Properties

Label 2160.11
Modulus $2160$
Conductor $432$
Order $36$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,9,26,0]))
 
pari: [g,chi] = znchar(Mod(11,2160))
 

Basic properties

Modulus: \(2160\)
Conductor: \(432\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{432}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2160.ep

\(\chi_{2160}(11,\cdot)\) \(\chi_{2160}(131,\cdot)\) \(\chi_{2160}(371,\cdot)\) \(\chi_{2160}(491,\cdot)\) \(\chi_{2160}(731,\cdot)\) \(\chi_{2160}(851,\cdot)\) \(\chi_{2160}(1091,\cdot)\) \(\chi_{2160}(1211,\cdot)\) \(\chi_{2160}(1451,\cdot)\) \(\chi_{2160}(1571,\cdot)\) \(\chi_{2160}(1811,\cdot)\) \(\chi_{2160}(1931,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: 36.36.5532004127928253705369187176396364210546696053048780432717505515499814912.1

Values on generators

\((271,1621,2081,1297)\) → \((-1,i,e\left(\frac{13}{18}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2160 }(11, a) \) \(1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{7}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2160 }(11,a) \;\) at \(\;a = \) e.g. 2