Properties

Label 2304.71
Modulus $2304$
Conductor $384$
Order $32$
Real no
Primitive no
Minimal no
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2304)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([16,13,16]))
 
pari: [g,chi] = znchar(Mod(71,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(384\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{384}(107,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.bm

\(\chi_{2304}(71,\cdot)\) \(\chi_{2304}(215,\cdot)\) \(\chi_{2304}(359,\cdot)\) \(\chi_{2304}(503,\cdot)\) \(\chi_{2304}(647,\cdot)\) \(\chi_{2304}(791,\cdot)\) \(\chi_{2304}(935,\cdot)\) \(\chi_{2304}(1079,\cdot)\) \(\chi_{2304}(1223,\cdot)\) \(\chi_{2304}(1367,\cdot)\) \(\chi_{2304}(1511,\cdot)\) \(\chi_{2304}(1655,\cdot)\) \(\chi_{2304}(1799,\cdot)\) \(\chi_{2304}(1943,\cdot)\) \(\chi_{2304}(2087,\cdot)\) \(\chi_{2304}(2231,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((1279,2053,1793)\) → \((-1,e\left(\frac{13}{32}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{15}{32}\right)\)\(-i\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{32})\)
Fixed field: 32.32.135104323545903136978453058557785670637514001130337144105502507008.1