Properties

Label 5185.26
Modulus $5185$
Conductor $1037$
Order $120$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(5185, base_ring=CyclotomicField(120))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,15,82]))
 
pari: [g,chi] = znchar(Mod(26,5185))
 

Basic properties

Modulus: \(5185\)
Conductor: \(1037\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1037}(26,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5185.jj

\(\chi_{5185}(26,\cdot)\) \(\chi_{5185}(246,\cdot)\) \(\chi_{5185}(331,\cdot)\) \(\chi_{5185}(376,\cdot)\) \(\chi_{5185}(1141,\cdot)\) \(\chi_{5185}(1226,\cdot)\) \(\chi_{5185}(1386,\cdot)\) \(\chi_{5185}(1396,\cdot)\) \(\chi_{5185}(1471,\cdot)\) \(\chi_{5185}(1481,\cdot)\) \(\chi_{5185}(1596,\cdot)\) \(\chi_{5185}(1641,\cdot)\) \(\chi_{5185}(1691,\cdot)\) \(\chi_{5185}(1726,\cdot)\) \(\chi_{5185}(1776,\cdot)\) \(\chi_{5185}(1946,\cdot)\) \(\chi_{5185}(2031,\cdot)\) \(\chi_{5185}(2361,\cdot)\) \(\chi_{5185}(2446,\cdot)\) \(\chi_{5185}(2491,\cdot)\) \(\chi_{5185}(2616,\cdot)\) \(\chi_{5185}(2701,\cdot)\) \(\chi_{5185}(2796,\cdot)\) \(\chi_{5185}(2841,\cdot)\) \(\chi_{5185}(2926,\cdot)\) \(\chi_{5185}(3691,\cdot)\) \(\chi_{5185}(4061,\cdot)\) \(\chi_{5185}(4146,\cdot)\) \(\chi_{5185}(4361,\cdot)\) \(\chi_{5185}(4666,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((3112,4576,2381)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{41}{60}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(-1\)\(1\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{79}{120}\right)\)\(e\left(\frac{103}{120}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{11}{120}\right)\)\(e\left(\frac{5}{6}\right)\)
value at e.g. 2