Properties

Label 49.f
Modulus $49$
Conductor $49$
Order $14$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(49, base_ring=CyclotomicField(14))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(6,49))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(49\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{7})\)
Fixed field: 14.0.1341068619663964900807.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(8\) \(9\) \(10\) \(11\) \(12\)
\(\chi_{49}(6,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{49}(13,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{49}(20,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{49}(27,\cdot)\) \(-1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{49}(34,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{49}(41,\cdot)\) \(-1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{13}{14}\right)\)