Properties

Label 3724.53
Modulus $3724$
Conductor $931$
Order $126$
Real no
Primitive no
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([0,30,77]))
 
pari: [g,chi] = znchar(Mod(53,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(931\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{931}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.ed

\(\chi_{3724}(53,\cdot)\) \(\chi_{3724}(261,\cdot)\) \(\chi_{3724}(333,\cdot)\) \(\chi_{3724}(345,\cdot)\) \(\chi_{3724}(401,\cdot)\) \(\chi_{3724}(585,\cdot)\) \(\chi_{3724}(697,\cdot)\) \(\chi_{3724}(793,\cdot)\) \(\chi_{3724}(865,\cdot)\) \(\chi_{3724}(877,\cdot)\) \(\chi_{3724}(933,\cdot)\) \(\chi_{3724}(1117,\cdot)\) \(\chi_{3724}(1229,\cdot)\) \(\chi_{3724}(1325,\cdot)\) \(\chi_{3724}(1397,\cdot)\) \(\chi_{3724}(1409,\cdot)\) \(\chi_{3724}(1465,\cdot)\) \(\chi_{3724}(1649,\cdot)\) \(\chi_{3724}(1761,\cdot)\) \(\chi_{3724}(1857,\cdot)\) \(\chi_{3724}(1997,\cdot)\) \(\chi_{3724}(2181,\cdot)\) \(\chi_{3724}(2293,\cdot)\) \(\chi_{3724}(2389,\cdot)\) \(\chi_{3724}(2461,\cdot)\) \(\chi_{3724}(2473,\cdot)\) \(\chi_{3724}(2825,\cdot)\) \(\chi_{3724}(2993,\cdot)\) \(\chi_{3724}(3005,\cdot)\) \(\chi_{3724}(3061,\cdot)\) ...

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

\((1863,3041,3137)\) → \((1,e\left(\frac{5}{21}\right),e\left(\frac{11}{18}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{23}{126}\right)\)\(e\left(\frac{43}{63}\right)\)\(e\left(\frac{23}{63}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{115}{126}\right)\)\(e\left(\frac{109}{126}\right)\)\(e\left(\frac{4}{63}\right)\)\(e\left(\frac{17}{63}\right)\)\(e\left(\frac{23}{63}\right)\)\(e\left(\frac{23}{42}\right)\)
value at e.g. 2