Properties

Label 2850.13
Modulus $2850$
Conductor $475$
Order $180$
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(2850, base_ring=CyclotomicField(180))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,171,50]))
 
pari: [g,chi] = znchar(Mod(13,2850))
 

Basic properties

Modulus: \(2850\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{475}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2850.cs

\(\chi_{2850}(13,\cdot)\) \(\chi_{2850}(67,\cdot)\) \(\chi_{2850}(97,\cdot)\) \(\chi_{2850}(127,\cdot)\) \(\chi_{2850}(223,\cdot)\) \(\chi_{2850}(337,\cdot)\) \(\chi_{2850}(433,\cdot)\) \(\chi_{2850}(523,\cdot)\) \(\chi_{2850}(547,\cdot)\) \(\chi_{2850}(553,\cdot)\) \(\chi_{2850}(583,\cdot)\) \(\chi_{2850}(637,\cdot)\) \(\chi_{2850}(667,\cdot)\) \(\chi_{2850}(697,\cdot)\) \(\chi_{2850}(763,\cdot)\) \(\chi_{2850}(877,\cdot)\) \(\chi_{2850}(1003,\cdot)\) \(\chi_{2850}(1117,\cdot)\) \(\chi_{2850}(1123,\cdot)\) \(\chi_{2850}(1153,\cdot)\) \(\chi_{2850}(1237,\cdot)\) \(\chi_{2850}(1267,\cdot)\) \(\chi_{2850}(1333,\cdot)\) \(\chi_{2850}(1363,\cdot)\) \(\chi_{2850}(1447,\cdot)\) \(\chi_{2850}(1477,\cdot)\) \(\chi_{2850}(1573,\cdot)\) \(\chi_{2850}(1663,\cdot)\) \(\chi_{2850}(1687,\cdot)\) \(\chi_{2850}(1723,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((1901,1027,1351)\) → \((1,e\left(\frac{19}{20}\right),e\left(\frac{5}{18}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{79}{180}\right)\)\(e\left(\frac{23}{180}\right)\)\(e\left(\frac{1}{180}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{37}{90}\right)\)\(e\left(\frac{25}{36}\right)\)
value at e.g. 2