Properties

Label 2175.1.b.c.2174.4
Level $2175$
Weight $1$
Character 2175.2174
Analytic conductor $1.085$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -87
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,1,Mod(2174,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2175.2174");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(1.08546640248\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.895152515625.1

Embedding invariants

Embedding label 2174.4
Root \(-1.87939i\) of defining polynomial
Character \(\chi\) \(=\) 2175.2174
Dual form 2175.1.b.c.2174.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.347296i q^{2} -1.00000i q^{3} +0.879385 q^{4} +0.347296 q^{6} +1.87939i q^{7} +0.652704i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+0.347296i q^{2} -1.00000i q^{3} +0.879385 q^{4} +0.347296 q^{6} +1.87939i q^{7} +0.652704i q^{8} -1.00000 q^{9} -1.53209 q^{11} -0.879385i q^{12} +0.347296i q^{13} -0.652704 q^{14} +0.652704 q^{16} +1.53209i q^{17} -0.347296i q^{18} +1.87939 q^{21} -0.532089i q^{22} +0.652704 q^{24} -0.120615 q^{26} +1.00000i q^{27} +1.65270i q^{28} +1.00000 q^{29} +0.879385i q^{32} +1.53209i q^{33} -0.532089 q^{34} -0.879385 q^{36} +0.347296 q^{39} +1.00000 q^{41} +0.652704i q^{42} -1.34730 q^{44} -1.87939i q^{47} -0.652704i q^{48} -2.53209 q^{49} +1.53209 q^{51} +0.305407i q^{52} -0.347296 q^{54} -1.22668 q^{56} +0.347296i q^{58} -1.87939i q^{63} +0.347296 q^{64} -0.532089 q^{66} -1.53209i q^{67} +1.34730i q^{68} -0.652704i q^{72} -2.87939i q^{77} +0.120615i q^{78} +1.00000 q^{81} +0.347296i q^{82} +1.65270 q^{84} -1.00000i q^{87} -1.00000i q^{88} +0.347296 q^{89} -0.652704 q^{91} +0.652704 q^{94} +0.879385 q^{96} -0.879385i q^{98} +1.53209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 6 q^{9} - 6 q^{14} + 6 q^{16} + 6 q^{24} - 12 q^{26} + 6 q^{29} + 6 q^{34} + 6 q^{36} + 6 q^{41} - 6 q^{44} - 6 q^{49} + 6 q^{56} + 6 q^{66} + 6 q^{81} + 12 q^{84} - 6 q^{91} + 6 q^{94} - 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(3\) − 1.00000i − 1.00000i
\(4\) 0.879385 0.879385
\(5\) 0 0
\(6\) 0.347296 0.347296
\(7\) 1.87939i 1.87939i 0.342020 + 0.939693i \(0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(8\) 0.652704i 0.652704i
\(9\) −1.00000 −1.00000
\(10\) 0 0
\(11\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(12\) − 0.879385i − 0.879385i
\(13\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(14\) −0.652704 −0.652704
\(15\) 0 0
\(16\) 0.652704 0.652704
\(17\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(18\) − 0.347296i − 0.347296i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 1.87939 1.87939
\(22\) − 0.532089i − 0.532089i
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0.652704 0.652704
\(25\) 0 0
\(26\) −0.120615 −0.120615
\(27\) 1.00000i 1.00000i
\(28\) 1.65270i 1.65270i
\(29\) 1.00000 1.00000
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.879385i 0.879385i
\(33\) 1.53209i 1.53209i
\(34\) −0.532089 −0.532089
\(35\) 0 0
\(36\) −0.879385 −0.879385
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0.347296 0.347296
\(40\) 0 0
\(41\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0.652704i 0.652704i
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −1.34730 −1.34730
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.87939i − 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(48\) − 0.652704i − 0.652704i
\(49\) −2.53209 −2.53209
\(50\) 0 0
\(51\) 1.53209 1.53209
\(52\) 0.305407i 0.305407i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −0.347296 −0.347296
\(55\) 0 0
\(56\) −1.22668 −1.22668
\(57\) 0 0
\(58\) 0.347296i 0.347296i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 1.87939i − 1.87939i
\(64\) 0.347296 0.347296
\(65\) 0 0
\(66\) −0.532089 −0.532089
\(67\) − 1.53209i − 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(68\) 1.34730i 1.34730i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 0.652704i − 0.652704i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.87939i − 2.87939i
\(78\) 0.120615i 0.120615i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0.347296i 0.347296i
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.65270 1.65270
\(85\) 0 0
\(86\) 0 0
\(87\) − 1.00000i − 1.00000i
\(88\) − 1.00000i − 1.00000i
\(89\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(90\) 0 0
\(91\) −0.652704 −0.652704
\(92\) 0 0
\(93\) 0 0
\(94\) 0.652704 0.652704
\(95\) 0 0
\(96\) 0.879385 0.879385
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) − 0.879385i − 0.879385i
\(99\) 1.53209 1.53209
\(100\) 0 0
\(101\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(102\) 0.532089i 0.532089i
\(103\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(104\) −0.226682 −0.226682
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0.879385i 0.879385i
\(109\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.22668i 1.22668i
\(113\) 1.87939i 1.87939i 0.342020 + 0.939693i \(0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.879385 0.879385
\(117\) − 0.347296i − 0.347296i
\(118\) 0 0
\(119\) −2.87939 −2.87939
\(120\) 0 0
\(121\) 1.34730 1.34730
\(122\) 0 0
\(123\) − 1.00000i − 1.00000i
\(124\) 0 0
\(125\) 0 0
\(126\) 0.652704 0.652704
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000i 1.00000i
\(129\) 0 0
\(130\) 0 0
\(131\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(132\) 1.34730i 1.34730i
\(133\) 0 0
\(134\) 0.532089 0.532089
\(135\) 0 0
\(136\) −1.00000 −1.00000
\(137\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(138\) 0 0
\(139\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(140\) 0 0
\(141\) −1.87939 −1.87939
\(142\) 0 0
\(143\) − 0.532089i − 0.532089i
\(144\) −0.652704 −0.652704
\(145\) 0 0
\(146\) 0 0
\(147\) 2.53209i 2.53209i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(152\) 0 0
\(153\) − 1.53209i − 1.53209i
\(154\) 1.00000 1.00000
\(155\) 0 0
\(156\) 0.305407 0.305407
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0.347296i 0.347296i
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0.879385 0.879385
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.22668i 1.22668i
\(169\) 0.879385 0.879385
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0.347296 0.347296
\(175\) 0 0
\(176\) −1.00000 −1.00000
\(177\) 0 0
\(178\) 0.120615i 0.120615i
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(182\) − 0.226682i − 0.226682i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.34730i − 2.34730i
\(188\) − 1.65270i − 1.65270i
\(189\) −1.87939 −1.87939
\(190\) 0 0
\(191\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(192\) − 0.347296i − 0.347296i
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.22668 −2.22668
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0.532089i 0.532089i
\(199\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(200\) 0 0
\(201\) −1.53209 −1.53209
\(202\) − 0.120615i − 0.120615i
\(203\) 1.87939i 1.87939i
\(204\) 1.34730 1.34730
\(205\) 0 0
\(206\) 0.347296 0.347296
\(207\) 0 0
\(208\) 0.226682i 0.226682i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) −0.652704 −0.652704
\(217\) 0 0
\(218\) − 0.120615i − 0.120615i
\(219\) 0 0
\(220\) 0 0
\(221\) −0.532089 −0.532089
\(222\) 0 0
\(223\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(224\) −1.65270 −1.65270
\(225\) 0 0
\(226\) −0.652704 −0.652704
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −2.87939 −2.87939
\(232\) 0.652704i 0.652704i
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0.120615 0.120615
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) − 1.00000i − 1.00000i
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(242\) 0.467911i 0.467911i
\(243\) − 1.00000i − 1.00000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0.347296 0.347296
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(252\) − 1.65270i − 1.65270i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.00000 −1.00000
\(262\) 0.652704i 0.652704i
\(263\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) −1.00000 −1.00000
\(265\) 0 0
\(266\) 0 0
\(267\) − 0.347296i − 0.347296i
\(268\) − 1.34730i − 1.34730i
\(269\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 1.00000i 1.00000i
\(273\) 0.652704i 0.652704i
\(274\) 0.347296 0.347296
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.53209i − 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(278\) 0.652704i 0.652704i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) − 0.652704i − 0.652704i
\(283\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0.184793 0.184793
\(287\) 1.87939i 1.87939i
\(288\) − 0.879385i − 0.879385i
\(289\) −1.34730 −1.34730
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 0.347296i − 0.347296i −0.984808 0.173648i \(-0.944444\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(294\) −0.879385 −0.879385
\(295\) 0 0
\(296\) 0 0
\(297\) − 1.53209i − 1.53209i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) − 0.347296i − 0.347296i
\(303\) 0.347296i 0.347296i
\(304\) 0 0
\(305\) 0 0
\(306\) 0.532089 0.532089
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) − 2.53209i − 2.53209i
\(309\) −1.00000 −1.00000
\(310\) 0 0
\(311\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(312\) 0.226682i 0.226682i
\(313\) − 1.87939i − 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(318\) 0 0
\(319\) −1.53209 −1.53209
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.879385 0.879385
\(325\) 0 0
\(326\) 0 0
\(327\) 0.347296i 0.347296i
\(328\) 0.652704i 0.652704i
\(329\) 3.53209 3.53209
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 1.22668 1.22668
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0.305407i 0.305407i
\(339\) 1.87939 1.87939
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 2.87939i − 2.87939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) − 0.879385i − 0.879385i
\(349\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(350\) 0 0
\(351\) −0.347296 −0.347296
\(352\) − 1.34730i − 1.34730i
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.305407 0.305407
\(357\) 2.87939i 2.87939i
\(358\) 0 0
\(359\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) 0.532089i 0.532089i
\(363\) − 1.34730i − 1.34730i
\(364\) −0.573978 −0.573978
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −1.00000 −1.00000
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0.815207 0.815207
\(375\) 0 0
\(376\) 1.22668 1.22668
\(377\) 0.347296i 0.347296i
\(378\) − 0.652704i − 0.652704i
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 0.694593i − 0.694593i
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 1.00000
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 1.65270i − 1.65270i
\(393\) − 1.87939i − 1.87939i
\(394\) 0 0
\(395\) 0 0
\(396\) 1.34730 1.34730
\(397\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) − 0.120615i − 0.120615i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) − 0.532089i − 0.532089i
\(403\) 0 0
\(404\) −0.305407 −0.305407
\(405\) 0 0
\(406\) −0.652704 −0.652704
\(407\) 0 0
\(408\) 1.00000i 1.00000i
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −1.00000 −1.00000
\(412\) − 0.879385i − 0.879385i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −0.305407 −0.305407
\(417\) − 1.87939i − 1.87939i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 1.87939i 1.87939i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −0.532089 −0.532089
\(430\) 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
\(436\) −0.305407 −0.305407
\(437\) 0 0
\(438\) 0 0
\(439\) −1.53209 −1.53209 −0.766044 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(440\) 0 0
\(441\) 2.53209 2.53209
\(442\) − 0.184793i − 0.184793i
\(443\) − 0.347296i − 0.347296i −0.984808 0.173648i \(-0.944444\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −0.532089 −0.532089
\(447\) 0 0
\(448\) 0.652704i 0.652704i
\(449\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(450\) 0 0
\(451\) −1.53209 −1.53209
\(452\) 1.65270i 1.65270i
\(453\) 1.00000i 1.00000i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.87939i 1.87939i 0.342020 + 0.939693i \(0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(458\) 0 0
\(459\) −1.53209 −1.53209
\(460\) 0 0
\(461\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) − 1.00000i − 1.00000i
\(463\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(464\) 0.652704 0.652704
\(465\) 0 0
\(466\) 0 0
\(467\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(468\) − 0.305407i − 0.305407i
\(469\) 2.87939 2.87939
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −2.53209 −2.53209
\(477\) 0 0
\(478\) 0 0
\(479\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 0.652704i − 0.652704i
\(483\) 0 0
\(484\) 1.18479 1.18479
\(485\) 0 0
\(486\) 0.347296 0.347296
\(487\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(492\) − 0.879385i − 0.879385i
\(493\) 1.53209i 1.53209i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 0.120615i − 0.120615i
\(503\) − 0.347296i − 0.347296i −0.984808 0.173648i \(-0.944444\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(504\) 1.22668 1.22668
\(505\) 0 0
\(506\) 0 0
\(507\) − 0.879385i − 0.879385i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.87939i 2.87939i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) − 0.347296i − 0.347296i
\(523\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(524\) 1.65270 1.65270
\(525\) 0 0
\(526\) −0.347296 −0.347296
\(527\) 0 0
\(528\) 1.00000i 1.00000i
\(529\) −1.00000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.347296i 0.347296i
\(534\) 0.120615 0.120615
\(535\) 0 0
\(536\) 1.00000 1.00000
\(537\) 0 0
\(538\) 0.532089i 0.532089i
\(539\) 3.87939 3.87939
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) − 1.53209i − 1.53209i
\(544\) −1.34730 −1.34730
\(545\) 0 0
\(546\) −0.226682 −0.226682
\(547\) − 1.53209i − 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(548\) − 0.879385i − 0.879385i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0.532089 0.532089
\(555\) 0 0
\(556\) 1.65270 1.65270
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −2.34730 −2.34730
\(562\) 0 0
\(563\) − 1.53209i − 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(564\) −1.65270 −1.65270
\(565\) 0 0
\(566\) 0.347296 0.347296
\(567\) 1.87939i 1.87939i
\(568\) 0 0
\(569\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(570\) 0 0
\(571\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) − 0.467911i − 0.467911i
\(573\) 2.00000i 2.00000i
\(574\) −0.652704 −0.652704
\(575\) 0 0
\(576\) −0.347296 −0.347296
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) − 0.467911i − 0.467911i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0.120615 0.120615
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 2.22668i 2.22668i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0.532089 0.532089
\(595\) 0 0
\(596\) 0 0
\(597\) 0.347296i 0.347296i
\(598\) 0 0
\(599\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 1.53209i 1.53209i
\(604\) −0.879385 −0.879385
\(605\) 0 0
\(606\) −0.120615 −0.120615
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 1.87939 1.87939
\(610\) 0 0
\(611\) 0.652704 0.652704
\(612\) − 1.34730i − 1.34730i
\(613\) − 1.87939i − 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 1.87939 1.87939
\(617\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(618\) − 0.347296i − 0.347296i
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0.652704i 0.652704i
\(623\) 0.652704i 0.652704i
\(624\) 0.226682 0.226682
\(625\) 0 0
\(626\) 0.652704 0.652704
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −0.120615 −0.120615
\(635\) 0 0
\(636\) 0 0
\(637\) − 0.879385i − 0.879385i
\(638\) − 0.532089i − 0.532089i
\(639\) 0 0
\(640\) 0 0
\(641\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(642\) 0 0
\(643\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0.652704i 0.652704i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 1.53209i − 1.53209i −0.642788 0.766044i \(-0.722222\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(654\) −0.120615 −0.120615
\(655\) 0 0
\(656\) 0.652704 0.652704
\(657\) 0 0
\(658\) 1.22668i 1.22668i
\(659\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(660\) 0 0
\(661\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(662\) 0 0
\(663\) 0.532089i 0.532089i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.53209 1.53209
\(670\) 0 0
\(671\) 0 0
\(672\) 1.65270i 1.65270i
\(673\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0.773318 0.773318
\(677\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(678\) 0.652704i 0.652704i
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 1.00000
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(692\) 0 0
\(693\) 2.87939i 2.87939i
\(694\) 0 0
\(695\) 0 0
\(696\) 0.652704 0.652704
\(697\) 1.53209i 1.53209i
\(698\) 0.347296i 0.347296i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) − 0.120615i − 0.120615i
\(703\) 0 0
\(704\) −0.532089 −0.532089
\(705\) 0 0
\(706\) 0 0
\(707\) − 0.652704i − 0.652704i
\(708\) 0 0
\(709\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.226682i 0.226682i
\(713\) 0 0
\(714\) −1.00000 −1.00000
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) − 0.347296i − 0.347296i
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 1.87939 1.87939
\(722\) 0.347296i 0.347296i
\(723\) 1.87939i 1.87939i
\(724\) 1.34730 1.34730
\(725\) 0 0
\(726\) 0.467911 0.467911
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) − 0.426022i − 0.426022i
\(729\) −1.00000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.34730i 2.34730i
\(738\) − 0.347296i − 0.347296i
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.87939i 1.87939i 0.342020 + 0.939693i \(0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −0.694593 −0.694593
\(747\) 0 0
\(748\) − 2.06418i − 2.06418i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) − 1.22668i − 1.22668i
\(753\) 0.347296i 0.347296i
\(754\) −0.120615 −0.120615
\(755\) 0 0
\(756\) −1.65270 −1.65270
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) − 0.652704i − 0.652704i
\(764\) −1.75877 −1.75877
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0.120615i 0.120615i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 1.00000i 1.00000i
\(784\) −1.65270 −1.65270
\(785\) 0 0
\(786\) 0.652704 0.652704
\(787\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(788\) 0 0
\(789\) 1.00000 1.00000
\(790\) 0 0
\(791\) −3.53209 −3.53209
\(792\) 1.00000i 1.00000i
\(793\) 0 0
\(794\) −0.347296 −0.347296
\(795\) 0 0
\(796\) −0.305407 −0.305407
\(797\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(798\) 0 0
\(799\) 2.87939 2.87939
\(800\) 0 0
\(801\) −0.347296 −0.347296
\(802\) 0 0
\(803\) 0 0
\(804\) −1.34730 −1.34730
\(805\) 0 0
\(806\) 0 0
\(807\) − 1.53209i − 1.53209i
\(808\) − 0.226682i − 0.226682i
\(809\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(810\) 0 0
\(811\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(812\) 1.65270i 1.65270i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 1.00000 1.00000
\(817\) 0 0
\(818\) 0 0
\(819\) 0.652704 0.652704
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) − 0.347296i − 0.347296i
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0.652704 0.652704
\(825\) 0 0
\(826\) 0 0
\(827\) − 1.00000i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) −1.53209 −1.53209
\(832\) 0.120615i 0.120615i
\(833\) − 3.87939i − 3.87939i
\(834\) 0.652704 0.652704
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) −0.652704 −0.652704
\(847\) 2.53209i 2.53209i
\(848\) 0 0
\(849\) −1.00000 −1.00000
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) − 0.184793i − 0.184793i
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 1.87939 1.87939
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −0.879385 −0.879385
\(865\) 0 0
\(866\) 0 0
\(867\) 1.34730i 1.34730i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0.532089 0.532089
\(872\) − 0.226682i − 0.226682i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(878\) − 0.532089i − 0.532089i
\(879\) −0.347296 −0.347296
\(880\) 0 0
\(881\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(882\) 0.879385i 0.879385i
\(883\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) −0.467911 −0.467911
\(885\) 0 0
\(886\) 0.120615 0.120615
\(887\) − 1.87939i − 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.53209 −1.53209
\(892\) 1.34730i 1.34730i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −1.87939 −1.87939
\(897\) 0 0
\(898\) − 0.652704i − 0.652704i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) − 0.532089i − 0.532089i
\(903\) 0 0
\(904\) −1.22668 −1.22668
\(905\) 0 0
\(906\) −0.347296 −0.347296
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0.347296 0.347296
\(910\) 0 0
\(911\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.652704 −0.652704
\(915\) 0 0
\(916\) 0 0
\(917\) 3.53209i 3.53209i
\(918\) − 0.532089i − 0.532089i
\(919\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.347296i 0.347296i
\(923\) 0 0
\(924\) −2.53209 −2.53209
\(925\) 0 0
\(926\) −0.120615 −0.120615
\(927\) 1.00000i 1.00000i
\(928\) 0.879385i 0.879385i
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 1.87939i − 1.87939i
\(934\) 0.347296 0.347296
\(935\) 0 0
\(936\) 0.226682 0.226682
\(937\) − 0.347296i − 0.347296i −0.984808 0.173648i \(-0.944444\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(938\) 1.00000i 1.00000i
\(939\) −1.87939 −1.87939
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0.347296 0.347296
\(952\) − 1.87939i − 1.87939i
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.53209i 1.53209i
\(958\) − 0.347296i − 0.347296i
\(959\) 1.87939 1.87939
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) −1.65270 −1.65270
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0.879385i 0.879385i
\(969\) 0 0
\(970\) 0 0
\(971\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(972\) − 0.879385i − 0.879385i
\(973\) 3.53209i 3.53209i
\(974\) 0.694593 0.694593
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) −0.532089 −0.532089
\(980\) 0 0
\(981\) 0.347296 0.347296
\(982\) 0.347296i 0.347296i
\(983\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(984\) 0.652704 0.652704
\(985\) 0 0
\(986\) −0.532089 −0.532089
\(987\) − 3.53209i − 3.53209i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) − 0.120615i − 0.120615i
\(999\) 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 2175.1.b.c.2174.4 6
3.2 odd 2 2175.1.b.d.2174.3 6
5.2 odd 4 2175.1.h.d.1826.2 yes 3
5.3 odd 4 2175.1.h.f.1826.2 yes 3
5.4 even 2 inner 2175.1.b.c.2174.3 6
15.2 even 4 2175.1.h.e.1826.2 yes 3
15.8 even 4 2175.1.h.c.1826.2 3
15.14 odd 2 2175.1.b.d.2174.4 6
29.28 even 2 2175.1.b.d.2174.3 6
87.86 odd 2 CM 2175.1.b.c.2174.4 6
145.28 odd 4 2175.1.h.c.1826.2 3
145.57 odd 4 2175.1.h.e.1826.2 yes 3
145.144 even 2 2175.1.b.d.2174.4 6
435.173 even 4 2175.1.h.f.1826.2 yes 3
435.347 even 4 2175.1.h.d.1826.2 yes 3
435.434 odd 2 inner 2175.1.b.c.2174.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2175.1.b.c.2174.3 6 5.4 even 2 inner
2175.1.b.c.2174.3 6 435.434 odd 2 inner
2175.1.b.c.2174.4 6 1.1 even 1 trivial
2175.1.b.c.2174.4 6 87.86 odd 2 CM
2175.1.b.d.2174.3 6 3.2 odd 2
2175.1.b.d.2174.3 6 29.28 even 2
2175.1.b.d.2174.4 6 15.14 odd 2
2175.1.b.d.2174.4 6 145.144 even 2
2175.1.h.c.1826.2 3 15.8 even 4
2175.1.h.c.1826.2 3 145.28 odd 4
2175.1.h.d.1826.2 yes 3 5.2 odd 4
2175.1.h.d.1826.2 yes 3 435.347 even 4
2175.1.h.e.1826.2 yes 3 15.2 even 4
2175.1.h.e.1826.2 yes 3 145.57 odd 4
2175.1.h.f.1826.2 yes 3 5.3 odd 4
2175.1.h.f.1826.2 yes 3 435.173 even 4