Properties

Label 116.l
Modulus $116$
Conductor $116$
Order $28$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(116, base_ring=CyclotomicField(28))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,5]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(3,116))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(116\)
Conductor: \(116\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{28})\)
Fixed field: \(\Q(\zeta_{116})^+\)

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(21\)
\(\chi_{116}(3,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(-i\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{1}{28}\right)\)
\(\chi_{116}(11,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(-i\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{5}{28}\right)\)
\(\chi_{116}(15,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(i\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{11}{28}\right)\)
\(\chi_{116}(19,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(-i\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{13}{28}\right)\)
\(\chi_{116}(27,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(i\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{3}{28}\right)\)
\(\chi_{116}(31,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(-i\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{17}{28}\right)\)
\(\chi_{116}(39,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(i\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{27}{28}\right)\)
\(\chi_{116}(43,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(-i\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{25}{28}\right)\)
\(\chi_{116}(47,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(i\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{19}{28}\right)\)
\(\chi_{116}(55,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(i\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{15}{28}\right)\)
\(\chi_{116}(79,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(-i\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{9}{28}\right)\)
\(\chi_{116}(95,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(i\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{23}{28}\right)\)