Properties

Label 3312.1.bp.a.2437.2
Level $3312$
Weight $1$
Character 3312.2437
Analytic conductor $1.653$
Analytic rank $0$
Dimension $12$
Projective image $D_{36}$
CM discriminant -23
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3312,1,Mod(229,3312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3312, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 4, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3312.229");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3312.bp (of order \(12\), degree \(4\), 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: \(1.65290332184\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{36}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{36} - \cdots)\)

Embedding invariants

Embedding label 2437.2
Root \(-0.642788 - 0.766044i\) of defining polynomial
Character \(\chi\) \(=\) 3312.2437
Dual form 3312.1.bp.a.1885.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.342020 + 0.939693i) q^{2} +(0.766044 + 0.642788i) q^{3} +(-0.766044 + 0.642788i) q^{4} +(-0.342020 + 0.939693i) q^{6} +(-0.866025 - 0.500000i) q^{8} +(0.173648 + 0.984808i) q^{9} +O(q^{10})\) \(q+(0.342020 + 0.939693i) q^{2} +(0.766044 + 0.642788i) q^{3} +(-0.766044 + 0.642788i) q^{4} +(-0.342020 + 0.939693i) q^{6} +(-0.866025 - 0.500000i) q^{8} +(0.173648 + 0.984808i) q^{9} -1.00000 q^{12} +(1.92450 + 0.515668i) q^{13} +(0.173648 - 0.984808i) q^{16} +(-0.866025 + 0.500000i) q^{18} +(-0.866025 + 0.500000i) q^{23} +(-0.342020 - 0.939693i) q^{24} +(-0.866025 - 0.500000i) q^{25} +(0.173648 + 1.98481i) q^{26} +(-0.500000 + 0.866025i) q^{27} +(0.469139 + 1.75085i) q^{29} +(-0.342020 - 0.592396i) q^{31} +(0.984808 - 0.173648i) q^{32} +(-0.766044 - 0.642788i) q^{36} +(1.14279 + 1.63207i) q^{39} +(-0.300767 + 0.173648i) q^{41} +(-0.766044 - 0.642788i) q^{46} +(-0.173648 + 0.300767i) q^{47} +(0.766044 - 0.642788i) q^{48} +(0.500000 + 0.866025i) q^{49} +(0.173648 - 0.984808i) q^{50} +(-1.80572 + 0.842020i) q^{52} +(-0.984808 - 0.173648i) q^{54} +(-1.48481 + 1.03967i) q^{58} +(0.500000 - 1.86603i) q^{59} +(0.439693 - 0.524005i) q^{62} +(0.500000 + 0.866025i) q^{64} +(-0.984808 - 0.173648i) q^{69} -0.684040i q^{71} +(0.342020 - 0.939693i) q^{72} -1.87939i q^{73} +(-0.342020 - 0.939693i) q^{75} +(-1.14279 + 1.63207i) q^{78} +(-0.939693 + 0.342020i) q^{81} +(-0.266044 - 0.223238i) q^{82} +(-0.766044 + 1.64279i) q^{87} +(0.342020 - 0.939693i) q^{92} +(0.118782 - 0.673648i) q^{93} +(-0.342020 - 0.0603074i) q^{94} +(0.866025 + 0.500000i) q^{96} +(-0.642788 + 0.766044i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{12} - 6 q^{27} + 6 q^{39} + 6 q^{49} - 6 q^{58} + 6 q^{59} - 6 q^{62} + 6 q^{64} - 6 q^{78} + 6 q^{82}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3312\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(2305\) \(2485\) \(2945\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.342020 + 0.939693i 0.342020 + 0.939693i
\(3\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(4\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(5\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(6\) −0.342020 + 0.939693i −0.342020 + 0.939693i
\(7\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) −0.866025 0.500000i −0.866025 0.500000i
\(9\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(10\) 0 0
\(11\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(12\) −1.00000 −1.00000
\(13\) 1.92450 + 0.515668i 1.92450 + 0.515668i 0.984808 + 0.173648i \(0.0555556\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.173648 0.984808i 0.173648 0.984808i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(19\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(24\) −0.342020 0.939693i −0.342020 0.939693i
\(25\) −0.866025 0.500000i −0.866025 0.500000i
\(26\) 0.173648 + 1.98481i 0.173648 + 1.98481i
\(27\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(28\) 0 0
\(29\) 0.469139 + 1.75085i 0.469139 + 1.75085i 0.642788 + 0.766044i \(0.277778\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(30\) 0 0
\(31\) −0.342020 0.592396i −0.342020 0.592396i 0.642788 0.766044i \(-0.277778\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(32\) 0.984808 0.173648i 0.984808 0.173648i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.766044 0.642788i −0.766044 0.642788i
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 1.14279 + 1.63207i 1.14279 + 1.63207i
\(40\) 0 0
\(41\) −0.300767 + 0.173648i −0.300767 + 0.173648i −0.642788 0.766044i \(-0.722222\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(42\) 0 0
\(43\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.766044 0.642788i −0.766044 0.642788i
\(47\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(48\) 0.766044 0.642788i 0.766044 0.642788i
\(49\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(50\) 0.173648 0.984808i 0.173648 0.984808i
\(51\) 0 0
\(52\) −1.80572 + 0.842020i −1.80572 + 0.842020i
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) −0.984808 0.173648i −0.984808 0.173648i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −1.48481 + 1.03967i −1.48481 + 1.03967i
\(59\) 0.500000 1.86603i 0.500000 1.86603i 1.00000i \(-0.5\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(60\) 0 0
\(61\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(62\) 0.439693 0.524005i 0.439693 0.524005i
\(63\) 0 0
\(64\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(68\) 0 0
\(69\) −0.984808 0.173648i −0.984808 0.173648i
\(70\) 0 0
\(71\) 0.684040i 0.684040i −0.939693 0.342020i \(-0.888889\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(72\) 0.342020 0.939693i 0.342020 0.939693i
\(73\) 1.87939i 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(74\) 0 0
\(75\) −0.342020 0.939693i −0.342020 0.939693i
\(76\) 0 0
\(77\) 0 0
\(78\) −1.14279 + 1.63207i −1.14279 + 1.63207i
\(79\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(82\) −0.266044 0.223238i −0.266044 0.223238i
\(83\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.766044 + 1.64279i −0.766044 + 1.64279i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.342020 0.939693i 0.342020 0.939693i
\(93\) 0.118782 0.673648i 0.118782 0.673648i
\(94\) −0.342020 0.0603074i −0.342020 0.0603074i
\(95\) 0 0
\(96\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(97\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) −0.642788 + 0.766044i −0.642788 + 0.766044i
\(99\) 0 0
\(100\) 0.984808 0.173648i 0.984808 0.173648i
\(101\) −1.86603 + 0.500000i −1.86603 + 0.500000i −0.866025 + 0.500000i \(0.833333\pi\)
−1.00000 \(1.00000\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) −1.40883 1.40883i −1.40883 1.40883i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) −0.173648 0.984808i −0.173648 0.984808i
\(109\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.48481 1.03967i −1.48481 1.03967i
\(117\) −0.173648 + 1.98481i −0.173648 + 1.98481i
\(118\) 1.92450 0.168372i 1.92450 0.168372i
\(119\) 0 0
\(120\) 0 0
\(121\) 0.866025 0.500000i 0.866025 0.500000i
\(122\) 0 0
\(123\) −0.342020 0.0603074i −0.342020 0.0603074i
\(124\) 0.642788 + 0.233956i 0.642788 + 0.233956i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.347296 −0.347296 −0.173648 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(128\) −0.642788 + 0.766044i −0.642788 + 0.766044i
\(129\) 0 0
\(130\) 0 0
\(131\) −0.816436 0.218763i −0.816436 0.218763i −0.173648 0.984808i \(-0.555556\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(138\) −0.173648 0.984808i −0.173648 0.984808i
\(139\) 0.296905 1.10806i 0.296905 1.10806i −0.642788 0.766044i \(-0.722222\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(140\) 0 0
\(141\) −0.326352 + 0.118782i −0.326352 + 0.118782i
\(142\) 0.642788 0.233956i 0.642788 0.233956i
\(143\) 0 0
\(144\) 1.00000 1.00000
\(145\) 0 0
\(146\) 1.76604 0.642788i 1.76604 0.642788i
\(147\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(148\) 0 0
\(149\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(150\) 0.766044 0.642788i 0.766044 0.642788i
\(151\) 1.32683 + 0.766044i 1.32683 + 0.766044i 0.984808 0.173648i \(-0.0555556\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −1.92450 0.515668i −1.92450 0.515668i
\(157\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −0.642788 0.766044i −0.642788 0.766044i
\(163\) 1.40883 1.40883i 1.40883 1.40883i 0.642788 0.766044i \(-0.277778\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(164\) 0.118782 0.326352i 0.118782 0.326352i
\(165\) 0 0
\(166\) 0 0
\(167\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(168\) 0 0
\(169\) 2.57176 + 1.48481i 2.57176 + 1.48481i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.366025 + 1.36603i 0.366025 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) −1.80572 0.157980i −1.80572 0.157980i
\(175\) 0 0
\(176\) 0 0
\(177\) 1.58248 1.10806i 1.58248 1.10806i
\(178\) 0 0
\(179\) 0.811160 0.811160i 0.811160 0.811160i −0.173648 0.984808i \(-0.555556\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(180\) 0 0
\(181\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.00000 1.00000
\(185\) 0 0
\(186\) 0.673648 0.118782i 0.673648 0.118782i
\(187\) 0 0
\(188\) −0.0603074 0.342020i −0.0603074 0.342020i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(193\) −0.173648 0.300767i −0.173648 0.300767i 0.766044 0.642788i \(-0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.939693 0.342020i −0.939693 0.342020i
\(197\) −0.811160 0.811160i −0.811160 0.811160i 0.173648 0.984808i \(-0.444444\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(201\) 0 0
\(202\) −1.10806 1.58248i −1.10806 1.58248i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.642788 0.766044i −0.642788 0.766044i
\(208\) 0.842020 1.80572i 0.842020 1.80572i
\(209\) 0 0
\(210\) 0 0
\(211\) −1.86603 0.500000i −1.86603 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0.439693 0.524005i 0.439693 0.524005i
\(214\) 0 0
\(215\) 0 0
\(216\) 0.866025 0.500000i 0.866025 0.500000i
\(217\) 0 0
\(218\) 0 0
\(219\) 1.20805 1.43969i 1.20805 1.43969i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(224\) 0 0
\(225\) 0.342020 0.939693i 0.342020 0.939693i
\(226\) 0 0
\(227\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(228\) 0 0
\(229\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.469139 1.75085i 0.469139 1.75085i
\(233\) 1.28558i 1.28558i 0.766044 + 0.642788i \(0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(234\) −1.92450 + 0.515668i −1.92450 + 0.515668i
\(235\) 0 0
\(236\) 0.816436 + 1.75085i 0.816436 + 1.75085i
\(237\) 0 0
\(238\) 0 0
\(239\) 0.642788 + 1.11334i 0.642788 + 1.11334i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(240\) 0 0
\(241\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(243\) −0.939693 0.342020i −0.939693 0.342020i
\(244\) 0 0
\(245\) 0 0
\(246\) −0.0603074 0.342020i −0.0603074 0.342020i
\(247\) 0 0
\(248\) 0.684040i 0.684040i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −0.118782 0.326352i −0.118782 0.326352i
\(255\) 0 0
\(256\) −0.939693 0.342020i −0.939693 0.342020i
\(257\) 0.342020 + 0.592396i 0.342020 + 0.592396i 0.984808 0.173648i \(-0.0555556\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.64279 + 0.766044i −1.64279 + 0.766044i
\(262\) −0.0736672 0.842020i −0.0736672 0.842020i
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.40883 1.40883i 1.40883 1.40883i 0.642788 0.766044i \(-0.277778\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(270\) 0 0
\(271\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0.866025 0.500000i 0.866025 0.500000i
\(277\) 1.75085 0.469139i 1.75085 0.469139i 0.766044 0.642788i \(-0.222222\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(278\) 1.14279 0.0999810i 1.14279 0.0999810i
\(279\) 0.524005 0.439693i 0.524005 0.439693i
\(280\) 0 0
\(281\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(282\) −0.223238 0.266044i −0.223238 0.266044i
\(283\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(284\) 0.439693 + 0.524005i 0.439693 + 0.524005i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.342020 + 0.939693i 0.342020 + 0.939693i
\(289\) 1.00000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 1.20805 + 1.43969i 1.20805 + 1.43969i
\(293\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(294\) −0.984808 + 0.173648i −0.984808 + 0.173648i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.92450 + 0.515668i −1.92450 + 0.515668i
\(300\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(301\) 0 0
\(302\) −0.266044 + 1.50881i −0.266044 + 1.50881i
\(303\) −1.75085 0.816436i −1.75085 0.816436i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0.366025 0.366025i 0.366025 0.366025i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.32683 + 0.766044i −1.32683 + 0.766044i −0.984808 0.173648i \(-0.944444\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(312\) −0.173648 1.98481i −0.173648 1.98481i
\(313\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.133975 0.500000i −0.133975 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.500000 0.866025i 0.500000 0.866025i
\(325\) −1.40883 1.40883i −1.40883 1.40883i
\(326\) 1.80572 + 0.842020i 1.80572 + 0.842020i
\(327\) 0 0
\(328\) 0.347296 0.347296
\(329\) 0 0
\(330\) 0 0
\(331\) 0.816436 0.218763i 0.816436 0.218763i 0.173648 0.984808i \(-0.444444\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 1.32683 + 1.11334i 1.32683 + 1.11334i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(338\) −0.515668 + 2.92450i −0.515668 + 2.92450i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −1.15846 + 0.811160i −1.15846 + 0.811160i
\(347\) 0.366025 1.36603i 0.366025 1.36603i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(348\) −0.469139 1.75085i −0.469139 1.75085i
\(349\) 0.424024 + 1.58248i 0.424024 + 1.58248i 0.766044 + 0.642788i \(0.222222\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(350\) 0 0
\(351\) −1.40883 + 1.40883i −1.40883 + 1.40883i
\(352\) 0 0
\(353\) 0.766044 1.32683i 0.766044 1.32683i −0.173648 0.984808i \(-0.555556\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(354\) 1.58248 + 1.10806i 1.58248 + 1.10806i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1.03967 + 0.484808i 1.03967 + 0.484808i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000i 1.00000i
\(362\) 0 0
\(363\) 0.984808 + 0.173648i 0.984808 + 0.173648i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) 0.342020 + 0.939693i 0.342020 + 0.939693i
\(369\) −0.223238 0.266044i −0.223238 0.266044i
\(370\) 0 0
\(371\) 0 0
\(372\) 0.342020 + 0.592396i 0.342020 + 0.592396i
\(373\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0.300767 0.173648i 0.300767 0.173648i
\(377\) 3.61144i 3.61144i
\(378\) 0 0
\(379\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(380\) 0 0
\(381\) −0.266044 0.223238i −0.266044 0.223238i
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) −0.984808 + 0.173648i −0.984808 + 0.173648i
\(385\) 0 0
\(386\) 0.223238 0.266044i 0.223238 0.266044i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000i 1.00000i
\(393\) −0.484808 0.692377i −0.484808 0.692377i
\(394\) 0.484808 1.03967i 0.484808 1.03967i
\(395\) 0 0
\(396\) 0 0
\(397\) 1.15846 1.15846i 1.15846 1.15846i 0.173648 0.984808i \(-0.444444\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.642788 + 0.766044i −0.642788 + 0.766044i
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) −0.352738 1.31644i −0.352738 1.31644i
\(404\) 1.10806 1.58248i 1.10806 1.58248i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.592396 + 0.342020i −0.592396 + 0.342020i −0.766044 0.642788i \(-0.777778\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0.500000 0.866025i 0.500000 0.866025i
\(415\) 0 0
\(416\) 1.98481 + 0.173648i 1.98481 + 0.173648i
\(417\) 0.939693 0.657980i 0.939693 0.657980i
\(418\) 0 0
\(419\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(420\) 0 0
\(421\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(422\) −0.168372 1.92450i −0.168372 1.92450i
\(423\) −0.326352 0.118782i −0.326352 0.118782i
\(424\) 0 0
\(425\) 0 0
\(426\) 0.642788 + 0.233956i 0.642788 + 0.233956i
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 1.76604 + 0.642788i 1.76604 + 0.642788i
\(439\) −1.11334 0.642788i −1.11334 0.642788i −0.173648 0.984808i \(-0.555556\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(440\) 0 0
\(441\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(442\) 0 0
\(443\) −1.92450 + 0.515668i −1.92450 + 0.515668i −0.939693 + 0.342020i \(0.888889\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1.70574 0.300767i −1.70574 0.300767i
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 1.00000 1.00000
\(451\) 0 0
\(452\) 0 0
\(453\) 0.524005 + 1.43969i 0.524005 + 1.43969i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.424024 1.58248i −0.424024 1.58248i −0.766044 0.642788i \(-0.777778\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(462\) 0 0
\(463\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(464\) 1.80572 0.157980i 1.80572 0.157980i
\(465\) 0 0
\(466\) −1.20805 + 0.439693i −1.20805 + 0.439693i
\(467\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(468\) −1.14279 1.63207i −1.14279 1.63207i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −1.36603 + 1.36603i −1.36603 + 1.36603i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −0.826352 + 0.984808i −0.826352 + 0.984808i
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.342020 + 0.939693i −0.342020 + 0.939693i
\(485\) 0 0
\(486\) 1.00000i 1.00000i
\(487\) 1.96962i 1.96962i 0.173648 + 0.984808i \(0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(488\) 0 0
\(489\) 1.98481 0.173648i 1.98481 0.173648i
\(490\) 0 0
\(491\) −0.424024 + 1.58248i −0.424024 + 1.58248i 0.342020 + 0.939693i \(0.388889\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(492\) 0.300767 0.173648i 0.300767 0.173648i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −0.642788 + 0.233956i −0.642788 + 0.233956i
\(497\) 0 0
\(498\) 0 0
\(499\) −0.168372 0.0451151i −0.168372 0.0451151i 0.173648 0.984808i \(-0.444444\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(500\) 0 0
\(501\) 1.70574 + 0.300767i 1.70574 + 0.300767i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.01567 + 2.79053i 1.01567 + 2.79053i
\(508\) 0.266044 0.223238i 0.266044 0.223238i
\(509\) −1.92450 0.515668i −1.92450 0.515668i −0.984808 0.173648i \(-0.944444\pi\)
−0.939693 0.342020i \(-0.888889\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 0 0
\(514\) −0.439693 + 0.524005i −0.439693 + 0.524005i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −0.597672 + 1.28171i −0.597672 + 1.28171i
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −1.28171 1.28171i −1.28171 1.28171i
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0.766044 0.357212i 0.766044 0.357212i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.500000 0.866025i 0.500000 0.866025i
\(530\) 0 0
\(531\) 1.92450 + 0.168372i 1.92450 + 0.168372i
\(532\) 0 0
\(533\) −0.668372 + 0.179090i −0.668372 + 0.179090i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.14279 0.0999810i 1.14279 0.0999810i
\(538\) 1.80572 + 0.842020i 1.80572 + 0.842020i
\(539\) 0 0
\(540\) 0 0
\(541\) 1.28171 1.28171i 1.28171 1.28171i 0.342020 0.939693i \(-0.388889\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(542\) −0.342020 0.939693i −0.342020 0.939693i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.424024 1.58248i −0.424024 1.58248i −0.766044 0.642788i \(-0.777778\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(553\) 0 0
\(554\) 1.03967 + 1.48481i 1.03967 + 1.48481i
\(555\) 0 0
\(556\) 0.484808 + 1.03967i 0.484808 + 1.03967i
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0.592396 + 0.342020i 0.592396 + 0.342020i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(564\) 0.173648 0.300767i 0.173648 0.300767i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −0.342020 + 0.592396i −0.342020 + 0.592396i
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 1.00000
\(576\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(577\) −1.28558 −1.28558 −0.642788 0.766044i \(-0.722222\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(578\) 0.342020 + 0.939693i 0.342020 + 0.939693i
\(579\) 0.0603074 0.342020i 0.0603074 0.342020i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.939693 + 1.62760i −0.939693 + 1.62760i
\(585\) 0 0
\(586\) 0 0
\(587\) −1.75085 + 0.469139i −1.75085 + 0.469139i −0.984808 0.173648i \(-0.944444\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(588\) −0.500000 0.866025i −0.500000 0.866025i
\(589\) 0 0
\(590\) 0 0
\(591\) −0.0999810 1.14279i −0.0999810 1.14279i
\(592\) 0 0
\(593\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −1.14279 1.63207i −1.14279 1.63207i
\(599\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(600\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(601\) −1.32683 0.766044i −1.32683 0.766044i −0.342020 0.939693i \(-0.611111\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.50881 + 0.266044i −1.50881 + 0.266044i
\(605\) 0 0
\(606\) 0.168372 1.92450i 0.168372 1.92450i
\(607\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.489282 + 0.489282i −0.489282 + 0.489282i
\(612\) 0 0
\(613\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(614\) 0.469139 + 0.218763i 0.469139 + 0.218763i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(618\) 0 0
\(619\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(620\) 0 0
\(621\) 1.00000i 1.00000i
\(622\) −1.17365 0.984808i −1.17365 0.984808i
\(623\) 0 0
\(624\) 1.80572 0.842020i 1.80572 0.842020i
\(625\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −1.10806 1.58248i −1.10806 1.58248i
\(634\) 0.424024 0.296905i 0.424024 0.296905i
\(635\) 0 0
\(636\) 0 0
\(637\) 0.515668 + 1.92450i 0.515668 + 1.92450i
\(638\) 0 0
\(639\) 0.673648 0.118782i 0.673648 0.118782i
\(640\) 0 0
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.87939i 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(648\) 0.984808 + 0.173648i 0.984808 + 0.173648i
\(649\) 0 0
\(650\) 0.842020 1.80572i 0.842020 1.80572i
\(651\) 0 0
\(652\) −0.173648 + 1.98481i −0.173648 + 1.98481i
\(653\) −1.10806 0.296905i −1.10806 0.296905i −0.342020 0.939693i \(-0.611111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.118782 + 0.326352i 0.118782 + 0.326352i
\(657\) 1.85083 0.326352i 1.85083 0.326352i
\(658\) 0 0
\(659\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(660\) 0 0
\(661\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(662\) 0.484808 + 0.692377i 0.484808 + 0.692377i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.28171 1.28171i −1.28171 1.28171i
\(668\) −0.592396 + 1.62760i −0.592396 + 1.62760i
\(669\) −1.62760 + 0.592396i −1.62760 + 0.592396i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.342020 + 0.592396i −0.342020 + 0.592396i −0.984808 0.173648i \(-0.944444\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(674\) 0 0
\(675\) 0.866025 0.500000i 0.866025 0.500000i
\(676\) −2.92450 + 0.515668i −2.92450 + 0.515668i
\(677\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.811160 + 0.811160i 0.811160 + 0.811160i 0.984808 0.173648i \(-0.0555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.500000 1.86603i −0.500000 1.86603i −0.500000 0.866025i \(-0.666667\pi\)
1.00000i \(-0.5\pi\)
\(692\) −1.15846 0.811160i −1.15846 0.811160i
\(693\) 0 0
\(694\) 1.40883 0.123257i 1.40883 0.123257i
\(695\) 0 0
\(696\) 1.48481 1.03967i 1.48481 1.03967i
\(697\) 0 0
\(698\) −1.34202 + 0.939693i −1.34202 + 0.939693i
\(699\) −0.826352 + 0.984808i −0.826352 + 0.984808i
\(700\) 0 0
\(701\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(702\) −1.80572 0.842020i −1.80572 0.842020i
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 1.50881 + 0.266044i 1.50881 + 0.266044i
\(707\) 0 0
\(708\) −0.500000 + 1.86603i −0.500000 + 1.86603i
\(709\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.592396 + 0.342020i 0.592396 + 0.342020i
\(714\) 0 0
\(715\) 0 0
\(716\) −0.0999810 + 1.14279i −0.0999810 + 1.14279i
\(717\) −0.223238 + 1.26604i −0.223238 + 1.26604i
\(718\) 0 0
\(719\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.939693 0.342020i 0.939693 0.342020i
\(723\) 0 0
\(724\) 0 0
\(725\) 0.469139 1.75085i 0.469139 1.75085i
\(726\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(727\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(728\) 0 0
\(729\) −0.500000 0.866025i −0.500000 0.866025i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(737\) 0 0
\(738\) 0.173648 0.300767i 0.173648 0.300767i
\(739\) −1.40883 + 1.40883i −1.40883 + 1.40883i −0.642788 + 0.766044i \(0.722222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(744\) −0.439693 + 0.524005i −0.439693 + 0.524005i
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) 0.266044 + 0.223238i 0.266044 + 0.223238i
\(753\) 0 0
\(754\) −3.39364 + 1.23518i −3.39364 + 1.23518i
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.11334 0.642788i 1.11334 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(762\) 0.118782 0.326352i 0.118782 0.326352i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.92450 3.33333i 1.92450 3.33333i
\(768\) −0.500000 0.866025i −0.500000 0.866025i
\(769\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(770\) 0 0
\(771\) −0.118782 + 0.673648i −0.118782 + 0.673648i
\(772\) 0.326352 + 0.118782i 0.326352 + 0.118782i
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) 0.684040i 0.684040i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.75085 0.469139i −1.75085 0.469139i
\(784\) 0.939693 0.342020i 0.939693 0.342020i
\(785\) 0 0
\(786\) 0.484808 0.692377i 0.484808 0.692377i
\(787\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(788\) 1.14279 + 0.0999810i 1.14279 + 0.0999810i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 1.48481 + 0.692377i 1.48481 + 0.692377i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.939693 0.342020i −0.939693 0.342020i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 1.11640 0.781713i 1.11640 0.781713i
\(807\) 1.98481 0.173648i 1.98481 0.173648i
\(808\) 1.86603 + 0.500000i 1.86603 + 0.500000i
\(809\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0 0
\(811\) 1.28171 + 1.28171i 1.28171 + 1.28171i 0.939693 + 0.342020i \(0.111111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(812\) 0 0
\(813\) −0.766044 0.642788i −0.766044 0.642788i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −0.524005 0.439693i −0.524005 0.439693i
\(819\) 0 0
\(820\) 0 0
\(821\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 1.32683 0.766044i 1.32683 0.766044i 0.342020 0.939693i \(-0.388889\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0.984808 + 0.173648i 0.984808 + 0.173648i
\(829\) −1.36603 + 1.36603i −1.36603 + 1.36603i −0.500000 + 0.866025i \(0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(830\) 0 0
\(831\) 1.64279 + 0.766044i 1.64279 + 0.766044i
\(832\) 0.515668 + 1.92450i 0.515668 + 1.92450i
\(833\) 0 0
\(834\) 0.939693 + 0.657980i 0.939693 + 0.657980i
\(835\) 0 0
\(836\) 0 0
\(837\) 0.684040 0.684040
\(838\) 0 0
\(839\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(840\) 0 0
\(841\) −1.97937 + 1.14279i −1.97937 + 1.14279i
\(842\) 0 0
\(843\) 0 0
\(844\) 1.75085 0.816436i 1.75085 0.816436i
\(845\) 0 0
\(846\) 0.347296i 0.347296i
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0.684040i 0.684040i
\(853\) 0.500000 0.133975i 0.500000 0.133975i 1.00000i \(-0.5\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.70574 0.984808i −1.70574 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
−0.766044 0.642788i \(-0.777778\pi\)
\(858\) 0 0
\(859\) 0.296905 1.10806i 0.296905 1.10806i −0.642788 0.766044i \(-0.722222\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.28558 −1.28558 −0.642788 0.766044i \(-0.722222\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(864\) −0.342020 + 0.939693i −0.342020 + 0.939693i
\(865\) 0 0
\(866\) 0 0
\(867\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 1.87939i 1.87939i
\(877\) 0.500000 + 0.133975i 0.500000 + 0.133975i 0.500000 0.866025i \(-0.333333\pi\)
1.00000i \(0.5\pi\)
\(878\) 0.223238 1.26604i 0.223238 1.26604i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −0.866025 0.500000i −0.866025 0.500000i
\(883\) −1.36603 + 1.36603i −1.36603 + 1.36603i −0.500000 + 0.866025i \(0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1.14279 1.63207i −1.14279 1.63207i
\(887\) 1.70574 0.984808i 1.70574 0.984808i 0.766044 0.642788i \(-0.222222\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −0.300767 1.70574i −0.300767 1.70574i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.80572 0.842020i −1.80572 0.842020i
\(898\) 0 0
\(899\) 0.876743 0.876743i 0.876743 0.876743i
\(900\) 0.342020 + 0.939693i 0.342020 + 0.939693i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −1.17365 + 0.984808i −1.17365 + 0.984808i
\(907\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(908\) 0 0
\(909\) −0.816436 1.75085i −0.816436 1.75085i
\(910\) 0 0
\(911\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0.515668 0.0451151i 0.515668 0.0451151i
\(922\) 1.34202 0.939693i 1.34202 0.939693i
\(923\) 0.352738 1.31644i 0.352738 1.31644i
\(924\) 0 0
\(925\) 0 0
\(926\) 0.642788 0.766044i 0.642788 0.766044i
\(927\) 0 0
\(928\) 0.766044 + 1.64279i 0.766044 + 1.64279i
\(929\) −0.984808 + 1.70574i −0.984808 + 1.70574i −0.342020 + 0.939693i \(0.611111\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.826352 0.984808i −0.826352 0.984808i
\(933\) −1.50881 0.266044i −1.50881 0.266044i
\(934\) 0 0
\(935\) 0 0
\(936\) 1.14279 1.63207i 1.14279 1.63207i
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(942\) 0 0
\(943\) 0.173648 0.300767i 0.173648 0.300767i
\(944\) −1.75085 0.816436i −1.75085 0.816436i
\(945\) 0 0
\(946\) 0 0
\(947\) −0.469139 1.75085i −0.469139 1.75085i −0.642788 0.766044i \(-0.722222\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(948\) 0 0
\(949\) 0.969139 3.61688i 0.969139 3.61688i
\(950\) 0 0
\(951\) 0.218763 0.469139i 0.218763 0.469139i
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.20805 0.439693i −1.20805 0.439693i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.266044 0.460802i 0.266044 0.460802i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.62760 + 0.939693i −1.62760 + 0.939693i −0.642788 + 0.766044i \(0.722222\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(968\) −1.00000 −1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(972\) 0.939693 0.342020i 0.939693 0.342020i
\(973\) 0 0
\(974\) −1.85083 + 0.673648i −1.85083 + 0.673648i
\(975\) −0.173648 1.98481i −0.173648 1.98481i
\(976\) 0 0
\(977\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(978\) 0.842020 + 1.80572i 0.842020 + 1.80572i
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −1.63207 + 0.142788i −1.63207 + 0.142788i
\(983\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(984\) 0.266044 + 0.223238i 0.266044 + 0.223238i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(992\) −0.439693 0.524005i −0.439693 0.524005i
\(993\) 0.766044 + 0.357212i 0.766044 + 0.357212i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.86603 0.500000i 1.86603 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000 \(0\)
\(998\) −0.0151922 0.173648i −0.0151922 0.173648i
\(999\) 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 3312.1.bp.a.2437.2 yes 12
9.4 even 3 3312.1.bp.b.229.1 yes 12
16.13 even 4 3312.1.bp.b.781.1 yes 12
23.22 odd 2 CM 3312.1.bp.a.2437.2 yes 12
144.13 even 12 inner 3312.1.bp.a.1885.2 12
207.22 odd 6 3312.1.bp.b.229.1 yes 12
368.45 odd 4 3312.1.bp.b.781.1 yes 12
3312.1885 odd 12 inner 3312.1.bp.a.1885.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3312.1.bp.a.1885.2 12 144.13 even 12 inner
3312.1.bp.a.1885.2 12 3312.1885 odd 12 inner
3312.1.bp.a.2437.2 yes 12 1.1 even 1 trivial
3312.1.bp.a.2437.2 yes 12 23.22 odd 2 CM
3312.1.bp.b.229.1 yes 12 9.4 even 3
3312.1.bp.b.229.1 yes 12 207.22 odd 6
3312.1.bp.b.781.1 yes 12 16.13 even 4
3312.1.bp.b.781.1 yes 12 368.45 odd 4