Properties

Label 49.h
Modulus $49$
Conductor $49$
Order $42$
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(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(3,49))
 
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: \(42\)
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: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{49})\)

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(8\) \(9\) \(10\) \(11\) \(12\)
\(\chi_{49}(3,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{11}{42}\right)\)
\(\chi_{49}(5,\cdot)\) \(-1\) \(1\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{25}{42}\right)\)
\(\chi_{49}(10,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{17}{42}\right)\)
\(\chi_{49}(12,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{37}{42}\right)\)
\(\chi_{49}(17,\cdot)\) \(-1\) \(1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{23}{42}\right)\)
\(\chi_{49}(24,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{29}{42}\right)\)
\(\chi_{49}(26,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{19}{42}\right)\)
\(\chi_{49}(33,\cdot)\) \(-1\) \(1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{31}{42}\right)\)
\(\chi_{49}(38,\cdot)\) \(-1\) \(1\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{41}{42}\right)\)
\(\chi_{49}(40,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{42}\right)\)
\(\chi_{49}(45,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{42}\right)\)
\(\chi_{49}(47,\cdot)\) \(-1\) \(1\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{13}{42}\right)\)