Properties

Label 1728.125
Modulus $1728$
Conductor $576$
Order $48$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1728, base_ring=CyclotomicField(48))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9,40]))
 
pari: [g,chi] = znchar(Mod(125,1728))
 

Basic properties

Modulus: \(1728\)
Conductor: \(576\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{576}(509,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1728.bz

\(\chi_{1728}(125,\cdot)\) \(\chi_{1728}(197,\cdot)\) \(\chi_{1728}(341,\cdot)\) \(\chi_{1728}(413,\cdot)\) \(\chi_{1728}(557,\cdot)\) \(\chi_{1728}(629,\cdot)\) \(\chi_{1728}(773,\cdot)\) \(\chi_{1728}(845,\cdot)\) \(\chi_{1728}(989,\cdot)\) \(\chi_{1728}(1061,\cdot)\) \(\chi_{1728}(1205,\cdot)\) \(\chi_{1728}(1277,\cdot)\) \(\chi_{1728}(1421,\cdot)\) \(\chi_{1728}(1493,\cdot)\) \(\chi_{1728}(1637,\cdot)\) \(\chi_{1728}(1709,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((703,325,1217)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{17}{48}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{23}{48}\right)\)\(-i\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1728 }(125,a) \;\) at \(\;a = \) e.g. 2