Properties

Label 3724.23
Modulus $3724$
Conductor $3724$
Order $126$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3724, base_ring=CyclotomicField(126))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([63,114,14]))
 
pari: [g,chi] = znchar(Mod(23,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(3724\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
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 3724.eh

\(\chi_{3724}(23,\cdot)\) \(\chi_{3724}(207,\cdot)\) \(\chi_{3724}(347,\cdot)\) \(\chi_{3724}(443,\cdot)\) \(\chi_{3724}(555,\cdot)\) \(\chi_{3724}(739,\cdot)\) \(\chi_{3724}(795,\cdot)\) \(\chi_{3724}(807,\cdot)\) \(\chi_{3724}(879,\cdot)\) \(\chi_{3724}(975,\cdot)\) \(\chi_{3724}(1087,\cdot)\) \(\chi_{3724}(1271,\cdot)\) \(\chi_{3724}(1327,\cdot)\) \(\chi_{3724}(1339,\cdot)\) \(\chi_{3724}(1411,\cdot)\) \(\chi_{3724}(1507,\cdot)\) \(\chi_{3724}(1619,\cdot)\) \(\chi_{3724}(1803,\cdot)\) \(\chi_{3724}(1859,\cdot)\) \(\chi_{3724}(1871,\cdot)\) \(\chi_{3724}(1943,\cdot)\) \(\chi_{3724}(2151,\cdot)\) \(\chi_{3724}(2335,\cdot)\) \(\chi_{3724}(2391,\cdot)\) \(\chi_{3724}(2403,\cdot)\) \(\chi_{3724}(2475,\cdot)\) \(\chi_{3724}(2571,\cdot)\) \(\chi_{3724}(2683,\cdot)\) \(\chi_{3724}(2867,\cdot)\) \(\chi_{3724}(2923,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((1863,3041,3137)\) → \((-1,e\left(\frac{19}{21}\right),e\left(\frac{1}{9}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{107}{126}\right)\)\(e\left(\frac{1}{63}\right)\)\(e\left(\frac{44}{63}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{26}{63}\right)\)\(e\left(\frac{109}{126}\right)\)\(e\left(\frac{46}{63}\right)\)\(e\left(\frac{13}{126}\right)\)\(e\left(\frac{2}{63}\right)\)\(e\left(\frac{23}{42}\right)\)
value at e.g. 2