\) 6.00000 0.244745 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) 0 0
\(603\) 15.0000 0.610847
\(604\) 22.0000 0.895167
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) −24.0000 −0.970143
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −31.0000 −1.24801 −0.624007 0.781419i \(-0.714496\pi\)
−0.624007 + 0.781419i \(0.714496\pi\)
\(618\) 0 0
\(619\) −17.0000 −0.683288 −0.341644 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 16.0000 0.642575
\(621\) 0 0
\(622\) 0 0
\(623\) 20.0000 0.801283
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) −36.0000 −1.43656
\(629\) −28.0000 −1.11643
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −17.0000 −0.674624
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) 18.0000 0.712069
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 11.0000 0.430463 0.215232 0.976563i \(-0.430949\pi\)
0.215232 + 0.976563i \(0.430949\pi\)
\(654\) 0 0
\(655\) −20.0000 −0.781465
\(656\) 44.0000 1.71791
\(657\) 42.0000 1.63858
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 2.00000 0.0774403
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 0 0
\(671\) 3.00000 0.115814
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 18.0000 0.692308
\(677\) −10.0000 −0.384331 −0.192166 0.981363i \(-0.561551\pi\)
−0.192166 + 0.981363i \(0.561551\pi\)
\(678\) 0 0
\(679\) 36.0000 1.38155
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 32.0000 1.22445 0.612223 0.790685i \(-0.290275\pi\)
0.612223 + 0.790685i \(0.290275\pi\)
\(684\) 12.0000 0.458831
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 12.0000 0.457496
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −14.0000 −0.532200
\(693\) −36.0000 −1.36753
\(694\) 0 0
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) −44.0000 −1.66662
\(698\) 0 0
\(699\) 0 0
\(700\) −32.0000 −1.20949
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) 14.0000 0.528020
\(704\) 24.0000 0.904534
\(705\) 0 0
\(706\) 0 0
\(707\) 48.0000 1.80523
\(708\) 0 0
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 0 0
\(711\) 24.0000 0.900070
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) −12.0000 −0.447214
\(721\) 20.0000 0.744839
\(722\) 0 0
\(723\) 0 0
\(724\) −20.0000 −0.743294
\(725\) 8.00000 0.297113
\(726\) 0 0
\(727\) 19.0000 0.704671 0.352335 0.935874i \(-0.385388\pi\)
0.352335 + 0.935874i \(0.385388\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 15.0000 0.552532
\(738\) 0 0
\(739\) 24.0000 0.882854 0.441427 0.897297i \(-0.354472\pi\)
0.441427 + 0.897297i \(0.354472\pi\)
\(740\) −14.0000 −0.514650
\(741\) 0 0
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) −2.00000 −0.0732743
\(746\) 0 0
\(747\) −30.0000 −1.09764
\(748\) −24.0000 −0.877527
\(749\) −56.0000 −2.04620
\(750\) 0 0
\(751\) −25.0000 −0.912263 −0.456131 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) −36.0000 −1.31278
\(753\) 0 0
\(754\) 0 0
\(755\) −11.0000 −0.400331
\(756\) 0 0
\(757\) −48.0000 −1.74459 −0.872295 0.488980i \(-0.837369\pi\)
−0.872295 + 0.488980i \(0.837369\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) −32.0000 −1.15848
\(764\) −50.0000 −1.80894
\(765\) 12.0000 0.433861
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −12.0000 −0.431889
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 0 0
\(775\) 32.0000 1.14947
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 22.0000 0.788232
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) 36.0000 1.28571
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) −24.0000 −0.854965
\(789\) 0 0
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) 15.0000 0.529999
\(802\) 0 0
\(803\) 42.0000 1.48215
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) −31.0000 −1.08856 −0.544279 0.838905i \(-0.683197\pi\)
−0.544279 + 0.838905i \(0.683197\pi\)
\(812\) −16.0000 −0.561490
\(813\) 0 0
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) 6.00000 0.209913
\(818\) 0 0
\(819\) 24.0000 0.838628
\(820\) −22.0000 −0.768273
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) 48.0000 1.67317 0.836587 0.547833i \(-0.184547\pi\)
0.836587 + 0.547833i \(0.184547\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −33.0000 −1.14752 −0.573761 0.819023i \(-0.694516\pi\)
−0.573761 + 0.819023i \(0.694516\pi\)
\(828\) −6.00000 −0.208514
\(829\) 42.0000 1.45872 0.729360 0.684130i \(-0.239818\pi\)
0.729360 + 0.684130i \(0.239818\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −16.0000 −0.554700
\(833\) −36.0000 −1.24733
\(834\) 0 0
\(835\) −4.00000 −0.138426
\(836\) 12.0000 0.415029
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 54.0000 1.85876
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 8.00000 0.274883
\(848\) 36.0000 1.23625
\(849\) 0 0
\(850\) 0 0
\(851\) −7.00000 −0.239957
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 15.0000 0.511793 0.255897 0.966704i \(-0.417629\pi\)
0.255897 + 0.966704i \(0.417629\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) 0 0
\(863\) −49.0000 −1.66798 −0.833990 0.551780i \(-0.813949\pi\)
−0.833990 + 0.551780i \(0.813949\pi\)
\(864\) 0 0
\(865\) 7.00000 0.238007
\(866\) 0 0
\(867\) 0 0
\(868\) −64.0000 −2.17230
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) 0 0
\(873\) 27.0000 0.913812
\(874\) 0 0
\(875\) 36.0000 1.21702
\(876\) 0 0
\(877\) 17.0000 0.574049 0.287025 0.957923i \(-0.407334\pi\)
0.287025 + 0.957923i \(0.407334\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −12.0000 −0.404520
\(881\) 28.0000 0.943344 0.471672 0.881774i \(-0.343651\pi\)
0.471672 + 0.881774i \(0.343651\pi\)
\(882\) 0 0
\(883\) −42.0000 −1.41341 −0.706706 0.707507i \(-0.749820\pi\)
−0.706706 + 0.707507i \(0.749820\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 68.0000 2.28065
\(890\) 0 0
\(891\) −27.0000 −0.904534
\(892\) −28.0000 −0.937509
\(893\) −18.0000 −0.602347
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 16.0000 0.533630
\(900\) −24.0000 −0.800000
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 12.0000 0.398234
\(909\) 36.0000 1.19404
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) −30.0000 −0.992855
\(914\) 0 0
\(915\) 0 0
\(916\) −8.00000 −0.264327
\(917\) 80.0000 2.64183
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) 0 0
\(927\) 15.0000 0.492665
\(928\) 0 0
\(929\) −28.0000 −0.918650 −0.459325 0.888268i \(-0.651909\pi\)
−0.459325 + 0.888268i \(0.651909\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) 0 0
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 18.0000 0.587095
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 0 0
\(943\) −11.0000 −0.358209
\(944\) 16.0000 0.520756
\(945\) 0 0
\(946\) 0 0
\(947\) 34.0000 1.10485 0.552426 0.833562i \(-0.313702\pi\)
0.552426 + 0.833562i \(0.313702\pi\)
\(948\) 0 0
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −56.0000 −1.81402 −0.907009 0.421111i \(-0.861640\pi\)
−0.907009 + 0.421111i \(0.861640\pi\)
\(954\) 0 0
\(955\) 25.0000 0.808981
\(956\) −54.0000 −1.74648
\(957\) 0 0
\(958\) 0 0
\(959\) 48.0000 1.55000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −42.0000 −1.35343
\(964\) 52.0000 1.67481
\(965\) 6.00000 0.193147
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −27.0000 −0.866471 −0.433236 0.901281i \(-0.642628\pi\)
−0.433236 + 0.901281i \(0.642628\pi\)
\(972\) 0 0
\(973\) 32.0000 1.02587
\(974\) 0 0
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 15.0000 0.479402
\(980\) −18.0000 −0.574989
\(981\) −24.0000 −0.766261
\(982\) 0 0
\(983\) −43.0000 −1.37149 −0.685744 0.727843i \(-0.740523\pi\)
−0.685744 + 0.727843i \(0.740523\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) 38.0000 1.20711 0.603555 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5.00000 0.158511
\(996\) 0 0
\(997\) −35.0000 −1.10846 −0.554231 0.832363i \(-0.686987\pi\)
−0.554231 + 0.832363i \(0.686987\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 269.2.a.a.1.1 1
3.2 odd 2 2421.2.a.b.1.1 1
4.3 odd 2 4304.2.a.a.1.1 1
5.4 even 2 6725.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
269.2.a.a.1.1 1 1.1 even 1 trivial
2421.2.a.b.1.1 1 3.2 odd 2
4304.2.a.a.1.1 1 4.3 odd 2
6725.2.a.a.1.1 1 5.4 even 2