Properties

Label 2205.2104
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([28,21,22]))
 
pari: [g,chi] = znchar(Mod(2104,2205))
 

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.dy

\(\chi_{2205}(184,\cdot)\) \(\chi_{2205}(499,\cdot)\) \(\chi_{2205}(529,\cdot)\) \(\chi_{2205}(844,\cdot)\) \(\chi_{2205}(1129,\cdot)\) \(\chi_{2205}(1159,\cdot)\) \(\chi_{2205}(1444,\cdot)\) \(\chi_{2205}(1474,\cdot)\) \(\chi_{2205}(1759,\cdot)\) \(\chi_{2205}(1789,\cdot)\) \(\chi_{2205}(2074,\cdot)\) \(\chi_{2205}(2104,\cdot)\)

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

\((1226,442,1081)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{11}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 2205 }(2104, a) \) \(1\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{31}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2205 }(2104,a) \;\) at \(\;a = \) e.g. 2