Properties

Label 2601.224
Modulus $2601$
Conductor $51$
Order $16$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2601, base_ring=CyclotomicField(16))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8,1]))
 
pari: [g,chi] = znchar(Mod(224,2601))
 

Basic properties

Modulus: \(2601\)
Conductor: \(51\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{51}(20,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2601.o

\(\chi_{2601}(224,\cdot)\) \(\chi_{2601}(503,\cdot)\) \(\chi_{2601}(827,\cdot)\) \(\chi_{2601}(998,\cdot)\) \(\chi_{2601}(1025,\cdot)\) \(\chi_{2601}(1196,\cdot)\) \(\chi_{2601}(1520,\cdot)\) \(\chi_{2601}(1799,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{51})^+\)

Values on generators

\((290,2026)\) → \((-1,e\left(\frac{1}{16}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(i\)\(e\left(\frac{1}{16}\right)\)\(-1\)
value at e.g. 2