Properties

Label 2205.709
Modulus $2205$
Conductor $2205$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([28,21,38]))
 
pari: [g,chi] = znchar(Mod(709,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(2205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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)
 

Galois orbit 2205.db

\(\chi_{2205}(4,\cdot)\) \(\chi_{2205}(319,\cdot)\) \(\chi_{2205}(394,\cdot)\) \(\chi_{2205}(634,\cdot)\) \(\chi_{2205}(709,\cdot)\) \(\chi_{2205}(1024,\cdot)\) \(\chi_{2205}(1264,\cdot)\) \(\chi_{2205}(1339,\cdot)\) \(\chi_{2205}(1579,\cdot)\) \(\chi_{2205}(1654,\cdot)\) \(\chi_{2205}(1894,\cdot)\) \(\chi_{2205}(1969,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((1226,442,1081)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{19}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 2205 }(709, a) \) \(1\)\(1\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{3}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2205 }(709,a) \;\) at \(\;a = \) e.g. 2