Properties

Label 3192.1.eb.a.2621.1
Level $3192$
Weight $1$
Character 3192.2621
Analytic conductor $1.593$
Analytic rank $0$
Dimension $12$
Projective image $D_{18}$
CM discriminant -152
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3192,1,Mod(341,3192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3192, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 5, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3192.341");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3192.eb (of order \(6\), degree \(2\), 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: \(1.59301552032\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

Embedding invariants

Embedding label 2621.1
Root \(0.984808 + 0.173648i\) of defining polynomial
Character \(\chi\) \(=\) 3192.2621
Dual form 3192.1.eb.a.341.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(-0.984808 - 0.173648i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.939693 - 0.342020i) q^{6} +(0.766044 + 0.642788i) q^{7} +1.00000i q^{8} +(0.939693 + 0.342020i) q^{9} +(-0.642788 + 0.766044i) q^{12} +1.53209i q^{13} +(-0.984808 - 0.173648i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.173648 + 0.300767i) q^{17} +(-0.984808 + 0.173648i) q^{18} +(0.866025 - 0.500000i) q^{19} +(-0.642788 - 0.766044i) q^{21} +(-1.70574 + 0.984808i) q^{23} +(0.173648 - 0.984808i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-0.766044 - 1.32683i) q^{26} +(-0.866025 - 0.500000i) q^{27} +(0.939693 - 0.342020i) q^{28} +1.87939i q^{29} +(0.866025 + 0.500000i) q^{32} -0.347296i q^{34} +(0.766044 - 0.642788i) q^{36} +(-0.866025 - 1.50000i) q^{37} +(-0.500000 + 0.866025i) q^{38} +(0.266044 - 1.50881i) q^{39} +(0.939693 + 0.342020i) q^{42} +(0.984808 - 1.70574i) q^{46} +(-0.500000 - 0.866025i) q^{47} +(0.342020 + 0.939693i) q^{48} +(0.173648 + 0.984808i) q^{49} -1.00000i q^{50} +(0.223238 - 0.266044i) q^{51} +(1.32683 + 0.766044i) q^{52} +(-0.300767 - 0.173648i) q^{53} +1.00000 q^{54} +(-0.642788 + 0.766044i) q^{56} +(-0.939693 + 0.342020i) q^{57} +(-0.939693 - 1.62760i) q^{58} +(0.984808 - 1.70574i) q^{59} +(0.500000 + 0.866025i) q^{63} -1.00000 q^{64} +(-0.342020 + 0.592396i) q^{67} +(0.173648 + 0.300767i) q^{68} +(1.85083 - 0.673648i) q^{69} +(-0.342020 + 0.939693i) q^{72} +(-1.11334 - 0.642788i) q^{73} +(1.50000 + 0.866025i) q^{74} +(0.642788 - 0.766044i) q^{75} -1.00000i q^{76} +(0.524005 + 1.43969i) q^{78} +(0.766044 + 0.642788i) q^{81} +(-0.984808 + 0.173648i) q^{84} +(0.326352 - 1.85083i) q^{87} +(-0.984808 + 1.17365i) q^{91} +1.96962i q^{92} +(0.866025 + 0.500000i) q^{94} +(-0.766044 - 0.642788i) q^{96} +(-0.642788 - 0.766044i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} - 6 q^{16} - 6 q^{25} - 6 q^{38} - 6 q^{39} - 6 q^{47} + 12 q^{54} + 6 q^{63} - 12 q^{64} + 18 q^{74} + 6 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3192\mathbb{Z}\right)^\times\).

\(n\) \(799\) \(913\) \(1009\) \(1597\) \(2129\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(3\) −0.984808 0.173648i −0.984808 0.173648i
\(4\) 0.500000 0.866025i 0.500000 0.866025i
\(5\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0.939693 0.342020i 0.939693 0.342020i
\(7\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(8\) 1.00000i 1.00000i
\(9\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) −0.642788 + 0.766044i −0.642788 + 0.766044i
\(13\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(14\) −0.984808 0.173648i −0.984808 0.173648i
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.500000 0.866025i
\(17\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(18\) −0.984808 + 0.173648i −0.984808 + 0.173648i
\(19\) 0.866025 0.500000i 0.866025 0.500000i
\(20\) 0 0
\(21\) −0.642788 0.766044i −0.642788 0.766044i
\(22\) 0 0
\(23\) −1.70574 + 0.984808i −1.70574 + 0.984808i −0.766044 + 0.642788i \(0.777778\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(24\) 0.173648 0.984808i 0.173648 0.984808i
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) −0.766044 1.32683i −0.766044 1.32683i
\(27\) −0.866025 0.500000i −0.866025 0.500000i
\(28\) 0.939693 0.342020i 0.939693 0.342020i
\(29\) 1.87939i 1.87939i 0.342020 + 0.939693i \(0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(33\) 0 0
\(34\) 0.347296i 0.347296i
\(35\) 0 0
\(36\) 0.766044 0.642788i 0.766044 0.642788i
\(37\) −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(38\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(39\) 0.266044 1.50881i 0.266044 1.50881i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.984808 1.70574i 0.984808 1.70574i
\(47\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(48\) 0.342020 + 0.939693i 0.342020 + 0.939693i
\(49\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(50\) 1.00000i 1.00000i
\(51\) 0.223238 0.266044i 0.223238 0.266044i
\(52\) 1.32683 + 0.766044i 1.32683 + 0.766044i
\(53\) −0.300767 0.173648i −0.300767 0.173648i 0.342020 0.939693i \(-0.388889\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(54\) 1.00000 1.00000
\(55\) 0 0
\(56\) −0.642788 + 0.766044i −0.642788 + 0.766044i
\(57\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(58\) −0.939693 1.62760i −0.939693 1.62760i
\(59\) 0.984808 1.70574i 0.984808 1.70574i 0.342020 0.939693i \(-0.388889\pi\)
0.642788 0.766044i \(-0.277778\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −0.342020 + 0.592396i −0.342020 + 0.592396i −0.984808 0.173648i \(-0.944444\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(68\) 0.173648 + 0.300767i 0.173648 + 0.300767i
\(69\) 1.85083 0.673648i 1.85083 0.673648i
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −0.342020 + 0.939693i −0.342020 + 0.939693i
\(73\) −1.11334 0.642788i −1.11334 0.642788i −0.173648 0.984808i \(-0.555556\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(74\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(75\) 0.642788 0.766044i 0.642788 0.766044i
\(76\) 1.00000i 1.00000i
\(77\) 0 0
\(78\) 0.524005 + 1.43969i 0.524005 + 1.43969i
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −0.984808 + 0.173648i −0.984808 + 0.173648i
\(85\) 0 0
\(86\) 0 0
\(87\) 0.326352 1.85083i 0.326352 1.85083i
\(88\) 0 0
\(89\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) −0.984808 + 1.17365i −0.984808 + 1.17365i
\(92\) 1.96962i 1.96962i
\(93\) 0 0
\(94\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(95\) 0 0
\(96\) −0.766044 0.642788i −0.766044 0.642788i
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −0.642788 0.766044i −0.642788 0.766044i
\(99\) 0 0
\(100\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) −0.0603074 + 0.342020i −0.0603074 + 0.342020i
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) −1.53209 −1.53209
\(105\) 0 0
\(106\) 0.347296 0.347296
\(107\) −1.32683 + 0.766044i −1.32683 + 0.766044i −0.984808 0.173648i \(-0.944444\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(108\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(109\) −0.642788 + 1.11334i −0.642788 + 1.11334i 0.342020 + 0.939693i \(0.388889\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(110\) 0 0
\(111\) 0.592396 + 1.62760i 0.592396 + 1.62760i
\(112\) 0.173648 0.984808i 0.173648 0.984808i
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0.642788 0.766044i 0.642788 0.766044i
\(115\) 0 0
\(116\) 1.62760 + 0.939693i 1.62760 + 0.939693i
\(117\) −0.524005 + 1.43969i −0.524005 + 1.43969i
\(118\) 1.96962i 1.96962i
\(119\) −0.326352 + 0.118782i −0.326352 + 0.118782i
\(120\) 0 0
\(121\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −0.866025 0.500000i −0.866025 0.500000i
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0.866025 0.500000i 0.866025 0.500000i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 0 0
\(133\) 0.984808 + 0.173648i 0.984808 + 0.173648i
\(134\) 0.684040i 0.684040i
\(135\) 0 0
\(136\) −0.300767 0.173648i −0.300767 0.173648i
\(137\) 0.592396 + 0.342020i 0.592396 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(138\) −1.26604 + 1.50881i −1.26604 + 1.50881i
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0.342020 + 0.939693i 0.342020 + 0.939693i
\(142\) 0 0
\(143\) 0 0
\(144\) −0.173648 0.984808i −0.173648 0.984808i
\(145\) 0 0
\(146\) 1.28558 1.28558
\(147\) 1.00000i 1.00000i
\(148\) −1.73205 −1.73205
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(151\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(152\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(153\) −0.266044 + 0.223238i −0.266044 + 0.223238i
\(154\) 0 0
\(155\) 0 0
\(156\) −1.17365 0.984808i −1.17365 0.984808i
\(157\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(158\) 0 0
\(159\) 0.266044 + 0.223238i 0.266044 + 0.223238i
\(160\) 0 0
\(161\) −1.93969 0.342020i −1.93969 0.342020i
\(162\) −0.984808 0.173648i −0.984808 0.173648i
\(163\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0.766044 0.642788i 0.766044 0.642788i
\(169\) −1.34730 −1.34730
\(170\) 0 0
\(171\) 0.984808 0.173648i 0.984808 0.173648i
\(172\) 0 0
\(173\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(174\) 0.642788 + 1.76604i 0.642788 + 1.76604i
\(175\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(176\) 0 0
\(177\) −1.26604 + 1.50881i −1.26604 + 1.50881i
\(178\) 0 0
\(179\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(182\) 0.266044 1.50881i 0.266044 1.50881i
\(183\) 0 0
\(184\) −0.984808 1.70574i −0.984808 1.70574i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −1.00000 −1.00000
\(189\) −0.342020 0.939693i −0.342020 0.939693i
\(190\) 0 0
\(191\) −0.592396 + 0.342020i −0.592396 + 0.342020i −0.766044 0.642788i \(-0.777778\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(192\) 0.984808 + 0.173648i 0.984808 + 0.173648i
\(193\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −1.70574 0.984808i −1.70574 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
−0.766044 0.642788i \(-0.777778\pi\)
\(200\) −0.866025 0.500000i −0.866025 0.500000i
\(201\) 0.439693 0.524005i 0.439693 0.524005i
\(202\) 0 0
\(203\) −1.20805 + 1.43969i −1.20805 + 1.43969i
\(204\) −0.118782 0.326352i −0.118782 0.326352i
\(205\) 0 0
\(206\) 0 0
\(207\) −1.93969 + 0.342020i −1.93969 + 0.342020i
\(208\) 1.32683 0.766044i 1.32683 0.766044i
\(209\) 0 0
\(210\) 0 0
\(211\) 1.28558 1.28558 0.642788 0.766044i \(-0.277778\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(212\) −0.300767 + 0.173648i −0.300767 + 0.173648i
\(213\) 0 0
\(214\) 0.766044 1.32683i 0.766044 1.32683i
\(215\) 0 0
\(216\) 0.500000 0.866025i 0.500000 0.866025i
\(217\) 0 0
\(218\) 1.28558i 1.28558i
\(219\) 0.984808 + 0.826352i 0.984808 + 0.826352i
\(220\) 0 0
\(221\) −0.460802 0.266044i −0.460802 0.266044i
\(222\) −1.32683 1.11334i −1.32683 1.11334i
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0.342020 + 0.939693i 0.342020 + 0.939693i
\(225\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(226\) 0 0
\(227\) 0.642788 1.11334i 0.642788 1.11334i −0.342020 0.939693i \(-0.611111\pi\)
0.984808 0.173648i \(-0.0555556\pi\)
\(228\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(229\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.87939 −1.87939
\(233\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(234\) −0.266044 1.50881i −0.266044 1.50881i
\(235\) 0 0
\(236\) −0.984808 1.70574i −0.984808 1.70574i
\(237\) 0 0
\(238\) 0.223238 0.266044i 0.223238 0.266044i
\(239\) 1.28558i 1.28558i 0.766044 + 0.642788i \(0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(240\) 0 0
\(241\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(242\) −0.866025 0.500000i −0.866025 0.500000i
\(243\) −0.642788 0.766044i −0.642788 0.766044i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.766044 + 1.32683i 0.766044 + 1.32683i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.00000 1.00000
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(257\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(258\) 0 0
\(259\) 0.300767 1.70574i 0.300767 1.70574i
\(260\) 0 0
\(261\) −0.642788 + 1.76604i −0.642788 + 1.76604i
\(262\) 0 0
\(263\) 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(267\) 0 0
\(268\) 0.342020 + 0.592396i 0.342020 + 0.592396i
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 1.11334 0.642788i 1.11334 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(272\) 0.347296 0.347296
\(273\) 1.17365 0.984808i 1.17365 0.984808i
\(274\) −0.684040 −0.684040
\(275\) 0 0
\(276\) 0.342020 1.93969i 0.342020 1.93969i
\(277\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −0.766044 0.642788i −0.766044 0.642788i
\(283\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.642788 + 0.766044i 0.642788 + 0.766044i
\(289\) 0.439693 + 0.761570i 0.439693 + 0.761570i
\(290\) 0 0
\(291\) 0 0
\(292\) −1.11334 + 0.642788i −1.11334 + 0.642788i
\(293\) 1.96962 1.96962 0.984808 0.173648i \(-0.0555556\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(294\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(295\) 0 0
\(296\) 1.50000 0.866025i 1.50000 0.866025i
\(297\) 0 0
\(298\) 0 0
\(299\) −1.50881 2.61334i −1.50881 2.61334i
\(300\) −0.342020 0.939693i −0.342020 0.939693i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −0.866025 0.500000i −0.866025 0.500000i
\(305\) 0 0
\(306\) 0.118782 0.326352i 0.118782 0.326352i
\(307\) 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
\(311\) −0.939693 + 1.62760i −0.939693 + 1.62760i −0.173648 + 0.984808i \(0.555556\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(312\) 1.50881 + 0.266044i 1.50881 + 0.266044i
\(313\) −0.592396 + 0.342020i −0.592396 + 0.342020i −0.766044 0.642788i \(-0.777778\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.32683 0.766044i 1.32683 0.766044i 0.342020 0.939693i \(-0.388889\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(318\) −0.342020 0.0603074i −0.342020 0.0603074i
\(319\) 0 0
\(320\) 0 0
\(321\) 1.43969 0.524005i 1.43969 0.524005i
\(322\) 1.85083 0.673648i 1.85083 0.673648i
\(323\) 0.347296i 0.347296i
\(324\) 0.939693 0.342020i 0.939693 0.342020i
\(325\) −1.32683 0.766044i −1.32683 0.766044i
\(326\) 0 0
\(327\) 0.826352 0.984808i 0.826352 0.984808i
\(328\) 0 0
\(329\) 0.173648 0.984808i 0.173648 0.984808i
\(330\) 0 0
\(331\) 0.642788 + 1.11334i 0.642788 + 1.11334i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(332\) 0 0
\(333\) −0.300767 1.70574i −0.300767 1.70574i
\(334\) 0 0
\(335\) 0 0
\(336\) −0.342020 + 0.939693i −0.342020 + 0.939693i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 1.16679 0.673648i 1.16679 0.673648i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(343\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(344\) 0 0
\(345\) 0 0
\(346\) −1.50000 0.866025i −1.50000 0.866025i
\(347\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(348\) −1.43969 1.20805i −1.43969 1.20805i
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0.642788 0.766044i 0.642788 0.766044i
\(351\) 0.766044 1.32683i 0.766044 1.32683i
\(352\) 0 0
\(353\) 0.939693 1.62760i 0.939693 1.62760i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(354\) 0.342020 1.93969i 0.342020 1.93969i
\(355\) 0 0
\(356\) 0 0
\(357\) 0.342020 0.0603074i 0.342020 0.0603074i
\(358\) 1.00000 1.00000
\(359\) 1.11334 0.642788i 1.11334 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(360\) 0 0
\(361\) 0.500000 0.866025i 0.500000 0.866025i
\(362\) −0.500000 0.866025i −0.500000 0.866025i
\(363\) −0.342020 0.939693i −0.342020 0.939693i
\(364\) 0.524005 + 1.43969i 0.524005 + 1.43969i
\(365\) 0 0
\(366\) 0 0
\(367\) 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(368\) 1.70574 + 0.984808i 1.70574 + 0.984808i
\(369\) 0 0
\(370\) 0 0
\(371\) −0.118782 0.326352i −0.118782 0.326352i
\(372\) 0 0
\(373\) 0.984808 + 1.70574i 0.984808 + 1.70574i 0.642788 + 0.766044i \(0.277778\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0.866025 0.500000i 0.866025 0.500000i
\(377\) −2.87939 −2.87939
\(378\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(379\) −1.96962 −1.96962 −0.984808 0.173648i \(-0.944444\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.342020 0.592396i 0.342020 0.592396i
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 0.684040i 0.684040i
\(392\) −0.984808 + 0.173648i −0.984808 + 0.173648i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) 1.96962 1.96962
\(399\) −0.939693 0.342020i −0.939693 0.342020i
\(400\) 1.00000 1.00000
\(401\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) −0.118782 + 0.673648i −0.118782 + 0.673648i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0.326352 1.85083i 0.326352 1.85083i
\(407\) 0 0
\(408\) 0.266044 + 0.223238i 0.266044 + 0.223238i
\(409\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(410\) 0 0
\(411\) −0.524005 0.439693i −0.524005 0.439693i
\(412\) 0 0
\(413\) 1.85083 0.673648i 1.85083 0.673648i
\(414\) 1.50881 1.26604i 1.50881 1.26604i
\(415\) 0 0
\(416\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −0.684040 −0.684040 −0.342020 0.939693i \(-0.611111\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(422\) −1.11334 + 0.642788i −1.11334 + 0.642788i
\(423\) −0.173648 0.984808i −0.173648 0.984808i
\(424\) 0.173648 0.300767i 0.173648 0.300767i
\(425\) −0.173648 0.300767i −0.173648 0.300767i
\(426\) 0 0
\(427\) 0 0
\(428\) 1.53209i 1.53209i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(432\) 1.00000i 1.00000i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.642788 + 1.11334i 0.642788 + 1.11334i
\(437\) −0.984808 + 1.70574i −0.984808 + 1.70574i
\(438\) −1.26604 0.223238i −1.26604 0.223238i
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(442\) 0.532089 0.532089
\(443\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(444\) 1.70574 + 0.300767i 1.70574 + 0.300767i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.766044 0.642788i −0.766044 0.642788i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0.342020 0.939693i 0.342020 0.939693i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 1.28558i 1.28558i
\(455\) 0 0
\(456\) −0.342020 0.939693i −0.342020 0.939693i
\(457\) 0.173648 + 0.300767i 0.173648 + 0.300767i 0.939693 0.342020i \(-0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(458\) 0 0
\(459\) 0.300767 0.173648i 0.300767 0.173648i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 1.62760 0.939693i 1.62760 0.939693i
\(465\) 0 0
\(466\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(467\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(468\) 0.984808 + 1.17365i 0.984808 + 1.17365i
\(469\) −0.642788 + 0.233956i −0.642788 + 0.233956i
\(470\) 0 0
\(471\) 0 0
\(472\) 1.70574 + 0.984808i 1.70574 + 0.984808i
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000i 1.00000i
\(476\) −0.0603074 + 0.342020i −0.0603074 + 0.342020i
\(477\) −0.223238 0.266044i −0.223238 0.266044i
\(478\) −0.642788 1.11334i −0.642788 1.11334i
\(479\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 2.29813 1.32683i 2.29813 1.32683i
\(482\) 0 0
\(483\) 1.85083 + 0.673648i 1.85083 + 0.673648i
\(484\) 1.00000 1.00000
\(485\) 0 0
\(486\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(487\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −0.565258 0.326352i −0.565258 0.326352i
\(494\) −1.32683 0.766044i −1.32683 0.766044i
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.87939 1.87939 0.939693 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(504\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(505\) 0 0
\(506\) 0 0
\(507\) 1.32683 + 0.233956i 1.32683 + 0.233956i
\(508\) 0 0
\(509\) −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(510\) 0 0
\(511\) −0.439693 1.20805i −0.439693 1.20805i
\(512\) 1.00000i 1.00000i
\(513\) −1.00000 −1.00000
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0.592396 + 1.62760i 0.592396 + 1.62760i
\(519\) −0.592396 1.62760i −0.592396 1.62760i
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) −0.326352 1.85083i −0.326352 1.85083i
\(523\) −0.300767 + 0.173648i −0.300767 + 0.173648i −0.642788 0.766044i \(-0.722222\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(524\) 0 0
\(525\) 0.984808 0.173648i 0.984808 0.173648i
\(526\) −1.73205 −1.73205
\(527\) 0 0
\(528\) 0 0
\(529\) 1.43969 2.49362i 1.43969 2.49362i
\(530\) 0 0
\(531\) 1.50881 1.26604i 1.50881 1.26604i
\(532\) 0.642788 0.766044i 0.642788 0.766044i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −0.592396 0.342020i −0.592396 0.342020i
\(537\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(542\) −0.642788 + 1.11334i −0.642788 + 1.11334i
\(543\) 0.173648 0.984808i 0.173648 0.984808i
\(544\) −0.300767 + 0.173648i −0.300767 + 0.173648i
\(545\) 0 0
\(546\) −0.524005 + 1.43969i −0.524005 + 1.43969i
\(547\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(548\) 0.592396 0.342020i 0.592396 0.342020i
\(549\) 0 0
\(550\) 0 0
\(551\) 0.939693 + 1.62760i 0.939693 + 1.62760i
\(552\) 0.673648 + 1.85083i 0.673648 + 1.85083i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0.984808 + 0.173648i 0.984808 + 0.173648i
\(565\) 0 0
\(566\) 0 0
\(567\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(568\) 0 0
\(569\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(570\) 0 0
\(571\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0.642788 0.233956i 0.642788 0.233956i
\(574\) 0 0
\(575\) 1.96962i 1.96962i
\(576\) −0.939693 0.342020i −0.939693 0.342020i
\(577\) −0.592396 0.342020i −0.592396 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(578\) −0.761570 0.439693i −0.761570 0.439693i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0.642788 1.11334i 0.642788 1.11334i
\(585\) 0 0
\(586\) −1.70574 + 0.984808i −1.70574 + 0.984808i
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −0.866025 0.500000i −0.866025 0.500000i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(593\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.50881 + 1.26604i 1.50881 + 1.26604i
\(598\) 2.61334 + 1.50881i 2.61334 + 1.50881i
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) −0.524005 + 0.439693i −0.524005 + 0.439693i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(608\) 1.00000 1.00000
\(609\) 1.43969 1.20805i 1.43969 1.20805i
\(610\) 0 0
\(611\) 1.32683 0.766044i 1.32683 0.766044i
\(612\) 0.0603074 + 0.342020i 0.0603074 + 0.342020i
\(613\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) −0.500000 0.866025i −0.500000 0.866025i
\(615\) 0 0
\(616\) 0 0
\(617\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(618\) 0 0
\(619\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(620\) 0 0
\(621\) 1.96962 1.96962
\(622\) 1.87939i 1.87939i
\(623\) 0 0
\(624\) −1.43969 + 0.524005i −1.43969 + 0.524005i
\(625\) −0.500000 0.866025i −0.500000 0.866025i
\(626\) 0.342020 0.592396i 0.342020 0.592396i
\(627\) 0 0
\(628\) 0 0
\(629\) 0.601535 0.601535
\(630\) 0 0
\(631\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(632\) 0 0
\(633\) −1.26604 0.223238i −1.26604 0.223238i
\(634\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(635\) 0 0
\(636\) 0.326352 0.118782i 0.326352 0.118782i
\(637\) −1.50881 + 0.266044i −1.50881 + 0.266044i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) −0.984808 + 1.17365i −0.984808 + 1.17365i
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) −1.26604 + 1.50881i −1.26604 + 1.50881i
\(645\) 0 0
\(646\) −0.173648 0.300767i −0.173648 0.300767i
\(647\) 0.766044 1.32683i 0.766044 1.32683i −0.173648 0.984808i \(-0.555556\pi\)
0.939693 0.342020i \(-0.111111\pi\)
\(648\) −0.642788 + 0.766044i −0.642788 + 0.766044i
\(649\) 0 0
\(650\) 1.53209 1.53209
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(654\) −0.223238 + 1.26604i −0.223238 + 1.26604i
\(655\) 0 0
\(656\) 0 0
\(657\) −0.826352 0.984808i −0.826352 0.984808i
\(658\) 0.342020 + 0.939693i 0.342020 + 0.939693i
\(659\) 1.53209i 1.53209i 0.642788 + 0.766044i \(0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(660\) 0 0
\(661\) 1.62760 + 0.939693i 1.62760 + 0.939693i 0.984808 + 0.173648i \(0.0555556\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(662\) −1.11334 0.642788i −1.11334 0.642788i
\(663\) 0.407604 + 0.342020i 0.407604 + 0.342020i
\(664\) 0 0
\(665\) 0 0
\(666\) 1.11334 + 1.32683i 1.11334 + 1.32683i
\(667\) −1.85083 3.20574i −1.85083 3.20574i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) −0.173648 0.984808i −0.173648 0.984808i
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0.866025 0.500000i 0.866025 0.500000i
\(676\) −0.673648 + 1.16679i −0.673648 + 1.16679i
\(677\) −0.342020 0.592396i −0.342020 0.592396i 0.642788 0.766044i \(-0.277778\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −0.826352 + 0.984808i −0.826352 + 0.984808i
\(682\) 0 0
\(683\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(684\) 0.342020 0.939693i 0.342020 0.939693i
\(685\) 0 0
\(686\) 1.00000i 1.00000i
\(687\) 0 0
\(688\) 0 0
\(689\) 0.266044 0.460802i 0.266044 0.460802i
\(690\) 0 0
\(691\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(692\) 1.73205 1.73205
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 1.85083 + 0.326352i 1.85083 + 0.326352i
\(697\) 0 0
\(698\) 0 0
\(699\) −1.62760 + 0.592396i −1.62760 + 0.592396i
\(700\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 1.53209i 1.53209i
\(703\) −1.50000 0.866025i −1.50000 0.866025i
\(704\) 0 0
\(705\) 0 0
\(706\) 1.87939i 1.87939i
\(707\) 0 0
\(708\) 0.673648 + 1.85083i 0.673648 + 1.85083i
\(709\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) −0.266044 + 0.223238i −0.266044 + 0.223238i
\(715\) 0 0
\(716\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(717\) 0.223238 1.26604i 0.223238 1.26604i
\(718\) −0.642788 + 1.11334i −0.642788 + 1.11334i
\(719\) 0.766044 + 1.32683i 0.766044 + 1.32683i 0.939693 + 0.342020i \(0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000i 1.00000i
\(723\) 0 0
\(724\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(725\) −1.62760 0.939693i −1.62760 0.939693i
\(726\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(727\) 0.684040i 0.684040i 0.939693 + 0.342020i \(0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(728\) −1.17365 0.984808i −1.17365 0.984808i
\(729\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(734\) −1.73205 −1.73205
\(735\) 0 0
\(736\) −1.96962 −1.96962
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(740\) 0 0
\(741\) −0.524005 1.43969i −0.524005 1.43969i
\(742\) 0.266044 + 0.223238i 0.266044 + 0.223238i
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.70574 0.984808i −1.70574 0.984808i
\(747\) 0 0
\(748\) 0 0
\(749\) −1.50881 0.266044i −1.50881 0.266044i
\(750\) 0 0
\(751\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(753\) 0 0
\(754\) 2.49362 1.43969i 2.49362 1.43969i
\(755\) 0 0
\(756\) −0.984808 0.173648i −0.984808 0.173648i
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 1.70574 0.984808i 1.70574 0.984808i
\(759\) 0 0
\(760\) 0 0
\(761\) −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i \(-0.777778\pi\)
−0.173648 0.984808i \(-0.555556\pi\)
\(762\) 0 0
\(763\) −1.20805 + 0.439693i −1.20805 + 0.439693i
\(764\) 0.684040i 0.684040i
\(765\) 0 0
\(766\) 0 0
\(767\) 2.61334 + 1.50881i 2.61334 + 1.50881i
\(768\) 0.642788 0.766044i 0.642788 0.766044i
\(769\) 1.96962i 1.96962i −0.173648 0.984808i \(-0.555556\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.642788 + 1.11334i −0.642788 + 1.11334i 0.342020 + 0.939693i \(0.388889\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −0.592396 + 1.62760i −0.592396 + 1.62760i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0.342020 + 0.592396i 0.342020 + 0.592396i
\(783\) 0.939693 1.62760i 0.939693 1.62760i
\(784\) 0.766044 0.642788i 0.766044 0.642788i
\(785\) 0 0
\(786\) 0 0
\(787\) −0.300767 0.173648i −0.300767 0.173648i 0.342020 0.939693i \(-0.388889\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(788\) 0 0
\(789\) −1.32683 1.11334i −1.32683 1.11334i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −1.70574 + 0.984808i −1.70574 + 0.984808i
\(797\) 0.684040 0.684040 0.342020 0.939693i \(-0.388889\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(798\) 0.984808 0.173648i 0.984808 0.173648i
\(799\) 0.347296 0.347296
\(800\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −0.233956 0.642788i −0.233956 0.642788i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.70574 0.984808i −1.70574 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
−0.766044 0.642788i \(-0.777778\pi\)
\(810\) 0 0
\(811\) 0.347296i 0.347296i 0.984808 + 0.173648i \(0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(812\) 0.642788 + 1.76604i 0.642788 + 1.76604i
\(813\) −1.20805 + 0.439693i −1.20805 + 0.439693i
\(814\) 0 0
\(815\) 0 0
\(816\) −0.342020 0.0603074i −0.342020 0.0603074i
\(817\) 0 0
\(818\) 0 0
\(819\) −1.32683 + 0.766044i −1.32683 + 0.766044i
\(820\) 0 0
\(821\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(822\) 0.673648 + 0.118782i 0.673648 + 0.118782i
\(823\) 0.939693 1.62760i 0.939693 1.62760i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −1.26604 + 1.50881i −1.26604 + 1.50881i
\(827\) 1.87939i 1.87939i −0.342020 0.939693i \(-0.611111\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(828\) −0.673648 + 1.85083i −0.673648 + 1.85083i
\(829\) −1.62760 0.939693i −1.62760 0.939693i −0.984808 0.173648i \(-0.944444\pi\)
−0.642788 0.766044i \(-0.722222\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.53209i 1.53209i
\(833\) −0.326352 0.118782i −0.326352 0.118782i
\(834\) 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
\(841\) −2.53209 −2.53209
\(842\) 0.592396 0.342020i 0.592396 0.342020i
\(843\) 0 0
\(844\) 0.642788 1.11334i 0.642788 1.11334i
\(845\) 0 0
\(846\) 0.642788 + 0.766044i 0.642788 + 0.766044i
\(847\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(848\) 0.347296i 0.347296i
\(849\) 0 0
\(850\) 0.300767 + 0.173648i 0.300767 + 0.173648i
\(851\) 2.95442 + 1.70574i 2.95442 + 1.70574i
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.766044 1.32683i −0.766044 1.32683i
\(857\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(858\) 0 0
\(859\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(864\) −0.500000 0.866025i −0.500000 0.866025i
\(865\) 0 0
\(866\) 0 0
\(867\) −0.300767 0.826352i −0.300767 0.826352i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −0.907604 0.524005i −0.907604 0.524005i
\(872\) −1.11334 0.642788i −1.11334 0.642788i
\(873\) 0 0
\(874\) 1.96962i 1.96962i
\(875\) 0 0
\(876\) 1.20805 0.439693i 1.20805 0.439693i
\(877\) −0.642788 1.11334i −0.642788 1.11334i −0.984808 0.173648i \(-0.944444\pi\)
0.342020 0.939693i \(-0.388889\pi\)
\(878\) 0 0
\(879\) −1.93969 0.342020i −1.93969 0.342020i
\(880\) 0 0
\(881\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) −0.342020 0.939693i −0.342020 0.939693i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −0.460802 + 0.266044i −0.460802 + 0.266044i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(888\) −1.62760 + 0.592396i −1.62760 + 0.592396i
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −0.866025 0.500000i −0.866025 0.500000i
\(894\) 0 0
\(895\) 0 0
\(896\) 0.984808 + 0.173648i 0.984808 + 0.173648i
\(897\) 1.03209 + 2.83564i 1.03209 + 2.83564i
\(898\) 0 0
\(899\) 0 0
\(900\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(901\) 0.104455 0.0603074i 0.104455 0.0603074i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −0.984808 + 1.70574i −0.984808 + 1.70574i −0.342020 + 0.939693i \(0.611111\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(908\) −0.642788 1.11334i −0.642788 1.11334i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(913\) 0 0
\(914\) −0.300767 0.173648i −0.300767 0.173648i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −0.173648 + 0.300767i −0.173648 + 0.300767i
\(919\) 0.173648 + 0.300767i 0.173648 + 0.300767i 0.939693 0.342020i \(-0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(920\) 0 0
\(921\) 0.173648 0.984808i 0.173648 0.984808i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.73205 1.73205
\(926\) 0.866025 0.500000i 0.866025 0.500000i
\(927\) 0 0
\(928\) −0.939693 + 1.62760i −0.939693 + 1.62760i
\(929\) 0.766044 + 1.32683i 0.766044 + 1.32683i 0.939693 + 0.342020i \(0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(930\) 0 0
\(931\) 0.642788 + 0.766044i 0.642788 + 0.766044i
\(932\) 1.73205i 1.73205i
\(933\) 1.20805 1.43969i 1.20805 1.43969i
\(934\) 0 0
\(935\) 0 0
\(936\) −1.43969 0.524005i −1.43969 0.524005i
\(937\) 1.28558i 1.28558i −0.766044 0.642788i \(-0.777778\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(938\) 0.439693 0.524005i 0.439693 0.524005i
\(939\) 0.642788 0.233956i 0.642788 0.233956i
\(940\) 0 0
\(941\) −0.342020 + 0.592396i −0.342020 + 0.592396i −0.984808 0.173648i \(-0.944444\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.96962 −1.96962
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(948\) 0 0
\(949\) 0.984808 1.70574i 0.984808 1.70574i
\(950\) −0.500000 0.866025i −0.500000 0.866025i
\(951\) −1.43969 + 0.524005i −1.43969 + 0.524005i
\(952\) −0.118782 0.326352i −0.118782 0.326352i
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0.326352 + 0.118782i 0.326352 + 0.118782i
\(955\) 0 0
\(956\) 1.11334 + 0.642788i 1.11334 + 0.642788i
\(957\) 0 0
\(958\) 2.00000i 2.00000i
\(959\) 0.233956 + 0.642788i 0.233956 + 0.642788i
\(960\) 0 0
\(961\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(962\) −1.32683 + 2.29813i −1.32683 + 2.29813i
\(963\) −1.50881 + 0.266044i −1.50881 + 0.266044i
\(964\) 0 0
\(965\) 0 0
\(966\) −1.93969 + 0.342020i −1.93969 + 0.342020i
\(967\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(968\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(969\) 0.0603074 0.342020i 0.0603074 0.342020i
\(970\) 0 0
\(971\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(972\) −0.984808 + 0.173648i −0.984808 + 0.173648i
\(973\) 0 0
\(974\) 0 0
\(975\) 1.17365 + 0.984808i 1.17365 + 0.984808i
\(976\) 0 0
\(977\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.984808 + 0.826352i −0.984808 + 0.826352i
\(982\) 0 0
\(983\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0.652704 0.652704
\(987\) −0.342020 + 0.939693i −0.342020 + 0.939693i
\(988\) 1.53209 1.53209
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 0 0
\(993\) −0.439693 1.20805i −0.439693 1.20805i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(998\) 0 0
\(999\) 1.73205i 1.73205i
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 3192.1.eb.a.2621.1 yes 12
3.2 odd 2 3192.1.eb.b.2621.4 yes 12
7.5 odd 6 3192.1.eb.b.341.4 yes 12
8.5 even 2 inner 3192.1.eb.a.2621.6 yes 12
19.18 odd 2 inner 3192.1.eb.a.2621.6 yes 12
21.5 even 6 inner 3192.1.eb.a.341.1 12
24.5 odd 2 3192.1.eb.b.2621.3 yes 12
56.5 odd 6 3192.1.eb.b.341.3 yes 12
57.56 even 2 3192.1.eb.b.2621.3 yes 12
133.75 even 6 3192.1.eb.b.341.3 yes 12
152.37 odd 2 CM 3192.1.eb.a.2621.1 yes 12
168.5 even 6 inner 3192.1.eb.a.341.6 yes 12
399.341 odd 6 inner 3192.1.eb.a.341.6 yes 12
456.341 even 2 3192.1.eb.b.2621.4 yes 12
1064.341 even 6 3192.1.eb.b.341.4 yes 12
3192.341 odd 6 inner 3192.1.eb.a.341.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.1.eb.a.341.1 12 21.5 even 6 inner
3192.1.eb.a.341.1 12 3192.341 odd 6 inner
3192.1.eb.a.341.6 yes 12 168.5 even 6 inner
3192.1.eb.a.341.6 yes 12 399.341 odd 6 inner
3192.1.eb.a.2621.1 yes 12 1.1 even 1 trivial
3192.1.eb.a.2621.1 yes 12 152.37 odd 2 CM
3192.1.eb.a.2621.6 yes 12 8.5 even 2 inner
3192.1.eb.a.2621.6 yes 12 19.18 odd 2 inner
3192.1.eb.b.341.3 yes 12 56.5 odd 6
3192.1.eb.b.341.3 yes 12 133.75 even 6
3192.1.eb.b.341.4 yes 12 7.5 odd 6
3192.1.eb.b.341.4 yes 12 1064.341 even 6
3192.1.eb.b.2621.3 yes 12 24.5 odd 2
3192.1.eb.b.2621.3 yes 12 57.56 even 2
3192.1.eb.b.2621.4 yes 12 3.2 odd 2
3192.1.eb.b.2621.4 yes 12 456.341 even 2