Properties

Label 2304.113
Modulus $2304$
Conductor $576$
Order $48$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2304, base_ring=CyclotomicField(48))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,40]))
 
pari: [g,chi] = znchar(Mod(113,2304))
 

Basic properties

Modulus: \(2304\)
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 \(\chi_{576}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.br

\(\chi_{2304}(113,\cdot)\) \(\chi_{2304}(209,\cdot)\) \(\chi_{2304}(401,\cdot)\) \(\chi_{2304}(497,\cdot)\) \(\chi_{2304}(689,\cdot)\) \(\chi_{2304}(785,\cdot)\) \(\chi_{2304}(977,\cdot)\) \(\chi_{2304}(1073,\cdot)\) \(\chi_{2304}(1265,\cdot)\) \(\chi_{2304}(1361,\cdot)\) \(\chi_{2304}(1553,\cdot)\) \(\chi_{2304}(1649,\cdot)\) \(\chi_{2304}(1841,\cdot)\) \(\chi_{2304}(1937,\cdot)\) \(\chi_{2304}(2129,\cdot)\) \(\chi_{2304}(2225,\cdot)\)

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

\((1279,2053,1793)\) → \((1,e\left(\frac{1}{16}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{29}{48}\right)\)\(i\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{1}{6}\right)\)
value at e.g. 2