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Properties

Label 345.r
Modulus 345345
Conductor 115115
Order 2222
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,11,15]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(19,345))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: 345345
Conductor: 115115
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 2222
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 115.i
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: Q(ζ11)\Q(\zeta_{11})
Fixed field: 22.0.1927323443393334271838358868310546875.1

Characters in Galois orbit

Character −1-1 11 22 44 77 88 1111 1313 1414 1616 1717 1919
χ345(19,⋅)\chi_{345}(19,\cdot) −1-1 11 e(1922)e\left(\frac{19}{22}\right) e(811)e\left(\frac{8}{11}\right) e(511)e\left(\frac{5}{11}\right) e(1322)e\left(\frac{13}{22}\right) e(322)e\left(\frac{3}{22}\right) e(122)e\left(\frac{1}{22}\right) e(722)e\left(\frac{7}{22}\right) e(511)e\left(\frac{5}{11}\right) e(311)e\left(\frac{3}{11}\right) e(522)e\left(\frac{5}{22}\right)
χ345(34,⋅)\chi_{345}(34,\cdot) −1-1 11 e(722)e\left(\frac{7}{22}\right) e(711)e\left(\frac{7}{11}\right) e(311)e\left(\frac{3}{11}\right) e(2122)e\left(\frac{21}{22}\right) e(1522)e\left(\frac{15}{22}\right) e(522)e\left(\frac{5}{22}\right) e(1322)e\left(\frac{13}{22}\right) e(311)e\left(\frac{3}{11}\right) e(411)e\left(\frac{4}{11}\right) e(322)e\left(\frac{3}{22}\right)
χ345(79,⋅)\chi_{345}(79,\cdot) −1-1 11 e(1722)e\left(\frac{17}{22}\right) e(611)e\left(\frac{6}{11}\right) e(111)e\left(\frac{1}{11}\right) e(722)e\left(\frac{7}{22}\right) e(522)e\left(\frac{5}{22}\right) e(922)e\left(\frac{9}{22}\right) e(1922)e\left(\frac{19}{22}\right) e(111)e\left(\frac{1}{11}\right) e(511)e\left(\frac{5}{11}\right) e(122)e\left(\frac{1}{22}\right)
χ345(109,⋅)\chi_{345}(109,\cdot) −1-1 11 e(322)e\left(\frac{3}{22}\right) e(311)e\left(\frac{3}{11}\right) e(611)e\left(\frac{6}{11}\right) e(922)e\left(\frac{9}{22}\right) e(1922)e\left(\frac{19}{22}\right) e(2122)e\left(\frac{21}{22}\right) e(1522)e\left(\frac{15}{22}\right) e(611)e\left(\frac{6}{11}\right) e(811)e\left(\frac{8}{11}\right) e(1722)e\left(\frac{17}{22}\right)
χ345(199,⋅)\chi_{345}(199,\cdot) −1-1 11 e(122)e\left(\frac{1}{22}\right) e(111)e\left(\frac{1}{11}\right) e(211)e\left(\frac{2}{11}\right) e(322)e\left(\frac{3}{22}\right) e(2122)e\left(\frac{21}{22}\right) e(722)e\left(\frac{7}{22}\right) e(522)e\left(\frac{5}{22}\right) e(211)e\left(\frac{2}{11}\right) e(1011)e\left(\frac{10}{11}\right) e(1322)e\left(\frac{13}{22}\right)
χ345(214,⋅)\chi_{345}(214,\cdot) −1-1 11 e(522)e\left(\frac{5}{22}\right) e(511)e\left(\frac{5}{11}\right) e(1011)e\left(\frac{10}{11}\right) e(1522)e\left(\frac{15}{22}\right) e(1722)e\left(\frac{17}{22}\right) e(1322)e\left(\frac{13}{22}\right) e(322)e\left(\frac{3}{22}\right) e(1011)e\left(\frac{10}{11}\right) e(611)e\left(\frac{6}{11}\right) e(2122)e\left(\frac{21}{22}\right)
χ345(244,⋅)\chi_{345}(244,\cdot) −1-1 11 e(922)e\left(\frac{9}{22}\right) e(911)e\left(\frac{9}{11}\right) e(711)e\left(\frac{7}{11}\right) e(522)e\left(\frac{5}{22}\right) e(1322)e\left(\frac{13}{22}\right) e(1922)e\left(\frac{19}{22}\right) e(122)e\left(\frac{1}{22}\right) e(711)e\left(\frac{7}{11}\right) e(211)e\left(\frac{2}{11}\right) e(722)e\left(\frac{7}{22}\right)
χ345(274,⋅)\chi_{345}(274,\cdot) −1-1 11 e(1522)e\left(\frac{15}{22}\right) e(411)e\left(\frac{4}{11}\right) e(811)e\left(\frac{8}{11}\right) e(122)e\left(\frac{1}{22}\right) e(722)e\left(\frac{7}{22}\right) e(1722)e\left(\frac{17}{22}\right) e(922)e\left(\frac{9}{22}\right) e(811)e\left(\frac{8}{11}\right) e(711)e\left(\frac{7}{11}\right) e(1922)e\left(\frac{19}{22}\right)
χ345(304,⋅)\chi_{345}(304,\cdot) −1-1 11 e(1322)e\left(\frac{13}{22}\right) e(211)e\left(\frac{2}{11}\right) e(411)e\left(\frac{4}{11}\right) e(1722)e\left(\frac{17}{22}\right) e(922)e\left(\frac{9}{22}\right) e(322)e\left(\frac{3}{22}\right) e(2122)e\left(\frac{21}{22}\right) e(411)e\left(\frac{4}{11}\right) e(911)e\left(\frac{9}{11}\right) e(1522)e\left(\frac{15}{22}\right)
χ345(319,⋅)\chi_{345}(319,\cdot) −1-1 11 e(2122)e\left(\frac{21}{22}\right) e(1011)e\left(\frac{10}{11}\right) e(911)e\left(\frac{9}{11}\right) e(1922)e\left(\frac{19}{22}\right) e(122)e\left(\frac{1}{22}\right) e(1522)e\left(\frac{15}{22}\right) e(1722)e\left(\frac{17}{22}\right) e(911)e\left(\frac{9}{11}\right) e(111)e\left(\frac{1}{11}\right) e(922)e\left(\frac{9}{22}\right)