Properties

Label 2205.554
Modulus $2205$
Conductor $2205$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

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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([35,21,30]))
 
pari: [g,chi] = znchar(Mod(554,2205))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.dp

\(\chi_{2205}(29,\cdot)\) \(\chi_{2205}(239,\cdot)\) \(\chi_{2205}(554,\cdot)\) \(\chi_{2205}(659,\cdot)\) \(\chi_{2205}(869,\cdot)\) \(\chi_{2205}(974,\cdot)\) \(\chi_{2205}(1184,\cdot)\) \(\chi_{2205}(1289,\cdot)\) \(\chi_{2205}(1499,\cdot)\) \(\chi_{2205}(1604,\cdot)\) \(\chi_{2205}(1919,\cdot)\) \(\chi_{2205}(2129,\cdot)\)

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

\((1226,442,1081)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{5}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 2205 }(554, a) \) \(-1\)\(1\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(1\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{17}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2205 }(554,a) \;\) at \(\;a = \) e.g. 2