Properties

Label 2205.2203
Modulus $2205$
Conductor $2205$
Order $84$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([56,63,10]))
 
pari: [g,chi] = znchar(Mod(2203,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(2205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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.ea

\(\chi_{2205}(157,\cdot)\) \(\chi_{2205}(187,\cdot)\) \(\chi_{2205}(283,\cdot)\) \(\chi_{2205}(502,\cdot)\) \(\chi_{2205}(598,\cdot)\) \(\chi_{2205}(628,\cdot)\) \(\chi_{2205}(787,\cdot)\) \(\chi_{2205}(817,\cdot)\) \(\chi_{2205}(943,\cdot)\) \(\chi_{2205}(1102,\cdot)\) \(\chi_{2205}(1132,\cdot)\) \(\chi_{2205}(1228,\cdot)\) \(\chi_{2205}(1258,\cdot)\) \(\chi_{2205}(1417,\cdot)\) \(\chi_{2205}(1447,\cdot)\) \(\chi_{2205}(1543,\cdot)\) \(\chi_{2205}(1573,\cdot)\) \(\chi_{2205}(1732,\cdot)\) \(\chi_{2205}(1762,\cdot)\) \(\chi_{2205}(1858,\cdot)\) \(\chi_{2205}(1888,\cdot)\) \(\chi_{2205}(2047,\cdot)\) \(\chi_{2205}(2173,\cdot)\) \(\chi_{2205}(2203,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1226,442,1081)\) → \((e\left(\frac{2}{3}\right),-i,e\left(\frac{5}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 2205 }(2203, a) \) \(1\)\(1\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{3}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2205 }(2203,a) \;\) at \(\;a = \) e.g. 2