Properties

Label 3800.1.b.d.949.2
Level $3800$
Weight $1$
Character 3800.949
Analytic conductor $1.896$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -152
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,1,Mod(949,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.949");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.b (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: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.419904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 6x^{4} + 9x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.8340544000000.1

Embedding invariants

Embedding label 949.2
Root \(1.53209i\) of defining polynomial
Character \(\chi\) \(=\) 3800.949
Dual form 3800.1.b.d.949.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +0.347296i q^{3} -1.00000 q^{4} +0.347296 q^{6} -1.87939i q^{7} +1.00000i q^{8} +0.879385 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +0.347296i q^{3} -1.00000 q^{4} +0.347296 q^{6} -1.87939i q^{7} +1.00000i q^{8} +0.879385 q^{9} -0.347296i q^{12} +1.53209i q^{13} -1.87939 q^{14} +1.00000 q^{16} +1.53209i q^{17} -0.879385i q^{18} +1.00000 q^{19} +0.652704 q^{21} -0.347296i q^{23} -0.347296 q^{24} +1.53209 q^{26} +0.652704i q^{27} +1.87939i q^{28} +1.53209 q^{29} -1.00000i q^{32} +1.53209 q^{34} -0.879385 q^{36} +1.00000i q^{37} -1.00000i q^{38} -0.532089 q^{39} -0.652704i q^{42} -0.347296 q^{46} -1.00000i q^{47} +0.347296i q^{48} -2.53209 q^{49} -0.532089 q^{51} -1.53209i q^{52} -1.87939i q^{53} +0.652704 q^{54} +1.87939 q^{56} +0.347296i q^{57} -1.53209i q^{58} +0.347296 q^{59} -1.65270i q^{63} -1.00000 q^{64} +1.87939i q^{67} -1.53209i q^{68} +0.120615 q^{69} +0.879385i q^{72} -0.347296i q^{73} +1.00000 q^{74} -1.00000 q^{76} +0.532089i q^{78} +0.652704 q^{81} -0.652704 q^{84} +0.532089i q^{87} +2.87939 q^{91} +0.347296i q^{92} -1.00000 q^{94} +0.347296 q^{96} +2.53209i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 6 q^{9} + 6 q^{16} + 6 q^{19} + 6 q^{21} + 6 q^{36} + 6 q^{39} - 6 q^{49} + 6 q^{51} + 6 q^{54} - 6 q^{64} + 12 q^{69} + 6 q^{74} - 6 q^{76} + 6 q^{81} - 6 q^{84} + 6 q^{91} - 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 1.00000i
\(3\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(4\) −1.00000 −1.00000
\(5\) 0 0
\(6\) 0.347296 0.347296
\(7\) − 1.87939i − 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(8\) 1.00000i 1.00000i
\(9\) 0.879385 0.879385
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) − 0.347296i − 0.347296i
\(13\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(14\) −1.87939 −1.87939
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(18\) − 0.879385i − 0.879385i
\(19\) 1.00000 1.00000
\(20\) 0 0
\(21\) 0.652704 0.652704
\(22\) 0 0
\(23\) − 0.347296i − 0.347296i −0.984808 0.173648i \(-0.944444\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(24\) −0.347296 −0.347296
\(25\) 0 0
\(26\) 1.53209 1.53209
\(27\) 0.652704i 0.652704i
\(28\) 1.87939i 1.87939i
\(29\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) − 1.00000i − 1.00000i
\(33\) 0 0
\(34\) 1.53209 1.53209
\(35\) 0 0
\(36\) −0.879385 −0.879385
\(37\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(38\) − 1.00000i − 1.00000i
\(39\) −0.532089 −0.532089
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) − 0.652704i − 0.652704i
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.347296 −0.347296
\(47\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(48\) 0.347296i 0.347296i
\(49\) −2.53209 −2.53209
\(50\) 0 0
\(51\) −0.532089 −0.532089
\(52\) − 1.53209i − 1.53209i
\(53\) − 1.87939i − 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(54\) 0.652704 0.652704
\(55\) 0 0
\(56\) 1.87939 1.87939
\(57\) 0.347296i 0.347296i
\(58\) − 1.53209i − 1.53209i
\(59\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 1.65270i − 1.65270i
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.87939i 1.87939i 0.342020 + 0.939693i \(0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(68\) − 1.53209i − 1.53209i
\(69\) 0.120615 0.120615
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0.879385i 0.879385i
\(73\) − 0.347296i − 0.347296i −0.984808 0.173648i \(-0.944444\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(74\) 1.00000 1.00000
\(75\) 0 0
\(76\) −1.00000 −1.00000
\(77\) 0 0
\(78\) 0.532089i 0.532089i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0.652704 0.652704
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −0.652704 −0.652704
\(85\) 0 0
\(86\) 0 0
\(87\) 0.532089i 0.532089i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 2.87939 2.87939
\(92\) 0.347296i 0.347296i
\(93\) 0 0
\(94\) −1.00000 −1.00000
\(95\) 0 0
\(96\) 0.347296 0.347296
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 2.53209i 2.53209i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0.532089i 0.532089i
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −1.53209 −1.53209
\(105\) 0 0
\(106\) −1.87939 −1.87939
\(107\) − 1.53209i − 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(108\) − 0.652704i − 0.652704i
\(109\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(110\) 0 0
\(111\) −0.347296 −0.347296
\(112\) − 1.87939i − 1.87939i
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0.347296 0.347296
\(115\) 0 0
\(116\) −1.53209 −1.53209
\(117\) 1.34730i 1.34730i
\(118\) − 0.347296i − 0.347296i
\(119\) 2.87939 2.87939
\(120\) 0 0
\(121\) 1.00000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −1.65270 −1.65270
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000i 1.00000i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) − 1.87939i − 1.87939i
\(134\) 1.87939 1.87939
\(135\) 0 0
\(136\) −1.53209 −1.53209
\(137\) − 1.87939i − 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(138\) − 0.120615i − 0.120615i
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0.347296 0.347296
\(142\) 0 0
\(143\) 0 0
\(144\) 0.879385 0.879385
\(145\) 0 0
\(146\) −0.347296 −0.347296
\(147\) − 0.879385i − 0.879385i
\(148\) − 1.00000i − 1.00000i
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 1.00000i 1.00000i
\(153\) 1.34730i 1.34730i
\(154\) 0 0
\(155\) 0 0
\(156\) 0.532089 0.532089
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0.652704 0.652704
\(160\) 0 0
\(161\) −0.652704 −0.652704
\(162\) − 0.652704i − 0.652704i
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0.652704i 0.652704i
\(169\) −1.34730 −1.34730
\(170\) 0 0
\(171\) 0.879385 0.879385
\(172\) 0 0
\(173\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(174\) 0.532089 0.532089
\(175\) 0 0
\(176\) 0 0
\(177\) 0.120615i 0.120615i
\(178\) 0 0
\(179\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(182\) − 2.87939i − 2.87939i
\(183\) 0 0
\(184\) 0.347296 0.347296
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 1.00000i 1.00000i
\(189\) 1.22668 1.22668
\(190\) 0 0
\(191\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(192\) − 0.347296i − 0.347296i
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.53209 2.53209
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(200\) 0 0
\(201\) −0.652704 −0.652704
\(202\) 0 0
\(203\) − 2.87939i − 2.87939i
\(204\) 0.532089 0.532089
\(205\) 0 0
\(206\) 0 0
\(207\) − 0.305407i − 0.305407i
\(208\) 1.53209i 1.53209i
\(209\) 0 0
\(210\) 0 0
\(211\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(212\) 1.87939i 1.87939i
\(213\) 0 0
\(214\) −1.53209 −1.53209
\(215\) 0 0
\(216\) −0.652704 −0.652704
\(217\) 0 0
\(218\) 1.87939i 1.87939i
\(219\) 0.120615 0.120615
\(220\) 0 0
\(221\) −2.34730 −2.34730
\(222\) 0.347296i 0.347296i
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −1.87939 −1.87939
\(225\) 0 0
\(226\) 0 0
\(227\) 1.87939i 1.87939i 0.342020 + 0.939693i \(0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(228\) − 0.347296i − 0.347296i
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.53209i 1.53209i
\(233\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(234\) 1.34730 1.34730
\(235\) 0 0
\(236\) −0.347296 −0.347296
\(237\) 0 0
\(238\) − 2.87939i − 2.87939i
\(239\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) − 1.00000i − 1.00000i
\(243\) 0.879385i 0.879385i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.53209i 1.53209i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.65270i 1.65270i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 1.87939 1.87939
\(260\) 0 0
\(261\) 1.34730 1.34730
\(262\) 0 0
\(263\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.87939 −1.87939
\(267\) 0 0
\(268\) − 1.87939i − 1.87939i
\(269\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) 0 0
\(271\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(272\) 1.53209i 1.53209i
\(273\) 1.00000i 1.00000i
\(274\) −1.87939 −1.87939
\(275\) 0 0
\(276\) −0.120615 −0.120615
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) − 0.347296i − 0.347296i
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) − 0.879385i − 0.879385i
\(289\) −1.34730 −1.34730
\(290\) 0 0
\(291\) 0 0
\(292\) 0.347296i 0.347296i
\(293\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(294\) −0.879385 −0.879385
\(295\) 0 0
\(296\) −1.00000 −1.00000
\(297\) 0 0
\(298\) 0 0
\(299\) 0.532089 0.532089
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 1.00000 1.00000
\(305\) 0 0
\(306\) 1.34730 1.34730
\(307\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(312\) − 0.532089i − 0.532089i
\(313\) − 1.53209i − 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 0.347296i − 0.347296i −0.984808 0.173648i \(-0.944444\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(318\) − 0.652704i − 0.652704i
\(319\) 0 0
\(320\) 0 0
\(321\) 0.532089 0.532089
\(322\) 0.652704i 0.652704i
\(323\) 1.53209i 1.53209i
\(324\) −0.652704 −0.652704
\(325\) 0 0
\(326\) 0 0
\(327\) − 0.652704i − 0.652704i
\(328\) 0 0
\(329\) −1.87939 −1.87939
\(330\) 0 0
\(331\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(332\) 0 0
\(333\) 0.879385i 0.879385i
\(334\) 0 0
\(335\) 0 0
\(336\) 0.652704 0.652704
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 1.34730i 1.34730i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) − 0.879385i − 0.879385i
\(343\) 2.87939i 2.87939i
\(344\) 0 0
\(345\) 0 0
\(346\) −1.00000 −1.00000
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) − 0.532089i − 0.532089i
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) −1.00000 −1.00000
\(352\) 0 0
\(353\) 1.87939i 1.87939i 0.342020 + 0.939693i \(0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(354\) 0.120615 0.120615
\(355\) 0 0
\(356\) 0 0
\(357\) 1.00000i 1.00000i
\(358\) 1.00000i 1.00000i
\(359\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) − 1.00000i − 1.00000i
\(363\) 0.347296i 0.347296i
\(364\) −2.87939 −2.87939
\(365\) 0 0
\(366\) 0 0
\(367\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) − 0.347296i − 0.347296i
\(369\) 0 0
\(370\) 0 0
\(371\) −3.53209 −3.53209
\(372\) 0 0
\(373\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 1.00000 1.00000
\(377\) 2.34730i 2.34730i
\(378\) − 1.22668i − 1.22668i
\(379\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 0.347296i − 0.347296i
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −0.347296 −0.347296
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0.532089 0.532089
\(392\) − 2.53209i − 2.53209i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) − 1.87939i − 1.87939i
\(399\) 0.652704 0.652704
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0.652704i 0.652704i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −2.87939 −2.87939
\(407\) 0 0
\(408\) − 0.532089i − 0.532089i
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0.652704 0.652704
\(412\) 0 0
\(413\) − 0.652704i − 0.652704i
\(414\) −0.305407 −0.305407
\(415\) 0 0
\(416\) 1.53209 1.53209
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(422\) 0.347296i 0.347296i
\(423\) − 0.879385i − 0.879385i
\(424\) 1.87939 1.87939
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 1.53209i 1.53209i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0.652704i 0.652704i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.87939 1.87939
\(437\) − 0.347296i − 0.347296i
\(438\) − 0.120615i − 0.120615i
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −2.22668 −2.22668
\(442\) 2.34730i 2.34730i
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0.347296 0.347296
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 1.87939i 1.87939i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 1.87939 1.87939
\(455\) 0 0
\(456\) −0.347296 −0.347296
\(457\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(458\) 0 0
\(459\) −1.00000 −1.00000
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(464\) 1.53209 1.53209
\(465\) 0 0
\(466\) 1.00000 1.00000
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) − 1.34730i − 1.34730i
\(469\) 3.53209 3.53209
\(470\) 0 0
\(471\) 0 0
\(472\) 0.347296i 0.347296i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −2.87939 −2.87939
\(477\) − 1.65270i − 1.65270i
\(478\) 0.347296i 0.347296i
\(479\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) −1.53209 −1.53209
\(482\) 0 0
\(483\) − 0.226682i − 0.226682i
\(484\) −1.00000 −1.00000
\(485\) 0 0
\(486\) 0.879385 0.879385
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 2.34730i 2.34730i
\(494\) 1.53209 1.53209
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 1.53209i − 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(504\) 1.65270 1.65270
\(505\) 0 0
\(506\) 0 0
\(507\) − 0.467911i − 0.467911i
\(508\) 0 0
\(509\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) −0.652704 −0.652704
\(512\) − 1.00000i − 1.00000i
\(513\) 0.652704i 0.652704i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) − 1.87939i − 1.87939i
\(519\) 0.347296 0.347296
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) − 1.34730i − 1.34730i
\(523\) − 1.87939i − 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −2.00000 −2.00000
\(527\) 0 0
\(528\) 0 0
\(529\) 0.879385 0.879385
\(530\) 0 0
\(531\) 0.305407 0.305407
\(532\) 1.87939i 1.87939i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −1.87939 −1.87939
\(537\) − 0.347296i − 0.347296i
\(538\) 1.00000i 1.00000i
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) − 0.347296i − 0.347296i
\(543\) 0.347296i 0.347296i
\(544\) 1.53209 1.53209
\(545\) 0 0
\(546\) 1.00000 1.00000
\(547\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(548\) 1.87939i 1.87939i
\(549\) 0 0
\(550\) 0 0
\(551\) 1.53209 1.53209
\(552\) 0.120615i 0.120615i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(564\) −0.347296 −0.347296
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.22668i − 1.22668i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0.120615i 0.120615i
\(574\) 0 0
\(575\) 0 0
\(576\) −0.879385 −0.879385
\(577\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(578\) 1.34730i 1.34730i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0.347296 0.347296
\(585\) 0 0
\(586\) 1.53209 1.53209
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0.879385i 0.879385i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.00000i 1.00000i
\(593\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.652704i 0.652704i
\(598\) − 0.532089i − 0.532089i
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 1.65270i 1.65270i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) − 1.00000i − 1.00000i
\(609\) 1.00000 1.00000
\(610\) 0 0
\(611\) 1.53209 1.53209
\(612\) − 1.34730i − 1.34730i
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 1.00000 1.00000
\(615\) 0 0
\(616\) 0 0
\(617\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0.226682 0.226682
\(622\) 1.87939i 1.87939i
\(623\) 0 0
\(624\) −0.532089 −0.532089
\(625\) 0 0
\(626\) −1.53209 −1.53209
\(627\) 0 0
\(628\) 0 0
\(629\) −1.53209 −1.53209
\(630\) 0 0
\(631\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(632\) 0 0
\(633\) − 0.120615i − 0.120615i
\(634\) −0.347296 −0.347296
\(635\) 0 0
\(636\) −0.652704 −0.652704
\(637\) − 3.87939i − 3.87939i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) − 0.532089i − 0.532089i
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0.652704 0.652704
\(645\) 0 0
\(646\) 1.53209 1.53209
\(647\) − 1.87939i − 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(648\) 0.652704i 0.652704i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) −0.652704 −0.652704
\(655\) 0 0
\(656\) 0 0
\(657\) − 0.305407i − 0.305407i
\(658\) 1.87939i 1.87939i
\(659\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(660\) 0 0
\(661\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(662\) 1.53209i 1.53209i
\(663\) − 0.815207i − 0.815207i
\(664\) 0 0
\(665\) 0 0
\(666\) 0.879385 0.879385
\(667\) − 0.532089i − 0.532089i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) − 0.652704i − 0.652704i
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.34730 1.34730
\(677\) 1.87939i 1.87939i 0.342020 + 0.939693i \(0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −0.652704 −0.652704
\(682\) 0 0
\(683\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(684\) −0.879385 −0.879385
\(685\) 0 0
\(686\) 2.87939 2.87939
\(687\) 0 0
\(688\) 0 0
\(689\) 2.87939 2.87939
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 1.00000i 1.00000i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) −0.532089 −0.532089
\(697\) 0 0
\(698\) 0 0
\(699\) −0.347296 −0.347296
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 1.00000i 1.00000i
\(703\) 1.00000i 1.00000i
\(704\) 0 0
\(705\) 0 0
\(706\) 1.87939 1.87939
\(707\) 0 0
\(708\) − 0.120615i − 0.120615i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 1.00000 1.00000
\(715\) 0 0
\(716\) 1.00000 1.00000
\(717\) − 0.120615i − 0.120615i
\(718\) 1.53209i 1.53209i
\(719\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 1.00000i − 1.00000i
\(723\) 0 0
\(724\) −1.00000 −1.00000
\(725\) 0 0
\(726\) 0.347296 0.347296
\(727\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(728\) 2.87939i 2.87939i
\(729\) 0.347296 0.347296
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 2.00000 2.00000
\(735\) 0 0
\(736\) −0.347296 −0.347296
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) −0.532089 −0.532089
\(742\) 3.53209i 3.53209i
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.347296 0.347296
\(747\) 0 0
\(748\) 0 0
\(749\) −2.87939 −2.87939
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) − 1.00000i − 1.00000i
\(753\) 0 0
\(754\) 2.34730 2.34730
\(755\) 0 0
\(756\) −1.22668 −1.22668
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 1.87939i 1.87939i
\(759\) 0 0
\(760\) 0 0
\(761\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(762\) 0 0
\(763\) 3.53209i 3.53209i
\(764\) −0.347296 −0.347296
\(765\) 0 0
\(766\) 0 0
\(767\) 0.532089i 0.532089i
\(768\) 0.347296i 0.347296i
\(769\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.652704i 0.652704i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) − 0.532089i − 0.532089i
\(783\) 1.00000i 1.00000i
\(784\) −2.53209 −2.53209
\(785\) 0 0
\(786\) 0 0
\(787\) − 1.53209i − 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(788\) 0 0
\(789\) 0.694593 0.694593
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −1.87939 −1.87939
\(797\) − 1.53209i − 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(798\) − 0.652704i − 0.652704i
\(799\) 1.53209 1.53209
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0.652704 0.652704
\(805\) 0 0
\(806\) 0 0
\(807\) − 0.347296i − 0.347296i
\(808\) 0 0
\(809\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(810\) 0 0
\(811\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(812\) 2.87939i 2.87939i
\(813\) 0.120615i 0.120615i
\(814\) 0 0
\(815\) 0 0
\(816\) −0.532089 −0.532089
\(817\) 0 0
\(818\) 0 0
\(819\) 2.53209 2.53209
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) − 0.652704i − 0.652704i
\(823\) − 0.347296i − 0.347296i −0.984808 0.173648i \(-0.944444\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −0.652704 −0.652704
\(827\) − 1.53209i − 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(828\) 0.305407i 0.305407i
\(829\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 1.53209i − 1.53209i
\(833\) − 3.87939i − 3.87939i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.34730 1.34730
\(842\) 0.347296i 0.347296i
\(843\) 0 0
\(844\) 0.347296 0.347296
\(845\) 0 0
\(846\) −0.879385 −0.879385
\(847\) − 1.87939i − 1.87939i
\(848\) − 1.87939i − 1.87939i
\(849\) 0 0
\(850\) 0 0
\(851\) 0.347296 0.347296
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.53209 1.53209
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0.652704 0.652704
\(865\) 0 0
\(866\) 0 0
\(867\) − 0.467911i − 0.467911i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −2.87939 −2.87939
\(872\) − 1.87939i − 1.87939i
\(873\) 0 0
\(874\) −0.347296 −0.347296
\(875\) 0 0
\(876\) −0.120615 −0.120615
\(877\) − 0.347296i − 0.347296i −0.984808 0.173648i \(-0.944444\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(878\) 0 0
\(879\) −0.532089 −0.532089
\(880\) 0 0
\(881\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) 2.22668i 2.22668i
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 2.34730 2.34730
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) − 0.347296i − 0.347296i
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 1.00000i − 1.00000i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.87939 1.87939
\(897\) 0.184793i 0.184793i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 2.87939 2.87939
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.87939i 1.87939i 0.342020 + 0.939693i \(0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(908\) − 1.87939i − 1.87939i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0.347296i 0.347296i
\(913\) 0 0
\(914\) 1.53209 1.53209
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 1.00000i 1.00000i
\(919\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(920\) 0 0
\(921\) −0.347296 −0.347296
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 1.00000 1.00000
\(927\) 0 0
\(928\) − 1.53209i − 1.53209i
\(929\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(930\) 0 0
\(931\) −2.53209 −2.53209
\(932\) − 1.00000i − 1.00000i
\(933\) − 0.652704i − 0.652704i
\(934\) 0 0
\(935\) 0 0
\(936\) −1.34730 −1.34730
\(937\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(938\) − 3.53209i − 3.53209i
\(939\) 0.532089 0.532089
\(940\) 0 0
\(941\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.347296 0.347296
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0.532089 0.532089
\(950\) 0 0
\(951\) 0.120615 0.120615
\(952\) 2.87939i 2.87939i
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) −1.65270 −1.65270
\(955\) 0 0
\(956\) 0.347296 0.347296
\(957\) 0 0
\(958\) − 1.00000i − 1.00000i
\(959\) −3.53209 −3.53209
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 1.53209i 1.53209i
\(963\) − 1.34730i − 1.34730i
\(964\) 0 0
\(965\) 0 0
\(966\) −0.226682 −0.226682
\(967\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 1.00000i 1.00000i
\(969\) −0.532089 −0.532089
\(970\) 0 0
\(971\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(972\) − 0.879385i − 0.879385i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.65270 −1.65270
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2.34730 2.34730
\(987\) − 0.652704i − 0.652704i
\(988\) − 1.53209i − 1.53209i
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) − 0.532089i − 0.532089i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 0 0
\(999\) −0.652704 −0.652704
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 3800.1.b.d.949.2 6
5.2 odd 4 3800.1.o.e.1101.2 yes 3
5.3 odd 4 3800.1.o.c.1101.2 3
5.4 even 2 inner 3800.1.b.d.949.5 6
8.5 even 2 3800.1.b.c.949.5 6
19.18 odd 2 3800.1.b.c.949.5 6
40.13 odd 4 3800.1.o.f.1101.2 yes 3
40.29 even 2 3800.1.b.c.949.2 6
40.37 odd 4 3800.1.o.d.1101.2 yes 3
95.18 even 4 3800.1.o.f.1101.2 yes 3
95.37 even 4 3800.1.o.d.1101.2 yes 3
95.94 odd 2 3800.1.b.c.949.2 6
152.37 odd 2 CM 3800.1.b.d.949.2 6
760.37 even 4 3800.1.o.e.1101.2 yes 3
760.189 odd 2 inner 3800.1.b.d.949.5 6
760.493 even 4 3800.1.o.c.1101.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.1.b.c.949.2 6 40.29 even 2
3800.1.b.c.949.2 6 95.94 odd 2
3800.1.b.c.949.5 6 8.5 even 2
3800.1.b.c.949.5 6 19.18 odd 2
3800.1.b.d.949.2 6 1.1 even 1 trivial
3800.1.b.d.949.2 6 152.37 odd 2 CM
3800.1.b.d.949.5 6 5.4 even 2 inner
3800.1.b.d.949.5 6 760.189 odd 2 inner
3800.1.o.c.1101.2 3 5.3 odd 4
3800.1.o.c.1101.2 3 760.493 even 4
3800.1.o.d.1101.2 yes 3 40.37 odd 4
3800.1.o.d.1101.2 yes 3 95.37 even 4
3800.1.o.e.1101.2 yes 3 5.2 odd 4
3800.1.o.e.1101.2 yes 3 760.37 even 4
3800.1.o.f.1101.2 yes 3 40.13 odd 4
3800.1.o.f.1101.2 yes 3 95.18 even 4