Properties

Label 1764.829
Modulus $1764$
Conductor $49$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1764, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,31]))
 
pari: [g,chi] = znchar(Mod(829,1764))
 

Basic properties

Modulus: \(1764\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(45,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1764.co

\(\chi_{1764}(73,\cdot)\) \(\chi_{1764}(145,\cdot)\) \(\chi_{1764}(397,\cdot)\) \(\chi_{1764}(577,\cdot)\) \(\chi_{1764}(649,\cdot)\) \(\chi_{1764}(829,\cdot)\) \(\chi_{1764}(1081,\cdot)\) \(\chi_{1764}(1153,\cdot)\) \(\chi_{1764}(1333,\cdot)\) \(\chi_{1764}(1405,\cdot)\) \(\chi_{1764}(1585,\cdot)\) \(\chi_{1764}(1657,\cdot)\)

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: \(\Q(\zeta_{49})\)

Values on generators

\((883,785,1081)\) → \((1,1,e\left(\frac{31}{42}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 1764 }(829, a) \) \(-1\)\(1\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{13}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1764 }(829,a) \;\) at \(\;a = \) e.g. 2