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Properties

Label 209.u
Modulus 209209
Conductor 209209
Order 4545
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,10]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(4,209))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: 209209
Conductor: 209209
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 4545
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: Q(ζ45)\Q(\zeta_{45})
Fixed field: 45.45.43679806300610465846484971330073185012597520004657724600953543350870941304329239684756561.1

Characters in Galois orbit

Character −1-1 11 22 33 44 55 66 77 88 99 1010 1212
χ209(4,⋅)\chi_{209}(4,\cdot) 11 11 e(1445)e\left(\frac{14}{45}\right) e(245)e\left(\frac{2}{45}\right) e(2845)e\left(\frac{28}{45}\right) e(2645)e\left(\frac{26}{45}\right) e(1645)e\left(\frac{16}{45}\right) e(115)e\left(\frac{1}{15}\right) e(1415)e\left(\frac{14}{15}\right) e(445)e\left(\frac{4}{45}\right) e(89)e\left(\frac{8}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(5,⋅)\chi_{209}(5,\cdot) 11 11 e(1345)e\left(\frac{13}{45}\right) e(3445)e\left(\frac{34}{45}\right) e(2645)e\left(\frac{26}{45}\right) e(3745)e\left(\frac{37}{45}\right) e(245)e\left(\frac{2}{45}\right) e(215)e\left(\frac{2}{15}\right) e(1315)e\left(\frac{13}{15}\right) e(2345)e\left(\frac{23}{45}\right) e(19)e\left(\frac{1}{9}\right) e(13)e\left(\frac{1}{3}\right)
χ209(9,⋅)\chi_{209}(9,\cdot) 11 11 e(245)e\left(\frac{2}{45}\right) e(2645)e\left(\frac{26}{45}\right) e(445)e\left(\frac{4}{45}\right) e(2345)e\left(\frac{23}{45}\right) e(2845)e\left(\frac{28}{45}\right) e(1315)e\left(\frac{13}{15}\right) e(215)e\left(\frac{2}{15}\right) e(745)e\left(\frac{7}{45}\right) e(59)e\left(\frac{5}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(16,⋅)\chi_{209}(16,\cdot) 11 11 e(2845)e\left(\frac{28}{45}\right) e(445)e\left(\frac{4}{45}\right) e(1145)e\left(\frac{11}{45}\right) e(745)e\left(\frac{7}{45}\right) e(3245)e\left(\frac{32}{45}\right) e(215)e\left(\frac{2}{15}\right) e(1315)e\left(\frac{13}{15}\right) e(845)e\left(\frac{8}{45}\right) e(79)e\left(\frac{7}{9}\right) e(13)e\left(\frac{1}{3}\right)
χ209(25,⋅)\chi_{209}(25,\cdot) 11 11 e(2645)e\left(\frac{26}{45}\right) e(2345)e\left(\frac{23}{45}\right) e(745)e\left(\frac{7}{45}\right) e(2945)e\left(\frac{29}{45}\right) e(445)e\left(\frac{4}{45}\right) e(415)e\left(\frac{4}{15}\right) e(1115)e\left(\frac{11}{15}\right) e(145)e\left(\frac{1}{45}\right) e(29)e\left(\frac{2}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(36,⋅)\chi_{209}(36,\cdot) 11 11 e(1645)e\left(\frac{16}{45}\right) e(2845)e\left(\frac{28}{45}\right) e(3245)e\left(\frac{32}{45}\right) e(445)e\left(\frac{4}{45}\right) e(4445)e\left(\frac{44}{45}\right) e(1415)e\left(\frac{14}{15}\right) e(115)e\left(\frac{1}{15}\right) e(1145)e\left(\frac{11}{45}\right) e(49)e\left(\frac{4}{9}\right) e(13)e\left(\frac{1}{3}\right)
χ209(42,⋅)\chi_{209}(42,\cdot) 11 11 e(3245)e\left(\frac{32}{45}\right) e(1145)e\left(\frac{11}{45}\right) e(1945)e\left(\frac{19}{45}\right) e(845)e\left(\frac{8}{45}\right) e(4345)e\left(\frac{43}{45}\right) e(1315)e\left(\frac{13}{15}\right) e(215)e\left(\frac{2}{15}\right) e(2245)e\left(\frac{22}{45}\right) e(89)e\left(\frac{8}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(47,⋅)\chi_{209}(47,\cdot) 11 11 e(1145)e\left(\frac{11}{45}\right) e(845)e\left(\frac{8}{45}\right) e(2245)e\left(\frac{22}{45}\right) e(1445)e\left(\frac{14}{45}\right) e(1945)e\left(\frac{19}{45}\right) e(415)e\left(\frac{4}{15}\right) e(1115)e\left(\frac{11}{15}\right) e(1645)e\left(\frac{16}{45}\right) e(59)e\left(\frac{5}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(80,⋅)\chi_{209}(80,\cdot) 11 11 e(4145)e\left(\frac{41}{45}\right) e(3845)e\left(\frac{38}{45}\right) e(3745)e\left(\frac{37}{45}\right) e(4445)e\left(\frac{44}{45}\right) e(3445)e\left(\frac{34}{45}\right) e(415)e\left(\frac{4}{15}\right) e(1115)e\left(\frac{11}{15}\right) e(3145)e\left(\frac{31}{45}\right) e(89)e\left(\frac{8}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(81,⋅)\chi_{209}(81,\cdot) 11 11 e(445)e\left(\frac{4}{45}\right) e(745)e\left(\frac{7}{45}\right) e(845)e\left(\frac{8}{45}\right) e(145)e\left(\frac{1}{45}\right) e(1145)e\left(\frac{11}{45}\right) e(1115)e\left(\frac{11}{15}\right) e(415)e\left(\frac{4}{15}\right) e(1445)e\left(\frac{14}{45}\right) e(19)e\left(\frac{1}{9}\right) e(13)e\left(\frac{1}{3}\right)
χ209(82,⋅)\chi_{209}(82,\cdot) 11 11 e(845)e\left(\frac{8}{45}\right) e(1445)e\left(\frac{14}{45}\right) e(1645)e\left(\frac{16}{45}\right) e(245)e\left(\frac{2}{45}\right) e(2245)e\left(\frac{22}{45}\right) e(715)e\left(\frac{7}{15}\right) e(815)e\left(\frac{8}{15}\right) e(2845)e\left(\frac{28}{45}\right) e(29)e\left(\frac{2}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(92,⋅)\chi_{209}(92,\cdot) 11 11 e(1945)e\left(\frac{19}{45}\right) e(2245)e\left(\frac{22}{45}\right) e(3845)e\left(\frac{38}{45}\right) e(1645)e\left(\frac{16}{45}\right) e(4145)e\left(\frac{41}{45}\right) e(1115)e\left(\frac{11}{15}\right) e(415)e\left(\frac{4}{15}\right) e(4445)e\left(\frac{44}{45}\right) e(79)e\left(\frac{7}{9}\right) e(13)e\left(\frac{1}{3}\right)
χ209(93,⋅)\chi_{209}(93,\cdot) 11 11 e(4345)e\left(\frac{43}{45}\right) e(1945)e\left(\frac{19}{45}\right) e(4145)e\left(\frac{41}{45}\right) e(2245)e\left(\frac{22}{45}\right) e(1745)e\left(\frac{17}{45}\right) e(215)e\left(\frac{2}{15}\right) e(1315)e\left(\frac{13}{15}\right) e(3845)e\left(\frac{38}{45}\right) e(49)e\left(\frac{4}{9}\right) e(13)e\left(\frac{1}{3}\right)
χ209(104,⋅)\chi_{209}(104,\cdot) 11 11 e(3845)e\left(\frac{38}{45}\right) e(4445)e\left(\frac{44}{45}\right) e(3145)e\left(\frac{31}{45}\right) e(3245)e\left(\frac{32}{45}\right) e(3745)e\left(\frac{37}{45}\right) e(715)e\left(\frac{7}{15}\right) e(815)e\left(\frac{8}{15}\right) e(4345)e\left(\frac{43}{45}\right) e(59)e\left(\frac{5}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(119,⋅)\chi_{209}(119,\cdot) 11 11 e(2245)e\left(\frac{22}{45}\right) e(1645)e\left(\frac{16}{45}\right) e(4445)e\left(\frac{44}{45}\right) e(2845)e\left(\frac{28}{45}\right) e(3845)e\left(\frac{38}{45}\right) e(815)e\left(\frac{8}{15}\right) e(715)e\left(\frac{7}{15}\right) e(3245)e\left(\frac{32}{45}\right) e(19)e\left(\frac{1}{9}\right) e(13)e\left(\frac{1}{3}\right)
χ209(130,⋅)\chi_{209}(130,\cdot) 11 11 e(3745)e\left(\frac{37}{45}\right) e(3145)e\left(\frac{31}{45}\right) e(2945)e\left(\frac{29}{45}\right) e(4345)e\left(\frac{43}{45}\right) e(2345)e\left(\frac{23}{45}\right) e(815)e\left(\frac{8}{15}\right) e(715)e\left(\frac{7}{15}\right) e(1745)e\left(\frac{17}{45}\right) e(79)e\left(\frac{7}{9}\right) e(13)e\left(\frac{1}{3}\right)
χ209(137,⋅)\chi_{209}(137,\cdot) 11 11 e(2345)e\left(\frac{23}{45}\right) e(2945)e\left(\frac{29}{45}\right) e(145)e\left(\frac{1}{45}\right) e(1745)e\left(\frac{17}{45}\right) e(745)e\left(\frac{7}{45}\right) e(715)e\left(\frac{7}{15}\right) e(815)e\left(\frac{8}{15}\right) e(1345)e\left(\frac{13}{45}\right) e(89)e\left(\frac{8}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(157,⋅)\chi_{209}(157,\cdot) 11 11 e(3145)e\left(\frac{31}{45}\right) e(4345)e\left(\frac{43}{45}\right) e(1745)e\left(\frac{17}{45}\right) e(1945)e\left(\frac{19}{45}\right) e(2945)e\left(\frac{29}{45}\right) e(1415)e\left(\frac{14}{15}\right) e(115)e\left(\frac{1}{15}\right) e(4145)e\left(\frac{41}{45}\right) e(19)e\left(\frac{1}{9}\right) e(13)e\left(\frac{1}{3}\right)
χ209(158,⋅)\chi_{209}(158,\cdot) 11 11 e(4445)e\left(\frac{44}{45}\right) e(3245)e\left(\frac{32}{45}\right) e(4345)e\left(\frac{43}{45}\right) e(1145)e\left(\frac{11}{45}\right) e(3145)e\left(\frac{31}{45}\right) e(115)e\left(\frac{1}{15}\right) e(1415)e\left(\frac{14}{15}\right) e(1945)e\left(\frac{19}{45}\right) e(29)e\left(\frac{2}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(168,⋅)\chi_{209}(168,\cdot) 11 11 e(145)e\left(\frac{1}{45}\right) e(1345)e\left(\frac{13}{45}\right) e(245)e\left(\frac{2}{45}\right) e(3445)e\left(\frac{34}{45}\right) e(1445)e\left(\frac{14}{45}\right) e(1415)e\left(\frac{14}{15}\right) e(115)e\left(\frac{1}{15}\right) e(2645)e\left(\frac{26}{45}\right) e(79)e\left(\frac{7}{9}\right) e(13)e\left(\frac{1}{3}\right)
χ209(169,⋅)\chi_{209}(169,\cdot) 11 11 e(3445)e\left(\frac{34}{45}\right) e(3745)e\left(\frac{37}{45}\right) e(2345)e\left(\frac{23}{45}\right) e(3145)e\left(\frac{31}{45}\right) e(2645)e\left(\frac{26}{45}\right) e(1115)e\left(\frac{11}{15}\right) e(415)e\left(\frac{4}{15}\right) e(2945)e\left(\frac{29}{45}\right) e(49)e\left(\frac{4}{9}\right) e(13)e\left(\frac{1}{3}\right)
χ209(180,⋅)\chi_{209}(180,\cdot) 11 11 e(2945)e\left(\frac{29}{45}\right) e(1745)e\left(\frac{17}{45}\right) e(1345)e\left(\frac{13}{45}\right) e(4145)e\left(\frac{41}{45}\right) e(145)e\left(\frac{1}{45}\right) e(115)e\left(\frac{1}{15}\right) e(1415)e\left(\frac{14}{15}\right) e(3445)e\left(\frac{34}{45}\right) e(59)e\left(\frac{5}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(196,⋅)\chi_{209}(196,\cdot) 11 11 e(1745)e\left(\frac{17}{45}\right) e(4145)e\left(\frac{41}{45}\right) e(3445)e\left(\frac{34}{45}\right) e(3845)e\left(\frac{38}{45}\right) e(1345)e\left(\frac{13}{45}\right) e(1315)e\left(\frac{13}{15}\right) e(215)e\left(\frac{2}{15}\right) e(3745)e\left(\frac{37}{45}\right) e(29)e\left(\frac{2}{9}\right) e(23)e\left(\frac{2}{3}\right)
χ209(207,⋅)\chi_{209}(207,\cdot) 11 11 e(745)e\left(\frac{7}{45}\right) e(145)e\left(\frac{1}{45}\right) e(1445)e\left(\frac{14}{45}\right) e(1345)e\left(\frac{13}{45}\right) e(845)e\left(\frac{8}{45}\right) e(815)e\left(\frac{8}{15}\right) e(715)e\left(\frac{7}{15}\right) e(245)e\left(\frac{2}{45}\right) e(49)e\left(\frac{4}{9}\right) e(13)e\left(\frac{1}{3}\right)