Properties

Label 2205.44
Modulus $2205$
Conductor $735$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2205, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,21,8]))
 
pari: [g,chi] = znchar(Mod(44,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(735\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{735}(44,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.dr

\(\chi_{2205}(44,\cdot)\) \(\chi_{2205}(179,\cdot)\) \(\chi_{2205}(359,\cdot)\) \(\chi_{2205}(494,\cdot)\) \(\chi_{2205}(674,\cdot)\) \(\chi_{2205}(809,\cdot)\) \(\chi_{2205}(989,\cdot)\) \(\chi_{2205}(1124,\cdot)\) \(\chi_{2205}(1619,\cdot)\) \(\chi_{2205}(1754,\cdot)\) \(\chi_{2205}(1934,\cdot)\) \(\chi_{2205}(2069,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.0.84210739520843868988898078615875297955777796101582267830777690578346867562770843505859375.1

Values on generators

\((1226,442,1081)\) → \((-1,-1,e\left(\frac{4}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(-1\)\(1\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{5}{21}\right)\)
value at e.g. 2